Table of Integrals, Series, and Products [8th Revised] 0123849330, 9780123849335

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Table of Integrals, Series, and Products Eighth Edition

Table of Integrals, Series, and Products Eighth Edition

I. S. Gradshteyn and I. M. Ryzhik Daniel Zwillinger, Editor Rensselaer Polytechnic Institute, USA Victor Moll (Scientific Editor) Tulane University, Department of Mathematics

Translated from Russian by Scripta Technica, Inc

Academic Press is an imprint of Elsevier 225 Wyman Street, Waltham, MA 02451, USA 525 B Street, Suite 1800, San Diego, CA 92101-4495, USA 32 Jamestown Road, London NW1 7BY, UK The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK Eighth Edition: c 2015, 2007, 2000, 1994 Elsevier Inc. All rights reserved. Copyright  No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. ISBN: 978-0-12-384933-5 Library of Congress Cataloging-in-Publication Data Gradshtein, I. S. (Izrail? Solomonovich) [Tablitsy integralov, summ, riadov i proizvedenii. English] Table of integrals, series, and products. – Eighth edition / Daniel Zwillinger. pages cm Includes bibliographical references and index. ISBN 978-0-12-384933-5 1. Mathematics–Tables. I. Zwillinger, Daniel, 1957- II. Title. QA55.G6613 2014 510.2 1–dc23 2014010276 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. For information on all Academic Press publications visit our web site at store.elsevier.com

Preface to the Eighth Edition Gradshteyn and Ryzhik continues to be a resource greatly used by mathematicians, scientists, and engineers in the theoretical, applied, and computational sciences. Since the publication in 2007 of the revised seventh edition, users have continued to submit corrections and new results that improve the book, and make suggestions for changes that improve the presentation of the material. We regret that the structure of the book makes it impossible to acknowledge these users by their individual contributions so, as usual, their names have been added to the acknowledgment list at the front of the book. This eighth edition includes corrections received since the publication of the seventh edition, together with a substantial amount of new material acquired from isolated sources. From among the many contributions we have included just those integrals that appear commonly in different contexts. Following our previous conventions, an amended entry has a superscript “12” added to its entry reference number, where the equivalent superscript number for the seventh edition was “11”. Similarly, an asterisk on an entry’s reference number indicates a new result. When, for technical reasons, an entry in a previous edition has been removed, the entry numbers will jump. This preserves the continuity of numbering between the new and older editions. This edition has also removed chapters present in the 7th edition that were not aligned with integrals, series, and products (e.g., the chapters on matrices and norms). We wish to express our gratitude to all who have been in contact with us with the object of improving and extending the book. Special thanks are extended to both Dr. Francis J. O’Brien, Jr. of the Naval Station in Newport, Rhode Island and Dr. Andrej Tenne-Sens of Ottowa, Ontario, Canada. They have each spent an unimaginably large number of hours helping with this edition. Experience over many years has shown that each new edition of Gradshteyn and Ryzhik generates many suggestions for new entries and new errata. Hence, we do not expect this new edition to be free from errors. All users who identify errors, or who wish to propose new entries, are invited to contact the authors whose email addresses are listed below. Corrections will be posted on the web site www.mathtable.com/ gr/errata. Finally, we mourn the passing of Professor Alan Jeffrey who, as Editor, guided this book from its 4th edition in 1965 to the 7th edition in 2007. Alan’s dedication to this book was evident both by his longevity—well over 40 years of improvements—and in the consideration he gave to all of our correspondents for whom he evaluated a huge number of integrals. Daniel Zwillinger [email protected] Victor Moll [email protected]

xvii

Acknowledgments The publisher and editors would like to take this opportunity to express their gratitude to the following users of the Table of Integrals, Series, and Products who either directly or through errata published in Mathematics of Computation have generously contributed corrections and addenda to the original printing. Anonymous Dr. Artem G. Abanov Dr. A. Abbas Dr. S. M. Abrarov Dr. P. B. Abraham Dr. Ari Abramson Dr. Jose Adachi Dr. R. J. Adler Dr. N. Agmon Dr. M. Ahmad Dr. S. A. Ahmad Dr. Rajai S. Alassar Dr. Luis Alvarez-Ruso Dr. Maarten H P Ambaum Dr. R. K. Amiet Dr. L. U. Ancarani Dr. M. Antoine Dr. Marian Apostol Dr. C. R. Appledorn Dr. D. R. Appleton Dr. Mitsuhiro Arikawa Dr. Ir. Luk R. Arnaut Dr. Peter Arnold Dr. P. Ashoshauvati Dr. C. L. Axness Dr. Scott Baalrud Dr. E. Badralexe Dr. S. B. Bagchi Dr. L. J. Baker Dr. R. Ball Dr. Ingo Barth

Dr. M. P. Barnett Dr. Fabio Bernardoni Dr. Florian Baumann Dr. Norman C. Beaulieu Dr. Jerome Benoit Mr. V. Bentley Dr. Laurent Berger Dr. M. van den Berg Dr. N. F. Berk Dr. C. A. Bertulani Dr. J. Betancort-Rijo Dr. P. Bickerstaff Dr. Iwo Bialynicki-Birula Dr. Chris Bidinosti Dr. G. R. Bigg Dr. Ian Bindloss Dr. L. Blanchet Dr. Mike Blaskiewicz Dr. R. D. Blevins Dr. Anders Blom Dr. L. M. Blumberg Dr. R. Blumel Dr. S. E. Bodner Dr. Simone Boi Dr. M. Bonsager Dr. George Boros Dr. S. Bosanac Dr. Ruben Van Boxem Dr. Christoph Bruder Dr. Patrick Bruno Dr. B. Van den Bossche

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Dr. A. Bostr¨ om Dr. J. E. Bowcock Dr. T. H. Boyer Dr. K. M. Briggs Dr. D. J. Broadhurst Dr. Chris Van Den Broeck Dr. W. B. Brower Dr. H. N. Browne Dr. Christoph Bruegger Dr. William J. Bruno Dr. Vladimir Bubanja Dr. D. J. Buch Dr. D. J. Bukman Dr. F. M. Burrows Dr. R. Caboz Dr. T. Calloway Dr. F. Calogero Dr. D. Dal Cappello Dr. David Cardon Dr. J. A. Carlson Gallos Dr. B. Carrascal Dr. A. R. Carr Dr. Neal Carron Dr. Florian Cartarius Dr. S. Carter Dr. Miguel Carvajal Dr. G. Cavalleri Mr. W. H. L. Cawthorne Dr. Alexandre Caz´e Dr. A. Cecchini Dr. B. Chan

xx

Dr. M. A. Chaudhry Dr. Sabino Chavez-Cerda Dr. Julian Cheng Dr. H. W. Chew Dr. D. Chin Dr. Young-seek Chung Dr. S. Ciccariello Dr. N. S. Clarke Dr. R. W. Cleary Dr. A. Clement Dr. Alain Cochard Dr. P. Cochrane Dr. D. K. Cohoon Dr. Howard S. Cohl Dr. L. Cole Dr. Filippo Colomo Donal Connon Dr. J. R. D. Copley Henry Corback, Esq. Dr. Daniel Benevides da Costa Dr. Barry J. Cox Dr. D. Cox Dr. J. Cox Dr. J. W. Criss Dr. A. E. Curzon Dr. D. Dadyburjor Dr. D. Dajaputra Dr. C. Dal Cappello Dr. P. Daly Dr. S. Dasgupta Dr. Charles E. Davidson Dr. John Davies Dr. C. L. Davis Dr. A. Degasperis Dr. Gustav Delius Dr. B. C. Denardo Dr. R. W. Dent Dr. E. Deutsch Dr. D. deVries Dr. Eran Dgani Dr. Enno Diekema Dr. P. Dita Dr. P. J. de Doelder Dr. Mischa Dohler Dr. G. Dˆome Dr. Shi-Hai Dong Dr. Balazs Dora

Acknowledgments

Dr. M. R. D’Orsogna Dr. Forrest Doss Dr. Adrian A. Dragulescu Dr. Zvi Drezner Dr. Eduardo Duenez Mr. Tommi J. Dufva Dr. Duc V. Duong Dr. E. B. Dussan, V Dr. Percy Dusek Dr. C. A. Ebner Dr. M. van der Ende Dr. Jonathan Engle Dr. G. Eng Dr. E. S. Erck Dr. Grant Erdmann Dr. Jan Erkelens Dr. Olivier Espinosa Dr. G. A. Est´evez Dr. K. Evans Dr. G. Evendon Dr. Valery I. Fabrikant Dr. L. A. Falkovsky Dr. Kambiz Farahmand Dr. Richard J. Fateman Dr. G. Fedele Dr. A. R. Ferchmin Dr. P. Ferrant Dr. Andr´e Ferrari Dr. H. E. Fettis Dr. W. B. Fichter Dr. George Fikioris Mr. J. C. S. S. Filho Dr. L. Ford Dr. Nicolao Fornengo Dr. J. France Dr. B. Frank Dr. S. Frasier Dr. Stefan Fredenhagen Dr. A. J. Freeman Dr. A. Frink Dr. Jason M. Gallaspy Dr. J. A. C. Gallas Dr. J. A. Carlson Gallas Dr. G. R. Gamertsfelder Dr. Jianliang Gao Dr. T. Garavaglia Dr. Jaime Zaratiegui Garcia

Dr. C. G. Gardner Dr. D. Garfinkle Dr. P. N. Garner Dr. F. Gasser Dr. E. Gath Dr. P. Gatt Dr. D. Gay Dr. M. P. Gelfand Dr. M. R. Geller Dr. Ali I. Genc Dr. Vincent Genot Dr. M. F. George Dr. Teschl Gerald Dr. P. Germain Dr. Ing. Christoph Gierull Dr. S. P. Gill Dr. Federico Girosi Dr. E. A. Gislason Dr. M. I. Glasser Dr. P. A. Glendinning Dr. L. I. Goldfischer Dr. Denis Golosov Dr. I. J. Good Dr. J. Good Mr. L. Gorin Dr. Martin G¨otz Dr. R. Govindaraj Dr. M. De Grauf Dr. Gabriele Gradoni Dr. L. Green Mr. Leslie O. Green Dr. R. Greenwell Dr. K. D. Grimsley Dr. Albert Groenenboom Dr. V. Gudmundsson Dr. J. Guillera Dr. K. Gunn Dr. D. L. Gunter Dr. Julio C. Guti´errez-Vega Dr. Roger Haagmans Dr. Howard Haber Dr. H. van Haeringen Dr. B. Hafizi Dr. Bahman Hafizi Dr. T. Hagfors Dr. M. J. Haggerty Dr. Timo Hakulinen

Acknowledgments

Dr. S. E. Hammel Dr. E. Hansen Dr. Wes Harker Dr. T. Harrett Dr. D. O. Harris Dr. Frank Harris Mr. Mazen D. Hasna Dr. Peter Hawkins Dr. Joel G. Heinrich Dr. Adam Dade Henderson Dr. Franck Hersant Dr. Sten Herlitz Dr. Chris Herzog Dr. A. Higuchi Dr. R. E. Hise Dr. Henrik Holm Dr. N. Holte Dr. R. W. Hopper Dr. P. N. Houle Dr. C. J. Howard Jie Hu Dr. Ben Yu-Kuang Hu Dr. J. H. Hubbell Dr. Felix Huber Dr. J. R. Hull Dr. W. Humphries Dr. Jean-Marc Hur´e Dr. Jamal A. Hussein Dr. Y. Iksbe Dr. Philip Ingenhoven Mr. L. Iossif Dr. Sean A. Irvine ´ ´Isberg Dr. Ottar Dr. Kazuhiro Ishida Dr. Cyril-Daniel Iskander Dr. S. A. Jackson Dr. John David Jackson Dr. Francois Jaclot Dr. B. Jacobs Dr. Pierre Jacobs Dr. E. C. James Dr. B. Jancovici Dr. D. J. Jeffrey Dr. H. J. Jensen Dr. Bin Jiang Dr. Edwin F. Johnson Dr. I. R. Johnson

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Dr. Steven Johnson Dr. Joel T. Johnson Dr. Fredrik Johansson Dr. I. Johnstone Dr. Y. P. Joshi Dr. Jae-Hun Jung Dr. Damir Juric Dr. Florian Kaempfer Dr. S. Kanmani Dr. Z. Kapal Dr. Peter Karadimas Dr. Dave Kasper Dr. M. Kaufman Dr. Eduardo Kausel Dr. B. Kay Louis Kempeneers Dr. Jack Kerlin Dr. Avinash Khare Dr. Karen T. Kohl Dr. Ilki Kim Dr. Youngsun Kim Dr. S. Klama Dr. L. Klingen Dr. C. Knessl Dr. M. J. Knight Dr. Yannis Kohninos Dr. D. Koks Dr. L. P. Kok Dr. K. S. K¨ olbig Dr. Yannis Komninos Dr. D. D. Konowalow Dr. Z. Kopal Dr. I. Kostyukov Dr. R. A. Krajcik Dr. Vincent Krakoviack Dr. Stefan Kr¨amer Dr. Tobias Kramer Dr. Hermann Krebs Dr. Chethan Krishnan Dr. J. W. Krozel Dr. E. D. Krupnikov Dr. Kun-Lin Kuo Dr. E. A. Kuraev Dr. Heinrich Kuttler Dr. Konstantinos Kyritsis Dr. Velimir Labinac Dr. Javier Navarro Laboulais

Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr.

A. D. J. Lambert A. Lambert A. Larraza K. D. Lee M. Howard Lee M. K. Lee P. A. Lee Todd Lee J. Legg Xianfu Lei Yefim Leifman Remigijus Leipus Armando Lemus Elvio Leonardo Eugene Lepelaars S. L. Levie D. Levi Michael Lexa Liang Li Kuo Kan Liang Sergey Liflandsky B. Linet Andrej Likar M. A. Lisa Donald Livesay H. Li Xian-Fang Li Georg Lohoefer I. M. Longman D. Long Sylvie Lorthois Wenzhou Lu Phil Lucht Stephan Ludwig Arpad Lukacs Y. L. Luke W. Lukosz T. Lundgren E. A. Luraev R. Lynch K. B. Ma Ilari Maasilta R. Mahurin R. Mallier G. A. Mamon A. Mangiarotti I. Manning

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Dr. Greg Marks Dr. J. Marmur Dr. A. Martin Dr. Carmelo P. Martin Sr. Yuzo Maruyama Dr. Richard Marthar Dr. David J. Masiello Dr. Richard J. Mathar Dr. H. A. Mavromatis Dr. M. Mazzoni Dr. P. McCullagh Dr. J. H. McDonnell Dr. J. R. McGregor Dr. Kim McInturff Dr. N. McKinney Dr. David McA McKirdy Dr. Andrew J. McHutchon Dr. Rami Mehrem Dr. W. N. Mei Dr. Angelo Melino Mr. Jos´e Ricardo Mendes Dr. Zvi Mendlowitz Dr. Andy Mennim Dr. Sarah Messer Dr. J. P. Meunier Dr. Haixing Miao Dr. Krys A. Michalski Dr. Gerard P. Michon Dr. D. F. R. Mildner Dr. D. L. Miller Dr. Steve Miller Dr. P. C. D. Milly Dr. S. P. Mitra Dr. K. Miura Dr. N. Mohankumar Dr. M. Moll Dr. Victor H. Moll Dr. D. Monowalow Mr. Tony Montagnese Dr. Thierry Montagu Dr. Jim Morehead Dr. J. Morice Dr. J. Guy Morgan Dr. W. Mueck Dr. C. Muhlhausen Dr. S. Mukherjee Dr. R. R. M¨ uller

Acknowledgments

Dr. Pablo Parmezani Munhoz Dr. Frank Namin Dr. Paul Nanninga Dr. A. Natarajan Dr. Sven Peter N¨asholm Dr. Javier Navarro Dr. Christian Netzel Dr. Stefan Neumeier Dr. C. T. Nguyen Dr. A. C. Nicol Dr. M. M. Nieto Dr. P. Noerdlinger Dr. Andrew N. Norris Dr. K. H. Norwich Dr. A. H. Nuttall Dr. F. O’Brien Dr. R. P. O’Keeffe Dr. A. Ojo Dr. O. Olendski Dr. Oleg Olendski Dr. P. Olsson Dr. Gilad Oren Dr. M. Ortner Dr. Matthew Orton Dr. S. Ostlund Dr. J. Overduin Dr. J. Pachner Mr. Robert A. Padgug Dr. D. Papadopoulos Dr. F. J. Papp Mr. Man Sik Park Dr. Jong-Do Park Dr. B. Patterson Dr. R. F. Pawula Dr. D. W. Peaceman Dr. Vittorio Peano Dr. Vincent Pegoraro Dr. D. Pelat Dr. L. Peliti Dr. Y. P. Pellegrini Dr. Thiago S. Pereira Dr. G. J. Pert Dr. Nicola Pessina Dr. J. B. Peterson Dr. Rickard Petersson Dr. Arnaud Pierens Dr. Emilio Pisanty

Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr.

Ralph Pixley Andrew Plumb Dror Porat E. A. Power E. Predazzi William S. Price Gunnar Pruessner Paul Radmore Carl E. Rasmussen F. Raynal X. R. Resende Guy A. Reynolds J. M. Riedler Sjoerd Rienstra Thomas Richard E. Ringel T. M. Roberts N. I. Robinson P. A. Robinson Elvira Romera D. M. Rosenblum R. A. Rosthal J. R. Roth Klaus Rottbrand Bastien Roucaries D. Roy E. Royer D. Rudermann Ahmad Rushdi Ali Rushdi Niall Ryan Sanjib Sabhapandit C. T. Sachradja J. Sadiku A. Sadiq Motohiko Saitoh Naoki Saito A. Salim Sherwood Samn J. H. Samson Miguel A. Sanchis-Lozano J. A. Sanders M. A. F. Sanjun P. Sarquiz Avadh Saxena Vito Scarola O. Sch¨arpf

Acknowledgments

Dr. A. Scherzinger Dr. B. Schizer Dr. Martin Schmid Dr. J. Scholes Dr. Mel Schopper Dr. H. J. Schulz Dr. Andreas Schulz Dr. Markus Schwarz Dr. G. J. Sears Dr. Kazuhiko Seki Dr. B. Seshadri Dr. Roger Sewell Dr. A. Shapiro Dr. Sihong Shao Dr. Masaki Shigemori Dr. J. S. Sheng Dr. Kenneth Ing Shing Dr. Tomohiro Shirai Dr. S. Shlomo Dr. D. Siegel Dr. Matthew Stapleton Dr. Leo Stein Dr. Michael L. Stein Dr. Steven H. Simon Dr. Ashok Kumar Singal Dr. Constantin Siriteanu Dr. C. Smith Dr. Richard Smith Dr. G. C. C. Smith Dr. Stefan Llewellyn Smith Dr. S. Smith Dr. Sasha Sodin Dr. G. Solt Dr. J. Sondow Dr. A. Sørenssen Dr. Marcus Spradlin Dr. Andrzej Staruszkiewicz Dr. Philip C. L. Stephenson Dr. Edgardo Stockmeyer Dr. J. C. Straton Mr. H. Suraweera Dr. N. F. Svaiter Dr. V. Svaiter Dr. R. Szmytkowski Dr. Sebastian S. Szyszkowicz Dr. S. Tabachnik

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Dr. Erik Talvila Dr. G. Tanaka Dr. C. Tanguy Dr. G. K. Tannahill Dr. B. T. Tan Dr. C. Tavard Dr. Gon¸calo Tavares Dr. Aba Teleki Andrej Tenne-Sens Dr. Gerald Teschl Dr. Arash Dahi Taleghani Dr. D. Temperley Dr. A. J. Tervoort Dr. Theodoros Theodoulidis Dr. D. J. Thomas Dr. Michael Thorwart Dr. S. T. Thynell Dr. Tamer Tlas Dr. D. C. Torney Dr. R. Tough Dr. Marwan Toutounji Dr. Dennis Trede Dr. B. F. Treadway Dr. Ming Tsai Dr. N. Turkkan Dr. Sandeep Tyagi Dr. J. J. Tyson Dr. Takahiro Ueda Dr. S. Uehara Dr. M. Vadacchino Dr. Stathis Vagenas Dr. O. T. Valls Dr. D. Vandeth Dr. Klaas Vantournhout Mr. Andras Vanyolos Dr. D. Veitch Mr. Jose Lopez Vicario Dr. Hari Vishnu Dr. K. Vogel Dr. J. M. M. J. Vogels Dr. Alexis De Vos Dr. Emanuel Voto Dr. Stuart Walsh Dr. Haiming Wang Dr. J. J. Wang Dr. Reinhold Wannemacher

Dr. S. Wanzura Dr. J. Ward Dr. S. I. Warshaw Dr. Alex Watson Dr. R. Weber Dr. Steffen Weissmann Dr. Wei Qian Dr. Detmar Welz Dr. Kyle Wendt Dr. D. H. Werner Dr. E. Wetzel Dr. Robert Whittaker Dr. Peter Widerin Dr. D. T. Wilton Dr. C. Wiuf Dr. K. T. Wong Dr. Rog´erio Nunes Wolff Mr. J. N. Wright Dr. J. D. Wright Dr. D. Wright Dr. Chong-shi Wu Dr. D. Wu Dr. Roahn Wynar Dr. Takashi Yanagisawa Dr. Michel Daoud Yacoub Dr. Yu S. Yakovlev Dr. H.-C. Yang Dr. J. J. Yang Dr. Z. J. Yang Dr. Mingwu Yao Dr. Yu-Min Yen Mr. Chun Kin Au Yeung Dr. Steve Young Dr. Kazuya Yuasa Dr. S. P. Yukon Dr. Zhuo-Quan Zeng Dr. B. Zhang Dr. Peng Zhang Dr. Y. C. Zhang Dr. Y. Zhao Dr. Ralf Zimmer Dr. Chunhui Zhu Dr. Zeng Zhuo-Quan

The Order of Presentation of the Formulas The question of the most expedient order in which to give the formulas, in particular, in what division to include particular formulas such as the definite integrals, turned out to be quite complicated. The thought naturally occurs to set up an order analogous to that of a dictionary. However, it is almost impossible to create such a system for the formulas of integral calculus. Indeed, in an arbitrary formula of the form  b f (x) dx = A a

one may make a large number of substitutions of the form x = ϕ(t) and thus obtain a number of “synonyms” of the given formula. We must point out that the table of definite integrals by Bierens de Haan and the earlier editions of the present reference both sin in the plethora of such “synonyms” and formulas of complicated form. In the present edition, we have tried to keep only the simplest of the “synonym” formulas. Basically, we judged the simplicity of a formula from the standpoint of the simplicity of the arguments of the “outer” functions that appear in the integrand. Where possible, we have replaced a complicated formula with a simpler one. Sometimes, several complicated formulas were thereby reduced to a single simpler one. We then kept only the simplest formula. As a result of such substitutions, we sometimes obtained an integral that could be evaluated by use of the formulas of chapter two and the Newton–Leibniz formula, or to an integral of the form  a f (x) dx, −a

where f (x) is an odd function. In such cases the complicated integrals have been omitted. Let us give an example using the expression  0

π/4

p−1

(cot x − 1) sin2 x

ln tan x dx = −

π cosec pπ. p

By making the natural substitution u = cot x − 1, we obtain  ∞ π up−1 ln(1 + u) du = cosec pπ. p 0 Integrals similar to formula (0.1) are omitted in this new edition. Instead, we have formula (0.2).

xxv

(0.1)

(0.2)

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The Order of Presentation of the Formulas

As a second example, let us take  I=

0

π/2

ln (tanp x + cotp x) ln tan x dx = 0.

The substitution u = tan x yields  I=

∞ 0

ln (up + u−p ) ln u du. 1 + u2

If we now set υ = ln u, we obtain  ∞  ∞   pυ υeυ ln (2 cosh pυ) −pυ dυ. I= dυ = ln e + e υ 2υ 2 cosh υ −∞ 1 + e −∞ The integrand is odd and, consequently, the integral is equal to 0. Thus, before looking for an integral in the tables, the user should simplify as much as possible the arguments (the “inner” functions) of the functions in the integrand. The functions are ordered as follows: First we have the elementary functions: 1. 2. 3. 4. 5. 6. 7.

The function f (x) = x. The exponential function. The hyperbolic functions. The trigonometric functions. The logarithmic function. The inverse hyperbolic functions. (These are replaced with the corresponding logarithms in the formulas containing definite integrals.) The inverse trigonometric functions.

Then follow the special functions: 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

Elliptic integrals. Elliptic functions. The logarithm integral, the exponential integral, the sine integral, and the cosine integral functions. Probability integrals and Fresnel’s integrals. The gamma function and related functions. Bessel functions. Mathieu functions. Legendre functions. Orthogonal polynomials. Hypergeometric functions. Degenerate hypergeometric functions. Parabolic cylinder functions. Meijer’s and MacRobert’s functions. Riemann’s zeta function.

The integrals are arranged in order of outer function according to the above scheme: the farther down in the list a function occurs, (i.e., the more complex it is) the later will the corresponding formula appear

The Order of Presentation of the Formulas

xxvii

in the tables. Suppose that several expressions have the same outer function. For example, consider sin ex , sin x, sin ln x. Here, the outer function is the sine function in all three cases. Such expressions are then arranged in order of the inner function. In the present work, these functions are therefore arranged in the following order: sin x, sin ex , sin ln x. Our list does not include polynomials, rational functions, powers, or other algebraic functions. An algebraic function that is included in tables of definite integrals can usually be reduced to a finite combination of roots of rational power. Therefore, for classifying our formulas, we can conditionally treat a power function as a generalization of an algebraic and, consequently, of a rational function.∗ We shall distinguish between all these functions and those listed above and we shall treat them as operators. Thus, in the expression sin2 ex , we shall think of the squaring operator as applied to the outer function, x+cos x namely, the sine. In the expression sin sin x−cos x , we shall think of the rational operator as applied to the trigonometric functions sine and cosine. We shall arrange the operators according to the following order: 1. 2. 3. 4.

Polynomials (listed in order of their degree). Rational operators. Algebraic operators (expressions of the form Ap/q , where q and p are rational, and q > 0; these are listed according to the size of q). Power operators.

Expressions with the same outer and inner functions are arranged in the order of complexity of the operators. For example, the following functions (whose outer functions are all trigonometric, and whose inner functions are all f (x) = x) are arranged in the order shown: sin x,

sin x cos x,

1 = cosec x, sin x

sin x = tan x, cos x

sin x + cos x , sin x − cos x

sinm x,

sinm x cos x.

Furthermore, if two outer functions ϕ1 (x) and ϕ2 (x), where ϕ1 (x) is more complex than ϕ2 (x), appear in an integrand and if any of the operations mentioned are performed on them, the corresponding integral will appear (in the order determined by the position of ϕ2 (x) in the list) after all integrals containing only the function ϕ1 (x). Thus, following the trigonometric functions are the trigonometric and power functions (that is, ϕ2 (x) = x). Then come • combinations of trigonometric and exponential functions, • combinations of trigonometric functions, exponential functions, and powers, etc., • combinations of trigonometric and hyperbolic functions, etc. Integrals containing two functions ϕ1 (x) and ϕ2 (x) are located in the division and order corresponding to the more complicated function of the two. However, if the positions of several integrals coincide because they contain the same complicated function, these integrals are put in the position defined by the complexity of the second function. To these rules of a general nature, we need to add certain particular considerations that will be easily 1 understood from the tables. For example, according to the above remarks, the function e x comes after 1 1 ex as regards complexity, but ln x and ln are equally complex since ln = − ln x. In the section on x x “powers and algebraic functions”, polynomials, rational functions, and powers of powers are formed from power functions of the form (a + bx)n and (α + βx)ν . ∗ For any natural number n, the involution (a + bx)n of the binomial a + bx is a polynomial. If n is a negative integer, (a + bx)n is a rational function. If n is irrational, the function (a + bx)n is not even an algebraic function.

Use of the Tables∗ For the effective use of the tables contained in this book it is necessary that the user should first become familiar with the classification system for integrals devised by the authors Gradshteyn and Ryzhik. This classification is described in detail in the section entitled The Order of Presentation of the Formulas (see page xxv) and essentially involves the separation of the integrand into inner and outer functions. The principal function involved in the integrand is called the outer function and its argument, which is itself usually another function, is called the inner function. Thus, if the integrand comprised the expression ln sin x, the outer function would be the logarithmic function while its argument, the inner function, would be the trigonometric function sin x. The desired integral would then be found in the section dealing with logarithmic functions, its position within that section being determined by the position of the inner function (here a trigonometric function) in Gradshteyn and Ryzhik’s list of functional forms. It is inevitable that some duplication of symbols will occur within such a large collection of integrals and this happens most frequently in the first part of the book dealing with algebraic and trigonometric integrands. The symbols most frequently involved are α, β, γ, δ, t, u, z, zk , and Δ. The expressions associated with these symbols are used consistently within each section and are defined at the start of each new section in which they occur. Consequently, reference should be made to the beginning of the section being used in order to verify the meaning of the substitutions involved. Integrals of algebraic functions are expressed as combinations of roots with rational power indices, and definite integrals of such functions are frequently expressed in terms of the Legendre elliptic integrals F (φ, k), E (φ, k) and Π(φ, n, k), respectively, of the first, second and third kinds. The four inverse hyperbolic functions arcsinh z, arccosh z, arctanh z and arccoth z are introduced through the definitions 1 arcsinh(iz) i 1 arccos z = arccosh(z) i 1 arctan z = arctanh(iz) i arccot z = i arccoth(iz) arcsin z =

∗ Prepared

by Alan Jeffrey for the English language edition.

xxix

xxx

Use of the Tables

or 1 arcsin(iz) i arccosh z = i arccos z 1 arctanh z = arctan(iz) i 1 arccoth z = arccot(−iz) i arcsinh z =

The numerical constants C and G which often appear in the definite integrals denote Euler’s constant and Catalan’s constant, respectively. Euler’s constant C is defined by the limit   s  1 C = lim − ln s = 0.577215 . . . . s→∞ m m=1 On occasions other writers denote Euler’s constant by the symbol γ, but this is also often used instead to denote the constant γ = eC = 1.781072 . . . . Catalan’s constant G is related to the complete elliptic integral  π/2 dx  K ≡ K(k) ≡ 0 1 − k 2 sin2 x by the expression  ∞  (−1)m 1 1 K dk = = 0.915965 . . . . G= 2 0 (2m + 1)2 m=0 Since the notations and definitions for higher transcendental functions that are used by different authors are by no means uniform, it is advisable to check the definitions of the functions that occur in these tables. This can be done by identifying the required function by symbol and name in the Index of Special Functions and Notation on page xxxvii, and by then referring to the defining formula or section number listed there. We now present a brief discussion of some of the most commonly used alternative notations and definitions for higher transcendental functions. Bernoulli and Euler Polynomials and Numbers Extensive use is made throughout the book of the Bernoulli and Euler numbers Bn and En that are defined in terms of the Bernoulli and Euler polynomials of order n, B n (x) and E n (x), respectively. These polynomials are defined by the generating functions ∞  text tn = for |t| < 2π B n (x) t e − 1 n=0 n! and ∞  2ext tn = for |t| < π. E n (x) t e + 1 n=0 n! The Bernoulli numbers are always denoted by Bn and are defined by the relation Bn = Bn (0)

for n = 0, 1, . . . ,

when B0 = 1,

1 B1 = − , 2

B2 =

1 , 6

B4 = −

1 ,.... 30

Use of the Tables

xxxi

The Euler numbers En are defined by setting

  1 for n = 0, 1, . . . En = 2 E n 2 The En are all integral and E0 = 1, E2 = −1, E4 = 5, E6 = −61, . . . . An alternative definition of Bernoulli numbers, which we shall denote by the symbol Bn∗ , uses the same generating function but identifies the Bn∗ differently in the following manner: t t2 t4 1 = 1 − t + B1∗ − B2∗ + . . . . t e −1 2 2! 4! This definition then gives rise to the alternative set of Bernoulli numbers n

B1∗ = 1/6, B2∗ = 1/30, B3∗ = 1/42, B4∗ = 1/30, B5∗ = 5/66, B6∗ = 691/2730, B7∗ = 7/6, B8∗ = 3617/510, . . . . These differences in notation must also be taken into account when using the following relationships that exist between the Bernoulli and Euler polynomials: n 1  n

Bn−k E k (2x) n = 0, 1, . . . 2n k k=0    x x+1 2n Bn − Bn E n−1 (x) = n 2 2

B n (x) =

or E n−1 (x) = and

x  2 B n (x) − 2n B n n 2

n = 1, 2, . . .

 n  n−k 2 − 1 Bn−k B k (x) n = 2, 3, . . . k 2 k=0 There are also alternative definitions of the Euler polynomial of order n, and it should be noted that some authors, using a modification of the third expression above, call   x  2 B n (x) − 2n B n n+1 2 the Euler polynomial of order n. E n−2 (x) = 2

n

−1

n−2 

Elliptic Functions and Elliptic Integrals The following notations are often used in connection with the inverse elliptic functions snu, cn u, and dn u: 1 snu snu sc u = cn u snu sd u = dn u

ns u =

1 cn u cn u cs u = snu cn u cd u = dn u nc u =

1 dn u dn u ds u = snu dn u dc u = cn u

nd u =

xxxii

Use of the Tables

The elliptic integral of the third kind is defined by Gradshteyn and Ryzhik to be  ϕ   da 2 Π ϕ, n , k =   2 2 0   1 − n sin a 1 − k 2 sin2 a −∞ < n2 < ∞  sin ϕ dx  = 2 2 (1 − n x ) (1 − x2 ) (1 − k 2 x2 ) 0 The Jacobi Zeta Function and Theta Functions The Jacobi zeta function zn(u, k), frequently written Z(u), is defined by the relation  u E E 2 dυ = E(u) − u. dn υ − zn(u, k) = Z(u) = K K 0 This is related to the theta functions by the relationship ∂ zn(u, k) = ln Θ(u) ∂u giving πu

 π ϑ1 2K cn u dn u πu − (i). zn(u, k) = 2K ϑ1 sn u 2K πu

 ϑ 2 dn u sn u π 2K + (ii). zn(u, k) = 2K ϑ2 πu cn u 2K πu

 ϑ sn u cn u π 3 2K

− k2 (iii). zn(u, k) = 2K ϑ3 πu dn u 2K πu

 π ϑ4 2K

(iv). zn(u, k) = 2K ϑ4 πu 2K Many different notations for the theta function are in current use. The most common variants are the replacement of the argument u by the argument u/π and, occasionally, a permutation of the identification of the functions ϑ1 to ϑ4 with the function ϑ4 replaced by ϑ. The Factorial (Gamma) Function In older reference texts the gamma function Γ(z), defined by the Euler integral  ∞ tz−1 e−t dt, Γ(z) = 0

is sometimes expressed in the alternative notation Γ(1 + z) = z! = Π(z). On occasions the related derivative of the logarithmic factorial function Ψ(z) is used where d(ln z!) (z!) = = Ψ(z + 1). dz z!

Use of the Tables

xxxiii

This function satisfies the recurrence relation Ψ(z) = Ψ(z − 1) + and is defined by the series Ψ(z) = −C +

∞   n=0

1 z−1

1 1 − n+1 z+n

 .

The derivative Ψ (z) satisfies the recurrence relation Ψ (z + 1) = Ψ (z) − and is defined by the series Ψ (z) =

1 z2

∞ 

1 . (z + n)2 n=0

Exponential and Related Integrals The exponential integrals E n (z) have been defined by Schloemilch using the integral  ∞ e−zt t−n dt (n = 0, 1, . . . , Re z > 0) E n (z) = 1

They should not be confused with the Euler polynomials already mentioned. The function E 1 (z) is related to the exponential integral Ei(z) through the expressions  ∞ E 1 (z) = − Ei(−z) = e−t t−1 dt z

and



z

dt = Ei (ln z) [z > 1] ln t 0 The functions E n (z) satisfy the recurrence relations  1  −z e − z E n−1 (z) E n (z) = [n > 1] n−1 and E n (z) = − E n−1 (z) with li(z) =

E 0 (z) = e−z /z. The function E n (z) has the asymptotic expansion    n n(n + 1) n(n + 1)(n + 2) e−z 3π 1− + − + · · · |arg z| < E n (z) ∼ z z z2 z3 2 while for large n,     n e−x n(n − 2x) n 6x2 − 8nx + n2 1+ + + + R(n, x) , E n (x) = x+n (x + n)2 (x + n)4 (x + n)6 where   1 −4 n−4 [x > 0] −0.36n ≤ R(n, x) ≤ 1 + x+n−1 The sine and cosine integrals si(x) and ci(x) are related to the functions Si(x) and Ci(x) by the integrals  x sin t π Si(x) = dt = si(x) + t 2 0 and

xxxiv

Use of the Tables



x

(cos t − 1) dt. t 0 The hyperbolic sine and cosine integrals shi(x) and chi(x) are defined by the relations  x sinh t dt shi(x) = t 0 and  x (cosh t − 1) chi(x) = C + ln x + dt. t 0 Some authors write  x (1 − cos t) Cin(x) = dt t 0 so that Ci(x) = C + ln x +

Cin(x) = − Ci(x) + ln x + C. The error function erf(x) is defined by the relation  x 2 2 e−t dt erf(x) = Φ(x) = √ π 0 and the complementary error function erfc(x) is related to the error function erfc(x) and to Φ(x) by the expression erfc(x) = 1 − erf(x). The Fresnel integrals S (x) and C (x) are defined by Gradshteyn and Ryzhik as  x 2 sin t2 dt S (x) = √ 2π 0 and  x 2 C (x) = √ cos t2 dt. 2π 0 Other definitions that are in use are  x  x πt2 πt2 S 1 (x) = dt, C 1 (x) = dt sin cos 2 2 0 0 and  x  x sin t cos t 1 1 √ dt, √ dt S 2 (x) = √ C 2 (x) = √ t t 2π 0 2π 0 These are related by the expressions      2 = S 2 x2 S (x) = S 1 x π and      2 = C 2 x2 C (x) = C 1 x π Hermite and Chebyshev Orthogonal Polynomials The Hermite polynomials H n (x) are related to the Hermite polynomials He n (x) by the relations   x −n/2 Hn √ He n (x) = 2 2 and √

H n (x) = 2n/2 He n x 2 .

Use of the Tables

xxxv

These functions satisfy the differential equations d2 Hn d Hn + 2n Hn = 0 − 2x 2 dx dx and d2 Hen d Hen + n Hen = 0. −x dx2 dx They obey the recurrence relations H n+1 = 2x H n −2n H n−1 and He n+1 = x He n −n He n−1 The first six orthogonal polynomials He n are He 0 = 1, He 1 = x, He 2 = x2 − 1, He 3 = x3 − 3x, He 4 = x4 − 6x2 + 3, He 5 = x5 − 10x3 + 15x. Sometimes the Chebyshev polynomial U n (x) of the second kind is defined as a solution of the equation d2 y dy + n(n + 2)y = 0. (1 − x2 ) 2 − 3x dx dx Bessel Functions A variety of different notations for Bessel functions are in use. Some common ones involve the replacement of Y n (z) by N n (z) and the introduction of the symbol  −n 1 z Γ(n + 1) J n (z). Λn (z) = 2 1 In the book by Gray, Mathews and MacRobert the symbol Y n (z) is used to denote π Y n (z) + 2 (ln 2 − C) J n (z) while Neumann uses the symbol Y (n) (z) for the identical quantity. (1) (2) The Hankel functions ν (z) and H ν (z) are sometimes denoted by Hsν (z) and Hiν (z) and some  H 1 πi H (1) authors write Gν (z) = ν (z). 2 The Neumann polynomial O n (t) is a polynomial of degree n + 1 in 1/t, with O 0 (t) = 1/t. The polynomials O n (t) are defined by the generating function ∞  1 = J 0 (z) O 0 (t) + 2 J k (z) O k (t), t−z k=1 giving  n−2k+1 [n/2] 1  n(n − k − 1)! 2 for n = 1, 2, . . . , O n (t) = 4 k! t k=0   where 12 n signifies the integral part of 12 n. The following relationship holds between three successive polynomials:   2 n2 − 1 nπ 2n (n − 1) O n+1 (t) + (n + 1) O n−1 (t) − O n (t) = sin2 . t t 2

xxxvi

Use of the Tables

The Airy functions Ai(z) and Bi(z) are independent solutions of the equation d2 u − zu = 0. dz 2 The solutions can be represented in terms of Bessel functions by the expressions         2 3/2 2 3/2 1 z 2 3/2 1√ z z K 1/3 z − I 1/3 = z I −1/3 Ai(z) = 3 3 3 π 3 3      2 3/2 2 3/2 1√ z z + J −1/3 z J 1/3 Ai(−z) = 3 3 3 and by

      z 2 3/2 2 3/2 I −1/3 z z + I 1/3 , 3 3 3       z 2 3/2 2 3/2 J −1/3 z z − J 1/3 . Bi(−z) = 3 3 3 Bi(z) =

Parabolic Cylinder Functions and Whittaker Functions The differential equation d2 y + (az 2 + bz + c)y = 0 dz 2 has associated with it the two equations     d2 y d2 y 1 2 1 2 + + a y = 0 and − + a y=0 z z dz 2 4 dz 2 4 the solutions of which are parabolic cylinder functions. The first equation can be derived from the second by replacing z by zeiπ/4 and a by −ia. The solutions of the equation   d2 y 1 2 z +a y =0 − dz 2 4 are sometimes written U (a, z) and V (a, z). These solutions are related to Whittaker’s function D p (z) by the expressions U (a, z) = D −a− 12 (z) and    1 1 +a D −a− 12 (−z) + (sin πa) D −a− 12 (z) . V (a, z) = Γ π 2 Mathieu Functions There are several accepted notations for Mathieu functions and for their associated parameters. The defining equation used by Gradshteyn and Ryzhik is  d2 y  + a − 2k 2 cos 2z y = 0 with k 2 = q. dz 2 Different notations involve the replacement of a and q in this equation by h and θ, λ and h2 and √ b and c = 2 q, respectively. The periodic solutions sen (z, q) and cen (z, q) and the modified periodic solutions Sen (z, q) and Cen (z, q) are suitably altered and, sometimes, re-normalized. A description of these relationships together with the normalizing factors is contained in: Tables relating to Mathieu functions. National Bureau of Standards, Columbia University Press, New York, 1951.

Index of Special Functions Notation β(x) Γ(z) γ(a, x), Γ(a, x) Δ(n − k) ξ(s) λ(x, y) μ(x, β), μ(x, β, α) ν(x), ν(x, α) Π(x) Π(ϕ, n, k) ζ(u) ζ(z, q), ζ(z)  πu   πu  , Θ1 (u) = ϑ⎫ Θ(u) ⎧ = ϑ4 2K 3 2K ⎪ ⎨ ϑ0 (υ | τ ) = ϑ4 (υ | τ ), ⎪ ⎬ ϑ1 (υ | τ ), ϑ2 (υ | τ ), ⎪ ⎪ ⎩ ⎭ ϑ3 (υ | τ ) σ(u) Φ(x) Φ(z, s, υ) Φ(α, γ; x) = 1 F 1 (α; γ; ⎧ ⎫ x) Φ (α, β, γ, x, y) ⎪ ⎪ ⎨ 1 ⎬  Φ2 (β, β , γ, x, y) ⎪ ⎪ ⎩ ⎭ Φ3 (β, γ, x, y) ψ(x) ℘(u) Ai(x) am(u, k) Bi(x) Bn B n (x) B(x, y)

Name of the function and the number of the formula containing its definition

Lobachevskiy angle of parallelism Elliptic integral of the third kind Weierstrass zeta function Riemann zeta functions Jacobi theta function

8.37 8.31–8.33 8.35 18.1 9.56 9.640 9.640 9.640 1.48 8.11 8.17 9.51–9.54 8.191–8.196

Elliptic theta functions

8.18, 8.19

Weierstrass sigma function Probability integral Lerch function Confluent hypergeometric function

8.17 8.25 9.55 9.21

Degenerate hypergeometric series in two variables

9.26

Euler psi function Weierstrass elliptic function Airy function Amplitude (of an elliptic function) Bairy function Bernoulli numbers Bernoulli polynomials Beta functions

8.36 8.16 page xxxvi 8.141 page xxxvi 9.61, 9.71 9.620 8.38

Gamma function Incomplete gamma functions Unit integer pulse function

continued on next page

xxxvii

xxxviii

Index of Special Functions

continued from previous page

Notation Bx (p, q) bei(z), ber(z) C C (x) C ν (a) C λn (t) C λn (x) ce2n (z, q),

ce2n+1 (z, q)

Ce2n (z, q),

Ce2n+1 (z, q)

chi(x) ci(x) cn(u) D(k) ≡ D D(ϕ, k) D n (z), D p (z) dn u e1 , e2 , e3 En E (ϕ, k)  E(k) = E 



E(k ) = E E(p; αr : q, ϕs : x) Eν (z) Ei(z) erf(x) erfc(x) = 1 − erf(x) F (ϕ, k) (α , . . . , αp ; β1 , . . . , βq ; z) F 1 p q (α, β; γ; z) = F (α, β; γ; z) F 2 1 (α; γ; z) = Φ(α, γ; z) F 1 1 FA (α :β1 , . . . , βn ; γ 1 , . . . . . . , γn : z 1 , . . . , z n ) F1 , F2 , F3 , F4  fen (z, q), Fen (z, q) . . . Feyn (z, q), Fekn (z, q) . . . G g2 , g3 gd x

Name of the function and the number of the formula containing its definition Incomplete beta functions 8.39 Thomson functions 8.56 Euler constant 9.73, 8.367 Fresnel cosine integral 8.25 Young functions 3.76 Gegenbauer polynomials 8.93 Gegenbauer functions 8.932 1 Periodic Mathieu functions (Mathieu 8.61 functions of the first kind) Associated (modified) Mathieu functions of 8.63 the first kind Hyperbolic cosine integral function 8.22 Cosine integral 8.23 Cosine amplitude 8.14 Elliptic integral 8.112 Elliptic integral 8.111 Parabolic cylinder functions 9.24–9.25 Delta amplitude 8.14 (used with the Weierstrass function) 8.162 Euler numbers 9.63, 9.72 Elliptic integral of the second kind 8.11–8.12 Complete elliptic integral of the second kind

8.11-8.12

MacRobert function Weber function Exponential integral function Error function Complementary error function Elliptic integral of the first kind Generalized hypergeometric series Gauss hypergeometric function Degenerate hypergeometric function Hypergeometric function of several variables Hypergeometric functions of two variables

9.4 8.58 8.21 8.25 8.25 8.11–8.12 9.14 9.10–9.13 9.21

Other nonperiodic solutions of Mathieu’s equation Catalan constant Invariants of the ℘(u)-function Gudermannian

8.64, 8.663

9.19 9.18

9.73 8.161 1.49

continued on next page

Index of Special Functions

xxxix

continued from previous page

Notation 

gen (z, q), Gen (z, q) Geyn (z, q), Gekn (z, q)

   a ,... ,a m,n G p,q x  b11,... ,bqp h(n) heiν (z), herν (z) H (1) ν (z),

H (2) ν (z)  πu  H (u) = ϑ1 2K  πu H 1 (u) = ϑ2 2K H n (z) Hν (z) I ν (z) I x (p, q) J ν (z) Jν (z) kν (x) K (k) = K, K (k ) = K K ν (z) kei(z), ker(z) L(x) Lν (z) Lα n (z) li(x) M λ,μ (z) O n (x) P μν (z),

P μν (x)

P ν (z), P ν (x) ⎫ ⎧ ⎨a b c ⎬ P α β γ z ⎩  ⎭ α β γ (α,β) P n (x) Q μν (z),

Q μν (x)

Q ν (z), Q ν (x) RC (x, y) RD (x, y, z) RF (x, y, z) RJ (x, y, z, p) S (x) S n (x) s μ,ν (z), S μ,ν (z)

Name of the function and the number of the formula containing its definition Other nonperiodic solutions of Mathieu’s equation

8.64, 8.663

Meijer functions

9.3

Unit integer function Thomson functions Hankel functions of the first and second kinds Theta function Theta function Hermite polynomials Struve functions Bessel functions of an imaginary argument Normalized incomplete beta function Bessel function Anger function Bateman function Complete elliptic integral of the first kind Bessel functions of imaginary argument Thomson functions Lobachevskiy function Modified Struve function Laguerre polynomials Logarithm integral Whittaker functions Neumann polynomials Associated Legendre functions of the first kind Legendre functions and polynomials

18.1 8.56

Riemann’s differential equation

9.160

Jacobi polynomials Associated Legendre functions of the second kind Legendre functions of the second kind Elliptic Function Elliptic Function Elliptic Function Elliptic Function Fresnel sine integral Schl¨afli polynomials Lommel functions

8.96

8.405, 8.42 8.192 8.192 8.95 8.55 8.406, 8.43 8.39 8.402, 8.41 8.58 9.210 3 8.11–8.12 8.407, 8.43 8.56 8.26 8.55 8.97 8.24 9.22, 9.23 8.59 8.7, 8.8 8.82, 8.83, 8.91

8.7, 8.8 8.82, 8.83 8.111 8.111 8.111 8.111 8.25 8.59 8.57 continued on next page

xl

Index of Special Functions

continued from previous page

Notation se2n+1 (z, q),

se2n+2 (z, q)

Se2n+1 (z, q),

Se2n+2 (z, q)

shi(x) si(x) snu T n (x) U n (x) U ν (w, z), V ν (w, z) W λ,μ (z) Y ν (z) Z ν (z) Zν (z)

Name of the function and the number of the formula containing its definition Periodic Mathieu functions 8.61 Mathieu functions of an imaginary 8.63 argument Hyperbolic sine integral 8.22 Sine integral 8.23 Sine amplitude 8.14 Chebyshev polynomial of the 1st kind 8.94 Chebyshev polynomials of the 2nd kind 8.94 Lommel functions of two variables 8.578 Whittaker functions 9.22, 9.23 Neumann functions 8.403, 8.41 Bessel functions 8.401 Bessel functions 5.5

Notation Symbol

Meaning The integral part of the real number x (also denoted by [x]).

x 



(b+)

a

(b−)

a

 C

PV

Contour integrals; the path of integration starting at the point a extends to the point b (along a straight line unless there is an indication to the contrary), encircles the point b along a small circle in the positive (negative) direction, and returns to the point a, proceeding along the original path in the opposite direction. Line integral along the curve C.



Principal value integral

z = x − iy

The complex conjugate of z = x + iy.

n!

= 1 · 2 · 3 . . . n,

(2n + 1)!!

= 1 · 3 . . . (2n + 1). (double factorial notation)

(2n)!!

= 2 · 4 . . . (2n). (double factorial notation)

0!! = 1 and (−1)!! = 1

(cf. 3.372 for n = 0.)

00 = 1

(cf. 0.112 and 0.113 for q = 0.)

p

=

p! p(p − 1) . . . (p − n + 1) = , 1 · 2...n n!(p − n)! [n = 1, 2, . . . , p ≥ n].

n

= a(a + 1) . . . (a + n − 1) =

k=m  n



uk ,

 m,n

,

p 0

= 1,

p n

=

p! n!(p − n)!

Γ(a+n) Γ(a)

(Pochhammer symbol). n  = um + um+1 + . . . + un . If n < m, we define uk = 0.

(a)n n 

0! = 1.



k=m

Summation over all integral values of n excluding n = 0, and summation over all integral values of n and m excluding m = n = 0, respectively.

An empty has value 0 and an empty has value 1. continued on next page

xli

xlii

Notation

continued from previous page

Symbol  δij =

Meaning i=j i = j

1 0

Kronecker delta

τ

Theta function parameter (cf. 8.18)

× and ∧

Vector product (cf. 10.11)

·

Scalar product (cf. 10.11)

∇ or “del”

Vector operator (cf. 10.21)

∇2

Laplacian (cf. 10.31)

∼

asymptotically equal to

arg z

The argument of the complex number z = x + iy.

curl or rot

Vector operator (cf. 10.21)

div

Vector operator (divergence) (cf. 10.21)

F

Fourier transform (cf. 17.21)

Fc

Fourier cosine transform (cf. 17.31)

Fs

Fourier sine transform (cf. 17.31)

grad

Vector operator (gradient) (cf. 10.21)

hi and gij

Metric coefficients (cf. 10.51)

H

Hermitian transpose of a vector or matrix (cf. 13.123)



H(x) =

0 1

x 0 such that |g(z)| ≤ M |f (z)| in some sufficiently small neighborhood of the point z0 , we write g(z) = O(f (z)). continued on next page

Notation

xliii

continued from previous page

Symbol

Meaning

q

The nome, a theta function parameter (cf. 8.18)

R

The real numbers

R(x)

A rational function

Re z ≡ x

The real part of the complex number z = x + iy.

(m)

Sn

(m)

Sn

⎧ ⎪ ⎨+1 x > 0 sign x = 0 x=0 ⎪ ⎩ −1 x < 0

Stirling number of the first kind (cf. 9.74) Stirling number of the second kind (cf. 9.74) The sign (signum) of the real number x.

T

Transpose of a vector or matrix (cf. 13.115)

Z

The integers (0, ±1, ±2, . . . )

Zb

Bilateral z transform (cf. 18.1)

Zu

Unilateral z transform (cf. 18.1)

f ∼g

f g

→ 1 in some appropriate limit

f = o(g)

f g

→ 0 in some appropriate limit

f = O(g)

|f | < Ag for constant A, in some appropriate limit

f ≈g

f is approximately equal to g

Note on the Bibliographic References The letters and numbers following equations refer to the sources used by Russian editors. The key to the letters will be found preceding each entry in the Bibliography beginning on page 1105. Roman numerals indicate the volume number of a multivolume work. Numbers without parentheses indicate page numbers, numbers in single parentheses refer to equation numbers in the original sources. Some formulas were changed from their form in the source material. In such cases, the letter a appears at the end of the bibliographic references. As an example we may use the reference to equation 3.354–5: ET I 118 (1) a The key on page 1105 indicates that the book referred to is: Erd´elyi, A. et al., Tables of Integral Transforms The Roman numeral denotes volume one of the work, 118 is the page on which the formula will be found, (1) refers to the number of the formula in this source, and the a indicates that the expression appearing in the source differs in some respect from the formula in this book. In several cases the editors have used Russian editions of works published in other languages. Under such circumstances, because the pagination and numbering of equations may be altered, we have referred the reader only to the original sources and dispensed with page and equation numbers.

xlv

Table of Integrals, Series, and Products. http://dx.doi.org/10.1016/B978-0-12-384933-5.00013-8 c 2015 Elsevier Inc. All rights reserved. Copyright 

0 Introduction 0.1 Finite sums 0.11 Progressions 0.11112 Arithmetic progression. n−1  n 1 n (a + l)(l − a + r) (a + kr) = [2a + (n − 1)r] = (a + l) = 2 2 2r k=0

[l = a + (n − 1)r is the last term] 0.112 Geometric progression. n  a (q n − 1) aq k−1 = q−1

[q = 1]

k=1

0.113 Arithmetic-geometric progression.   n−1  rq 1 − q n−1 a − [a + (n − 1)r]q n k + (a + kr)q = 1−q (1 − q)2 k=0

0.1148



n−1 

k 2 xk =

2



n+2

−n + 2n − 1 x

k=1

[q = 1, n > 1]  2  n+1 + 2n − 2n − 1 x − n2 xn + x2 + x (1 − x)3

JO (5)

0.12 Sums of powers of natural numbers 0.121

n  k=1

1.

n  k=1

2.

n  k=1

nq+1 1 q 1 q nq 1 q B2 nq−1 + B4 nq−3 + B6 nq−5 + · · · + + q+1 2 2 1 4 3 6 5 nq qnq−1 q(q − 1)(q − 2) q−3 q(q − 1)(q − 2)(q − 3)(q − 4) q−5 nq+1 + + − n n = + − ··· q+1 2 12 720 30, 240 2 [last term contains either n or n ] CE 332

kq =

k=

n(n + 1) 2

k2 =

CE 333

n(n + 1)(2n + 1) 6

CE 333

1

2

Finite sums

3.

n 

k3 =

k=1

4.

n 

n 

n 

n 

k5 =

1 2 n (n + 1)2 (2n2 + 2n − 1) 12

CE 333

k6 =

1 n(n + 1)(2n + 1)(3n4 + 6n3 − 3n + 1) 42

CE 333

k7 =

1 2 n (n + 1)2 (3n4 + 6n3 − n2 − 4n + 2) 24

CE 333

k=1

0.122

CE 333

CE 333

k=1

7.

2

1 n(n + 1)(2n + 1)(3n2 + 3n − 1) 30

k=1

6.

n(n + 1) 2

k4 =

k=1

5.



0.122

n 

 2q q+1 1  q  q−1 1  q  q−3  3 n 2 B2 nq−1 − 2 2 − 1 B4 nq−3 − · · · − q+1 2 1 4 3

(2k − 1)q =

k=1

[last term contains either n or n2 .] 1.

n 

(2k − 1) = n2

k=1

2.

n 

1 n(4n2 − 1) 3

(2k − 1)2 =

k=1

3.

n 

JO (32a)

(2k − 1)3 = n2 (2n2 − 1)

JO (32b)

k=1

4.11

n 

(mk − 1) =

k=1

5.10

n 

n [m(n + 1) − 2] 2

(mk − 1)2 =

1 n[m2 (n + 1)(2n + 1) − 6m(n + 1) + 6] 6

(mk − 1)3 =

1 n[m3 n(n + 1)2 − 2m2 (n + 1)(2n + 1) + 6m(n + 1) − 4] 4

k=1

6.10

n  k=1

0.123 0.124 1.

n 

k(k + 1)2 =

k=1

1 n(n + 1)(n + 2)(3n + 5) 12

q    1   k n2 − k 2 = q(q + 1) 2n2 − q 2 − q 4 k=1

2.10

n  k=1

k(k + 1)3 =

  1 n(n + 1) 12n3 + 63n2 + 107n + 58 60

[q = 1, 2, . . .]

0.134

0.125 0.126

Sums of products of reciprocals of natural numbers n  k=1 n  k=1

3

k! · k = (n + 1)! − 1 (n + k)! = k!(n − k)!



e K 1 π n+ 2

AD (188.1)

 1 2

WA 94

0.13 Sums of reciprocals of natural numbers ∞ n   1 Ak 1 = C + ln n + − , k 2n n(n + 1) . . . (n + k − 1)

0.13112

k=1

where

JO (59), AD (1876)

k=2

1 1 x(1 − x)(2 − x)(3 − x) · · · (k − 1 − x) dx k 0 1 1 , A3 = , A2 = 12 12 19 9 A4 = , A5 = , 120 20 863 1375 , A7 = A6 = 504 168  3  n  1 2 − 1 B4 1 B2 = (C + ln n) + ln 2 + 2 + 0.1327 + ... 2k − 1 2 8n 64n4 k=1 n  1 3 2n + 1 = − 0.133 2 k −1 4 2n(n + 1) k=2 ∞  ln n ln 2 ln 3 ln 4 (ln 2)2 = − + − · · · = C ln 2 − 0.134∗ (−1)n n 2 3 4 2 n=2 Ak =

JO (71a)a JO (184f)

0.14 Sums of products of reciprocals of natural numbers 1.

n  k=1

2.

n  k=1

3.

n  k=1

4.12

n  k=1

1 n = [p + (k − 1)q](p + kq) p(p + nq)

GI III (64)a

1 n(2p + nq + q) = [p + (k − 1)q](p + kq)[p + (k + 1)q] 2p(p + q)(p + nq)[p + (n + 1)q]

GI III (65)a

1 [p + (k − 1)q](p + kq) . . . [p + (k + l)q]

 1 1 1 − = (l + 1)q p(p + q) . . . (p + lq) (p + nq)[p + (n + 1)q] . . . [p + (n + l)q] 1 1 = [1 + (k − 1)q][1 + (k − 1)q + p] p



n  k=1

1 − 1 + (k − 1)q

n  k=1

1 1 + (k − 1)q + p



AD (1856)a

GI III (66)a

4

Finite sums

5.∗

n  k=1

0.142

1 (k + x)(k + x + p) . . . (k + x + mp)  p  n+p   1 1 1 − = mp [k + x] . . . [k + x + (m − 1)p] [k + x] . . . [k + x + (m − 1)p] k=1

k=n+1

[m > 0, 6.∗

∞  k=1

1 1 = (k + x)(k + x + p) . . . (k + x + mp) mp

k=1

[m > 0, 0.142

n  k2 + k − 1 k=1

(k + 2)!

=

p > 0,

p 

n > 0,

x = −n − mp, . . . , −2, −1]

1 [k + x] . . . [k + x + (m − 1)p]

p > 0,

x = −1, −2, . . . ]

1 n+1 − 2 (n + 2)!

JO (157)

0.15 Sums of the binomial coefficients Notation: n is a natural number 

 m

 n+k n+m+1 = 1. n n+1 k=0 n n 2. 1+ + + . . . = 2n−1 2 4 n n n 3. + + + . . . = 2n−1 1 3 5

 m n  n−1 [n ≥ 1] = (−1)m 4. (−1)k m k k=0  



 n  Γ(k + 12 )Γ(n + 2) 2j 2n + 2 = 2−(2n+1) n+1− 5.∗ 2−2j j n+1 Γ(k)Γ(n + 32 ) j=k 0.152 1. 2. 3. 0.153 1. 2. 3.

1 n nπ  2 + 2 cos 3 3 0 3 6

 n n n 1 (n − 2)π 2n + 2 cos + + + ... = 3 3 1 4 7

 n n n 1 (n − 4)π n 2 + 2 cos + + + ... = 3 3 2 5 8 n

+

n

+

n

+ ... =

1  n−1 nπ  n 2 + 2 2 cos 2 4 0 4 8 n n n n 1  n−1 nπ  2 + + + ... = + 2 2 sin 2 4 1 5 9 n n  n  n 1  n−1 nπ  2 + + + ... = − 2 2 cos 2 4 2 6 10

n

+

n

+

n

+ ... =

KR 64 (70.1) KR 62 (58.1) KR 62 (58.1) KR 64 (70.2)

KR 62 (59.1) KR 62 (59.2) KR 62 (59.3)

KR 63 (60.1) KR 63 (60.2) KR 63 (60.3)

0.157

4.

Sums of the binomial coefficients

n 3

+

n 7

0.154 1.

n 

n 

n 1  n−1 nπ  n 2 + ... = − 2 2 sin 2 4 11

n

(k + 1)

k

k=0

2.

+

(−1)k+1 k

5

= 2n−1 (n + 2)

n

=0 k

 n  N k k n−1 = 0 (−1) k

KR 63 (60.4)

[n ≥ 0]

KR 63 (66.1)

[n ≥ 2]

KR 63 (66.2)

k=1

3.

[N ≥ n ≥ 1,

00 ≡ 1]

k=0

4.

n 

(−1)k

n k

k=0

5.

n 

(−1)k

k=0

6.

n 

(−1)k

n k

k=0

0.155 1.

N k

k n = (−1)n n!

[n ≥ 0,

00 ≡ 1]

(α + k)n = (−1)n n!

[n ≥ 0,

00 ≡ 1]



(α + k)n−1 = 0

[N ≥ n ≥ 1,

00 ≡ 1,

N, n ∈ N + ]

n  (−1)k+1  n  n = k+1 k n+1

KR 63 (67)

1  n  2n+1 − 1 = k+1 k n+1

KR 63 (68.1)

k=1

2.

n  k=0

3.

n  αk+1  n  (α + 1)n+1 − 1 = k+1 k n+1

KR 63 (68.2)

k=0

4.

n n  (−1)k+1  n   1 = k k m m=1

KR 64 (69)

k=1

0.156 1.



 p  

 n m n+m = k p−k p k=0

2.

n−p  k=0

0.157 1.

n k

n    n k=0

k



2

n p+k

=



2n n

=

(2n)! (n − p)!(n + p)!

[m is a natural number]

KR 64 (71.1)

KR 64 (71.2)

 KR 64 (72.1)

6

Finite sums 2n 

2.



k

(−1)

k=0

3.

12

2n+1 

k

(−1)

k=0

4. 0.15810 1.

2.

3.

4.

2n k





2

n



= (−1)

2n + 1 k

2n n

0.158

 KR 64 (72.2)

m =0

KR 64 (72.3)

n n  (2n − 1)! 2 k = 2 k [(n − 1)!] k=1

n  





KR 64 (72.4)





 2n n k=1





 n   2n − k 2n − k − 1 2n 2k − 2k+1 k 2 = 4n − 3 · 4n n−k n−k−1 n k=1





 n   2n − k 2n − k − 1 2n k k+1 3 n 2 −2 k = (6n + 13)4 − 18n n−k n−k−1 n k=1 





 n  2n − k 2n − k − 1 2n 2k − 2k+1 k 4 = (32n2 + 104n) − (60n + 75)4n n−k n−k−1 n 2k

2n − k n−k

− 2k+1

2n − k − 1 n−k−1

k = 4n −



k=1

0.15910 1.

2.

3.

 



 n 

 2n 2n 1 n 2n 4 − − k= 2 n−k n−k−1 n k=0 

 



  n  2n 2n 2n 1 2 n (2n + 1) −4 − k = 2 n n−k n−k−1 k=0

 

 n 

 2n 2n (3n + 2) n 1 2n 3 ·4 − (3n + 1) − k = 4 2 n n−k n−k−1 k=0

0.16010 1.

   k 2n

n−1

 1 2n (1 + α)2n−1 (1 − α)  2k 1 2n α k n α + α + = (1 + α)2n 2 n 2 (1 + α)2 2 k k

k=n+1

2.

 n  Γ (r + b) B (n + a − b, b) = (−1)r Γ (r + a) Γ (a − b) r r=0

n 

k=0

0.225

Convergence tests

7

0.2 Numerical series and infinite products 0.21 The convergence of numerical series The series∞  0.211 uk = u1 + u2 + u3 + . . . k=1

is said to converge absolutely if the series ∞  0.212 |uk | = |u1 | + |u2 | + |u3 | + · · · , k=1

composed of the absolute values of its terms converges. If the series 0.211 converges and the series 0.212 diverges, the series 0.211 is said to converge conditionally. Every absolutely convergent series converges.

0.22 Convergence tests Suppose that lim |uk |

k→∞

1/k

=q

If q < 1, the series 0.211 converges absolutely. On the other hand, if q > 1, the series 0.211 diverges. (Cauchy) 0.222 Suppose that

   uk+1   =q lim k→∞ uk 

   uk+1   Here, if q < 1, the series 0.211 converges absolutely. If q > 1, the series 0.211 diverges. If  uk  approaches 1 but remains greater than unity, then the series 0.211 diverges. (d’Alembert) 0.223 Suppose that  

  uk    lim k  −1 =q k→∞ uk+1  Here, if q > 1, the series 0.211 converges absolutely. If q < 1, the series 0.211 diverges. (Raabe) 0.224 Suppose that f (x) is a positive decreasing function and that   ek f ek =q lim k→∞ f (k) ∞ for natural k. If q < 1, the series k=1 f (k) converges. If q > 1, this series diverges. (Ermakov) 0.225 Suppose that    uk  |vk | q    uk+1  = 1 + k + k p , where p > 1 and the |vk | are bounded, that is, the |vk | are all less than some M , which is independent of k. Here, if q > 1, the series 0.211 converges absolutely. If q ≤ 1, this series diverges. (Gauss)

8

Numerical series and infinite products

0.226

0.226 Suppose that a function f (x) defined for x ≥ q ≥ 1 is continuous, positive, and decreasing. Under these conditions, the series ∞  f (k) converges or diverges according as the integral

k=1 ∞

q

f (x) dx

converges or diverges (the Cauchy integral test.) 0.227 Suppose that all terms of a sequence u1 , u2 , . . . , un are positive. In such a case, the series 1.

∞ 

(−1)k+1 uk = u1 − u2 + u3 − . . .

k=1

is called an alternating series. If the terms of an alternating series decrease monotonically in absolute value and approach zero, that is, if 2.

3.12

uk+1 < uk and lim uk = 0, k→∞

the series 0.227 1 converges. Here, the remainder of the series is   ∞ n ∞       (−1)k−n+1 uk =  (−1)k+1 uk − (−1)k+1 uk  < un+1   k=n+1

k=1

(Leibniz)

k=1

0.228 If the series 1.

∞ 

vk = v1 + v2 + . . . + vk + . . .

k=1

converges and the numbers uk form a monotonic bounded sequence, that is, if |uk | < M for some number M and for all k, the series 2.

∞ 

uk vk = u1 v1 + u2 v2 + . . . + uk vk + . . .

FI II 354

k=1

converges. (Abel) 0.229 If the partial sums of the series 0.228 1 are bounded and if the numbers uk constitute a monotonic sequence that approaches zero, that is, if  n      [n = 1, 2, . . .] and lim uk = 0, vk  < M FI II 355    k→∞ k=1

then the series 0.228 2 converges. (Dirichlet)

0.234

Examples of numerical series

9

0.23–0.24 Examples of numerical series 0.231 Progressions 1.

∞  k=0

2.

∞ 

aq k =

a 1−q

(a + kr)q k =

k=0

0.232 1.

∞ 

∞ 

rq a + 1−q (1 − q)2

(−1)k+1

1 = ln 2 k

(−1)k+1

 1 1 π =1−2 = 2k − 1 (4k − 1)(4k + 1) 4

k=1

2.

[|q| < 1]

k=1

(cf. 0.113)

(cf. 1.511) ∞

k=1

(cf. 1.643) ⎤

⎡

3.

[|q| < 1]

∞ a i    ka (−1)j (a + 1)!(i − j)a ⎦ 1 ⎣ 1 = k a+1 a−i b (b − 1) b j!(a + 1 − j)! i=1 j=0 k=1

[a = 1, 2, 3, . . . , 0.233 1.

∞  1 1 1 = 1 + p + p + . . . = ζ(p) kp 2 3

b = 1]

[Re p > 1]

WH

[Re p > 0]

WH

k=1

2.

∞ 

(−1)k+1

k=1

3.10

1 = (1 − 21−p ) ζ(p) kp

∞  1 22n−1 π 2n |B2n |, = 2n k (2n)! k=1

4.

∞ 

(−1)k+1

k=1

5.

∞  k=1

6.

∞ 

1.

∞  k=1

FI II 721

k=1

1 (22n−1 − 1)π 2n |B2n | = 2n k (2n)!

1 (22n − 1)π 2n |B2n | = (2k − 1)2n 2 · (2n)! (−1)k+1

1 π 2n+1 |E2n | = (2k − 1)2n+1 22n+2 (2n)!

(−1)k+1

1 π2 = k2 12

k=1

0.234

∞  1 π2 = 2 k 6

JO (165)

JO (184b)

JO (184d)

EU

10

2.

Numerical series and infinite products ∞  k=1

3.12

∞  k=1

4.

1 π2 = 2 (2k − 1) 8

0.235

EU

(−1)k = G−1 (2k + 1)2

FI II 482

∞  (−1)k+1 π3 = (2k − 1)3 32

EU

k=1

5.

∞  k=1

6.

1 π4 = (2k − 1)4 96

EU

∞  (−1)k+1 5π 5 = (2k − 1)5 1536

EU

k=1

7.

∞ 

(−1)k+1

k=1

8.6

9.

k π2 − ln 2 = 2 (k + 1) 12

∞ 

1 = 2 − 2 ln 2 k(2k + 1) k=1   ∞  √ Γ n + 12 = π ln 4 2 n Γ (n) n=1

0.235 Sn =

∞  k=1

S1 =

1 n (4k 2 − 1) 1 , 2

S2 =

π2 − 8 , 16

S3 =

32 − 3π 2 , 64

S4 =

π 4 + 30π 2 − 384 768 JO (186)

0.236 1.

∞  k=1

2.

∞  k=1

3.

∞  k=1

4.

∞  k=1

5.

∞  k=1

6.

∞  k=1

1 = 2 ln 2 − 1 k (4k 2 − 1)

BR 51a

1 3 = (ln 3 − 1) k (9k 2 − 1) 2

BR 51a

1 3 = −3 + ln 3 + 2 ln 2 k (36k 2 − 1) 2 k

=

2

(4k 2 − 1) 1 k (4k 2

2

− 1)

12k 2 − 1

2

k (4k 2 − 1)

1 8

=

BR 52, AD (6913.3)

BR 52

3 − 2 ln 2 2

= 2 ln 2

BR 52

AD (6917.3), BR 52

0.239

7.6

Examples of numerical series ∞  k=1

11

1 π2 − 2 ln 2 = 4− 2 k(2k + 1) 4

0.237 1.

∞  k=1

2.

∞  k=1

3.

∞  k=2

4.

1 1 = (2k − 1)(2k + 1) 2 1 1 π = − (4k − 1)(4k + 1) 2 8 1 3 = (k − 1)(k + 1) 4

∞   k=1,k=m

5.

AD (6917.2), BR 52

∞   k=1,k=m

[see 0.133],

1 3 =− (m + k)(m − k) 4m2

[m is an integer]

AD (6916.1)

(−1)k−1 3 = (m − k)(m + k) 4m2

[m is an even number]

AD (6916.2)

0.238 1.

∞  k=1

2.

∞  k=1

3.

∞  k=0

1 1 = ln 2 − (2k − 1)2k(2k + 1) 2 (−1)k+1 1 = (1 − ln 2) (2k − 1)2k(2k + 1) 2 1 1 1 π = − ln 3 + √ (3k + 1)(3k + 2)(3k + 3)(3k + 4) 6 4 12 3

0.239 1.11

∞ 

∞ 

∞  k=1

4.

∞  k=1

5.

∞  k=1

π √ + ln 2 3

(−1)k+1

1 1 = 3k − 1 3

(−1)k+1

√  1 1  = √ π + 2 ln 2+1 4k − 3 4 2



π √ − ln 2 3

k+3 1 π 1 (−1)[ 2 ] = + ln 2 k 4 2

k+3 (−1)[ 2 ]

π 1 = √ 2k − 1 2 2

GI III (94)a

GI III (95)



1 1 = 3k − 2 3

k=1

3.



(−1)k+1

k=1

2.7

GI III (93)

GI III (85), BR∗ 161 (1)

 BR∗ 161 (1)

BR∗ 161 (1)

GI III (87)

12

Numerical series and infinite products ∞ 

6.

k+5 (−1)[ 3 ]

k=1 ∞ 

7.

k=1

0.241 1.

0.241

5π 1 = 2k − 1 12

GI III (88)

 1 1 π √ = − 2+1 (8k − 1)(8k + 1) 2 16

∞  1 = ln 2 2k k

JO (172g)

k=1 ∞ 

2. 3.12

π2 1 1 2 = − (ln 2) 2k k 2 12 2 k=1  ∞

 1 2k pk = √ k 1 − 4p

JO (174)

 1  − 4 < p < 14

k=0 12

4.

p ∞  pk ln(1 − x) π2 − dx = k2 6 x 1

[0 ≤ p ≤ 1]

k=1

5.12

n  

2k



k=1

6.12

n  

2k

k=1

7.12 8.12

n  



2n − k n−k 2n − k n−k



 

− 2k+1

− 2k+1





2n − (k + 1) n − (k + 1) 2n − (k + 1) n − (k + 1)



 

k = 4n −

k 2 = 4n

2n − k n−k

− 2k+1

2n − (k + 1) n − (k + 1)

 n

 n+k

13.12

k

 n

 n+k

2n−k = 4n n−k

k = (n + 1)4n − (2n + 1)

k  

 n

 2n 1 n 2n 4 + = 2 k n k=0

 n  2n n k = 4n 2 k k=0

12.12



− 3 · 4n



   n n−1

2n

 2n k 1 2n (1 + k)2n−1 (1 − k)  2n 1 kn + kk + = (1 + k)2n 2 2 n 2 (1 + k) 2 k n n=0

k=0

11.12

2n n



k 3 = (6n + 13)4n − 18n

k=n+1

10.12



2n n

 2n n k=1





 n     2n 2n − k 2n − (k + 1) 2k − 2k+1 k 4 = 32n2 + 104n − (60n + 75)4n n−k n − (k + 1) n 2k

k=1

9.12



k=0



2n n



0.245

14.12

Examples of numerical series

 n

 2n k

k=0

15.∗

n 

n 

17.

n  k=0

0.24212 0.243 1.

 n(n + 1) n n+1 k = (−1)n n! k 2



[n = 1, 2, . . . ]

 k2 n   n n+m n k (−1) = (−1) n! km . . . k1 k k

km=0

(−1)k

k=0

k=1

2n n

(−1)k

∞ 

∞ 



n!(x − k)n = n! k!(n − k)!

k=0 ∗

n2 2

(−1)k

k=0

16.∗

k 2 = (2n + 1)n4n−1 −

13

[m, n = 1, 2, . . . ]

k1 =0

n2 1 = 2 2k n n +1

[|n| > 1]

1 1 1 = [p + (k − 1)q](p + kq) . . . [p + (k + l)q] (l + 1)q p(p + q) . . . (p + lq) (see also 0.141 3)

12

2.

∞  k=1

∞ 

3.

k=0

0.244 1.

∞  k=1 ∞ 

2.

x 1 1 tp−1 (1 − t)t = dt [p + (k − 1)q][p + (k − 1)q + 1][p + (k − 1)q + 2] . . . [p + (k − 1)q + l] l! 0 1 − xtq   p > 0, x2 < 1 BR∗ 161 (2), AD (6.704) k−1

1 (2k + 1)3

3.

∞  k=1

    (2k + 1)πx π3 (2k + 1)π 1 tanh + x tanh = x 2 2x 16

1 1 = (k + p)(k + q) q−p k+1

(−1)

k=1 10



1 = p + (k − 1)q



1

0

0

1

xp − xq dx 1−x

[p > −1,

tp−1 dt 1 + tq

[p > 0,

q  1 1 1 = (k + p)(k + q) q − p m=p+1 m

Summations of reciprocals of factorials 0.245 1.

∞  1 =e k! k=0

2.12

∞  (−1)k 1 = k! e k=0

q > −1, q > 0]

[q > p > −1,

p = q]

GI III (90)

BR∗ 161 (1)

p and q integers]

14

3.12

4.

5.

6.

7. 8.12

Numerical series and infinite products ∞  k=1 ∞ 

1.

2.

3.

4.

5.

6.

k=0 ∞  k=0 ∞ 

∞  k=0 ∞  k=0 ∞  k=0 ∞  k=0 ∞  k=0 ∞  k=0

0.247 0.248

k 1 = (2k + 1)! 2e

k =1 (k + 1)! k=1

 ∞  1 1 1 = e+ (2k)! 2 e k=0

 ∞  1 1 1 = e− (2k + 1)! 2 e

k=1

0.246

∞  k=1 ∞  k=1

(−1)k = cos 1 (2k)! (−1)k−1 = sin 1 (2k − 1)! 1 2

(k!)

= I 0 (2) ≈ 2.27958530

1 = I 1 (2) ≈ 1.590636855 k!(k + 1)! 1 = I n (2) k!(k + n)! (−1)k 2

(k!)

= J 0 (2) ≈ 0.22389078

(−1)k = J 1 (2) ≈ 0.57672481 k!(k + 1)! (−1)k = J n (2) k!(k + n)! k! 1 = (n + k − 1)! (n − 2) · (n − 1)! kn = Sn , k! S1 = e, S5 = 52e,

0.2497

0.246

∞  (k + 1)3 k=0

k!

= 15e

S2 = 2e, S6 = 203e,

S3 = 5e, S7 = 877e,

S4 = 15e S8 = 4140e

0.263

Examples of infinite products

15

0.25 Infinite products 0.250 Suppose that a sequence of numbers a1 , a2 , . . . , ak , . . . is given. If the limit lim

n→∞

n 

(1 + ak )

k=1

exists, whether finite or infinite (but of definite sign), this limit is called the value of the infinite product ∞  (1 + ak ) and we write k=1

1.

lim

n→∞

n 

(1 + ak ) =

k=1

∞ 

(1 + ak )

k=1

If an infinite product has a finite nonzero value, it is said to converge. Otherwise, the infinite product is said to diverge. We assume that no ak is equal to −1. FI II 400 0.251 For the infinite product 0.250 1. to converge, it is necessary that lim ak = 0. k→∞

FI II 403

0.252 If ak > 0 or ak < 0 for all values of the index k starting with some particular value, then, for the ∞ product 0.250 1 to converge, it is necessary and sufficient that the series k=1 ak converges. ∞ ∞   (1 + ak ) is said to converge absolutely if the product (1 + |ak |) converges. 0.253 The product k=1

k=1

FI II 403

0.254 Absolute convergence of an infinite product implies its convergence. ∞ ∞   0.255 The product (1 + ak ) converges absolutely if, and only if, the series ak converges absok=1

lutely.

k=1

FI II 406

0.26 Examples of infinite products 0.261 0.262 1.

2.

3.

 ∞

 √ (−1)k+1 = 2 1+ 2k − 1

 ∞

 1 1 1− 2 = k 2 k=2

 ∞  1 2 1− = 2 (2k) π k=1

 ∞  1 π 1− = (2k + 1)2 4 k=1

0.263 1.

EU

k=1

2 e= · 1

1/4

1/8

1/2

6·8 10 · 12 · 14 · 16 4 ... 3 5·7 9 · 11 · 13 · 15

FI II 401

FI II 401

FI II 401

16

3.

Functional series

π = 2

0.264

1/2 2 1/4 3 1/8 4 4 1/16 1 2 2 ·4 2 ·4 ··· 2 1·3 1 · 33 1 · 36 · 5

where the nth factor is the (n + 1)th root of the product 0.264 1.

2.

∞ 

0.2668

+ 1)(−1)

k+1

( nk ) .

FI II 402

where the nth factor is the (n + 1)th root of the product Euler constant, denoted in other works by γ. 0.265

k=0 (k

√ k

e 1 k=1 1 + k

1/2 2 1/3 3 1/4 4 4 1/5 2 2 ·4 2 ·4 2 eC = ··· 1 1·3 1 · 33 1 · 36 · 5 eC =

n

  ! ! 2 1 1 1 1 "1 1 1 1 1 = · + · + + ... π 2 2 2 2 2 2 2 2 2 ∞    k 1 1 + x2 = 1−x

n

k=0 (k

+ 1)(−1)

k+1

( nk ) . Here C is the

FI II 402

[0 < x < 1]

FI II 401

k=0

0.3 Functional series 0.30 Definitions and theorems 0.301 The series 1.

∞ 

fk (x),

k=1

the terms of which are functions, is called a functional series. The set of values of the independent variable x for which the series 0.301 1 converges constitutes what is called the region of convergence of that series. 0.302 A series that converges for all values of x in a region M is said to converge uniformly in that region if, for every ε ≥ 0, there exists a number N such that, for n > N , the inequality  ∞       fk (x) < ε    k=n+1

holds for all x in M .

0.303 If the terms of the functional series 0.301 1 satisfy the inequalities: |fk (x)| < uk (k = 1, 2, 3, . . .) , throughout the region M , where the uk are the terms of some convergent numerical series ∞  uk = u1 + u2 + . . . + uk + . . . , k=1

the series 0.301 1 converges uniformly in M . (Weierstrass)

0.311

Power series

17

0.304 Suppose that the series 0.301 1 converges uniformly in a region M and that a set of functions gk (x) constitutes (for each x) a monotonic sequence, and that these functions are uniformly bounded, that is, suppose that a number L exists such that the inequalities 1.

|gn (x)| ≤ L

2.

hold for all n and x. Then, the series ∞  fk (x)gk (x) k=1

converges uniformly in the region M . (Abel)

FI II 451

0.305 Suppose that the partial sums of the series 0.301 1 are uniformly bounded; that is, suppose that, for some L and for all n and x in M , the inequalities   n     fk (x) < L    k=1

hold. Suppose also that for each x the functions gn (x) constitute a monotonic sequence that approaches zero uniformly in the region M . Then, the series 0.304 2 converges uniformly in the region M . (Dirichlet) FI II 451 6

0.306 If the functions fk (x) (for k = 1, 2, 3, . . . ) are integrable on the interval [a, b] and if the series 0.301 1 made up of these functions converges uniformly on that interval, this series may be integrated termwise; that is, $ b # ∞ ∞ b  fk (x) dx = fk (x) dx [a ≤ x ≤ b] FI II 459 a

k=1

k=1

a

 0.307 Suppose that the functions fk (x) (for k = 1, 2, 3, . . . ) have continuous derivatives ∞ fk (x) on the interval [a, b]. If the series 0.301 1 converges on this interval and if the series k=1 fk (x) of these derivatives converges uniformly, the series 0.301 1 may be differentiated termwise; that is, ∞  ∞   fk (x) = fk (x) FI II 460 k=1

k=1

0.31 Power series 0.311 A functional series of the form 1.

∞ 

ak (x − ξ)k = a0 + a1 (x − ξ) + a2 (x − ξ)2 + . . .

k=0

is called a power series. The following is true of any power series: if it is not everywhere convergent, the region of convergence is a circle with its center at the point ξ and a radius equal to R; at every interior point of this circle, the power series 0.311 1 converges absolutely and outside this circle, it diverges. This circle is called the circle of convergence and its radius is called the radius of convergence. If the series converges at all points of the complex plane, we say that the radius of convergence is infinite (R = +∞).

18

Functional series

0.312

0.312 Power series may be integrated and differentiated termwise inside the circle of convergence; that is,  x  ∞ ∞  ak k (x − ξ)k+1 , dx = ak (x − ξ) k+1 ξ k=0 k=0 ∞  ∞   d ak (x − ξ)k = kak (x − ξ)k−1 . dx k=0

k=1

The radius of convergence of a series that is obtained from termwise integration or differentiation of another power series coincides with the radius of convergence of the original series. Operations on power series 0.313 Division of power series.

∞  k=0 ∞ 

bk xk = ak xk

∞ 1  ck xk , a0 k=0

k=0

where

n

cn + or

⎡

1 cn−k ak − bn = 0, a0 k=1

a1 b 0 − a0 b 1 a2 b 0 − a 0 b 2 a3 b 0 − a0 b 3 .. .

a0 a1 a2 .. .

0 a0 a1 .. .

···

⎢ ⎢ (−1) ⎢ ⎢ cn = ⎢ .. an0 ⎢ . ⎢ ⎣an−1 b0 − a0 bn−1 an−2 an−3 · · · an b 0 − a0 b n an−1 an−2 · · · 0.314 Power series raised to powers. $n #∞ ∞   k ak x = ck xk , n

k=0

where c0 =

an0 ,

cm

m 1  = (kn − m + k)ak cm−k ma0

⎤ 0 0⎥ ⎥ 0⎥ ⎥ ⎥ ⎥ ⎥ a0 ⎦ a1

AD (6360)

k=0

for m ≥ 1

[n is a natural number]

AD (6361)

k=1

0.315 The substitution of one series into another. ∞ ∞ ∞    bk y k = ck xk y= ak xk ; k=1

c1 = a 1 b 1 ,

k=1

c2 = a2 b1 + a21 b2 ,

k=1

c3 = a3 b1 + 2a1 a2 b2 + a31 b3 ,

c4 = a4 b1 + a22 b2 + 2a1 a3 b2 + 3a21 a2 b3 + a41 b4 ,

...

AD (6362)

0.319

Power series

0.316 Multiplication of power series ∞ ∞ ∞    ak xk bk xk = ck xk k=0

k=0

cn =

k=0

19

n 

ak bn−k

FI II 372

k=0

Taylor series 0.317 If a function f (x) has derivatives of all orders throughout a neighborhood of a point ξ, then we may write the series 1.

2.

(x − ξ)2  (x − ξ)3  (x − ξ)  f (ξ) + f (ξ) + f (ξ) + . . . , 1! 2! 3! which is known as the Taylor series of the function f (x).

f (ξ) +

The Taylor series converges to the function f (x) if the remainder n  (x − ξ)k (k) f (ξ) Rn (x) = f (x) − f (ξ) − k! k=1

approaches zero as n → ∞. The following are different forms for the remainder of a Taylor series. 3. 4. 5.

Rn (x) =

(x − ξ)n+1 (n+1) f (ξ + θ(x − ξ)) (n + 1)!

(Lagrange)

(x − ξ)n+1 (1 − θ)n f (n+1) (ξ + θ(x − ξ)) [0 < θ < 1] (Cauchy) n! ψ(x − ξ) − ψ(0) (x − ξ)n (1 − θ)n (n+1) f Rn (x) =  (ξ + θ(x − ξ)) ψ [(x − ξ)(1 − θ)] n! [0 < θ < 1] , (Schl¨ omilch) where ψ(x) is an arbitrary function satisfying the following two conditions: (1) It and its derivative ψ  (x) are continuous in the interval (0, x − ξ); and (2) the derivative ψ  (x) does not change sign in that interval. If we set ψ(x) = xp+1 , we obtain the following form for the remainder: Rn (x) =

(x − ξ)n+1 (1 − θ)n−p−1 (n+1) f (ξ + θ(x − ξ)) (p + 1)n! 1 x (n+1) Rn (x) = f (t)(x − t)n dt n! ξ Rn (x) =

6.

[0 < θ < 1]

[0 < p ≤ n,

0 < θ < 1]

(Rouch´e)

0.318 Other forms in which a Taylor series may be written: ∞  xk (k) x x2 f (a) = f (a) + f  (a) + f  (a) + . . . 1.11 f (a + x) = k! 1! 2! k=0

2.12

f (x) =

∞  xk k=0

k!

f (k) (0) = f (0) +

x  x2 f (0) + f  (0) + . . . 1! 2!

(Maclaurin series)

0.319 The Taylor series of functions of several variables: ∂f (ξ, η) ∂f (ξ, η) + (y − η) f (x, y) = f (ξ, η) + (x − ξ) ∂x ∂y

 2 2 ∂ f (ξ, η) ∂ 2 f (ξ, η) 1 2 2 ∂ f (ξ, η) (x − ξ) + (y − η) + ... + 2(x − ξ)(y − η) + 2! ∂x2 ∂x ∂y ∂y 2

20

Functional series

0.320

0.32 Fourier series 0.320 Suppose that f (x) is a periodic function of period 2l and that it is absolutely integrable (possibly improperly) over the interval (−l, l). The following trigonometric series is called the Fourier series of f (x): 1.

 ∞

a0  kπx kπx ak cos + + bk sin , 2 l l k=1

2. 3.11

the coefficients of which (the Fourier coefficients) are given by the formulas 1 α+2l 1 l kπt kπt dt = dt (k = 0, 1, 2, . . .) f (t) cos f (t) cos ak = l −l l l α l 1 α+2l 1 l kπt kπt dt = dt (k = 1, 2, . . .) bk = f (t) sin f (t) sin l −l l l α l

Convergence tests 0.321 The Fourier series of a function f (x) at a point x0 converges to the number f (x0 + 0) + f (x0 − 0) , 2 if, for some h > 0, the integral h |f (x0 + t) + f (x0 − t) − f (x0 + 0) − f (x0 − 0)| dt t 0 exists. Here, it is assumed that the function f (x) either is continuous at the point x0 or has a discontinuity of the first kind (a saltus) at that point and that both one-sided limits f (x0 + 0) and f (x0 − 0) exist. (Dini) FI III 524 0.322 The Fourier series of a periodic function f (x) that satisfies the Dirichlet conditions on the interval [a, b] converges at every point x0 to the value 12 [f (x0 + 0) + f (x0 − 0)]. (Dirichlet) We say that a function f (x) satisfies the Dirichlet conditions on the interval [a, b] if it is bounded on that interval and if the interval [a, b] can be partitioned into a finite number of subintervals inside each of which the function f (x) is continuous and monotonic. 0.323 The Fourier series of a function f (x) at a point x0 converges to 12 [f (x0 + 0) + f (x0 − 0)] if f (x) is of bounded variation in some interval (x0 − h, x0 + h) with center at x0 . (Jordan–Dirichlet) FI III 528 The definition of a function of bounded variation. Suppose that a function f (x) is defined on some interval [a, b], where a < b. Let us partition this interval in an arbitrary manner into subintervals with the dividing points a = x0 < x1 < x2 < . . . < xn−1 < xn = b and let us form the sum n  |f (xk ) − f (xk−1 )| k=1

Different partitions of the interval [a, b] (that is, different choices of points of division xi ) yield, generally speaking, different sums. If the set of these sums is bounded above, we say that the function f (x) is of bounded variation on the interval [a, b]. The least upper bound of these sums is called the total variation of the function f (x) on the interval [a, b].

0.330

Asymptotic series

21

0.324 Suppose that a function f (x) is piecewise-continuous on the interval [a, b] and that in each interval of continuity it has a piecewise-continuous derivative. Then, at every point x0 of the interval [a, b], the Fourier series of the function f (x) converges to 12 [f (x0 + 0) + f (x0 − 0)]. 0.325 A function f (x) defined in the interval (0, l) can be expanded in a cosine series of the form ∞

1.

kπx a0  + , ak cos 2 l k=1

where 2.

2 ak = l



l

0

f (t) cos

kπt dt l

0.326 A function f (x) defined in the interval (0, l) can be expanded in a sine series of the form 1.

∞ 

bk sin

k=1

where 2.

bk =

2 l

0

kπx , l l

f (t) sin

kπt dt l

The convergence tests for the series 0.325 1 and 0.326 1 are analogous to the convergence tests for the series 0.320 1 (see 0.321–0.324). 0.327 The Fourier coefficients ak and bk (given by formulas 0.320 2 and 0.320 3) of an absolutely integrable function approach zero as k → ∞. If a function f (x) is square-integrable on the interval (−l, l), the equation of closure is satisfied: ∞  1 l 2 a20   2 2 a = + b f (x) dx (A. M. Lyapunov) FI III 705 + k k 2 l −l k=1

0.328 Suppose that f (x) and ϕ(x) are two functions that are square-integrable on the interval (−l, l) and that ak , bk and αk , βk are their Fourier coefficients. For such functions, the generalized equation of closure (Parseval’s equation) holds: ∞ a0 α0  1 l (ak αk + bk βk ) = f (x)ϕ(x) dx FI III 709 + 2 l −l k=1 For examples of Fourier series, see 1.44 and 1.45.

0.33 Asymptotic series 0.330 Included in the collection of all divergent series is the broad class of series known as asymptotic or semiconvergent series. Despite the fact that these series diverge, the values of the functions that they represent can be calculated with a high degree of accuracy if we take the sum of a suitable number of terms of such series. In the case of alternating asymptotic series, we obtain greatest accuracy if we break off the series in question at whatever term is of lowest absolute value. In this case, the error (in absolute value) does not exceed the absolute value of the first of the discarded terms (cf. 0.227 3). Asymptotic series have many properties that are analogous to the properties of convergent series and, for that reason, they play a significant role in analysis.

22

Certain formulas from differential calculus

0.331

The asymptotic expansion of a function is denoted as follows: ∞  An f (z) ∼ zn n=0 ∞  An The definition of an asymptotic expansion. The divergent series is called the asymptotic expansion zn n=0

of a function f (z) in a given region of values of arg z if the expression Rk (z) = z k [f (z) − Sk (z)], where k  An Sk (z) = , satisfies the condition lim Rk (z) = 0 for fixed k. FI II 820 zn |z|→∞ n=0 A divergent series that represents the asymptotic expansion of some function is called an asymptotic series. 0.331 Properties of asymptotic series 1.

The operations of addition, subtraction, multiplication, and raising to a power can be performed on asymptotic series just as on absolutely convergent series. The series obtained as a result of these operations will also be asymptotic.

2.

One asymptotic series can be divided by another provided that the first term A0 of the divisor is not equal to zero. The series obtained as a result of division will also be asymptotic. FI II 823-825

3.

An asymptotic series can be integrated termwise, and the resultant series will also be asymptotic. In contrast, differentiation of an asymptotic series is, in general, not permissible. FI II 824

4.

A single asymptotic expansion can represent different functions. On the other hand, a given function can be expanded in an asymptotic series in only one manner.

0.4 Certain formulas from differential calculus 0.41 Differentiation of a definite integral with respect to a parameter ϕ(a) d dψ(a) d ϕ(a) dϕ(a) − f (ψ(a), a) + f (x, a) dx f (x, a) dx = f (ϕ(a), a) da ψ(a) da da da ψ(a) 0.411 In particular, d a 1. f (x) dx = f (a) da b a d f (x) dx = −f (b) 2. db b 0.410

0.42 The nth derivative of a product (Leibniz’s rule) Suppose that u and v are n-times-differentiable functions of x. Then,



 2 n−2

 3 n−3 v v dn (uv) dn v dn u n du dn−1 v n d ud n d ud = u + + + + ···+ v n n n n−1 2 n−2 3 n−3 1 dx dx 2 dx dx 3 dx dx dx dx dx

FI II 680

The nth derivative of a composite function

0.432

23

or, symbolically, dn (uv) = (u + v)n dxn

FI I 272

0.43 The nth derivative of a composite function 0.430 If f (x) = F (y)and y = ϕ(x), then 1.

2.

U1  U2  U3  Un (n) dn F (y) + F (y) + F (y) + . . . + F (y), f (x) = dxn 1! 2! 3! n! where n dn k k dn k−1 k(k − 1) 2 dn k−2 k−1 k−1 d y y y y − y + y − . . . + (−1) ky AD (7361) GO Uk = dxn 1! dxn 2! dxn dxn

i  j  h

(l) k  n! dm F y  y y dn y f (x) = · · · , dxn i!j!h! . . . k! dy m 1! 2! 3! l!  Here, the symbol indicates summation over all solutions in non negative integers of the equation i + 2j + 3h + . . . + lk = n and m = i + j + h + . . . + k.

0.431 1.

(−1)n

dn F dxn





 1 n−1 n 1 1 1 = 2n F (n) + 2n−1 F (n−1) x x x x 1!

x (n − 1)(n − 2) n(n − 1) (n−2) 1 F + ... + x2n−2 2! x ⎡

2.

(−1)n

AD (7362.1)

 n   a n−1  n   a n−2 dn a 1 a  a n ex = n ex ⎣ + (n − 1) + (n − 1)(n − 2) n dx x x x x 1 2 ⎤  n   a n−3 + (n − 1)(n − 2)(n − 3) + . . .⎦ x 3 AD (7362.2)

0.432 1.

  n(n − 1)   dn  2  (2x)n−2 F (n−1) x2 F x = (2x)n F (n) x2 + n dx 1!   n(n − 1)(n − 2)(n − 3) (2x)n−4 F (n−2) x2 + + 2!   n(n − 1)(n − 2)(n − 3)(n − 4)(n − 5) (2x)n−6 F (n−3) x2 + . . . + 3! ⎡

2.12

AD (7363.1)

2 dn ax2 n(n − 1) n(n − 1)(n − 2)(n − 3) + e = (2ax)n eax ⎣1 + 2 dxn 1! (4ax2 ) 2! (4ax2 ) ⎤ n(n − 1)(n − 2)(n − 3)(n − 4)(n − 5) + + · · ·⎦ [there are n2 terms] 3 3! (4ax2 )

AD (7363.2)

24

3.

Certain formulas from differential calculus

0.433

 dn  p(p − 1)(p − 2) . . . (p − n + 1)(2ax)n 2 p 1 + ax = n−p dxn (1 + ax2 )  

2 n(n − 1) 1 + ax2 n(n − 1)(n − 2)(n − 3) 1 + ax2 × 1+ + + ... , 1!(p − n + 1) 4ax2 2!(p − n + 1)(p − n + 2) 4ax2 AD (7363.3)

4. 5.

6. 0.433 1.

 1 dm−1  (2m − 1)!! 2 m− 2 1 − x = (−1)m−1 sin (m arccos x) dxm−1 m





 2k n+1 n  a a n+1 b n ∂ k = n! (−1) (−1) n 2 2 2 2 ∂a a +b a +b a 2k (−1)n

∂n ∂an



b a2 + b 2



= n!

0≤2k≤n+1

a a2 + b 2

n+1 

0≤2k≤n

(−1)k



n+1 2k + 1

 2k+1 b a

AD (7363.4) (3.944.12)

(3.944.11)

√ √ √ dn √  F (n) ( x) n(n − 1) F (n−1) ( x) (n + 1)n(n − 1)(n − 2) F (n−2) ( x) F x = √ n − √ n+1 + √ n+2 − . . . dxn 1! 2! (2 x) (2 x) (2 x) AD (7364.1)

2. 0.43412 0.43512

n−1 √ 2n−1 dn  (2n − 1)!! a 1 2 √ a 1 + a x = − dxn 2n x x



 n 1 n 1 n n 2 dn p n−p p−1 d y p−2 d (y ) y y y =p − + − ... dxn dxn dxn n 1 p−1 2 p−2 dn ln y = dxn

   n 1 dn y  n  1 dn (y 2 )  n  1 dn (y 3 ) n − + x − . . . 1 1 · y dxn 2 2 · y 2 dxn 3 3 · y 3 dxn

AD (7364.2)

[n < p] AD (737.1) AD (737.2)

0.44 Integration by substitution 0.44011 Let f (g(x)) and g(x) be continuous in [a, b]. Further, let g  (x) exist and be continuous there. b g(b)  Then f [g(x)]g (x) dx = f (u) du. a

g(a)

Table of Integrals, Series, and Products. http://dx.doi.org/10.1016/B978-0-12-384933-5.00001-1 c 2015 Elsevier Inc. All rights reserved. Copyright 

1 Elementary Functions 1.1 Power of Binomials 1.11 Power series ∞    q q(q − 1) 2 q(q − 1) . . . (q − k + 1) k x + ··· + x + ··· = xk 2! k! k k=0 If q is neither a natural number nor zero, the series converges absolutely for |x| < 1 and diverges for |x| > 1. For x = 1, the series converges for q > −1 and diverges for q ≤ −1. For x = 1, the series converges absolutely for q > 0. For x = −1, it converges absolutely for q > 0 and diverges for q < 0. If q = n is a natural number, the series 1.110 is reduced to the finite sum 1.111. FI II 425 n    n xk an−k 1.111 (a + x)n = k

1.110 (1 + x)q = 1 + qx +

1.112 1.

k=0

(1 + x)−1 = 1 − x + x2 − x3 + · · · =

∞ 

(−1)k−1 xk−1

k=1

(see also 1.121 2) 2.

(1 + x)−2 = 1 − 2x + 3x2 − 4x3 + · · · =

∞ 

(−1)k−1 kxk−1

k=1

3.11 4.

1·1 2 1·1·3 3 1·1·3·5 4 1 x + x − x + ... (1 + x)1/2 = 1 + x − 2 2·4 2·4·6 2·4·6·8 1·3 2 1·3·5 3 1 x − x + ... (1 + x)−1/2 = 1 − x + 2 2·4 2·4·6 ∞

1.113 1.114 1.

 x = kxk (1 − x)2

 2  x 1

AD (6350.3)

[x < 2π]

FI II 520

k=1

∞

2.

 2  x 1 and x < 0 or 0 < a < 1 and x > 0]

(−1)k e−2kx

[x > 0]

k=1

2.

3.

sech x = 2

∞ 

(−1)k e−(2k+1)x

k=0 ∞ 

cosech x = 2

e−(2k+1)x

k=0



4.

∞  cos2n x sin x = exp − 2n n=1

[x > 0] [x > 0]

 [0 ≤ x ≤ π]

MO 216

28

Trigonometric and Hyperbolic Functions

1.311

1.3–1.4 Trigonometric and Hyperbolic Functions 1.30 Introduction The trigonometric and hyperbolic sines are related by the identities sinh x =

1 sin(ix), i

sin x =

1 sinh(ix). i

The trigonometric and hyperbolic cosines are related by the identities cosh x = cos(ix),

cos x = cosh(ix).

Because of this duality, every relation involving trigonometric functions has its formal counterpart involving the corresponding hyperbolic functions, and vice-versa. In many (though not all) cases, both pairs of relationships are meaningful. The idea of matching the relationships is carried out in the list of formulas given below. However, not all the meaningful “pairs” are included in the list.

1.31 The basic functional relations 1.311

 1  ix e − e−ix 2i = −i sinh(ix)

1.

sin x =

2.

sinh x =

3.

cos x =

4.

cosh x =

5. 6. 7. 8.

 1 x e − e−x 2 = −i sin(ix)

 1  ix e + e−ix 2 = cosh(ix)  1 x e + e−x 2 = cos(ix)

1 sin x = tanh(ix) cos x i 1 sinh x = tan(ix) tanh x = cosh x i 1 cos x = = i coth(ix) cot x = sin x tan x 1 cosh x = = i cot (ix) coth x = sinh x tanh x tan x =

1.312 1.

cos2 x + sin2 x = 1

1.314

2.

The basic functional relations

cosh2 x − sinh2 x = 1

1.313 1. sin (x ± y) = sin x cos y ± sin y cos x 2.

sinh (x ± y) = sinh x cosh y ± sinh y cosh x

3.

sin (x ± iy) = sin x cosh y ± i sinh y cos x

4.

sinh (x ± iy) = sinh x cos y ± i sin y cosh x

5.

cos (x ± y) = cos x cos y ∓ sin x sin y

6.

cosh (x ± y) = cosh x cosh y ± sinh x sinh y

7.

cos (x ± iy) = cos x cosh y ∓ i sin x sinh y

8.

cosh (x ± iy) = cosh x cos y ± i sinh x sin y tan x ± tan y tan (x ± y) = 1 ∓ tan x tan y tanh x ± tanh y tanh (x ± y) = 1 ± tanh x tanh y tan x ± i tanh y tan (x ± iy) = 1 ∓ i tan x tanh y tanh x ± i tan y tanh (x ± iy) = 1 ± i tanh x tan y

9. 10. 11. 12. 1.314 1. 2. 3. 4. 5. 6. 7. 8. 9.

1 1 (x ± y) cos (x ∓ y) 2 2 1 1 sinh x ± sinh y = 2 sinh (x ± y) cosh (x ∓ y) 2 2 1 1 cos x + cos y = 2 cos (x + y) cos (x − y) 2 2 1 1 cosh x + cosh y = 2 cosh (x + y) cosh (x − y) 2 2 1 1 cos x − cos y = 2 sin (x + y) sin (y − x) 2 2 1 1 cosh x − cosh y = 2 sinh (x + y) sinh (x − y) 2 2 sin (x ± y) tan x ± tan y = cos x cos y sinh (x ± y) tanh x ± tanh y = cosh x cosh y



π 1 π 1 sin (x − y) ± sin x ± cos y = ±2 sin (x + y) ± 4

4

2 2 π 1 π 1 cos (x − y) ∓ = ±2 cos (x + y) ∓

4 2

4 2 π 1 π 1 cos (x ∓ y) ∓ = 2 sin (x ± y) ± 2 4 2 4 sin x ± sin y = 2 sin

29

30

Trigonometric and Hyperbolic Functions

 10.

a sin x ± b cos x = a

1+

2

b b sin x ± arctan a a [a = 0]

 11.

±a sin x + b cos x = b

1+

 a 2 b

  a  cos x ∓ arctan b

[b = 0]

2 r r 1 a sin x ± b sin y = q 1 + sin (x ± y) + arctan q 2 q



1 1 [q = 0] q = (a + b) cos (x ∓ y) , r = (a − b) sin (x ∓ y) 2 2   s 

 s 2 1 [t = 0] a cos x + b cos y = t 1 + cos (x ∓ y) + arctan t 2 t 

2 t t 1 = −s 1 + [s = 0] sin (x ∓ y) − arctan s 2 s



1 1 s = (a − b) sin (x ± y) , t = (a + b) cos (x ± y) 2 2 

12.12

13.12

1.315 1.

sin2 x − sin2 y = sin(x + y) sin(x − y) = cos2 y − cos2 x

2.

sinh2 x − sinh2 y = sinh(x + y) sinh(x − y) = cosh2 x − cosh2 y

3.

cos2 x − sin2 y = cos(x + y) cos(x − y) = cos2 y − sin2 x

4.

sinh2 x + cosh2 y = cosh(x + y) cosh(x − y) = cosh2 x + sinh2 y

1.316 1. 2.

n

(cos x + i sin x) = cos nx + i sin nx n

(cosh x + sinh x) = sinh nx + cosh nx

1.317 1. 2. 3. 4. 5.

1.315

x sin = ± 2



1 (1 − cos x) 2  x 1 sinh = ± (cosh x − 1) 2 2  x 1 cos = ± (1 + cos x) 2 2  x 1 cosh = (cosh x + 1) 2 2 x 1 − cos x sin x tan = = 2 sin x 1 + cos x

[n is an integer] [n is an integer]

1.321

6.

Trigonometric and hyperbolic functions: expansion in multiple angles

tanh

31

cosh x − 1 sinh x x = = 2 sinh x cosh x + 1

The signs in front of the radical in formulas 1.317 1, 1.317 2, and 1.317 3 are taken so as to agree with the signs of the left hand members. The sign of the left hand members depends in turn on the value of x.

1.32 The representation of powers of trigonometric and hyperbolic functions in terms of functions of multiples of the argument (angle) 1.320 1.

2.

n−1   2n 2n n−k (−1) 2 cos 2(n − k)x + sin k n k=0 n−1  (−1)n  2n 2n 2n n−k sinh x = 2n (−1) 2 cosh 2(n − k)x + 2 k n 2n

1 x = 2n 2

k=0

3.

4.

5.

6.

sin2n−1 x =

n−1 

1 22n−2

(−1)n+k−1

k=0

8.

2n − 1 k

cos2n−1 x = 2n−1

cosh

n−1 

1 22n−2

x=

k=0

1 22n−2

2n − 1 k

3. 4.

KR 56 (10, 4)

n−1  k=0

2n − 1 k

cos(2n − 2k − 1)x

cosh(2n − 2k − 1)x

1 (− cos 2x + 1) 2 1 sin3 x = (− sin 3x + 3 sin x) 4 1 sin4 x = (cos 4x − 4 cos 2x + 3) 8 1 (sin 5x − 5 sin 3x + 10 sin x) sin5 x = 16

sin2 x =

KR 56 (10, 1)

1.321

2.

sin(2n − 2k − 1)x



n−1 (−1)n−1  n+k−1 2n − 1 sinh(2n − 2k − 1)x sinh x = 2n−2 (−1) 2 k k=0 n−1   1 2n 2n 2n cos x = 2n 2 cos 2(n − k)x + 2 k n k=0 n−1   1 2n 2n 2n cosh x = 2n 2 cosh 2(n − k)x + 2 k n

Special cases

1.

2n−1

k=0

7.



KR 56 (10, 2)

KR 56 (10, 3)

32

5. 6.

Trigonometric and Hyperbolic Functions

1 (− cos 6x + 6 cos 4x − 15 cos 2x + 10) 32 1 (− sin 7x + 7 sin 5x − 21 sin 3x + 35 sin x) sin7 x = 64

sin6 x =

1.322 1.

sinh2 x =

2.

sinh3 x =

3.

sinh4 x =

4.

sinh5 x =

5.12

sinh6 x =

6.12

sinh7 x =

1 (cosh 2x − 1) 2 1 (sinh 3x − 3 sinh x) 4 1 (cosh 4x − 4 cosh 2x + 3) 8 1 (sinh 5x − 5 sinh 3x + 10 sinh x) 16 1 (cosh 6x − 6 cosh 4x + 15 cosh 2x − 10) 32 1 (sinh 7x − 7 sinh 5x + 21 sinh 3x − 35 sinh x) 64

1.323 1.

cos2 x =

2.

cos3 x =

3.

cos4 x =

4.

cos5 x =

5.

cos6 x =

6.

cos7 x =

1 (cos 2x + 1) 2 1 (cos 3x + 3 cos x) 4 1 (cos 4x + 4 cos 2x + 3) 8 1 (cos 5x + 5 cos 3x + 10 cos x) 16 1 (cos 6x + 6 cos 4x + 15 cos 2x + 10) 32 1 (cos 7x + 7 cos 5x + 21 cos 3x + 35 cosh x) 64

1.324 1.

cosh2 x =

2.

cosh3 x =

3.

cosh4 x =

4.

cosh5 x =

5.

cosh6 x =

6.

cosh7 x =

1 (cosh 2x + 1) 2 1 (cosh 3x + 3 cosh x) 4 1 (cosh 4x + 4 cosh 2x + 3) 8 1 (cosh 5x + 5 cosh 3x + 10 cosh x) 16 1 (cosh 6x + 6 cosh 4x + 15 cosh 2x + 10) 32 1 (cosh 7x + 7 cosh 5x + 21 cosh 3x + 35 cosh x) 64

1.322

1.332

Trigonometric and hyperbolic functions: expansion in powers

33

1.33 The representation of trigonometric and hyperbolic functions of multiples of the argument (angle) in terms of powers of these functions 1.331 1.7

n n sin nx = n cosn−1 x sin x − cosn−3 x sin3 x + cosn−5 x sin5 x − . . . ; 3 5 ⎧

⎨ n − 2 n−3 2 cosn−3 x = sin x 2n−1 cosn−1 x − ⎩ 1 ⎫



⎬ n − 4 n−7 n − 3 n−5 cosn−5 x − 2 cosn−7 x + . . . + 2 ⎭ 3 2 AD (3.175) (n+1)/2

2.

12

n sinh2k−2 x coshn−2k+1 x sinh nx = sinh x 2k − 1 k=1

(n−1)/2  n − k − 1 n−2k−1 2 (−1)k coshn−2k−1 x = sinh x k 

k=0

3.

cos nx = cosn x −

n

n

cosn−4 x sin4 x − . . . ; 4

n n−3 n n − 3 n−5 n−1 n n−2 2 =2 cos x − 2 cos x+ cosn−4 x 1 2 1

n n − 4 n−7 2 − cosn−6 x + . . . 3 2 2

cosn−2 x sin2 x +

AD (3.175)

4.3

n sinh2k x coshn−2k x 2k k=0

n/2  n n−1 k1 n−k −1 2n−2k−1 coshn−2k x cosh x + n (−1) =2 k k−1

cosh nx =

n/2 



k=1

1.332 1.



  2   4n − 22 4n2 − 42 4n2 − 22 3 5 sin x + sin x − . . . sin 2nx = 2n cos x sin x − 3! 5!  2n − 2 2n−3 2n−3 2 = (−1)n−1 cos x 22n−1 sin2n−1 x − sin x 1! (2n − 3)(2n − 4) 2n−5 2n−5 2 + sin x 2!  (2n − 4)(2n − 5)(2n − 6) 2n−7 2n−7 2 − sin x + ... 3!

AD (3.171)

AD (3.173)

34

2.

3.12

Trigonometric and Hyperbolic Functions

1.333

 (2n − 1)2 − 12 sin3 x sin(2n − 1)x = (2n − 1) sin x − 3!     (2n − 1)2 − 12 (2n − 1)2 − 32 sin5 x − . . . AD (3.172) + 5!  2n − 1 2n−4 2n−3 2 = (−1)n−1 22n−2 sin2n−1 x − sin x 1! (2n − 1)(2n − 4) 2n−6 2n−5 2 + sin x 2!  (2n − 1)(2n − 5)(2n − 6) 2n−8 2n−7 2 − sin x + ... AD (3.174) 3!      4n2 4n2 − 22 4n2 4n2 − 22 4n2 − 42 4n2 2 4 sin x + sin x − sin6 x + . . . cos 2nx = 1 − 2! 4! 6! 

= (−1)n 22n−1 sin2n x −

+

AD (3.171)

2n 2n−3 2n−2 2 sin x 1!

 2n(2n − 3) 2n−5 2n−4 2n(2n − 4)(2n − 5) 2n−7 2n−6 2 2 sin x− sin x + ... 2! 3! AD (3.173)a

4.

5.∗ 6.∗

 (2n − 1)2 − 12 sin2 x cos(2n − 1)x = cos x 1 − 2!     (2n − 1)2 − 12 (2n − 1)2 − 32 4 sin x − . . . +  4! 2n − 3 2n−4 2n−4 2 = (−1)n−1 cos x 22n−2 sin2n−2 x − sin x 1! (2n − 4)(2n − 5) 2n−6 2n−6 2 + sin x 2!  (2n − 5)(2n − 6)(2n − 7) 2n−8 2n−8 2 − sin x + ... 3! sinh(n − 1)x + sinh(n + 1)x 2 cosh x cosh(n − 1)x + cosh(n + 1)x cosh nx = 2 cosh x sinh nx =

Special cases 1.333 1.

sin 2x = 2 sin x cos x

2.

sin 3x = 3 sin x − 4 sin3 x   sin 4x = cos x 4 sin x − 8 sin3 x

3. 4. 5.

sin 5x = 5 sin x − 20 sin3 x + 16 sin5 x   sin 6x = cos x 6 sin x − 32 sin3 x + 32 sin5 x

[n = 1, 2, . . . ] [n = 1, 2, . . . ]

AD (3.172)

AD (3.174)

1.337

6.

Trigonometric and hyperbolic functions: expansion in powers

sin 7x = 7 sin x − 56 sin3 x + 112 sin5 x − 64 sin7 x

1.334 1.

sinh 2x = 2 sinh x cosh x

2.

sinh 3x = 3 sinh x + 4 sinh3 x   sinh 4x = cosh x 4 sinh x + 8 sinh3 x

3.11 5.11

sinh 5x = 5 sinh x + 20 sinh3 x + 16 sinh5 x   sinh 6x = cosh x 6 sinh x + 32 sinh3 x + 32 sinh5 x

6.

sinh 7x = 7 sinh x + 56 sinh3 x + 112 sinh5 x + 64 sinh7 x

4.

1.335 1.

cos 2x = 2 cos2 x − 1

2.

cos 3x = 4 cos3 x − 3 cos x

3.

cos 4x = 8 cos4 x − 8 cos2 x + 1

4.

cos 5x = 16 cos5 x − 20 cos3 x + 5 cos x

5.

cos 6x = 32 cos6 x − 48 cos4 x + 18 cos2 x − 1

6.

cos 7x = 64 cos7 x − 112 cos5 x + 56 cos3 x − 7 cos x

1.336 1.

cosh 2x = 2 cosh2 x − 1

2.

cosh 3x = 4 cosh3 x − 3 cosh x

3.

cosh 4x = 8 cosh4 x − 8 cosh2 x + 1

4.

cosh 5x = 16 cosh5 x − 20 cosh3 x + 5 cosh x

5.

cosh 6x = 32 cosh6 x − 48 cosh4 x + 18 cosh2 x − 1

6.

cosh 7x = 64 cosh7 x − 112 cosh5 x + 56 cosh3 x − 7 cosh x

1.337 1. 2. 3. 4. 5. 6.

cos 3x cos3 x cos 4x cos4 x cos 5x cos5 x cos 6x cos6 x sin 3x cos3 x sin 4x cos4 x

= 1 − 3 tan2 x = 1 − 6 tan2 x + tan4 x = 1 − 10 tan2 x + 5 tan4 x = 1 − 15 tan2 x + 15 tan4 x − tan6 x = 3 tan x − tan3 x = 4 tan x − 4 tan3 x

35

36

7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

Trigonometric and Hyperbolic Functions

1.341

sin 5x = 5 tan x − 10 tan3 x + tan5 x cos5 x sin 6x = 6 tan x − 20 tan3 x + 6 tan5 x cos6 x cos 3x = cot3 x − 3 cot x sin3 x cos 4x = cot4 x − 6 cot2 x + 1 sin4 x cos 5x = cot5 x − 10 cot3 x + 5 cot x sin5 x cos 6x = cot6 x − 15 cot4 x + 15 cot2 x − 1 sin6 x sin 3x = 3 cot2 x − 1 sin3 x sin 4x = 4 cot3 x − 4 cot x sin4 x sin 5x = 5 cot4 x − 10 cot2 x + 1 sin5 x sin 6x = 6 cot5 x − 20 cot3 x + 6 cot x sin6 x

1.34 Certain sums of trigonometric and hyperbolic functions 1.341 1.

n−1  k=0

2.

3.

ny y n−1 y sin cosec sin(x + ky) = sin x + 2 2 2

ny 1 n−1 y sinh sinh(x + ky) = sinh x + 2 2 sinh y k=0 2

n−1  ny y n−1 y sin cosec cos(x + ky) = cos x + 2 2 2

AD (361.8)

n−1 

AD (361.9)

k=0

4.

5.

ny 1 n−1 y sinh cosh(x + ky) = cosh x + 2 2 sinh y k=0 2

2n−1  y 2n − 1 y sin ny sec (−1)k cos(x + ky) = sin x + 2 2 n−1 

JO (202)

k=0

6.

n(y + π) y n−1 (y + π) sin sec (−1)k sin(x + ky) = sin x + 2 2 2

n−1  k=0

AD (202a)

1.351

Sums of powers of trigonometric functions of multiple angles

37

Special cases 1.342 1. 2.10

n  k=1 n 

sin kx = sin

nx x n+1 x sin cosec 2 2 2

AD (361.1)

nx x n+1 x sin cosec + 1 2 2 2    sin n + 12 x n+1 x 1 nx sin x cosec = 1+ = cos 2 2 2 2 sin x2

cos kx = cos

k=0

AD (361.2)

3.

4.

n  k=1 n 

sin(2k − 1)x = sin2 nx cosec x 1 sin 2nx cosec x 2

cos(2k − 1)x =

k=1

1.343 1.

2.

3.

  1 (−1)n cos 2n+1 2 x (−1) cos kx = − + x 2 2 cos k=1 2 n  sin 2nx (−1)k+1 sin(2k − 1)x = (−1)n+1 2 cos x n 

k=1 n 

k

cos(4k − 3)x +

k=1

AD (361.7)

n 

JO (207)

AD (361.11)

AD (361.10)

sin(4k − 1)x = sin 2nx (cos 2nx + sin 2nx) (cos x + sin x) cosec 2x

k=1

JO (208)

1.344 1.

n−1  k=1

2.

n−1  k=1

3.

n−1  k=0

sin

π πk = cot n 2n

AD (361.19)

√  n nπ nπ  2πk 2 = 1 + cos − sin sin n 2 2 2

AD (361.18)

√  n nπ nπ  2πk 2 = 1 + cos + sin cos n 2 2 2

AD (361.17)

1.35 Sums of powers of trigonometric functions of multiple angles 1.351 1.

n  k=1

1 [(2n + 1) sin x − sin(2n + 1)x] cosec x 4 n cos(n + 1)x sin nx = − 2 2 sin x

sin2 kx =

AD (361.3)

38

Trigonometric and Hyperbolic Functions n 

2.

k=1

n 

3.

n−1 1 + cos nx sin(n + 1)x cosec x 2 2 n cos(n + 1)x sin nx = + 2 2 sin x

cos2 kx =

n 

n+1 nx x 1 3(n + 1)x 3nx 3x 3 sin x sin cosec − sin sin cosec 4 2 2 2 4 2 2 2

JO (210)

cos3 kx =

n+1 nx x 1 3(n + 1) 3nx 3x 3 cos x sin cosec + cos x sin cosec 4 2 2 2 4 2 2 2

JO (211)a

sin4 kx =

1 [3n − 4 cos(n + 1)x sin nx cosec x + cos 2(n + 1)x sin 2nx cosec 2x] 8

JO (212)

cos4 kx =

1 [3n + 4 cos(n + 1)x sin nx cosec x + cos 2(n + 1)x sin 2nx cosec 2x] 8

JO (213)

k=1 n 

5.

k=1 n 

6.

k=1

1.352 1.

11

2.11

AD (361.4)a

sin3 kx =

k=1

4.

1.352

  n cos 2n−1 sin nx 2 x k sin kx = − 2 sin x2 4 sin2 x2 k=1   n−1  n sin 2n−1 1 − cos nx 2 x k cos kx = − 2 sin x2 4 sin2 x2 n−1 

AD (361.5)

AD (361.6)

k=1

1.353 1.

n−1 

pk sin kx =

k=1

2.

n−1 

p sin x − pn sin nx + pn+1 sin(n − 1)x 1 − 2p cos x + p2

pk sinh kx =

k=1

3.

n−1  k=0

4.

n−1  k=0

pk cos kx =

p sinh x − pn sinh nx + pn+1 sinh(n − 1)x 1 − 2p cosh x + p2

1 − p cos x − pn cos nx + pn+1 cos(n − 1)x 1 − 2p cos x + p2

pk cosh kx =

AD (361.12)a

1 − p cosh x − pn cosh nx + pn+1 cosh(n − 1)x 1 − 2p cosh x + p2

AD (361.13)a¡

JO (396)

1.36 Sums of products of trigonometric functions of multiple angles 1.361 1.

n  k=1

sin kx sin(k + 1)x =

1 [(n + 1) sin 2x − sin 2(n + 1)x] cosec x 4

JO (214)

1.381

2.

3.

Sums leading to hyperbolic tangents and cotangents n 

1 n cos 2x − cos(n + 3)x sin nx cosec x 2 2 k=1

n  n(x + 2y) x + 2y n+1 x sin cosec 2 sin kx cos(2k − 1)y = sin ny + 2 2 2

k=1 n(2y − x) 2y − x n+1 x sin cosec − sin ny − 2 2 2 sin kx sin(k + 2)x =

39

JO (216)

JO (217)

1.362 1.

2.

n  

x 2  n x 2 = 2 sin − sin2 x 2k 2n k=1

2

2 n  1 x 1 x 2 sec = cosec x − cosec 2k 2k 2n 2n 2k sin2

AD (361.15)

AD (361.14)

k=1

1.37 Sums of tangents of multiple angles 1.371 1.

n  x 1 x 1 tan k = n cot n − 2 cot 2x 2k 2 2 2

AD (361.16)

n  1 x 22n+2 − 1 1 2 x tan = + 4 cot2 2x − 2n cot2 n 22k 2k 3 · 22n−1 2 2

AD (361.20)

k=0

2.

k=0

1.38 Sums leading to hyperbolic tangents and cotangents 1.381 1.12

n−1  k=0

2.12

n−1  k=1

3.12

n−1 

tanh x

n sin2

1  2k+1  4n π

= tanh(2nx)

tanh2 x   1+ tan2 2k+1 4n π tanh x

1   n sin2 kπ 2n

tanh2 x   1+ tan2 kπ 2n 2 tanh x

= coth (2nx) −

1 2



(2n + 1) sin

tanh x 

2k+1 2(2n+1) π

2

k=0

1+

tan2

2k+1 2(2n+1) π



JO (402)a

1 (tanh x + coth x) 2n

JO (403)

 = tanh (2n + 1) x −

tanh x 2n + 1

JO (404)

40

4.12

Trigonometric and Hyperbolic Functions

n 

2 tanh x

1.

n−1 

1+



k=0

2.

n−1 



k=1

3.

n−1  k=0

4.

(2n + 1) sin2



kπ 2n+1

 = coth (2n + 1) x −

tanh x   kπ tan2 2n+1 2

k=1

1.382

1

n  k=1

1  2k+1  x sin 1 4n π + tanh sinh x 2 2  kπ 

1

x sin 2n 1 + tanh sinh x 2 2

⎛



⎛



coth x 2n + 1

JO (406)

= 2n coth (nx) − 2 coth x



1 

⎞ = (2n + 1) tanh 2k+1   π sin2 2(2n+1) 1 x ⎠ ⎝ + tanh sinh x 2 2

⎝

2

sin

kπ 2n+1



sinh x



1

JO (405)

= 2n tanh (nx)

2

2

1.382

⎞ = (2n + 1) coth

x 1 ⎠ + tanh 2 2

JO (407)

(2n + 1)x 2

(2n + 1)x 2

− tanh

− coth

x 2

x 2

JO (408)

JO (409)

1.39 The representation of cosines and sines of multiples of the angle as finite products 1.391 1.

sin nx = n sin x cos x

n−2 2  k=1

2.

3.

4.

n 2 

⎛

⎞

⎛

sin x ⎟ ⎜ ⎝1 − kπ ⎠ sin2 n ⎞ 2

sin2 x ⎟ ⎠ (2k − 1)π 2 k=1 sin 2n ⎞ ⎛ n−1 2 2  sin x ⎟ ⎜ sin nx = n sin x ⎝1 − kπ ⎠ k=1 sin2 n ⎛ ⎞ n−1 2 2  sin x ⎜ ⎟ cos nx = cos x ⎝1 − ⎠ (2k − 1)π k=1 sin2 2n cos nx =

⎜ ⎝1 −

[n is even]

JO (568)

[n is even]

JO (569)

[n is odd]

JO (570)

[n is odd]

JO (571)a

1.396

1.392 1.

2.

Representing sines and cosines as finite products

kπ sin x + n k=0

n  2k − 1 π cos nx = 2n−1 sin x + 2n sin nx = 2n−1

n−1 

41

JO (548)

JO (549)

k=1

1.393 1.

1 2k π = n−1 cos nx cos x + n 2 k=0  n 1  = n−1 (−1) 2 − cos nx 2

n−1 

[n odd] [n even] JO (543)

2.11

n−1 (−1) 2 2k π = sin x + sin nx n 2n−1 n k=0 (−1) 2 = n−1 (1 − cos nx) 2

n−1 

[n odd] [n even] JO (544)

1.394 1.395 1.

 2kπ 2 +y x − 2xy cos α + = x2n − 2xn y n cos nα + y 2n n

n−1  k=0

2

cos nx − cos ny = 2n−1

k=0

2.

cosh nx − cos ny = 2n−1

1.

n−1 

x2 − 2x cos

k=1

2.

3.

n  2kπ x2 − 2x cos 2n + 1 k=1 n  2kπ x2 + 2x cos 2n + 1 k=1

4.

kπ +1 n

n−1 

x2 − 2x cos

k=0

 2kπ cosh x − cos y + n

n−1  k=0

1.396

 2kπ cos x − cos y + n

n−1 

JO (573)

JO (573)

JO (538)

x2n − 1 x2 − 1

x2n+1 − 1 +1 = x−1

x2n+1 − 1 +1 = x+1 =

(2k + 1)π +1 2n

KR 58 (28.1)

KR 58 (28.2)

KR 58 (28.3)

= x2n + 1

KR 58 (28.4)

42

Trigonometric and Hyperbolic Functions

1.411

1.41 The expansion of trigonometric and hyperbolic functions in power series 1.411 sin x =

1.

∞ 

(−1)k

k=0

sinh x =

2.

x2k+1 (2k + 1)!

∞  x2k+1 (2k + 1)!

k=0

cos x =

3.

∞ 

(−1)k

k=0

x2k (2k)!

∞  x2k (2k)! k=0   ∞  22k 22k − 1 |B2k |x2k−1 tan x = (2k)!

cosh x =

4.

5.



π2 x < 4 

k=1

6.

11

FI II 523

 ∞  22k 22k − 1 2x5 17 7 x3 + − x + ··· = B2k x2k−1 tanh x = x − 3 15 315 (2k)! k=1

π2 x2 < 4 ∞

7.



2

cot x =

1  22k |B2k | 2k−1 − x x (2k)!

 2  x < π2

FI II 523a

k=1

∞

8.

9.

2x5 1  22k B2k 2k−1 1 x x3 + − + − ··· = + x x 3 45 945 x (2k)! k=1  2  x < π2

∞  |E2k | 2k π2 2 x x < sec x = (2k)! 4 coth x =

FI II 522a CE 330a

k=0

∞

10.

sech x = 1 −

 E2k 5x4 61x6 x2 + − + ··· = 1 + x2k 2 24 720 (2k)! k=1

x2
0,

xy < −1]

[x < 0,

xy < −1] NV 59(6)

1.629

1.626 1.

Functional relations

   2 arcsin x = arcsin 2x 1 − x2    = π − arcsin 2x 1 − x2    = −π − arcsin 2x 1 − x2

59



1 |x| ≤ √ 2

1 √ 1] [x < −1] NV 61 (9)

  π arctan x ± x2 + 1 = 2 arctan(x) ± 2

arctan x + arctan

arctan x + arctan

1 π = 2 x π =− 2 1−x π = 4 1+x 3 =− π 4

NV 61 (8)

[|x| < 1]



1.627 1.

[0 ≤ x ≤ 1]

[x > 0] [x < 0]

GI I (878)

[x > −1] [x < −1]

NV 62, GI I (881)

1.628 1.12

arcsin

2x = −π − 2 arctan x 1 + x2 = 2 arctan x = π − 2 arctan x

[x < −1] [−1 ≤ x < 1] [x ≥ 1] NV 65

2.

1.629

arccos

1 − x2 = 2 arctan x 1 + x2 = −2 arctan x



1 2x − 1 2x − 1 − arctan tan π = E (x) 2 π 2

[x ≥ 0] [x ≤ 0]

NV 66

GI (886)

60

The Inverse Trigonometric and Hyperbolic Functions

1.631 Relations between the inverse hyperbolic functions.  x 1. arcsinh x = arccosh x2 + 1 = arctanh √ 2 x +1 √  x2 − 1 2. arccosh x = arcsinh x2 − 1 = arctanh x x 1 1 3. arctanh x = arcsinh √ = arccosh √ = arccoth 2 2 x 1−x 1−x     2 4. arcsinh x ± arcsinh y = arcsinh x 1 + y ± y 1 + x2    5. arccosh x ± arccosh y = arccosh xy ± (x2 − 1) (y 2 − 1) 6.

arctanh x ± arctanh y = arctanh

x±y 1 ± xy

1.631

JA JA JA JA JA JA

1.64 Series representations 1.641 1.12

2.12

1.642 1.12

1 3 π 1·3 5 1·3·5 7 − arccos x = x + x + x + x + ... 2 2·3 2 · 4 ·5 2 · 4 · 6 · 7 ∞  (2k)! 1 1 3 2 , ; ;x = x2k+1 = x F 2 2k 2 2 2 2 (k!) (2k + 1) k=0   2 x 1

AD (6480.3)a

  2 x ≤1

FI II 479

1.646

Series representations ∞

2.12

arctanh x = x +

 x2k+1 x5 x3 + + ··· = =xF 3 5 2k + 1



k=0

61

1 3 , 1; ; x2 2 2



x2 < 1



AD (6480.4)

1.644 1.12

k ∞  (2k)! x x2 arctan x = √ 1 + x2 k=0 22k (k!)2 (2k + 1) 1 + x2

x2 x 1 1 3 , ; ; = √ F 2 2 2 1 + x2 1 + x2 AD (641.3)

2.12

arctan x =

1 1 π 1 1 − + 3 − 5 + 7 − ··· = 2 x 3x 5x 7x

π − 2

∞ 

(−1)k

k=0

1 [x > 0] (2k + 1)x2k+1

∞

1 1 1 1 1 π  π (−1)k = − − + 3 − 5 + 7 − ··· = − − 2 x 3x 5x 7x 2 (2k + 1)x2k+1

[x < 0]

k=0

AD (641.4)

1.645 1.

∞

1 1 1·3 π  (2k)!x−(2k+1) π − − − · · · = arcsec x = − − 2 2 x 2 · 3x3 2 · 4 · 5x5 2 (k!) 22k (2k + 1) k=0

1 1 1 3 1 π , ; ; = − F 2 x 2 2 2 x2

 2  x >1 AD (641.5)

2. 3.

∞ 2   2  22k (k!) x2k+2 x ≤1 AD (642.2), GI III (152)a (2k + 1)!(k + 1) k=0



1 1 3! 3! 1 (arcsin x)3 = x3 + 32 1 + 2 x5 + 32 · 52 1 + 2 + 2 x7 + . . . 5! 3 7! 3 5  x2 ≤ 1 2

(arcsin x) =

BR* 188, AD (642.2), GI III (153)a

1.646 1.12

2.12

∞

arcsinh

arccosh

 (−1)k (2k)! 1 = arcosech x = x−2k−1 2k (k!)2 (2k + 1) x 2 k=0 2 1 = arcsech x = ln − x x

∞  k=1

(2k)! 22k

2

(k!) 2k

x2k

 2  x >1

AD (6480.5)

[0 < x < 1]

AD (6480.6)

[0 < x < 1]

AD (6480.7)a

∞

3.12

arcsinh

2  (−1)k+1 (2k)! 2k 1 = arcosech x = ln + x 2 x x 22k (k!) 2k k=1

∞

4.

arctanh

 x−(2k+1) 1 = arccoth x = x 2k + 1 k=0

 2  x >1

AD (6480.8)

62

1.647 1.12

The Inverse Trigonometric and Hyperbolic Functions

⎛    ∗ n ∞ ∗  B4n−2j+3 tanh(2k − 1) (π/2) π 4n+3 ⎝  (−1)j−1 22j − 1 24n−2j+4 − 1 B2j−1 = (2k − 1)4n+3 2 (2j)!(4n − 2j + 4)! j=1 k=1 ⎞   2 ∗ 2 (−1)n 22n+2 − 1 B2n+1 ⎠ + 2 [(2n + 2)!] ⎛

2.12

1.647

[n = 0, 1, 2, . . .]

⎞ n−1 ∞ j ∗ ∗ k−1 4n+1 ∗ n ∗ 2   (−1) B B (−1) sech(2k − 1) (π/2) π 2B4n (−1) B2n ⎠ 2j 4n−2j + + = 4n+3 ⎝ , (2k − 1)4n+1 2 (2j)!(4n − 2j)! (4n)! [(2n)]!2 j=1 k=1

[n = 1, 2, . . .]

(The summation term on the right is to be omitted for n = 1.) (See page xxxi for the definition of Br∗ .)

Table of Integrals, Series, and Products. http://dx.doi.org/10.1016/B978-0-12-384933-5.00002-3 c 2015 Elsevier Inc. All rights reserved. Copyright 

2 Indefinite Integrals of Elementary Functions 2.0 Introduction 2.00 General remarks We omit the constant of integration in all the formulas of this chapter. Therefore, the equality sign (=) means that the functions on the left and right of this symbol differ by a constant. For example (see integral 2.01 15), we write  dx = arctan x = − arccot x 1 + x2 although π arctan x = − arccot x ± 2 we  integrate certain functions, we obtain the logarithm of the absolute value (for example,  √  When √ dx = ln x + 1 + x2 ). In such formulas, the absolute-value bars in the argument of the logarithm 1+x2 are omitted for simplicity in writing. In certain cases, it is important to give the complete form of the primitive function. Such primitive functions, written in the form of definite integrals, are given in Chapter 2 and in other chapters. Closely related to these formulas are formulas in which the limits of integration and the integrand depend on the same parameter. A number of formulas lose their meaning for certain values of the constants (parameters) or for certain relationships between these constants (for example, formula 2.02 8 for n = −1 or formula 2.02 15 for a = b). These values of the constants and the relationships between them are for the most part completely clear from the very structure of the right hand member of the formula (the one not containing an integral sign). Therefore, throughout the chapter, we omit remarks to this effect. However, if the value of the integral is given by means of some other formula for those values of the parameters for which the formula in question loses meaning, we accompany this second formula with the appropriate explanation. The letters x, y, t, . . . denote independent variables; f , g, ϕ, . . . denote functions of x, y, t, . . . ; f  ,  g , ϕ , . . . , f  , g  , ϕ , . . . denote their first, second, etc., derivatives; a, b, m, p, . . . denote constants, by which we generally mean arbitrary real numbers. If a particular formula is valid only for certain values of the constants (for example, only for positive numbers or only for integers), an appropriate remark is made provided the restriction that we make does not follow from the form of the formula itself. Thus, in formulas 2.148 4 and 2.424 6, we make no remark since it is clear from the form of these formulas themselves that n must be a natural number (that is, a positive integer).

63

64

Introduction

2.01 The basic integrals 

1. 2. 3. 4. 5. 11

6. 7.

8.11

xn+1 (n = −1) x dx = n+1  dx = ln x x  ex dx = ex  ax ax dx = ln a  sin x dx = − cos x  cos x dx = sin x  dx = − cot x sin2 x  dx = tan x cos2 x 

16. 17. 18. 19. 20. 21. 22.11 23. 24. 25. 26.

n

1 1+x dx = arctanh x = ln 2 1−x 2 1−x  dx √ = arcsin x = − arccos x 1 − x2     dx √ = arcsinh x = ln x + x2 + 1 x2 + 1     dx √ = arccosh x = ln x + x2 − 1 x2 − 1  sinh x dx = cosh x  cosh x dx = sinh x  dx = − coth x sinh2 x  dx = tanh x cosh2 x  tanh x dx = ln cosh x  coth x dx = ln sinh x  dx x = ln tanh sinh x 2

 9.  10. 

sin x dx = sec x cos2 x cos x dx = − cosec x sin2 x tan x dx = − ln cos x

11. 

cot x dx = ln sin x

12.  13. 14. 15.12

dx x = ln tan sin x 2  π x dx = ln (sec x + tan x) = ln tan + cos x 4 2  dx = arctan x = − arccot x 1 + x2

General formulas

65

2.02 General formulas 



af dx = a

1.

f dx

 2. 3. 4. 5. 6. 7. 8.

9. 10. 11. 12. 13. 14. 15.

16. 17.





[af ± bϕ ± cψ ± . . .] dx = a f dx ± b ϕ dx ± c ψ dx ± . . .  d f dx = f dx  f  dx = f    f ϕ dx = f ϕ − f ϕ dx [integration by parts]   (n+1) (n)  (n−1)  (n−2) n (n) n+1 f ϕ(n+1) f dx ϕ dx = ϕf −ϕf +ϕ f − . . . + (−1) ϕ f + (−1)   f (x) dx = f [ϕ(y)]ϕ (y) dy [x = ϕ(y)] [change of variable] 

11



(f )n f  dx =

(f )n+1 n+1

For n = −1   f dx = ln f f  (af + b)n+1 (af + b)n f  dx = a(n + 1) √   2 af + b f dx √ = a af + b    f ϕ−ϕf f dx = ϕ2 ϕ   f f ϕ − ϕ f dx = ln fϕ ϕ    dx dx dx =± ∓ f (f ± ϕ) fϕ ϕ (f ± ϕ)      f dx  = ln f + f 2 + a f2 + a    a dx b dx f dx = − (f + a)(f + b) a − b (f + a) a − b (f + b) For a = b    dx f dx dx −a = (f + a)2 f +a (f + a)2    f dx dx ϕ dx = − (f + ϕ)n (f + ϕ)n−1 (f + ϕ)n  qf f  dx 1 arctan = 2 2 2 p +q f pq p

[n = −1]

66

Rational Functions

2.101



18. 19. 20. 21. 22. 23. 24. 25.

qf − p f  dx 1 ln = q 2 f 2 − p2 2pq qf + p   dx f dx = −x + 1−f 1−f    1 f dx f 2 dx 1 f dx + = f 2 − a2 2 f −a 2 f +a  f  dx f  = arcsin a a2 − f 2   1 f f dx = ln 2 af + bf b af + b  f f  dx 1  = arcsec 2 2 a a f f −a    (f ϕ − f ϕ ) dx f = arctan 2 2 f +ϕ ϕ  (f  ϕ − f ϕ ) dx 1 f −ϕ = ln f 2 − ϕ2 2 f +ϕ

2.1 Rational Functions 2.10 General integration rules F (x) , where F (x) and f (x) are polynomials with f (x) no common factors, we first need to separate out the integral part E(x) (where E(x) is a polynomial), if  there is an integral part, and then to integrate separately the integral part and the remainder, thus: F (x) dx ϕ(x) = E(x) dx + dx. f (x) f (x) Integration of the remainder, which is then a proper rational function (that is, one in which the degree of the numerator is less than the degree of the denominator) is based on the decomposition of the fraction into elementary fractions, the so-called partial fractions. 2.102 If a, b, c, . . . , m are roots of the equation f (x) = 0 and if α, β, γ, . . . , μ are their corresponding ϕ(x) can be decomposed into the following multiplicities, so that f (x) = (x−a)α (x−b)β . . . (x−m)μ then, f (x) partial fractions: ϕ(x) Aα Bβ Aα−1 A1 Bβ−1 B1 = + + ... + + ...+ + + ...+ α α−1 β β−1 f (x) (x − a) (x − a) x − a (x − b) (x − b) x−b Mμ Mμ−1 M1 , + + + ...+ μ μ−1 (x − m) (x − m) x−m 2.101 To integrate an arbitrary rational function

where the numerators of the individual fractions are determined by the following formulas: (k−1)

Aα−k+1 = ψ1 (x) =

(a) ψ1 , (k − 1)! ϕ(x)(x − a)α , f (x)

(k−1)

Bβ−k+1 = ψ2 (x) =

(k−1)

(b) ψ2 , (k − 1)!

...,

Mμ−k+1 =

ϕ(x)(x − b)β , f (x)

...,

ψm (x) =

ψm (m) , (k − 1)! ϕ(x)(x − m)μ f (x)

2.104

General integration rules

67

TI 51a

If a, b, . . . , m are simple roots, that is, if α = β = . . . = μ=1, then ϕ(x) A B M = + + ···+ , f (x) x−a x−b x−m where ϕ(b) ϕ(m) ϕ(a) B=  , ... , M=  . A=  , f (a) f (b) f (m) If some of the roots of the equation f (x) = 0 are imaginary, we group together the fractions that represent conjugate roots of the equation. Then, after certain manipulations, we represent the corresponding pairs of fractions in the form of real fractions of the form M 1 x + N1 M2 x + N2 M p x + Np + + ...+ 2 p. x2 + 2Bx + C (x + 2Bx + C) (x2 + 2Bx + C)2  g dx ϕ(x) reduces to integrals of the form 2.103 Thus, the integration of a proper rational fraction f (x) (x − a)α  Mx + N or dx. Fractions of the first form yield rational functions for α > 1 and logarithms (A + 2Bx + Cx2 )p for α = 1. Fractions of the second form yield rational functions and logarithms or arctangents:   g dx d(x − a) g 1. =g =− (x − a)α (x − a)α (α − 1)(x − a)α−1   d(x − a) g dx =g = g ln |x − a| 2. x−a x−a  Mx + N N B − M A + (N C − M B)x 3. dx = p−1 (A + 2Bx + Cx2 )p 2(p − 1) (AC − B 2 ) (A + 2Bx + Cx2 ) dx (2p − 3)(N C − M B) + 2(p − 1) (AC − B 2 ) (A + 2Bx + Cx2 )p−1   dx 1 Cx + B 4. = √ arctan √ for AC > B 2 2 2 A + 2Bx + Cx2 AC − B AC − B    Cx + B − √B 2 − AC   1   √ = √ ln   for AC < B 2 2 2   2 B − AC Cx + B + B − AC  (M x + N ) dx 5. A + 2Bx + Cx2   NC − MB M  Cx + B ln A + 2Bx + Cx2  + √ = arctan √ for AC > B 2 2 2 2C C AC − B AC − B    Cx + B − √B 2 − AC    M  N C − M B   √ √ ln A + 2Bx + Cx2  + = ln   for AC < B 2 2 2   2C 2C B − AC Cx + B + B − AC

The Ostrogradskiy–Hermite method 

ϕ(x) dx f (x) without finding the roots of the equation f (x) = 0 and without decomposing the integrand into partial fractions: 2.104 By means of the Ostrogradskiy–Hermite method, we can find the rational part of

68

Rational Functions

2.110



 M ϕ(x) N dx FI II 49 dx = + f (x) D Q Here, M , N , D, and Q are rational functions of x. Specifically, D is the greatest common divisor of f (x) the function f (x) and its derivative f  (x); Q = ; M is a polynomial of degree no higher than m − 1, D where m is the degree of the polynomial D; N is a polynomial of degree no higher than n − 1, where n is the degree of the polynomial Q. The coefficients of the polynomials M and N are determined by equating the coefficients of like powers of x in the following identity: ϕ(x) = M  Q − M (T − Q ) + N D f  (x) and M  and Q are the derivatives of the polynomials M and Q. where T = D

2.11–2.13 Forms containing the binomial a + bxk 2.110 Reduction formulas for zk = a + bxk and an explicit expression for the general case.   amk xn+1 zkm + xn zkm−1 dx 1. xn zkm dx = km + n + 1 km + n + 1 p (ak)s (m + 1)m(m − 1) . . . (m − s + 1)zkm−s xn+1

= m + 1 s=0 [mk + n + 1][(m − 1)k + n + 1] . . . [(m − s)k + n + 1]  (ak)p+1 m(m − 1) . . . (m − p + 1)(m − p) xn zkm−p−1 dx + [mk + n + 1][(m − 1)k + n + 1] . . . [(m − p)k + n + 1] LA 126(4)

 2.  3.  4.  5.  6. 7.∗



 8.

−xn+1 zkm+1



km + k + n + 1 xn zkm+1 dx ak(m + 1) ak(m + 1)  bkm xn+1 zkm n m − xn+k zkm−1 dx x zk dx = n+1 n+1  xn+1−k zkm+1 n+1−k − xn−k zkm+1 dx xn zkm dx = bk(m + 1) bk(m + 1)  xn+1−k zkm+1 a(n + 1 − k) − xn−k zkm dx xn zkm dx = b(km + n + 1) b(km + n + 1)  xn+1 zkm+1 b(km + k + n + 1) n m − xn+k zkm dx x zk dx = a(n + 1) a(n + 1)  b c k−i  k i  b k n +n nk (−1) k! Γ a+1 n b  a+1 xa+1+ib x nx + c dx = b i=0 (k − i)! Γ b + i + 1 xn zkm dx =

+



xn zkm dx =

a m−i

LA 126 (6)

LA 125 (2)

LA 126 (3)

LA 126 (5)

[a, b, k ≥ 0 are all integers]

m xk(J+i+1) bm (−1)i m!J! xk + b k i=0 (m − i)!(J + i + 1)!

J=

n+1 −1 k

[a, b, k, m, n real,

k = 0,

m ≥ 0 an integer]

Forms containing the binomial a + bxk

2.114

Forms containing the binomial z1 = a + bx 2.111 1.

2.

3.8

4. 5. 6. 2.113



z1m+1 b(m + 1) For m = −1  dx 1 = ln z1 z1 b  n−1  n dx x na x dx xn − = m−1 z1m z1m z1 (n + 1 − m)b (n + 1 − m)b For n = m − 1, we may use the formula   m−1 dx 1 xm−2 dx x xm−1 + = − m−1 z1m z1 (m − 1)b b z1m−1 For m = 1  n an−1 x (−1)n an x dx axn−1 xn a2 xn−2 − = + − . . . + (−1)n−1 + ln z1 2 3 z1 nb (n − 1)b (n − 2)b 1 · bn bn+1  n n−1 n−1 kak−1 xn−k an x dx

n−1 n+1 na = (−1)k−1 + (−1) + (−1) ln z1 2 z1 (n − k)bk+1 bn+1 z1 bn+1 k=1  a x dx x = − 2 ln z1 z1 b b  2 ax a2 x dx x2 − 2 + 3 ln z1 = z1 2b b b



1.  2.  3. 2.114 1. 2. 3. 4.6



z1m dx =

1 dx =− 2 z1 bz1 x dx x 1 a 1 =− + ln z1 = 2 + 2 ln z1 z12 bz1 b2 b z1 b x2 dx x a2 2a = − − 3 ln z1 2 2 3 z1 b b z1 b

1 dx =− z13 2bz12 

x a  1 x dx + = − z13 b 2b2 z12    2 x dx 2ax 3a2 1 1 = + + 3 ln z1 z13 b2 2b3 z12 b   3  3 a 2 x dx x a2 5 a3 1 a + 2 2x − 2 3 x − = − 3 4 ln z1 3 2 4 z1 b b b 2 b z1 b

69

70

2.115 1. 2. 3. 4. 2.116 1. 2. 3. 4. 2.117

Rational Functions



1 dx =− z14 3bz13 

x a  1 x dx + 2 3 =− 4 z1 2b 6b z1   2  2 ax x dx a2 1 x + = − + z14 b b2 3b3 z13    3 x dx 3ax2 9a2 x 11a3 1 1 = + + + ln z1 z14 b2 2b2 6b4 z13 b4



1 dx =− 5 z1 4bz14 

x a  1 x dx + = − z15 3b 12b2 z14   2  2 ax 1 x dx x a2 + =− + z15 2b 3b2 12b3 z14   3  3 3ax2 x dx x a2 x a3 1 + = − + + z15 b 2b2 b3 4b4 z14



1.  2.  3.  4.

dx −1 b(2 − n − m) = + xn z1m a(n − 1) (n − 1)axn−1 z1m−1 dx 1 =− m z1 (m − 1)bz1m−1 1 dx 1 + = m−1 xz1m z1 a(m − 1) a



dx xz1m−1

n−1

(−1)k bk−1 dx (−1)n bn−1 z1 = + ln xn z1 (n − k)ak xn−k an x k=1

2.118 1. 2. 3. 2.119 1.



1 z1 dx = − ln , xz1 a x  b dx 1 z1 + 2 ln =− 2 x z1 ax a x  b2 z1 1 b dx =− + 2 − 3 ln 3 2 x z1 2ax a x a x 



dx 1 1 z1 = − 2 ln 2 xz1 az1 a x

dx xn−1 z1m

2.115

Forms containing the binomial a + bxk

2.124

  2b 1 dx 1 2b z1 + =− + 3 ln x2 z12 ax a2 z1 a x    3b2 1 1 3b 3b2 z1 dx = − + 2 + 3 − 4 ln 2 3 2 x z1 2ax 2a x a z1 a x

 2. 3. 2.121 1. 2. 3. 2.122 1. 2. 3. 2.123 1.11 2. 3.

  bx 1 3 1 z1 dx + = − 3 ln 3 2 2 xz1 2a a z1 a x    2 9b 1 3b x 1 3b z1 dx + =− + 3 + ln x2 z13 ax 2a2 a z12 a4 x    1 2b 6b3 x 1 6b2 z1 dx 9b2 = − + 2 + 3 + 4 − 5 ln 3 2 3 2 x z1 2ax a x a a z1 a x



  1 z1 dx 11 5bx b2 x2 1 + = + 3 − 4 ln xz14 6a 2a2 a z13 a x    22b 10b2 x 4b3 x2 1 1 4b z1 dx + = − + + + ln x2 z14 ax 3a2 a3 a4 z13 a5 x    2 3 4 2 55b 1 1 5b 25b x 10b x 10b2 z1 dx + = − + + + − ln x3 z14 2ax2 2a2 x 3a3 a4 a5 z13 a6 x



  13bx 7b2 x2 dx 25 b3 x3 1 1 z1 + = + + − 5 ln xz15 12a 3a2 2a3 a4 z14 a x    2 3 2 4 3 125b 65b x 35b x 1 5b x 5b z1 dx 1 − = − − − − + 6 ln 5 4 2 2 3 4 5 x z1 ax 12a 3a 2a a z1 a x    2 3 4 2 5 3 125b 1 1 3b 65b x 105b x 15b x 15b2 z1 dx = − + 2 + + + + − 7 ln 5 4 3 2 3 4 5 6 x z1 2ax a x 4a a 2a a z1 a x 

2.124 Forms containing the binomial z2 = a + bx2 .     ab dx 1 12 [if ab > 0] 1. = √ arctan x z2 a ab √ 1 a + xi ab √ = √ ln [if ab < 0] 2i ab a − xi ab  x dx 1 2. =− z2m 2b(m − 1)z2m−1

(see also 2.141 2) (see also 2.143 2 and 2.143 3) (see also 2.145 2, 2.145 6, and 2.18)

71

72

Rational Functions

2.125

Forms containing the binomial z3 = a + bx3  a Notation: α = 3 b 2.125  n−3  n dx x (n − 2)a xn−2 x dx − = 1. m m−1 m z3 z3 z3 (n + 1 − 3m)b b(n + 1 − 3m)  n  n n+1 x dx x n + 4 − 3m x dx 2. = − m m−1 z3 3a(m − 1) 3a(m − 1)z3 z3m−1 2.126



1.

 2.



3. 4. 5. 2.127

 

2.  3.  4. 2.128 1. 2.

 √  √ (x + α)2 dx α 1 x 3 ln = + 3 arctan z3 3a 2 x2 − αx + α2 2α − x   2 √ (x + α) α 1 2x − α √ ln 2 = + 3 arctan 3a 2 x − αx + α2 α 3 1 x dx =− z3 3bα



(see also 2.141 3 and 2.143)  √ (x + α) 2x − α 1 √ ln − 3 arctan 2 x2 − αx + α2 α 3 2

(see also 2.145 3. and 2.145 7)

 1 1 x dx ln 1 + x3 α−3 = ln z3 = z3 3b 3b   3 x dx x a dx = − z3 b b z3   4 2 x dx x a x dx = − z3 2b b z3

1.

LA 133 (1)

2

dx x 2 = + z32 3az3 3a



x dx x2 1 = + 2 z3 3az3 3a



dx z3

x2 dx 1 =− z32 3bz3 x3 dx x 1 =− + 2 z3 3bz3 3b

(see 2.126 2)

(see 2.126 1)

x dx z3



(see 2.126 1)

dx z3

(see 2.126 2)

(see 2.126 1)

 dx 1 b(3m + n − 4) dx = − − m m−1 n n−3 n−1 x z3 a(n − 1) x z3m (n − 1)ax z3   dx dx 1 n + 3m − 4 = + xn z3m 3a(m − 1) xn z3m−1 3a(m − 1)xn−1 z3m−1 

LA 133 (2)

Forms containing the binomial a + bxk

2.133

2.129 1. 2. 3. 2.131 1. 2. 3.



x3 dx 1 ln = xz3 3a z3   b x dx dx 1 − = − x2 z3 ax a z3   1 b dx dx =− − x3 z3 2ax2 a z3

(see 2.126 2) (see 2.126 1)



dx 1 1 x3 = + ln xz32 3az3 3a2 z3     4bx2 1 dx 1 4b x dx + = − − x2 z32 ax 3a2 z3 3a2 z3     dx dx 5bx 1 5b 1 =− + 2 − 2 2 3 2 x z3 2ax 6a z3 3a z3

(see 2.126 2) (see 2.126 1)

Forms containing the binomial z4 = a + bx4   a a 4  α = 4 − Notation: α = b b 2.132  √ √   2 2 + αx 2 + α 2 dx x α αx √ for ab > 0 ln = √ + 2 arctan 2 1.8 z4 α − x2 4a 2 x2 − αx 2 + α2   α x + α x = ln for ab < 0 + 2 arctan 4a x − α α   x dx b 1 2 2. for ab > 0 (see = √ arctan x z4 a 2 ab √ 1 a + x2 i ab √ = √ ln for ab < 0 (see 4i ab a − x2 i ab  √ √   2 x2 − αx 2 + α2 1 αx 2 x dx √ √ for ab > 0 ln = + 2 arctan 2 3. z4 α − x2 4bα 2 x2 + αx 2 + α2   x + α x 1 ln for ab < 0 − 2 arctan  =− 4bα x − α α  3 1 x dx ln z4 = 4. z4 4b 2.133 1. 2.

73



xn dx xn+1 4m − n − 5 = m−1 + 4a(m − 1) z4m 4a(m − 1)z4

(see also 2.141 4) (see also 2.143 5) also 2.145 4) also 2.145 8)



xn dx z4m−1  n−4  n dx x (n − 3)a x dx xn−3 − = m−1 z4m z4m z4 (n + 1 − 4m)b b(n + 1 − 4m)

LA 134 (1)

74

2.134

Rational Functions



1.  2.  3.  4. 

x 3 dx = + z42 4az4 4a



x dx x2 1 = + z42 4az4 2a



x2 dx x3 1 = + 2 z4 4az4 4a

dx z4



2.134

(see 2.132 1)

x dx z4

(see 2.132 2)

x2 dx z4

(see 2.132 3)

x3 dx x4 1 = =− z42 4az4 4bz4

dx 1 b(4m + n − 5) 2.135 =− − m m−1 n n−1 x z4 (n − 1)a (n − 1)ax z4 For n = 1   dx dx dx 1 b = − m m−1 −3 xz4 a xz4 a x z4m 2.136  1 x4 ln x ln z4 dx − = ln = 1. xz4 a 4a 4a z4  2  b x dx dx 1 − 2. =− 2 x z4 ax a z4



dx xn−4 z4m

(see 2.132 3)

2.14 Forms containing the binomial 1 ± xn 2.141  1.  2.11

4.

dx = arctan x = − arctan 1 + x2

  1 x

(see also 2.124 1)

√ 1+x x 3 dx 1 1 (see also 2.126 1) = ln √ + √ arctan 1 + x3 3 2−x 3 1 − x + x2 √ √  1 1 1 + x 2 + x2 x 2 dx √ √ √ = + ln arctan 1 + x4 1 − x2 4 2 1 − x 2 + x2 2 2 

3.

dx = ln(1 + x) 1+x

(see also 2.132 1)  2.142

n

−1

2 dx 2

= − Pk cos 1 + xn n

k=0



n    2 −1 2

2k + 1 2k + 1 Qk sin π + π n n n

k=0

n−3 2

=

2

1 ln(1 + x) − Pk cos n n k=0



for n a positive even number 

n−3 2

2

2k + 1 π + Qk sin n n k=0





TI (43)a

2k + 1 π n for n a positive odd number TI (45)

Forms containing the binomial 1 ± xn

2.145

where

2.143 1. 2. 3. 4. 5.

75

    1 2k + 1 2 π +1 Pk = ln x − 2x cos 2 n   x − cos 2k+1 x sin 2k+1 π π n2k+1 = arctan  2k+1n Qk = arctan 1 − x cos n π sin n π



dx = − ln(1 − x) 1−x  dx 1 1+x = arctanh x [−1 < x < 1] (see also 2.141 1) = ln 2 1−x 2 1−x  1 x−1 dx = ln = − arccoth x [x > 1, x < −1] 2 x −1 2 x+1 √ √  dx 1 + x + x2 1 x 3 1 + √ arctan (see also 2.126 1) = ln 1 − x3 3 1−x 2+x 3  1 dx 1 1+x 1 + arctan x = (arctanh x + arctan x) = ln 4 1−x 4 1−x 2 2 (see also 2.132 1)

2.144



1.

n

−1

n

−1

2 2 dx 2

2

1 1+x 2k 2k ln − π + = P cos Qk sin π k 1 − xn n 1−x n n n n

k=1

k=1

for n a positive even number

  1 2k + 1 π+1 , where Pk = ln x2 + 2x cos 2 n  2.

n−3

n−3

k=0

k=0

1. 2. 3.

x + cos 2k+1 n π sin 2k+1 n π

2 2 2

2

1 2k + 1 2k + 1 dx ln(1 − x) + π + π = − P cos Qk sin k n 1−x n n n n n

  1 2k 2 where Pk = ln x − 2x cos π + 1 , 2 n 2.145

Qk = arctan

TI (47)

for n a positive odd number Qk = arctan

x − cos

2k n π

sin 2k n π



x dx = x − ln(1 + x) 1+x  x dx 1  = ln 1 + x2 2 1+x 2  (1 + x)2 x dx 2x − 1 1 1 ln = − + √ arctan √ 1 + x3 6 1 − x + x2 3 3

(see also 2.126 2)

TI (49)

76

Rational Functions

2.146

 4. 5. 6. 7. 8.

x dx 1 = arctan x2 1 + x4 2  x dx = − ln(1 − x) − x 1−x  x dx 1  = − ln 1 − x2 2 1−x 2  (1 − x)2 1 1 2x + 1 x dx ln = − − √ arctan √ 1 − x3 6 1 + x + x2 3 3  x dx 1 1 + x2 = ln 1 − x4 4 1 − x2

(see also 2.126 2) (see also 2.132 2)

2.146 For m and n natural numbers.    m−1 n dx 2k − 1 1

mπ(2k − 1) x 2 ln 1 − 2x cos π + x = − cos 1. 1 + x2n 2n 2n 2n k=1   n 2k−1 x − cos 2n π 1

mπ(2k − 1) arctan + sin n 2n sin 2k−1 2n π k=1

2.

3.11

TI (44)a

[m < 2n]

TI (48)

n

m−1

k=1

 4.

[m < 2n]

  dx 1 2k − 1 x mπ(2k − 1) m+1 ln(1 + x) 2 − ln 1 − 2x cos π+x = (−1) cos 1 + x2n+1 2n + 1 2n + 1 2n + 1 2n + 1 k=1   n 2k−1 x − cos 2n+1 π 2

mπ(2k − 1) arctan + sin 2k−1 2n + 1 2n + 1 sin 2n+1 π k=1 TI (46)a [m ≤ 2n]    m−1 n−1  dx kπ x 1  1

kmπ m+1 2 (−1) ln 1 − 2x cos + x = ln(1 + x) − ln(1 − x) − cos 1 − x2n 2n 2n n n k=1   n−1 kπ x − cos n 1

kmπ arctan + sin n n sin kπ n



m−1

dx x 1 ln(1 − x) =− 1 − x2n+1 2n + 1   n 1

2k − 1 mπ(2k − 1) m+1 2 +(−1) ln 1 + 2x cos π+x cos 2n + 1 2n + 1 2n + 1 k=1   n 2k−1 x + cos 2n+1 π 2

mπ(2k − 1) m+1 arctan +(−1) sin 2k−1 2n + 1 2n + 1 sin 2n+1 π k=1 [m ≤ 2n]

2.147    xm dx 1 xm dx 1 xm dx 1. = + 1 − x2n 2 1 − xn 2 1 + xn  m−2  dx xm−1 x xm dx 1 m−1 · 2. + n =− 2n − m − 1 (1 + x2 )n−1 2n − m − 1 (1 + x2 )n (1 + x2 )

TI (50)

LA 139 (28)

Forms containing the binomial 1 ± xn

2.149

 3.  4.

 5. 2.148



1.

2. 3.

4.

xm xm−1 − dx = 1 + x2 m−1



77

xm−2 dx 1 + x2

 m−2 dx xm−1 x 1 m−1 xm dx = − n n−1 2 2 2 2n − m − 1 (1 − x ) 2n − m − 1 (1 − x )n (1 − x )  m−1 x xm−2 dx 1 m−1 = − 2n − 2 (1 − x2 )n−1 2n − 2 (1 − x2 )n−1 m

m−1

x dx x + =− 2 1−x m−1



LA 139 (33) m−2

dx x 1 − x2

1 2n + m − 3 dx 1 − n = − m − 1 xm−1 (1 + x2 )n−1 m−1 xm (1 + x2 )



dx n xm−2 (1 + x2 )

LA 139 (29)

For   m=1 1 1 dx dx = + LA 139 (31) n−1 x (1 + x2 )n 2n − 2 (1 + x2 )n−1 x (1 + x2 ) For m = 1 and n = 1  x dx = ln √ 2 x (1 + x ) 1 + x2   1 dx dx = − − m 2 m−1 m−2 x (1 + x ) (m − 1)x x (1 + x2 )   x dx dx 1 2n − 3 = + FI II 40 n n−1 2 2n − 2 (1 + x2 ) 2n − 2 (1 + x2 )n−1 (1 + x )  n−1 dx x (2n − 1)(2n − 3)(2n − 5) · · · (2n − 2k + 1) (2n − 3)!! arctan x = + n−1 n n−k 2 k 2 2n − 1 2 (n − 1)! (1 + x ) k=1 2 (n − 1)(n − 2) . . . (n − k) (1 + x ) TI (91)

2.149  1.

2.

3.

1 2n + m − 3 dx n = − n−1 + m−1 2 m−1 xm (1 − x2 ) (m − 1)x (1 − x )



dx n xm−2 (1 − x2 )

LA 139 (34)

For   m=1 1 dx dx = LA 139 (36) n−1 + n−1 2 x (1 − x2 )n 2(n − 1) (1 − x ) x (1 − x2 ) For m = 1 and n = 1  x dx = ln √ 2 x (1 − x ) 1 − x2   dx dx 1 2n − 3 x = + LA 139 (35) n n−1 2 2 (1 − x ) 2n − 2 (1 − x ) 2n − 2 (1 − x2 )n−1  n−1 1+x dx x (2n − 1)(2n − 3)(2n − 5) . . . (2n − 2k + 1) (2n − 3)!! ln + n n = n−k k 2 2n − 1 2 · (n − 1)! 1−x (1 − x2 ) k=1 2 (n − 1)(n − 2) . . . (n − k) (1 − x ) TI (91)

78

Rational Functions

2.151

2.15 Forms containing pairs of binomials: a + bx and α + βx Notation: t = α + βx; Δ = aβ − αb   z = a + bx; n+1 tm mΔ z n m − z n tm−1 dx 2.151 z t dx = (m + n + 1)b (m + n + 1)b 2.152  bx Δ z dx = + 2 ln t 1. t β β  βx Δ t dx = − 2 ln z 2. z b b  m−1  m dx tm t 1 mΔ t dx = − 2.153 zn (m − n + 1)b z n−1 (m − n +  1)b zn tm+1 1 (m − n + 2)β tm dx = − n−1 (n − 1)Δ z n−1 (n − 1)Δ  m−1 z t 1 mβ tm =− + dx (n − 1)b z n−1 (n − 1)b z n−1  1 t dx = ln 2.154 zt Δ z   1 dx dx 1 (m + n − 2)b 2.155 = − − z n tm (m − 1)Δ tm−1 z n−1 (m − 1)Δ tm−1 z n 1 dx 1 (m + n − 2)β = + m z n−1 (n− 1)Δ tm−1 z n−1 (n − 1)Δ t   1 a α x dx = ln z − ln t 2.156 zt Δ b β

2.16 Forms containing the trinomial a + bxk + cx2k 2.160 Reduction formulas for Rk = a + bxk + cx2k .    xm Rkn+1 (m + k + nk)b (m + 2k + 2kn)c m−1 n m+k−1 n 1. x − x xm+2k−1 Rkn dx Rk dx = Rk dx − ma ma ma    bkn xm Rkn 2ckn m−1 n m+k−1 n−1 − x xm+2k−1 Rkn−1 dx Rk dx = Rk dx − 2. x m m m    xm−2k Rkn+1 (m − 2k)a (m − k + kn)b − xm−2k−1 Rkn dx − xm−k−1 Rkn dx 3. xm−1 Rkn dx = (m + 2kn)c (m + 2kn)c (m + 2kn)c   2kna xm Rkn bkn + xm−1 Rkn−1 dx + xm+k−1 Rkn−1 dx = m + 2kn m + 2kn m + 2kn 2.161 Forms containing the trinomial R2 = a + bx2 + cx4 . b 1 2 b 1 2 − b − 4ac, g = + b − 4ac, 2 2 2 2    a h = b2 − 4ac, q = 4 , l = 2a(n − 1) b2 − 4ac , c

Notation: f =

b cos α = − √ 2 ac

Forms containing a + bx + cx2 and powers of x

2.172

 1.

dx R2

   2 dx dx − h >0 LA 146 (5) 2 2 cx + f cx + g    2 α x2 + 2qx cos α2 + q 2 x2 − q 2 1 α h < 0 LA 146 (8)a sin ln arctan = + 2 cos α 4cq 3 sin α 2 x2 − 2qx cos 2 + q 2 2 2qx sin α2   2 x dx cx2 + f 1 h >0 LA 146 (6) ln 2 = R2 2h cx + g 2 2  2 x − q cos α 1 h 0 = LA 146 (7) 2 2 R2 h cx + g h cx + f     bcx3 + b2 − 2ac x b2 − 6ac dx bc x2 dx dx = + + R22 lR2 l R2 l R2  2  2   3 bcx + b − 2ac x (4n − 7)bc x dx 2(n − 1)h2 + 2ac − b2 dx dx = + + n n−1 n−1 R2 l l lR2 R2 R2n−1 =

2.

3. 4. 5.12

79

 6.12

c h



dx 1 (m + 2n − 3)b =− − xm R2n (m − 1)a (m − 1)axm−1 R2n−1



[n > 1] dx (m + 4n − 5)b − xm−2 R2n (m − 1)a



LA 146

dx xm−4 R2n LA 147 (12)a

2.17 Forms containing the quadratic trinomial a + bx + cx2 and powers of x Notation: R = a + bx + cx2 ; Δ = 4ac − b2 2.171    am xm Rn+1 b(m + n + 1) m+1 n m−1 n − x xm Rn dx 1. x R dx = R dx − c(m + 2n + 2) c(m + 2n + 2) c(m + 2n + 2)  2. 3. 4.12

n



n+1

n



TI (97) n

R dx R b(n − m + 1) R dx c(2n − m + 2) R dx =− + + LA 142(3), TI (96)a m+1 m x amx am xm am xm−1   dx b + 2cx (4n − 2)c dx = + TI (94)a Rn+1 nΔRn nΔ Rn  n−1 n dx dx (2cx + b) 2k (2n + 1)(2n − 1)(2n − 3) . . . (2n − 2k + 1)ck n (2n − 1)!!c = + 2 n+1 k+1 n−k n R 2n + 1 n(n − 1) · · · (n − k)Δ R n!Δ R k=0

 2.17212

√ 1 (b + 2cx) − −Δ −2 b + 2cx dx √ =√ ln =√ arctanh √ R −Δ (b + 2cx) + −Δ −Δ −Δ −2 = b + 2cx 2 b + 2cx = √ arctan √ Δ Δ

TI (96)a

[for Δ < 0] [for Δ = 0, b and c non-zero] [for Δ > 0]

80

Rational Functions

2.173

2.173   b + 2cx 2c dx dx + = 1. R2 ΔR Δ R     1 6c2 dx dx b + 2cx 3c + 2. = + R3 Δ 2R2 ΔR Δ2 R 2.174 1.

2. 2.175 1. 2. 3. 4. 5. 6.

 m−1  m−2 dx dx x x xm dx xm−1 (n − m)b (m − 1)a = − − + n n−1 n n R (2n − m − 1)cR (2n − m − 1)c R (2n − m − 1)c R For m = 2n − 1, this formula is inapplicable. Instead, we may use     2n−1 dx 1 x2n−3 dx a x2n−3 dx b x2n−2 dx x = − − Rn c Rn−1 c Rn c Rn

 1 b dx x dx = ln R − R 2c 2c R   b dx 2a + bx x dx − =− R2 ΔR Δ R   x dx 2a + bx 3b(b + 2cx) 3bc dx − 2 =− − R3 2ΔR2 2Δ2 R Δ R  2  x b x dx b2 − 2ac dx = − 2 ln R + R c 2c 2c2 R  2  2  ab + b − 2ac x 2a dx x dx + = R2 cΔR Δ R    2 2 ab + b − 2ac x 2ac + b2 (b + 2cx) x dx + = + R3 2cΔR2 2cΔ2 R



7.

 8.

 9.  2.176

1.

(see 2.172)





2.177

(see 2.172)

(see 2.172) (see 2.172) (see 2.172) (see 2.172) 2ac + b2 Δ2

  b b2 − 3ac dx x3 dx x2 bx b2 − ac ln R − = − 2 + 3 3 R 2c c 2c 2c R



dx R

(see 2.172)

(see 2.172)     2 2 a 2ac − b + b 3ac − b x b 6ac − b2 dx 1 x dx − = 2 ln R + 2 2 2 R 2c c ΔR 2c Δ R 3

x dx =− R3 3

dx xm Rn



(see 2.172)

=



abx 2a x + + c cΔ cΔ 2

2



−1 (m − 1)axm−1 Rn−1

1 x2 b dx = ln − xR 2a R 2a



dx R

1 3ab − 2R2 2cΔ



(see 2.172)

dx (see 2.173 1) R2   dx dx b(m + n − 2) c(m + 2n − 3) − − m−1 n m−2 a(m − 1) x R a(m − 1) x Rn (see 2.172)

2.177

Quadratic trinomials and binomials

 2.

 3.

1 dx 1 x2 + = ln 2 2 xR 2a R 2aR

4. 5.

    b(b + 2cx) b dx 2ac 1− − 2 1+ Δ 2a Δ R

1 b 1 1 x dx − = + 2 + 3 ln xR3 4aR2 2a R 2a R 2a 2



81



(see 2.172)   dx dx dx b b − 2 − 3 R3 2a R2 2a R (see 2.172, 2.173)



b 1 b − 2ac dx x dx = − 2 ln − + (see 2.172) x2 R 2a R ax 2a2 R  2    b − 3ac (b + 2cx) dx dx b x2 6b2 c 6c2 a + bx 1 b4 = − ln − + − + − 2 2 3 2 2 3 2 x R a R a xR a ΔR Δ a a a R

 6.  7.

 8.

 9.

2

2

dx 1 3b =− − x2 R3 axR2 a



dx 5c − xR3 a



(see 2.172) dx R3

(see 2.173 and 2.177 3) 

b 3ac − b dx ac − b2 x2 b 1 =− + 2 − ln + 3 3 2 x R 2a R a x 2ax 2a3

2 

dx R

(see 2.172)   2    1 3b dx 1 3b 2c 9bc dx dx + = − + − + x3 R2 2ax2 2a2 x R a2 a xR2 2a2 R2

dx = x3 R3



−1 2b + 2 2ax2 a x



1 + R2



2

6b 3c − a2 a



(see 2.173 1 and 2.177 2)  dx 10bc dx + 2 xR3 a R3 (see 2.173 2 and 2.177 3)

2.18 Forms containing the quadratic trinomial a + bx + cx2 and the binomial α + βx Notation: R = a + bx + cx2 ; B = bβ − 2cα;

2.

A = aβ 2 − αbβ + cα2 ;

Δ = 4ac − b2

 (m − 1)A z z m−2 Rn dx R dx − (m + 2n + 1)c  n  n−1 dx Rn R dx R 1 2nA = − − zm (m − 2n − 1)β z m−1 (m − 2n − 1)β 2 zm Rn−1 dx nB − ; LA 184 (4)a (m − 2n − 1)β 2 z m−1  n  n n+1 R −β (m − n − 2)B R dx (m − 2n − 3)c R dx = − − LA 148 (5) m−1 (m − 1)A z (m − 1)A  z m−1 (m − 1)A z m−2 Rn Rn−1 dx Rn−1 dx 1 nB 2nc =− + + LA 418 (6) m−1 2 m−1 2 (m − 1)β z (m − 1)β z (m − 1)β z m−2

 1.

z = α + βx;

βz m−1 Rn+1 (m + n)B z R dx = − (m + 2n + 1)c (m + 2n + 1)c m

n



m−1

n

82

Algebraic functions

 3.

z m dx β (m − n)B z m−1 = − n n−1 R (m − 2n + 1)c R (m − 2n + 1)c m

=

b + 2cx z 2(m − 2n + 3)c − (n − 1)Δ Rn−1 (n − 1)Δ





2.201

z m−1 dx (m − 1)A − n R (m − 2n + 1)c

m

z dx Bm − Rn−1 (n − 1)Δ





z m−2 dx Rn LA 147 (1)

z

m−1

dx Rn−1 LA 148 (3)

 4.3

1 dx β (m + n − 2)B =− − m n m−1 n−1 z R (m − 1)A z R (m − 1)A

=

1 β B − 2(n − 1)A z m−1 Rn−1 2A





dx (m + 2n − 3)c − m−1 n z R (m − 1)A

dx (m + 2n − 3)β + z m−1 Rn 2(n − 1)A

2



dx z m−2 Rn LA 148 (7)

dx z m Rn−1 LA 148 (8)

For m = 1 and n = 1   β z2 B dx dx = ln − zR 2A R 2A R For A = 0   1 dx dx β (m + 2n − 2)c =− − z m Rn (m + n − 1)B z m Rn−1 (m + n − 1)B z m−1 Rn

LA 148 (9)

2.2 Algebraic functions 2.20 Introduction

r  s    αx + β αx + β 2.201 The integrals R x, , , . . . dx, where r, s, . . . are rational numbers, γx + δ γx + δ can be reduced to integrals of rational functions by means of the substitution αx + β = tm , FI II 57 γx + δ where m is the common denominator of the fractions r, s, . . . .  2.202 Integrals of the form xm (a + bxn )p dx,∗ where m, n, and p are rational numbers, can be 

expressed in terms of elementary functions only in the following cases: (a)

When p is an integer; then, this integral takes the form of a sum of the integrals shown in 2.201;

(b)

is an integer: by means of the substitution xn = z, this integral can be transformed m+1 1 (a + bz)p z n −1 dz, which we considered in 2.201; to the form n

(c)

+ p is an integer; by means of the same substitution xn = z, this integral can be p   m+1 a + bz 1 reduced to an integral of the form z n +p−1 dz, considered in 2.201; n z

When

When

m+1 n

m+1 n

For reduction formulas for integrals of binomial differentials, see 2.110. ∗ Translator:

The authors term such integrals “integrals of binomial differentials”.

Forms containing the binomial a + bxk and

2.214

2.21 Forms containing the binomial a + bxk and

√

x

√ x

Notation: z1 = a + bx.   bx 2 dx √ = √ arctan [ab > 0] 2.211 a √ z1 x ab 1 a − bx + 2i xab = √ ln [ab < 0] z1 i ab  m√ m m+1  √ (−1)k ak xm−k x x dx m+1 a √ 2.212 dx = 2 x + (−1) z1 (2m − 2k + 1)bk+1 bm+1 z1 x k=0

(see 2.211) 2.213

 √

√  x dx dx 2 x a √ − (see 2.211) 1. = z1 b b z1 x  √  x x dx  x a √ dx a2 √ 2. − 2 2 x+ 2 (see 2.211) = z1 3b b b z1 x   2  2√  √ x x dx xa dx x a2 a3 √ 3. − (see 2.211) = + 3 2 x− 3 z1 5b 3b2 b b z1 x √   dx x dx 1 √ √ 4. = (see 2.211) + z12 x az1 2a z1 x √  √  x dx x dx 1 √ 5. (see 2.211) =− + 2 z1 bz1 2b z1 x √  √  √ x x dx x dx 2x x 3a 6. = − (see 2.213 5) z12 bz1 b z12  √  2  2√  √ x dx 5ax 2 x 5a2 x x dx x − 2 7. = + 2 (see 2.213 5) 2 z1 3b 3b z1 b z12     √ 1 dx dx 3 3 √ √ = (see 2.211) 8. + x + 2az12 4a2 z1 8a2 z1 x z13 x    √  √ x dx dx 1 1 1 √ 9. (see 2.211) = − + x + 3 2 z1 2bz1 4abz1 8ab z1 x √  √  √ x x dx x dx 2x x 3a 10. =− + (see 2.213 9) z13 bz12 b z13  √  2  2√  √ x dx 5ax 2 x 15a2 x x dx x + 11. = − (see 2.213 9) 3 2 2 2 z1 b b z1 b z13   a a  4 2 , α = 4 − . Notation: z2 = a + bx , α = b b  √  √   2 dx a 1 α 2x x + α 2x + α √ = √ ln 2.214 >0 + arctan 2 √ z2 α −x b z2 x bα3 2  √ √  

 x α − x a 1 0 0] [a < 0]

 √

 √ dx z dx √ =2 z+a x x z √  √  z dx z b dx √ + =− 2 x x 2 x z √ √   √ z dx z3 b2 dx b z √ − =− + x3 2ax2 4ax 8a x z

(see 2.224 4) (see 2.224 4) (see 2.224 4)

85

86

2.226 1. 2. 3.

Algebraic functions

2.226

 √

  √ z 3 dx  z dx √ = + a 2 z + a2 x 3 x z √  √ 3  √ 3 z dx z5 z dx 3b + =− x2 ax 2a x √   √ 3  √ 3 z dx z dx 1 b 3b2 5+ = − + z x3 2ax2 4a2 x 8a2 x

(see 2.224 4) (see 2.226 1)

(see 2.226 1)  2.227 2.228 1. 2. 2.229 1. 2. 3.

 m−1

1 dx dx 2 √ √ = √ + m m m−k k xz z (2k + 1)a z z a x z

(see 2.224 4)

k=0

√  z b dx dx √ √ = − − 2 x z ax 2a x z     √ dx 1 dx 3b 3b2 √ √ = − + 2 z+ 2 x3 z 2ax2 4a x 8a x z



(see 2.224 4) (see 2.224 4)

 dx 1 dx 2 √ √ = √ + (see 2.224 4) a x z a z x z3     dx 3b 3b 1 dx 1 √ √ = − − 2 √ − 2 (see 2.224 4) 2 3 ax a z 2a x z x z     dx 15b2 15b2 1 dx 1 5b √ √ = − √ + + + 2ax2 4a2 x 4a3 8a3 x z z x3 z 3



(see 2.224 4)

Cube root 2.231 1.

2.

3.

4.

 √  3 (−1)k nk z n−k ak 3 z 3(m+1)+1 3n − 3k + 3(m + 1) + 1 bn+1 k=0   n  

(−1)k n z n−k ak 3 xn dx k √ √ = 3 3n − 3k − 3(m − 1) − 2 bn+1 3 z 3(m−1)+2 z 3m+2 k=0  √  n   √ 3

(−1)k n z n−k ak 3 z 3(m+1)+2 3 n k z 3m+2 x dx = 3n − 3k + 3(m + 1) + 2 bn+1 k=0   n  

(−1)k n z n−k ak 3 xn dx k √ √ = 3 3 3m+1 n+1 3n − 3k − 3(m − 1) − 1 z b z 3(m−1)+1  n  √

3 n 3m+1 z x dx =

k=0

2.236

Forms containing

 n

(a + bx)k

 1 z n dx z n dx z n+ 3 3n − 3m + 4 b √ √ = − + 3 m−1 (m − 1)ax 3(m − 1) a xm−1 3 z 2 xm x2 For m = 1  n  n−1 dx z dx 3z n z √ √ √ = + a 3 3 3 2 2 x z (3n − 2) z x z2 √ √   3 z dx dx 33z 1 √ 6. = + 3 n (3n − 1)az a xz n xz n z 2   √ √ √ √  3 dx 1 z− 3a √ 33z 3 √ √ √ − 3 arctan √ 2.232 = √ ln 3 3 3 3 x z+23a x z2 a2 2 

5.

2.233 1. 2. 3.

 √  3 √ z dx dx 3 √ =3 z+a (see 2.232) 3 x x z2 √   √ 3 z dx b√ dx z3z b 3 √ + = − z + (see 2.232) x2 ax a 3 x 3 z2     √ 3 √ dx 1 b b2 √ b2 z dx 3 3 √ z = − + z − z − 3 2 2 2 x 2ax 3a x 3a 9a x 3 z 2 √  3 z 2b dx dx √ √ − =− 3 3 2 2 ax 3a x z x z2     dx dx 1 5b √ 5b2 3 √ √ = − + z + 3 2ax2 6a2 x 9a2 x 3 z 2 x3 z 2

 4. 5. 2.234

(see 2.232) (see 2.232) (see 2.232)

√  zn 3 z z n dx z n dx 3n − 3m + 5 b √ √ 1. = − + 3 xm z (m − 1)axm−1 3(m − 1) a xm−1 3 z For m = 1:  n−1  n dx 3z n z z dx √ √ √ = + a 2. 3 3 3 x z (3n − 1) z x z √  √  3 3 3 z 1 dx z dx √ = + 3. 3 xz n z (3n − 2)az n a xz n   √ √ √ √  3 z− 3a √ 33z 1 3 dx √ = √ √ √ ln + 3 arctan √ 2.235 3 3 3 x3z x z+23a a2 2 

12

2.236 1. 2.

  √ 3 z 2 dx 3√ dx 3 2 √ = z +a x 2 x3z √   √ 3 3 z 2 dx z5 b√ dx 2b 3 2+ √ + = − z x2 ax a 3 x3z

(see 2.235) (see 2.235)

87

88

Algebraic functions

2.241

    √ 3 z 2 dx dx 1 b b2 √ b2 3 5/3 2− √ z = − + − z 3 2 2 2 x 2ax 6a x 6a 9a x 3 z

3.

 4.  5.

dx √ x2 3 z dx √ 3 z

x3

√  3 z2 b dx √ =− − ax 3a x 3 z    1 dx 2b √ 2b2 3 √ = − + 2 z+ 2 2 2ax 3a x 9a x3z

2.24 Forms containing

√

(see 2.235) (see 2.235) (see 2.235)

a + bx and the binomial α + βx

Notation: z = a + bx, t = α + βx, Δ = aβ − bα. 2.241  m−1 n  m n √ 2 z (2m − 1)Δ t dx z t dx √ √ = tn+1 z m−1 z + 1. z (2n + 2m + 1)β (2n + 2m + 1)β z    n m n   n−k k

k

√ t z dx β k z k−p ap n α p √ = 2 z 2m+1 2. (−1) z bk+1 p=0 p 2k − 2p + 2m + 1 k k=0

2.242



11

1.

 2.  3.12

 4.  5.  6.

 7.  8.

√ z  2√z 2α z t dx √ = +β −a b 3 b2 z  2  √ √ z  2√z 2α2 z 2 z 2 z t2 dx 2 2 √ = + 2αβ −a − za + a + β z b 3 b2 5 3 b3   √ √ √ 2 z z 2 t3 dx 2α3 z z2 2 z 2 2 2 √ = + 3α β −a − za + a + 3αβ z b 3 b2  √ 5 3 b3 3 2 3z a z 2 z +β 3 − + za2 − a3 7 5 b4 √  z a  2√z 3 2α z 3 tz dx √ = +β − z 3b 5 3 b2 √ √  √  2 z a  2 z3 2za a2 2 z 3 2α2 z 3 z t2 z dx 2 √ + 2αβ − − + = +β 3b 5 3 b2 7 5 3 b3 z √ √  √  2 z a  2 z3 2za a2 2 z 3 t3 z dx 2α3 z 3 z 2 2 √ = + 3α β − − + + 3αβ z 3b 5 3 b2 7 5 3 b3 √   3 2 2 3 3z a 3za a 2 z3 z − + − +β 3 9 7 5 3 b4 √ √  z a  2 z5 2α z 5 tz 2 dx √ = +β − z 5b 7 5 b2 √ √  √  2 z 2α2 z 5 a  2 z5 2za a2 2 z 5 z t2 z 2 dx 2 √ = + 2αβ − − + +β z 5b 7 5 b2 9 7 5 b3

LA 176 (1)

2.244

Forms with

√

a + bx and α + βx

89

√ √  √  2 z a  2 z5 2za a2 2 z 5 z t3 z 2 dx 2α3 z 5 2 2 √ + 3α β − − + = + 3αβ 5b 7 5 b2 9 7 5 b3 z √   3 2 2 3 5 3z a 3za a 2 z z − + − +β 3 11 9 7 5 b4 √ √  3  z a  2 z7 2α z 7 tz dx √ +β − = 7b 9 7 b2 z √ √  √  2  2 3 z 2α2 z 7 a  2 z7 2za a2 2 z 7 z t z dx 2 √ = + 2αβ − − + +β z 7b 9 7 b2 11 9 7 b3 √ √ √    3 3 z a  2 z7 2za a2 2 z 7 z2 t z dx 2α3 z 7 2 2 √ = + 3α β − − + + 3αβ z 7b 9 7 b2 11 9 7 b3 √   3 2 2 3 7 3z a 3za a 2 z z − + − +β 3 13 11 9 7 b4 

9.

10. 11. 12.

2.243



1.

 tn dx 2 tn+1 √ tn dx (2n − 2m + 3)β √ √ = z − z m z (2m − 1)Δ z m (2m − 1)Δ z m−1 z  tn √ tn−1 dx 2nβ 2 √ z+ =− m (2m − 1)b z (2m − 1)b z m−1 z LA 176 (2)

 2. 2.244



1.  2.  3.  4.  5.  6.  7.

2 tn dx √ = √ 2m−1 zm z z

  n   n−k k

k

β k z k−p ap n α p (−1) bk+1 p=0 p 2k − 2p − 2m + 1 k

k=0

2α 2β(z + a) t dx √ =− √ + √ z z b z b2 z

 2 − 2za − a 2α 4αβ(z + a) t dx √ =− √ + √ √ + z z b z b2 z b3 z     2 z2 2 3 z3 2 2 3 6αβ 2β 3 3 2 − 2za − a − z a + 3za + a 3 5 2α 6α β(z + a) t dx √ =− √ + √ √ √ + + z z b z b2 z b3 z b4 z  2β z − a3 2α t dx √ √ √ = − − z2 z 3b z 3 b2 z 3    a2 2 2 a z 2β 2 2 + 2az − 4αβ z − 3 3 2α t dx √ =− √ − √ √ + 3 2 3 3 3 z2 z 3b z b z b z    3 2  2 6αβ z 2 + 2za − a3 2β 3 z3 − 3z 2 a − 3za2 + 6α2 β z − a3 2α3 t3 dx √ =− √ − √ √ √ + + z2 z 3b z 3 b2 z 3 b3 z 3 b4 z 3  2β z3 − a5 2α t dx √ =− √ − √ z3 z 5b z 5 b2 z 5 2

2

2β 2



z2 3

a3 3



90

Algebraic functions

 8.  9.

2.245 1.

2.12

t dx √ z3 z 2

t3 dx √ z3 z

2.245

  2za a2 2 2 z 2β − + 4αβ 3 − 3 5 2α √ √ =− √ − − 5 2 5 3 5 5b z b z b z    2za a2 2 2 z a 2 z 6αβ 3 − + 6α β 3 − 5 3 5 2α √ √ =− √ − − 5 2 5 3 5 5b  z b z b z  3 2β 3 z 3 + 3z 2 a − za2 + a5 √ + b4 z 5 z

2

a 5





 m−1 dx z m dx z m−1 √ z 2 (2m − 1)Δ √ =− √ z− (2n − 2m − 1)β tn−1 (2n− 2m − 1)β tn z tn z z m−1 √ z m−1 1 (2m − 1)b √ dx =− z + (n − 1)β tn−1 2(n − 1)β tn−1  zm m √ 1 z z dx (2n − 2m − 3)b √ =− z− n−1 (n − 1)Δ t 2(n − 1)Δ tn−1 z ⎡  m √ 1 1 z dz √ = −z m z ⎣ n n−1 t z (n − 1)Δ t

⎤ (2n − 2m − 3)(2n − 2m − 5) . . . (2n − 2m − 2k + 1)(−b)k−1 1 ⎦ + 2k−1 (n − 1)(n − 2) . . . (n − k)Δk tn−k k=2  (2n − 2m − 3)(2n − 2m − 5) . . . (−2m + 3)(−2m + 1)(−b)n−1 z m dx √ − 2n−1 · (n − 1)!Δn t z n−1

3. 4.

For n = 1   m zm 2 Δ z m−1 dx z dx √ = √ √ + (2m − 1)β z β t z t z   m m−1

Δk z m−k Δm dx z dx √ =2 √ √ + k+1 m (2m − 2k − 1)β β t z z t z 

2.24612

2.247

k=0

√ √ 1 β z − βΔ dx √ =√ √ ln √ [βΔ > 0] t z βΔ β z + √ βΔ β z 2 arctan √ [βΔ < 0] =√ −βΔ −βΔ √ 2 z =− [Δ = 0] bt   m

β k−1 z k 2 βm dx dx √ √ = m−1 √ + + m m k tz z z z Δ (2m − 2k + 1) Δ t z k=1

(see 2.246) 2.248 1.



2 β dx √ = √ + tz z Δ z Δ



dx √ t z

(see 2.246)

LA 176 (3)

2.248

Forms with

 2.  3.

 4.  5.  6.

 7.

 8.  9.

 10.12  11.  12.  13.  14.  15.  16.

2 2β β2 dx √ √ √ = + + tz 2 z 3Δz z Δ2 z Δ2



√

a + bx and α + βx

dx √ t z

dx 2 2β 2β 2 β3 √ + 3√ + 3 √ = √ + 3 2 2 tz z 5Δz z 3Δ z z Δ z Δ √  z b dx dx √ √ = − − 2 t z Δt 2Δ t z 1 3b 3bβ dx √ = − √ − 2√ − t2 z z Δt z Δ z 2Δ2

91

(see 2.246) 

dx √ t z (see 2.246) (see 2.246)



dx √ t z

dx 1 5b 5bβ 5bβ 2 √ √ √ √ = − − − − t2 z 2 z Δtz 2 z 3Δ2 z z Δ3 z 2Δ3

(see 2.246) 

dx √ t z

(see 2.246)  1 7b 7bβ 7bβ 7bβ 3 dx dx √ − 4√ − √ √ =− √ − √ − 2Δ4 t z t2 z 3 z Δtz 2 z 5Δ2 z 2 z 3Δ3 z z Δ z 2

(see 2.246)

√

√  z 3b2 dx 3b z dx √ √ + = − + 2Δt2 4Δ2 t 8Δ2 t z t3 z dx 1 5b 15b2 15b2 β √ √ √ √ = − + + + t3 z z 2Δt2 z 4Δ2 t z 4Δ3 z 8Δ3

(see 2.246) 

dx √ t z

(see 2.246) √  2 1 7b z 35b 35b2 β 35b2 β 2 dx dx √ + √ + √ + √ √ =− √ + 3 2 2 2 3 4 4 t z z 2Δt z z 4Δ tz z 12Δ z z 4Δ z 8Δ t z (see 2.246) 1 9b 63b2 21b2 β 63b2 β 2 63b2 β 3 dx √ √ √ √ √ √ = − + + + + + t3 z 3 z 2Δt2 z 2 z 4Δ2 tz 2 z 20Δ3 z 2 z 4Δ4 z z 4Δ5 z 8Δ5 (see 2.246) √  2 z Δ dx z dx √ = √ + (see 2.246) t z β β t z √ √  z 2 dx 2z z 2Δ z dx Δ2 √ = √ + (see 2.246) + 2 2 t z 3β β β t z √ √ √  z 3 dx 2Δz z 2z 2 z dx 2Δ2 z Δ3 √ = √ + (see 2.246) + + 5β 3β 2 β3 β3 t z t z √ √  z dx z z b z b dx √ √ =− + + (see 2.246) 2 t z Δt βΔ 2β t z √ √ √  z 2 dx z2 z bz z 3b z 3bΔ dx √ √ = − + + (see 2.246) + t2 z Δt βΔ β2 2β 2 t z



dx √ t z

92

Algebraic functions

 17.

2.249

√ √ √ √  z3 z bz 2 z 5bz z dx z 3 dx 5bΔ z 5Δ2 b √ √ =− + + + + 2 2 3 3 t z Δt βΔ 3β β 2β t z

(see 2.246) √ √ √ 2 2  b z dx z z z dx bz z b √ √ =− + (see 2.246) − + 2Δt2 4Δ2 t 4βΔ2 8βΔ t z t3 z √ √ √ √  2  z dx b2 z z 3b2 z2 z dx bz 2 z 3b2 z √ √ + + = − + + 2 2 2 2 2 3 2Δt 4Δ t 4βΔ 4β Δ 8β t z t z

 18.12 19.

 20.

(see 2.246) √ √ √ √ √  3 2 2 2 z z dx 3bz z 3b z z 5b z z 15b2 z 15b2 Δ z dx √ √ = − + + + + + 3 2 2 2 2 3 3 t z 2Δt Δ t 4βΔ 4β Δ 4β 8β t z 3

3

(see 2.246) 2.249 1.

2.

√  2 (2n + 2m − 3)β z dx dx √ = √ + m n z m−1 z (2m − 1)Δ tn−1 z (2m − 1)Δ z m tn z t √  z dx 1 (2n + 2m − 3)b √ =− − m n−1 n−1 (n − 1)Δ z t 2(n − 1)Δ t zm z ⎡ √  z ⎣ −1 1 dx √ = m z m tn z z (n − 1)Δ tn−1



LA 177 (4)

⎤ k−1 (2n + 2m − 3)(2n + 2m − 5) . . . (2n + 2m − 2k + 1)b 1 + (−1)k · n−k ⎦ 2k−1 (n − 1)(n − 2) . . . (n − k)Δk t k=2 n−1  (2n + 2m − 3)(2n + 2m − 5) . . . (−2m + 3)(−2m + 1)b dx √ +(−1)n−1 n−1 n−1 m 2 (n − 1)!Δ tz z n−1

For n = 1   2 1 β dx dx √ √ √ = + m m−1 m−1 z t z (2m − 1)Δ z z Δ tz z

2.25 Forms containing

√ a + bx + cx2

Integration techniques  2.251 It is possible to rationalize the integrand in integrals of the form

   R x, a + bx + cx2 dx by

using one or more of the following three substitutions, known as the “Euler substitutions”. √ √ 1. a + bx + cx2 = xt ± a for a > 0; √ √ 2. a + bx + cx2 = t ± x c for c > 0;  3. c (x − x1 ) (x − x2 ) = t (x − x1 ) when x1 and x2 are real roots of the equation a + bx + cx2 = 0.

2.252

Forms containing

√

a + bx + cx2

93

2.252 Besides Euler substitutions, there is also the following method of calculating integrals of   the   the form R x, a + bx + cx2 dx. By removing the irrational expressions in the denominator and performing simple algebraic operations, we can reduce the integrand to the sum of some rational function P (x) √ 1 of x and an expression of the form , where P 1 (x) and P 2 (x) are both polynomials. P 2 (x) a + bx + cx2 P 1 (x) from the remainder and decomposing By separating the integral portion of the rational function P 2 (x) the latter into partial fractions, we can reduce the integral of these partial fractions to the sum of integrals each of which is in one of the following three forms:  P (x) dx √ 1. , where P (x) is a polynomial of some degree r; a + bx + cx2  dx √ ; 2. k (x + p) a + bx + cx2    a b (M x + N ) dx . , a1 = , b 1 = 3. m c c (a + bx + x2 ) c (a1 + b1 x + x2 ) In more detail:    dx P (x) dx √ = Q(x) a + bx + cx2 + λ √ , where Q(x) is a polynomial of 1. a + bx + cx2 a + bx + cx2 degree (r − 1). Its coefficients, and also the number λ, can be calculated by the method of undetermined coefficients from the identity 1  LI II 77 P (x) = Q (x) a + bx + cx2 + Q(x)(b + 2cx) + λ 2  P (x) dx Integrals of the form √ (where r ≤ 3) can also be calculated by use of formulas a + bx + cx2 2.26.  P (x) dx √ , where the degree n of the polynomial P (x) is 2. Integrals of the form k (x + p) a + bx + cx2 1 lower than k can, by means of the substitution t = , be reduced to an integral of the form x + p  P (t) dt  . (See also 2.281). α + βt + γt2  (M x + N ) dx can be calculated by the following 3.12 Integrals of the form m (α + βx + x2 ) c (a1 + b1 x + x2 ) procedure. • If b1 = β, by using the substitution x=

a1 − α t − 1 + β − b1 t+1

(

2

(a1 − α) − (αb1 − a1 β) (β − b1 ) β − b1

94

Algebraic functions

2.260



P (t) dt , where P (t) m + p) c (t2 + q)  P (t) dt  is a polynomial of degree no higher than 2m − 1. The integral can be 2 + p)m (t t2 + q   dt t dt   and . reduced to the sum of integrals of the forms k k 2 2 2 (t + p) t +q (t + p) t2 + q  P (t) dt • If b1 = β, we can reduce it to integrals of the form by means of the m 2 (t + p) c (t2 + q) b1 substitution t = x + . 2  t dt The integral can be evaluated by means of the substitution t2 + q = k 2 (t + p) c (t2 + q) u2 .  t dt = can be evaluated by means of the substitution  The integral k 2 2 2 t +q (t + p) c (t + q) FI II 78-82 υ (see also 2.283). we can reduce this integral to an integral of the form

2.26 Forms containing

√

(t2

a + bx + cx2 and integral powers of x

Notation: R = a + bx + cx2 , Δ = 4ac − b2 For simplified formulas for the case b = 0, see 2.27. 2.260  √  √ √ (2m + 2n + 1)b xm−1 R2n+3 m 2n+1 − xm−1 R2n+1 dx 1. x R dx = (m + 2n + 2)c  2(m + 2n + 2)c √ (m − 1)a xm−2 R2n+1 dx − (m + 2n + 2)c

2. 3.

TI (192)a

 2cx + b √ 2n+1 2n + 1 Δ √ 2n−1 2n+1 R dx = R + R dx TI (188) 4(n + 1)c 8(n + 1) c  √   √ n−1

(2n + 1)(2n − 1) . . . (2n − 2k + 1)  Δ k+1 (2cx + b) R n n−k−1 2n+1 R + R dx = R 4(n + 1)c 8k+1 n(n − 1) . . . (n − k) c k=0  n+1  dx (2n + 1)!! Δ √ + n+1 8 (n + 1)! c R  √

TI (190)

2.26112



For n = −1  √  1 dx √ = √ ln 2 cR + 2cx + b c R  √  1 = − √ ln 2 cR − 2cx − b c   1 2cx + b √ = √ arcsinh c Δ 1 √ ln(2cx + b) = c   2cx + b −1 = √ arcsin √ −c −Δ

c > 0,

c > 0,

 √ −Δ, Δ < 0  √ 2cx + b < − −Δ, Δ < 0

[c > 0,

Δ > 0]

DW

[c > 0,

Δ = 0]

DW

[c < 0,

Δ < 0]

TI (128)

2cx + b >

TI (127)

2.263

2.262 1. 2. 3.

Forms containing

√  dx (2cx + b) R Δ √ R dx = + 4c 8c R √   √ 3 R (2cx + b)b √ bΔ − R− x R dx = 3c 8c2 16c2 √   2  √ 5b x 5b − R3 + − x2 R dx = 4c 24c2 16c2

5.

6.

2.263 1.

(see 2.261) dx √ (see 2.261) R √   2   5b dx a (2cx + b) R a Δ √ + − 4c 4c 16c2 4c 8c R

√ x3 R dx =



2

2

(see 2.261) √    √ R5 b √ 3 dx 3Δb √ 3Δ2 b √ x R3 dx = − (2cx + b) R + R − 5c 16c2 128c3 256c3 R

(see 2.261)     √ 3   √ R 7b √ 5 b 3Δ √ x 7b2 a 2 3 − 2x + + x R dx = R + − R 6c 60c2 24c2 6c c 8 64c   2  dx Δ2 7b √ −a + 3 4c 256c R (see 2.261)   2  √ 3bx x 3b2 2a √ 5 − + − R x3 R3 dx = 2 3 7c 28c 40c 35c2   √    3 R3 3Δ √ b 3b ab + 2x + − − 2 R 16c3 4c c 8 64c   2  dx 3Δ2 b 3b √ −a − 4c 512c4 R (see 2.261) 

8.

95

(see 2.261) √ √   3 7bx x 7b 2a 7b 3ab (2cx + b) R 3 − + − R − − 2 5c 40c2  48c3  15c2 32c3 8c 4c  3 dx 3ab Δ 7b √ − 2 − 32c3 8c 8c R (see 2.261)     √ √ 3Δ dx R 3Δ2 √ + (2cx + b) R + R3 dx = 2 8c 64c 128c2 R



7.

a + bx + cx2 and xn

 √

 4.

√





xm−1 (2m − 2n − 1)b xm dx √ √ = − 2n+1 2n−1 2(m − 2n)c R (m − 2n)c R



xm−1 dx (m − 1)a √ − 2n+1 (m − 2n)c R



xm−2 dx √ R2n+1 TI (193)a

2.

For m = 2n    x2n−1 x2n−2 x2n−1 b 1 x2n dx √ √ √ √ =− − dx + dx 2c c R2n+1 (2n − 1)c R2n−1 R2n+1 R2n−1

TI (194)a

96

Algebraic functions

 √

3. 

√

4.

dx R2n+1 dx R2n+1

2.264

 dx 2(2cx + b) 8(n − 1)c √ √ = + 2n−1 (2n − 1)Δ (2n − 1)Δ R R2n−1   n−1

2(2cx + b) 8k (n − 1)(n − 2) . . . (n − k) ck k √ 1+ = R (2n − 3)(2n − 5) . . . (2n − 2k − 1) Δk (2n − 1)Δ R2n−1 k=1

[n ≥ 1] . 2.264



1.

dx √ R

TI (189)

TI (191)

(see 2.261)

√  R b dx x dx √ = √ − (see 2.261) c 2c R R    2   2 3b √ dx x dx 3b a x √ √ − 2 R+ − = (see 2.261) 2c 4c 8c2 2c R R    3  2  3 5bx x dx dx 5b2 2a √ 5b 3ab x √ √ − + 3− 2 R− − 2 = 2 3 3c 12c 8c 3c 16c 4c R R

 2. 3. 4.

(see 2.261)



2(2cx + b) dx √ √ = 3 Δ R R  x dx 2(2a + bx) √ √ =− 3 Δ R R    2 Δ − b2 x − 2ab 1 dx x dx √ √ √ =− + (see 2.261) 3 c cΔ R R R     3 cΔx2 + b 10ac − 3b2 x + a 8ac − 3b2 dx 3b x dx √ √ = − 2 √ 2c c2 Δ R R R3

5. 6. 7. 8.

 √ 2.265

R2n+1 xm

(see 2.261) √  √ 2n+1 R2n+3 R (2n − 2m + 5)b dx = − + dx (m − 1)axm−1 √ 2(m − 1)a xm−1  (2n − m + 4)c R2n+1 + dx (m − 1)a xm−2 TI (195)

For =1 √  √m 2n+1   √ 2n−1 R R2n+1 R b √ 2n−1 dx = + dx R dx + a x 2n + 1 2 x For(a = 0 ( ( 2n+1 2n+3 2n+1   (bx + cx2 ) (bx + cx2 ) 2 (bx + cx2 ) 2(m − 2n − 3)c dx = + xm (2n − 2m + 3)bxm (2n − 2m + 3)b xm−1 For m = 0 see 2.260 2 and 2.260 3. For n = −1 and m = 1:

TI (198)

LA 169 (3)

2.267

Forms containing

 2.26612

  √ 2 aR + 2a + bx dx 1 √ = − √ ln |x| a x R  √  1 2 aR − 2a − bx = − √ ln a |x| 2a + bx 1 arcsin √ = √ −a x −Δ 2a + bx 1 arctan √ √ = √ −a 2 −a R 2a + bx 1 = − √ arcsinh √ a x Δ 2a + bx 1 = − √ arctanh √ √ a 2 a R x 1 = √ ln a√ 2a + bx 2 bx + cx2 =− bx   2a + bx 1 √ = √ arccosh a x −Δ

√

a + bx + cx2 and xn

a > 0,

2a + bx >

a > 0,

√ 2a + bx < − −Δ|x|,

[a < 0,

Δ < 0]

√ −Δ|x|,

97

Δ 0]

DW

[a > 0] [a > 0,

Δ = 0]

[a = 0,

b = 0]

[a > 0,

Δ < 0]

LA 170 (16)

2.267  √   dx dx b R dx √ √ √ + 1. = R+a (see 2.261 and 2.266) x x R 2 R √   √  dx dx R dx R b √ 2. =− +c √ (see 2.261 and 2.266) + x2 x 2 x R R For√a = 0 √   bx + cx2 dx 2 bx + cx2 +c √ dx = − (see 2.261) x2 x bx + cx2 √     2  √ R dx c dx 1 b b √ − 3. = − + R − 3 2 x 2x 4ax 8a 2 x R

4.

(see 2.266) For a = 0 ( 3  √ 2 (bx + cx2 ) bx + cx2 dx = − x3 3bx3 √     √ 3 b 12ac − b2 R R3 2bcx + b2 + 8ac √ dx dx 2 √ √ + dx = + R+a x 3 8c 16c x R R (see 2.261 and 2.266) √    √ 3 4ac + b2 R3 R5 cx + b √ 3 3 dx dx 3 √ √ + + dx = − R + (2cx + 3b) R + ab x2 ax a 4 2 8 x R R

 √ 5.

For(a = 0 3  (bx + cx2 ) x2

(see 2.261 and 2.266) ( =

(bx + cx2 ) 2x

3

3b  3b2 + bx + cx2 + 4 8

 √

dx bx + cx2

(see 2.261)

98

Algebraic functions

 √

 √ 1 b bcx + 2ac + b2 √ 3 3 bcx + 2ac + b2 √ 5 + 2 R + R + R 2 2ax2 4ax 4a  4a dx 3 dx 3 √ + bc √ + 4ac + b2 8 x R 2 R (see 2.261 and 2.266)

R3 dx = − x3

6.



For a = 0 (  (bx + cx2 )3 x3

 dx =

c−

2b x



bx + cx2 +

3bc 2



dx √ bx + cx2 (see 2.261)

 2.268

2.268

xm

dx 1 √ √ =− 2n+1 m−1 2n−1 R (m − 1)ax R  dx dx (2n + 2m − 3)b (2n + m − 2)c √ √ − − 2(m − 1)a (m − 1)a xm−1 R2n+1 xm−2 R2n+1 TI (196)

For m = 1



For a = 0 

2.269



1.

3.

dx 1 √ + 2n+1 a R



dx √ x R2n−1

dx 2 ( ( =− xm (bx + cx2 )2n+1 (2n + 2m − 1)bxm (bx + cx2 )2n−1  dx (4n + 2m − 2)c ( − (2n + 2m − 1)b 2n+1 xm−1 (bx + cx2 ) (see 2.265)

dx √ x R

(see 2.266)

√  R b dx dx √ =− √ − (see 2.266) 2 ax 2a x R x R For a = 0    1 dx 2 2c  √ − 2+ 2 = bx + cx2 3 bx b x x2 bx + cx2    2   √ dx dx 3b 3b c 1 √ = − √ + R + − 2ax2 4a2 x 8a2 2a x3 R x R

 2.

dx 1 b √ √ = − 2n+1 2n−1 2a x R (2n − 1)a R



(see 2.266)

4.

For a = 0    dx 1 2 4c 8c2  √ − 3+ 2 2− 3 = bx + cx2 5 bx 3b x 3b x x3 bx + cx2    2 bcx − 2ac + b2 dx dx 1 √ √ √ =− + (see 2.266) 3 a x R aΔ R x R For a = 0

TI (199)

2.271

Forms containing

  5.11

dx

( 3 x (bx + cx2 )

2 = 3

a + cx2 and xn

99

  1 4c 8c2 x 1 √ − + 2 + 3 bx b b bx + cx2

dx A 3b √ = −√ − 2 2 3 R 2a x R



dx √ x R

where A =

6.

√



   b 10ac − 3b2 c 8ac − 3b2 x 1 − − − ax a2 Δ a2 Δ

(see 2.266)

For a = 0    dx 1 8c2 1 2 2c 16c3 x ( √ − 2+ 2 − 3 − = 4 5 bx b x b b 3 bx + cx2 x2 (bx + cx2 )  dx √ 3 x R3     bc 15b2 − 52ac x 15b4 − 62acb2 + 24a2 c2 1 dx 1 5b 15b2 − 12ac √ √ − = − 2+ 2 − + ax 2a x 2a3 Δ 2a3 Δ 8a3 2 R x R (see 2.266) For a = 0    1 64c3 1 2 8c 16c2 128c4x dx ( √ − 3+ 2 2− 3 + = + 4 5 7 bx 5b x 5b x 5b 5b 3 bx + cx2 x3 (bx + cx2 )

2.2712 Forms containing

√ a + cx2 and integral powers of x

√ Notation: u = a + cx2 . 1  √ I1 = √ ln x c + u c  √ 1 = − √ ln −x c + u c  1 c arcsin x − = √ a −c √ u− a 1 √ I2 = √ ln 2 a √ u+ a 1 a−u = √ ln √ 2 a a + u  1 1 1 c a arcsec x − = √ arccos = √ − a x c −a −a 2.271  1.  2.  3.  4.

1 5 5 5 5 xu + axu3 + a2 xu + a3 I1 6 24 16 16 3 1 3 u3 dx = xu3 + axu + a2 I1 4 8 8 1 1 u dx = xu + aI1 2 2 dx = I1 u u5 dx =



 

c > 0, c > 0,

 a − , a 0 and c > 0] [a > 0 and c < 0] [a < 0 and c > 0]

DW DW DW DW

100

Algebraic functions

 5.  6.  7. 2.272

dx 1x = u3 au dx u2n+1

  n−1 1 (−1)k n − 1 ck x2k+1 = n a 2k + 1 k u2k+1

x2 u3 dx = x2 u dx =

2.  3.  4.  5.  6.  7.

 4.  5.  6. 

DW

1 x2 x dx = − + I1 u3 cu c

DW

x2 1 x3 dx = u5 3 au3

DW

  n−2 x2 dx 1 (−1)k n − 2 ck x2k+3 = n−1 u2n+1 a 2k + 3 k u2k+3 k=0

x dx 1 a =− + u2n+1 (2n − 3)c2 u2n−3 (2n − 1)c2 u2n−1 3

x4 u dx =

3.

DW

DW

 

DW

1 xu 1 a x2 dx = − I1 u 2 c 2c

x4 u3 dx =

2.

1 axu3 1 a2 xu 1 a3 1 xu5 − − − I1 6 c 24 c 16 c 16 c

1 axu 1 a2 1 xu3 − − I1 4 c 8 c 8 c



1.

7.

k=0

x dx 1 =− 2n+1 u (2n − 1)cu2n−1



2.273

DW



1.

2.272

axu5 1 x3 u5 a2 xu3 3a3 xu 3a4 − + + + I1 8 c 16c2 64c2 128c2 128c2

axu3 1 x3 u3 a2 xu a3 − + + I1 2 2 6 c 8c 16c 16c2

DW

DW DW

1 x3 u 3 axu 3 a2 x4 dx = − + I1 u 4 c 8 c2 8 c2

DW

3 a x4 1 xu ax dx = + 2 − I1 3 2 u 2c c u 2 c2

DW

1 x3 x4 x 1 dx = − 2 − + 2 I1 5 u c u 3 cu3 c

DW

x4 1 x5 dx = u7 5 au5

DW

  n−3 x4 dx 1 (−1)k n − 3 ck x2k+5 = n−2 u2n+1 a 2k + 5 k u2k+5 k=0

2.275

Forms containing

 8.

12

2.274

x6 u3 dx =  x6 u dx =

2.  3.  4.  5.  6.  7.  8.  9. 2.275



1.  2. 3. 4. 5. 6. 7. 8.12

a + cx2 and xn

x5 dx 1 2a a2 = − + − u2n+1 (2n − 5)c3 u2n−5 (2n − 3)c3 u2n−3 (2n − 1)c3 u2n−1



1.

√

101

DW

ax3 u5 1 x5 u5 a2 xu5 a3 xu3 3a4 xu 3 a5 − + − − − I1 2 3 3 3 10 c 16c 32c 128c 256c 256 c3

5 ax3 u3 1 x5 u3 5a2 xu3 5a3 xu 5 a4 − + − − I1 8 c 48 c2 64c3 128c3 128 c3

1 x5 u 5 ax3 u x6 5 a2 xu 5 a3 dx = − + − I1 u 6 c 24 c2 16 c3 16 c3

DW

1 x5 5 ax3 15 a2 x 15 a2 x6 dx = I1 − − + 3 2 u 4 cu 8 c u 8 c3 u 8 c3

DW

x6 1 x5 10 ax3 5 a2 x 5 a dx = + + − I1 u5 2 cu3 3 c2 u 3 2 c3 u 3 2 c3

DW

x6 23 x5 7 ax3 a2 x 1 dx = − − − + 3 I1 7 5 2 5 3 5 u 15 cu 3c u c u c

DW

x6 1 x7 dx = 9 u 7 au7

DW

  n−4 x6 dx 1 (−1)k n − 4 ck x2k+7 = u2n+1 an−3 2k + 7 k u2k+7 k=0

x dx 1 3a 3a2 a3 = − + − + u2n+1 (2n − 7)c4 u2n−7 (2n − 5)c4 u2n−5 (2n − 3)c4 u2n−3 (2n − 1)c4 u2n−1

DW

u5 1 u5 dx = + au3 + a2 u + a3 I2 x 5 3

DW

7

u3 u3 dx = + au + a2 I2 x 3  u dx = u + aI2 x  dx = I2 xu  n−1

1 dx 1 = I + 2 2n+1 n xu a (2k + 1)an−k u2k+1 k=0  5 5 15 u u5 15 dx = − + cxu3 + acxu + a2 I1 2 x x 4 8 8  3 3 3 3 u u dx = − + cxu + aI1 x2 x 2 2  u u dx = − + I1 x2 x

DW DW DW

DW DW DW

102

Algebraic functions

 9. 2.276



1.  2.  3.  4.  5.  6.  7.  8.  9.  10. 2.277



1.  2.  3.  4.

dx 1 = − n+1 x2 u2n+1 a



2.276

n u (−1)k+1  n  k  x 2k−1 + c x 2k − 1 k u



k=1

5 u5 5 5 u5 dx = − + cu3 + acu + a2 cI2 3 2 x 2x 6 2 2

DW

3 u3 u3 3 dx = − 2 + cu + acI2 3 x 2x 2 2 u u c dx = − 2 + I2 x3 2x 2 u dx c =− − I2 x3 u 2ax2 2a 3c 3c dx 1 − − =− I2 x3 u3 2ax2 u 2a2 u 2a2 5 c dx 1 5 c 5 c − =− − − I2 x3 u5 2ax2 u3 6 a2 u 3 2 a3 u 2 a3

DW DW DW DW DW

u5 au3 2acu c2 xu 5 + + acI1 dx = − − x4 3x3 x 2 2

DW

u3 u3 cu + cI1 dx = − 3 − 4 x 3x x

DW

u u3 dx = − x4 3ax3 dx 1 = n+2 x4 u2n+1 a

DW



n+1

u3 cu (−1)k − 3 + (n + 1) + 3x x 2k − 3 k=2



n+1 k



ck

 x 2k−3



u

u3 3 cu3 3 c2 u 3 2 u3 + c I2 dx = − 4 − + 5 2 x 4x 8 ax 8 a 8

DW

u u 1 cu 1 c2 I2 dx = − − − x5 4x4 8 ax2 8 a

DW

u dx 3 cu 3 c2 = − + + I2 x5 u 4ax4 8 a2 x2 8 a2

DW

5 c 15 c2 15 c2 dx 1 + + + =− I2 5 3 4 2 2 3 x u 4ax u 8 a x u 8 a u 8 a3 DW

2.278



1.  2.

u3 u5 dx = − x6 5ax5

DW

u u3 2 cu3 dx = − + 6 5 x 5ax 15 a2 x3

DW

2.284

Forms containing

 3.  4.

√

a + bx + cx2 and polynomials

103

  u5 2 cu3 c2 u − 5+ − DW 5x 3 x3 x         n+2 u5 1 1 n + 2 cu3 n + 2 c2 u (−1)k n + 2 k  x 2k−5 + c = n+3 − 5 + − a 5x 3 1 x3 x 2k − 5 k u 2

1 dx = 3 6 x u a dx x6 u2n+1

k=3

2.28 Forms containing

√

a + bx + cx2 and first-and second-degree polynomials

Notation: R = a + bx + cx2 See also 2.252   dx tn−1 dt 3 √ 2.281 =−  (x + p)n R c + (b − 2pc)t + (a − bp + cp2 ) t2  t= 2.282 1.

 √

R dx =c x+p

3



   x dx dx dx √ + (b − cp) √ + a − bp + cp2 √ R R (x + p) R



2. 3. 4. 5.





[x + p > 0]

dx dx dx 1 1 √ = √ + √ q − p p − q (x + p)(x + q) R (x + p) R (x + q) R √  √  √  R dx R dx R dx 1 1 = + (x + p)(x + q) q−p x+p p−q x+q √   √  √ (x + p) R dx R dx R dx + (p − q) = x+q x+q    dx dx (rx + s) dx s − pr s − qr √ = √ + √ q − p p − q (x + p)(x + q) R (x + p) R (x + q) R 

2.283

1 >0 x+p

A (Ax + B) dx √ = n c (p + R) R



du 2Bc − Ab + (p + u2 )n 2c





1 − cυ 2

n−1

dυ  n , 2 b − cpυ 2 p+a− 4c

√ b + 2cx √ . where u = R and υ = 2c R  Ax + B 2Bc − Ab A √ dx = I1 +  2.28412 I2 , c (p + R) R c p [b2 − 4(a + p)c] where ) R 1 I1 = √ arctan [p > 0] p p √ √ −p − R 1 √ ln √ = √ [p < 0] 2 −p −p + R



104

Algebraic functions



p b + 2cx √ b2 − 4(a + p)c R p b + 2cx √ = − arctan  2 R    p [b − 4(a + p)c] √ 2 p[4(a + p)c − b ] R + p(b + 2cx) i √ = ln  2 p[4(a + p)c − b2 ] R − p(b + 2cx)

I2 = arctan

2.290

   2 p b − 4(a + p)c > 0,

p 0,

p>0



  2  p b − 4(a + p)c < 0

2.29 Integrals that can be reduced to elliptic or pseudo-elliptic integrals  2.290 Integrals of the form

   R x, P (x) dx, where P (x) is a third-or fourth-degree polynomial can,

by means of algebraic transformations, be reduced to a sum of integrals expressed in terms of elementary functions and elliptic integrals (see 8.11). Since the substitutions that transform the given integral into an elliptic integral in the normal Legendre form are different for different intervals of integration, the corresponding formulas are given in the chapter on definite integrals (see 3.13, 3.17).     2.291 Certain integrals of the form R x, P (x) dx, where Pn (x) is a polynomial of not more than     fourth degree, can be reduced to integrals of the form R x, k Pn (x) dx with k ≥2. Below are examples of this procedure.     dx dz 1 2 √ 1. x = =− √ 1 + z2 1 − x6 3 + 3z 2 + z 4   dz dx 1 √ √ 2. = 2 a + bx2 + cx4 + dx6 az + bz 2 + cz 3 + dz 4  2 x =z   1  3 z 2 A± 3 dz 2 3 ±1/3 dx = 3. a + 2bx + cx + gx 2 B ⎤ ⎡ 3   2 + (z 3 − a) c  b −b + ⎣a + 2bx + cx2 = z 3 , A = g + z 3 , B = b2 + (z 3 − a) c⎦ c

4.12

This integral involving a one-third power cannot be reduced to an elliptic  dx √ 6 a + bx + cx2 + dx3 + cx4 + bx5 + ax   1 dx dz 1   = −√ −√ 2 2 (z + 1)p (z − 1)p dx dz 1 1   +√ = −√ 2 2 (z + 1)p (z − 1)p   where p = 2a 4z 3 − 3z + 2b 2z 2 − 1 + 2cz + d.

integral. 

 x = z + z2 − 1 

 x = z − z2 − 1

2.292

Integrals reducible to elliptic integrals



105



dy √ [x = y] √  2 3 4 y a + by + cy + by + ay 

 dz dz 1 1   =− √ − √ y = z + z2 − 1 2 2 (z + 1)p 2 2  (z − 1)p 

 dz dz 1 1   + √ y = z − z2 − 1 =− √ 2 2 (z + 1)p 2 2 (z − 1)p  where p = 2a 2z 2 − 1 + 2bz + c.       a√ dt dx 18 a 8 12 √ √√ 6. x = = t ; 2 c c t a + b1 t2 + at4 a + bx4 + cx8    

  a 1 dz dz  t = z + z2 − 1 =− √ 8 −  2 2 c (z + 1)p (z − 1)p     

 dz dz 1 8 a  +  t = z − z2 − 1 =− √ 2 2 c (z + 1)p (z − 1)p  2  where p = 2a 2z − 1 + b1 ; b1 = b ac .    x dx z 2 dz √ √ 7. a + bx2 + cx4 = z 4 , A = b2 − 4ac, B = 4c = 2 4 a + bx2 + cx4 A + Bz 4        4 2 R2 z 4 z 2 dz b2 − a (c − z 4 ) + b dx 2 √  = z dz = R1 z z dz +  , 8. 4 a + 2bx2 + cx4 (c − z 4 ) b2 − a (c − z 4 ) b2 − a (c − z 4 )   where R1 z 4 and R2 z 4 are rational functions of z 4 and a + 2bx2 + cx4 = x4 z 4 .     2.29212 In certain cases, integrals of the form R x, P (x) dx, where P (x) is a third-or fourth-degree polynomial, can be expressed in terms of elementary functions. Such integrals are called pseudo-elliptic integrals. Thus, if the relations       1 1 − k2 x 1−x f1 (x) = −f1 , f , f , (x) = −f (x) = −f 2 2 3 3 k2 x k 2 (1 − x) 1 − k2 x 5.

12

√

dx 1 = 2 4 6 8 2 a + bx + cx + bx + ax

hold, then   f1 (x) dx  1. = R1 (z) dz x(1 − x) (1 − k 2 x)   f2 (x) dx  2. = R2 (z) dz x(1 − x) (1 − k 2 x)   f3 (x) dx  = R3 (z) dz 3. x(1 − x) (1 − k 2 x) where R1 (z), R2 (z), and R3 (z) are rational functions of z.

  zx = x(1 − x) (1 − k 2 x) 



x (1 − k 2 x) √ z= 1−x    x(1 − x) z= √ 1 − k2 x



106

The Exponential Function

2.311

2.3 The Exponential Function 2.31 Forms containing eax 

eax a 2.312 ax in an integrand should be replaced with ex ln a = ax 2.313  1 dx [mx − ln (a + bemx )] = 1. a + bemx am  dx ex 2. = ln = x − ln (1 + ex ) x 1+e 1 + ex     dx 1 a mx [ab > 0] 2.314 = √ arctan e mx −mx ae + be b m ab √    b + emx −ab  1  [ab < 0] √ √ ln  = 2m −ab b − emx −ab  √ √  a + bemx − a 1 dx √ √ √ ln [a > 0] = 2.315 √ mx a + bemx m a a + be √ + a a + bemx 2 √ arctan [a < 0] = √ m −a −a  Ei (ax ) 2.316* exp (ax ) dx = ln a 2.311

eax dx =

PE (410) PE (409) PE (411)

2.32 The exponential combined with rational functions of x 2.321   n xn eax n ax 12 − xn−1 eax dx x e dx = 1. a a  n   

(−1)k k! n n ax ax n−k k 11 x e dx = e x 2. ak+1

[a = 0]

k=0

2.322    1 x − 2 1. xeax dx = eax a a  2   2x x 2 2 ax ax − 2 + 3 2. x e dx = e a a a  3   2 3x x 6x 6 − 2 + 3 − 4 3. x3 eax dx = eax a a a a  4   3 2 12x 24x 24 4x x 4 ax ax 10 4. x e dx = e − 2 + 3 − 4 + 5 a a a a a  m P (k) (x) eax

2.323 P m (x)eax dx = (−1)k , a ak k=0

where P m (x) is a polynomial in x of degree m and P (k) (x) is the k-th derivative of P m (x) with respect to x.

Exponentials and rational functions of x

2.325

2.324



1.12  2.

  ax  eax eax dx 1 e dx − = + a xm m−1 xm−1 xm−1

107

[m = 1]

n−1

ak−1 an−1 eax ax Ei(ax) dx = −e + xn (n − 1)(n − 2) . . . (n − k)xn−k (n − 1)! k=1

2.325 1. 2. 3. 4. 5. 6.12

7. 8.12

9. 10. 11.

12.



eax dx = Ei(ax) x  ax e eax + a Ei(ax) dx = − 2 x x  ax e a2 eax aeax + Ei(ax) dx = − − x3 2x2 2x 2  ax e a3 eax aeax a2 eax + Ei(ax) dx = − − − x4 3x3 6x2 6x 6  ±axn  ±axn  ±axn  e e 1 e − m−1 ± na dx = dx [m = 1] xm m−1 x xm−n  axn e (−1)z+1γ az Γ (−z, −axn ) dx = m x n (−1)z+1 az ∞ e−t dt = z+1 n −axn t m−1 , for Γ(α, x) see 8.350.2 [a = 0, n = 0] z= n  axn Ei (axn ) e dx = [a = 0, n = 0] x n *z−1 az−k−1  axn az Ei (axn ) e k=0 k! xn(k+1) axn + dx = −e xm nz! nz!   m−1 = 1, 2, . . . , m = 2, 3, . . . a = 0, n = 0, z = n    axn n m−1 e eax a Ei (axn ) a =  0, z = = 1 dx = − + xm nxn n n    axn axn axn 2 n m−1 e e ae a Ei (ax ) a = 0, z = =2 dx = − − + xm 2nx2n 2nxn 2n n  axn n n n e eax eax a2 eax a3 Ei (axn ) dx = − − − + xm 3nx3n 6nx2n 6nxn 6n   m−1 =3 a = 0, z = n  ax2 2 √ √ e eax erf(iz) + aπ erfi ax dx = − where erfi(z) = 2 x x i

108

The Exponential Function

 2.326 2.33 1.

 e

8

xeax dx eax = (1 + ax)2 a2 (1 + ax)

3.12



[a = 0]

π exp a



b2 − ac a



  √ b √ erf ax + a [a = 0]



 2.

1 dx = 2

−(ax2 +2bx+c)

2.326

√ 2 1 π erf(iz) erfi ax where erfi(z) = eax dx = 2 a i       2 √ 2 1 π b 4ac − b erfi eax +bx+c dx = ax + √ exp 2 a 4a 2 a where erfi(z) = 

4.12

 5.12

n

xm e±ax dx = ±

n

xm eax dx =

eax n

xm+1−n ±axn m + 1 − n e ∓ na na

n

 (γ − 1)!

γ−1

(−1)k+1−γ

k=0



x e

6.  7.  8. 

xnk k!aγ−k

n



[a = 0,

n = 0]



 m+1 = 1, 2, . . . n = 0, γ = n  m+1 if =1 n  m+1 =2 if n  m+1 =3 if n  m+1 =4 if n

 10.

n

eax n

n

eax dx = n

n

n

xm eax dx =

n

a = 0,   

xn 1 − 2 a a



x2n 2xn 2 − 2 + 3 a a a

a = 0, 

x3n 3x2n 6xn 6 − 2 + 3 − 4 a a a a

Γ (γ, βxn ) γ nβ  ∞ 1 =− γ tγ−1 e−t dt nβ βxn

xm e−βx dx = −

  a = 0, 

 a = 0,

for Γ(α, x) see 8.350.2   m+1 , β = 0, n = 0 γ= n 

γ−1

xnk (γ − 1)! n exp (−βx ) x exp (−βx ) dx = − n k!β γ−k k=0   m+1 = 1, 2, . . . γ= n    m+1 exp (−βxn ) xm exp (−βxn ) dx = − if =1 nβ n



12.

eax n

n

m axn



n

eax dx = na

xm eax dx =

x e

9.

11.

m axn

m

n

[a = 0]

xm−n e±ax dx

a = 0, 

erf(iz) i

[a = 0]

Exponentials and rational functions of x

2.33

 13.  14.

exp (−βxn ) x exp (−βx ) dx = − n m

n

xm exp (−βxn ) dx = −

exp (−βxn ) n

 

xn 1 + 2 β β





109

m+1 =2 if n 



x2n 2xn 2 + 2 + 3 β β β 

 m+1 =3 n  n 6x 6 + 3 + 4 β β   m+1 =4 if n if

 15.

 16.12  17.

12

exp (−βxn ) x exp (−βx ) dx = − n m

n

2

e−βx dx =

1 2





3x2n x3n + 2 β β

  π exp βx β

[β = 0]

exp (−βxn ) β z Γ (−z, βxn ) dx = − xm  n β z ∞ e−t dt =− n βxn tz+1

 n = 0,

 18. 

19.12

20. 21.

22.

23.

β = 0,

m−1 z= n



exp (−βxn ) Ei (−βxn ) dx = x n z−1

Ei (−βxn )

βz exp (−βxn ) β z−k−1 Ei (−βxn ) dx = (−1)z (−1)k k! n(k+1) + (−1)z m x nz! nz! x k=0   m−1 n = 0, β = 0, z = = 1, 2, . . . , m = 2, 3, . . . n    exp (−βxn ) m−1 exp (−βxn ) β Ei (−βxn ) z = = 1 dx = − − xm nxn n n  n n n 2 exp (−βx ) β exp (−βx ) β Ei (−βxn ) exp (−βx ) dx = − + + xm 2nx2n 2nxn 2n   m−1 =2 z= n  n n n 2 exp (−βx ) β exp (−βx ) β exp (−βxn ) β 3 Ei (−βxn ) exp (−βx ) dx = − + − − xm 3nx3n 6nx2n 6nxn 6n   m−1 =3 z= n      β exp −βx2 exp (−βxn ) − dx = − βπ erf βx x2 x

110

Hyperbolic Functions

2.411

2.4 Hyperbolic Functions 2.41–2.43 Powers of sinh x, cosh x, tanh x, and coth x 

2.411

2.412 1.



 sinhp+1 x coshq−1 x q − 1 + sinhp x coshq−2 x dx sinh x cosh x dx = p+q p+q  sinhp−1 x coshq+1 x p − 1 − sinhp−2 x coshq x dx = p+q p+q  sinhp−1 x coshq+1 x p − 1 − sinhp−2 x coshq+2 x dx = q+1 q+1  sinhp+1 x coshq−1 x q − 1 − sinhp+2 x coshq−2 x dx = p+1 p+1  sinhp+1 x coshq+1 x p + q + 2 − sinhp+2 x coshq x dx = p+1 p+1  sinhp+1 x coshq+1 x p + q + 2 + sinhp x coshq+2 x dx =− q+1 q+1 p

q

⎡ p+1 x sinh ⎣ cosh2n−1 x sinhp x cosh2n x dx = 2n + p

3.

4.

⎤

(2n − 1)(2n − 3) . . . (2n − 2k + 1) cosh2n−2k−1 x⎦ (2n + p − 2)(2n + p − 4) . . . (2n + p − 2k) k=1  (2n − 1)!! + sinhp x dx (2n + p)(2n + p − 2) . . . (p + 2) This formula is applicable for arbitrary real p except for the following negative even integers: −2, −4, . . . , −2n. If p is a natural number and n = 0, we have      m−1

2m 2m sinh(2m − 2k)x x 1 2m m k sinh x dx = (−1) + 2m−1 (−1) TI (543) 2 2m − 2k m 22m k k=0    m 2m + 1 cosh(2m − 2k + 1)x 1

; sinh2m+1 x dx = 2m (−1)k TI (544) 2 2m − 2k + 1 k k=0 m  m  cosh2k+1 x

= (−1)n (−1)k GU (351) (5) 2k + 1 k k=0  sinhp x cosh2n+1 x dx   n

2k n(n − 1) . . . (n − k + 1) cosh2n−2k x sinhp+1 x 2n cosh x + = 2n + p + 1 (2n + p − 1)(2n + p − 3) . . . (2n + p − 2k + 1) k=1 This formula is applicable for arbitrary real p except for the following negative odd integers: −1, −3, . . . , −(2n + 1). +

2.

n−1

2.414

2.413

Powers of hyperbolic functions



1.

⎡ p+1 cosh x ⎣ sinh2n−1 x coshp x sinh2n x dx = 2n + p

3.

4.

n−1

⎤

(2n − 1)(2n − 3) . . . (2n − 2k + 1) sinh x⎦ (2n + p − 2)(2n + p − 4) . . . (2n + p − 2k) k=1  (2n − 1)!! n coshp x dx +(−1) (2n + p)(2n + p − 2) . . . (p + 2) This formula is applicable for arbitrary real p except for the following negative even integers: −2, −4, . . . , −2n. If p is a natural number and n = 0, we have    m−1

 2m  sinh(2m − 2k)x 2m x 1 2m cosh x dx = + 2m−1 TI (541) 2 2m − 2k k m 22m k=0   m  1 2m + 1 sinh(2m − 2k + 1)x cosh2m+1 x dx = 2m TI (542) 2 2m − 2k + 1 k k=0 m 

m  sinh2k+1 x = GU (351) (8) 2k + 1 k k=0 ⎡  p+1 x cosh ⎣ sinh2n x coshp x sinh2n+1 x dx = 2n + p + 1 ⎤ n 2n−2k k

2 n(n − 1) . . . (n − k + 1) sinh x ⎦ (−1)k + (2n + p − 1)(2n + p − 3) . . . (2n + p − 2k + 1) +

2.

111

(−1)k

2n−2k−1

k=1

This formula is applicable for arbitrary real p except for the following negative odd integers: −1, −3, . . . , −(2n + 1). 2.414

 

2.  3.  4.  5.  6.

1 cosh ax a x 1 sinh2 ax dx = sinh 2ax − 4a 2 1 1 3 cosh 3x = cosh3 x − cosh x sinh3 x dx = − cosh x + 4 12 3 1 1 3 3 1 3 sinh 4x = x − sinh x cosh x + sinh3 x cosh x sinh4 x dx = x − sinh 2x + 8 4 32 8 8 4 5 1 5 sinh5 x dx = cosh x − cosh 3x + cosh 5x 8 48 80 1 4 4 cosh3 x = cosh x + sinh4 x cosh x − 5 5 15 15 3 1 5 sinh 2x − sinh 4x + sinh 6x sinh6 x dx = − x + 16 64 64 192 1 5 5 5 sinh3 x cosh x + sinh x cosh x = − x + sinh5 x cosh x − 16 6 24 16 sinh ax dx =

1.

112

Hyperbolic Functions

2.415

 7.  8. 9. 10. 11. 12.

35 sinh7 x dx = − 64 cosh x +

7 64

cosh 3x −

7 320

24 cosh x + = − 35

8 35

cosh3 x −

6 35

1 448

cosh 7x

cosh x sinh4 x +

1 7

cosh x sinh6 x

1 sinh ax a  1 x sinh 2ax cosh2 ax dx = + 2 4a  1 cosh3 x dx = 34 sinh x + 12 sinh 3x = sinh x + 13 sinh3 x  1 cosh4 x dx = 38 x + 14 sinh 2x + 32 sinh 4x = 38 x + 38 sinh x cosh x +  5 1 sinh 3x + 80 sinh 5x cosh5 x dx = 58 sinh x + 48 cosh ax dx =

=  13.

4 5

sinh x + +

15 64

sinh 2x +

=

5 16 x

+

5 16

sinh x cosh x +

4 15

sinh 4x +

sinh x +

7 64

sinh 3x +

7 320

=

24 35

sinh x +

8 35

sinh3 x +

6 35

1 192

sinh 5x +

1 448

cosh(a + b)x cosh(a − b)x + 2(a + b) 2(a − b)

sinh ax cosh ax dx =

1 cosh 2ax 4a

sinh2 x cosh x dx =

1 3

sinh3 x

sinh3 x cosh x dx =

1 4

sinh4 x

sinh4 x cosh x dx =

1 5

sinh5 x

sinh x cosh2 x dx =

1 3

cosh3 x

sinh x cosh5 x

sinh 7x

sinh x cosh4 x +

sinh ax cosh bx dx =

1 6

1 7

sinh x cosh6 x

 5.  6.  sinh2 x cosh2 x dx = − 81 x +

7.  8.

1 32



sinh3 x cosh2 x dx =

1 5

sinh4 x cosh2 x dx =

1 16 x

sinh 4x

sinh2 x −

2 3



cosh3 x

 9.

−

1 64

sinh 2x −

1 64

sinh x cosh3 x

sinh 6x

 4.

1 4

sinh3 x

sinh x cosh3 x +

5 24

35 64

 3.

3 64

cosh7 x dx =

 2.

cosh4 x sinh x +

5 16 x



1.

1 5

cosh6 x dx = 

14.

2.415

cosh 5x +

sinh 4x +

1 192

sinh 6x

2.416

Powers of hyperbolic functions

113

 sinh x cosh3 x dx =

10. 

1 4

sinh2 x cosh3 x dx =

11.

cosh4 x 1 5





cosh2 x +

2 3

3 sinh3 x cosh3 x dx = − 64 cosh 2x +

12.



sinh3 x

1 192 4

cosh 6x =

1 48

cosh3 2x −

1 16

cosh 2x

cosh6 x cosh4 x sinh6 x sinh x + = − 6 4 6 4    sinh4 x cosh3 x dx = 17 sinh3 x cosh4 x − 35 cosh2 x − 25 = 17 cosh2 x + 25 sinh5 x  sinh x cosh4 x dx = 15 cosh5 x  1 1 1 1 sinh2 x cosh4 x dx = − 16 x − 64 sinh 2x + 64 sinh 4x + 192 sinh 6x    sinh3 x cosh4 x dx = 17 cosh3 x sinh4 x + 35 sinh2 x − 25 = 17 sinh2 x − 25 cosh5 x  3 1 1 sinh4 x cosh4 x dx = 128 x − 128 sinh 4x + 1024 sinh 8x =

13. 14. 15. 16. 17. 2.416



1.10

p

p+1

⎡

x⎣ sinh x sinh dx = sech2n−1 x 2n − 1 cosh2n x

⎤ (2n − p − 2)(2n − p − 4) . . . (2n − p − 2k) sech2n−2k−1 x⎦ + (2n − 3)(2n − 5) . . . (2n − 2k − 1) k=1  (2n − p − 2)(2n − p − 4) . . . (−p + 2)(−p) + sinhp x dx (2n − 1)!!  This formula is applicable for arbitrary real p. For sinhp x dx, where p is a natural number, see 2.412 2 and 2.412 3. For n = 0 and p a negative integer, we have for this integral: ⎡  cosh x ⎣ dx = − cosech2m−1 x sinh2m x 2m − 1 ⎤ m−1 k

2 (m − 1)(m − 2) . . . (m − k) cosech2m−2k−1 x⎦ (−1)k−1 · + (2m − 3)(2m − 5) . . . (2m − 2k − 1) n−1

2.

k=1

 3.

⎡ cosh x ⎣ = − cosech2m x 2m sinh2m+1 x dx

+

⎤ (2m − 1)(2m − 3) . . . (2m − 2k + 1) cosech2m−2k x⎦ (−1)k−1 · 2k (m − 1)(m − 2) . . . (m − k)

m−1

k=1

+(−1)m

(2m − 1)!! x ln tanh (2m)!! 2

114

2.417

Hyperbolic Functions



1.

⎡ sinhp+1 x ⎣ sinhp x dx = sech2n x 2n cosh2n+1 x

 3.

k=1

sinh x dx = cosh x 2m

4. 

1.



k=1

2k − 1

dx (−1)k cosech2m−2k+2 x + (−1)m ln tanh x = 2m − 2k + 2 x cosh x k=1

2m+1

m

(−1)k cosech2m−2k+2 x dx + (−1)m arctan sinh x = 2m 2m − 2k + 1 sinh x cosh x k=1 ⎡ coshp+1 x ⎣ coshp x dx = − cosech2n−1 x 2n − 1 sinh2n x

⎤ (−1)k (2n − p − 2)(2n − p − 4) . . . (2n − p − 2k) cosech2n−2k−1 x⎦ + (2n − 3)(2n − 5) . . . (2n − 2k − 1) k=1  (−1)n (2n − p − 2)(2n − p − 4) . . . (−p + 2)(−p) + coshp x dx (2n − 1)!!  This formula is applicable for arbitrary real p. For the integral coshp x dx, where p is a natural number, see 2.413 2 and 2.413 3. If p is a negative integer, we have for this integral:    m−1

2k (m − 1)(m − 2) . . . (m − k) dx sinh x 2m−1 2m−2k−1 sech sech = x+ x 2m − 1 (2m − 3)(2m − 5) . . . (2m − 2k − 1) cosh2m x k=1    m−1

(2m − 1)(2m − 3) . . . (2m − 2k + 1) sinh x dx sech2m−2k x sech2m x + = 2m 2k (m − 1)(m − 2) . . . (m − k) cosh2m+1 x k=1 (2m − 1)!! arctan sinh x + (2m)!! n−1

2.

3.

sinh2k−1 x + (−1)m arctan (sinh x)

m

sinh

5.

m

(−1)m+k

[m ≥ 1]



2.418

⎤

n−1

(2n − p − 1)(2n − p − 3) . . . (2n − p − 2k + 1) sech2n−2k x⎦ 2k (n − 1)(n − 2) . . . (n − k) k=1  (2n − p − 1)(2n − p − 3) . . . (3 − p)(1 − p) sinhp x + dx 2n n! cosh x This formula is applicable for arbitrary real p. For n = 0 and p integral, we have  m

sinh2m+1 x (−1)m+k dx = sinh2k x + (−1)m ln cosh x cosh x 2k k=1 m

(−1)m+k  m  cosh2k x + (−1)m ln cosh x [m ≥ 1] = 2k k +

2.

2.417

2.423

2.419

Powers of hyperbolic functions



1.

115

⎡ coshp x coshp+1 x ⎣ dx = − cosech2n x 2n sinh2n+1 x

⎤ (−1)k (2n − p − 1)(2n − p − 3) . . . (2n − p − 2k + 1) cosech2n−2k x⎦ + 2k (n − 1)(n − 2) . . . (n − k) k=1  (−1)n (2n − p − 1)(2n − p − 3) . . . (3 − p)(1 − p) coshp x + dx 2n n! sinh x This formula is applicable for arbitrary real p. For n = 0 and p an integer  m

cosh2m x cosh2k−1 x x dx = + ln tanh sinh x 2k − 1 2 k=1  m

cosh2m+1 x cosh2k x dx = + ln sinh x sinh x 2k k=1 m 

m  sinh2k x + ln sinh x = 2k k n−1

2.

3.

 4.  5.

k=1

m

sech2m−2k+1 x x dx + ln tanh = 2m 2 sinh x cosh x k=1 2m − 2k + 1 m

sech2m−2k+2 x dx + ln tanh x = sinh x cosh2m+1 x k=1 2m − 2k + 2

2.421 In formulas 2.421 1 and 2.421 2, s = 1 for m odd and m < 2n + 1; in all other cases, s = 0. GI (351)(11, 13)

 1.10  2.

n 2k−m+1  

m−1 x sinh2n+1 x n+k n cosh dx = + s(−1)n+ 2 (−1) m cosh x 2k − m + 1 k k=0 k= m−1 2



n

m−1 2

 ln cosh x

  n  n  sinh2k−m+1 x

n cosh2n+1 x dx = + s ln sinh x m−1 sinhm x 2k − m + 1 k 2 k=0 k= m−1 2

2.422 1.

2.

  m+n−1

(−1)k+1 m+n−1 dx tanh2k−2m+1 x = 2m − 2k − 1 k sinh2m x cosh2n x k=0    m+n

(−1)k+1  m + n  m+n dx 2k−2m m tanh ln tanh x = x + (−1) 2m − 2k k m sinh2m+1 x cosh2n+1 x k=0



k=m

GI (351)(15)

2.423  1.

x 1 cosh x − 1 dx = ln tanh = ln sinh x 2 2 cosh x + 1

116

Hyperbolic Functions

 2.  3.  4.  5.  6.

 7. 

8. 9.

dx = − coth x sinh2 x x cosh x 1 dx =− − ln tanh 3 2 2 2 sinh x 2 sinh x 1 cosh x 2 dx =− + coth x = − coth3 x + coth x 4 3 3 3 sinh x 3 sinh x x cosh x 3 cosh x 3 dx =− + + ln tanh 5 4 2 8 8 2 sinh x 4 sinh x sinh x cosh x 4 dx 4 coth3 x − coth x =− + 6 5 15 5 sinh x 5 sinh x 1 2 5 3 = − coth x + coth x − coth x 5 3   5 x cosh x 1 5 15 dx − ln tanh = − − + 8 16 2 sinh7 x 6 sinh2 x sinh4 x 4 sinh2 x

dx 3 1 = coth x − coth3 x + coth5 x − coth7 x 8 5 7 sinh x  dx = arctan (sinh x) cosh x = arcsin (tanh x) = 2 arctan (ex ) = gd x 

10.  11.  12.  13.  14.

 15.12 

16. 17.

dx = tanh x cosh2 x sinh x 1 dx = + arctan (sinh x) cosh3 x 2 cosh2 x 2 sinh x 2 dx = + tanh x cosh4 x 3 cosh3 x 3 1 = − tanh3 x + tanh x 3 dx sinh x 3 sinh x 3 = + + arctan (sinh x) 5 4 2 cosh x 4 cosh x 8 cosh x 8 sinh x 4 dx 4 tanh3 x + tanh x = − 6 5 5 cosh x 5 cosh x 15 1 2 = tanh5 x − tanh3 x + tanh x 5 3   5 dx sinh x 1 5 15 + arctan (sinh x) = + + 8 16 cosh7 x 6 cosh2 x cosh4 x 4 cosh6 x

dx 1 3 = − tanh7 x + tanh5 x − tanh3 x + tanh x 8 7 5 cosh x  sinh x dx = ln cosh x cosh x

2.423

2.423

Powers of hyperbolic functions

 18.  19.



20. 21. 22. 23. 24. 25.

26. 27.

28. 29. 30. 31. 32. 33. 34.

sinh2 x dx = sinh x − arctan (sinh x) cosh x sinh3 x 1 dx = sinh2 x − ln cosh x cosh x 2 1 = cosh2 x − ln cosh x 2

1 sinh4 x dx = sinh3 x − sinh x + arctan (sinh x) cosh x 3  1 sinh x dx = − cosh x cosh2 x  sinh2 x dx = x − tanh x cosh2 x  sinh3 x 1 dx = cosh x + cosh x cosh2 x  1 3 sinh4 x dx = − x + sinh 2x + tanh x 2 2 4 cosh x  1 sinh x dx = − cosh3 x 2 cosh2 x 1 = tanh2 x 2  2 sinh x sinh x 1 dx = − + arctan (sinh x) 3 2 cosh x 2 cosh x 2  3 sinh x 1 dx = − tanh2 x + ln cosh x 3 2 cosh x 1 + ln cosh x = 2 cosh2 x  sinh4 x 3 sinh x + sinh x − arctan (sinh x) dx = 2 cosh x 2 cosh3 x  1 sinh x dx = − 4 cosh x 3 cosh3 x  sinh2 x 1 dx = tanh3 x 4 3 cosh x  3 1 sinh x 1 + dx = − 4 cosh x cosh x 3 cosh3 x  4 sinh x 1 dx = − tanh3 x − tanh x + x 3 cosh4 x  cosh x dx = ln sinh x sinh x  cosh2 x x dx = cosh x + ln tanh sinh x 2

117

118

Hyperbolic Functions

 35. 

36. 37. 38. 39. 40. 41.

42. 43.

44. 45. 46. 47. 48. 49. 50. 51.

1 cosh3 x dx = cosh2 x + ln sinh x sinh x 2

1 cosh4 x x dx = cosh3 x + cosh x + ln tanh sinh x 3 2  1 cosh x dx = − 2 sinh x sinh x  2 cosh x dx = x − coth x sinh2 x  cosh3 x 1 dx = sinh x − 2 sinh x sinh x  1 3 cosh4 x dx = x + sinh 2x − coth x 2 4 sinh2 x  1 cosh x dx = − sinh3 x 2 sinh2 x 1 = − coth2 x 2  2 cosh x cosh x x dx = − + ln tanh 2 sinh3 x 2 sinh2 x  3 1 cosh x dx = − + ln sinh x 3 sinh x 2 sinh2 x 1 = − coth2 x + ln sinh x 2  4 x cosh x cosh x 3 dx = − + cosh x + ln tanh 3 2 2 2 sinh x 2 sinh x  1 cosh x dx = − 4 sinh x 3 sinh3 x  cosh2 x 1 dx = − coth3 x 3 sinh4 x  3 1 cosh x 1 − dx = − 4 sinh x 3 sinh3 x sinh x  cosh4 x 1 dx = − coth3 x − coth x + x 3 sinh4 x  dx = ln tanh x sinh x cosh x  x 1 dx + ln tanh = 2 cosh x 2 sinh x cosh x  1 dx = + ln tanh x 3 sinh x cosh x 2 cosh2 x 1 = − tanh2 x + ln tanh x 2

2.423

2.424

Powers of hyperbolic functions

 52.  53.  54.  55.  56.  57.  58.  59.  60.  61.  62.  63.  64. 2.424 1. 2.

1 1 x dx + = + ln tanh cosh x 3 cosh3 x 2 sinh x cosh4 x 1 dx − arctan sinh x =− sinh x sinh2 x cosh x dx = −2 coth 2x sinh2 x cosh2 x dx 3 sinh x 1 − arctan sinh x =− − sinh2 x cosh3 x 2 cosh2 x sinh x 2 1 8 dx = − coth 2x sinh2 x cosh4 x 3 sinh x cosh3 x 3 dx 1 =− − ln tanh x sinh3 x cosh x 2 sinh2 x 1 = − coth2 x + ln coth x 2 cosh x x dx 1 3 − =− − ln tanh 3 2 2 cosh x 2 2 sinh x cosh x 2 sinh x 2 cosh 2x dx =− − 2 ln tanh x 3 3 sinh x cosh x sinh2 2x 1 1 = tanh2 x − coth2 x − 2 ln tanh x 2 2 1 x dx 2 cosh x 5 − =− − − ln tanh 3 4 2 2 cosh x 3 cosh x 2 sinh x 2 2 sinh x cosh x 1 1 dx − = + arctan sinh x sinh x 3 sinh3 x sinh4 x cosh x dx 1 8 =− + coth 2x 4 2 3 sinh x cosh x 3 cosh x sinh x 3 1 2 sinh x 5 dx − = + + arctan sinh x 4 3 3 2 sinh x 3 sinh x 2 cosh x 2 sinh x cosh x dx 8 = 8 coth 2x − coth3 2x 4 4 3 sinh x cosh x

 tanhp−1 x + tanhp−2 x dx [p = 1] p−1  n

(−1)k−1  n  1 + ln cosh x tanh2n+1 x dx = 2k k cosh2k x k=1 n

tanh2n−2k+2 x + ln cosh x =− 2n − 2k + 2



tanhp x dx = −

k=1



3.

4.

119

n

tanh2n−2k+1 x +x 2n − 2k + 1 k=1   cothp−1 x + cothp−2 x dx cothp x dx = − p−1 tanh2n x dx = −

GU (351)(12)

[p = 1]

120

Hyperbolic Functions



2.425

n

1 n 1 + ln sinh x 2n k sinh2k x k=1 n

coth2n−2k+2 x + ln sinh x =− 2n − 2k + 2

coth2n+1 x dx = −

5.

k=1

 coth2n x dx = −

6.

n

k=1

coth2n−2k+1 x +x 2n − 2k + 1

GU (351)(14)

For formulas containing powers of tanh x and coth x equal to n = 1, 2, 3, 4, see 2.423 17, 2.423 22, 2.423 27, 2.423 32, 2.423 33, 2.423 38, 2.423 43, 2.423 48. Powers of hyperbolic functions and hyperbolic functions of linear functions of the argument 2.425

 sinh(ax + b) sinh(cx + d) dx =

1.

 sinh(ax + b) cosh(cx + d) dx =

2.

 cosh(ax + b) cosh(cx + d) dx =

3.

4. 5. 6. 2.426 1.

1 sinh[(a + c)x + b + d] 2(a + c) 1 sinh[(a − c)x + b − d] − 2(a − c)  2 a = c2

GU (352)(2a)

1 cosh[(a + c)x + b + d] 2(a + c) 1 cosh[(a − c)x + b − d] + 2(a − c)  2 a = c2

GU (352)(2c)

1 sinh[(a + c)x + b + d] 2(a + c) 1 sinh[(a − c)x + b − d] + 2(a − c)  2 a = c2

GU (352)(2b)

When a = c:  1 x sinh(2ax + b + d) sinh(ax + b) sinh(ax + d) dx = − cosh(b − d) + 2 4a  1 x cosh(2ax + b + d) sinh(ax + b) cosh(ax + d) dx = sinh(b − d) + 2 4a  1 x sinh(2ax + b + d) cosh(ax + b) cosh(ax + d) dx = cosh(b − d) + 2 4a  sinh ax sinh bx sinh cx dx =

GU (352)(3a) GU (352)(3c) GU (352)(3b)

cosh(a + b + c)x cosh(−a + b + c)x − 4(a + b + c) 4(−a + b + c) cosh(a − b + c)x cosh(a + b − c)x − − 4(a − b + c) 4(a + b − c) GU (352)(4a)

2.428

Powers of hyperbolic functions

 sinh ax sinh bx cosh cx dx =

2.

sinh ax cosh bx cosh cx dx =

cosh(a + b + c)x cosh(−a + b + c)x − 4(a + b + c) 4(−a + b + c) cosh(a − b + c)x cosh(a + b − c)x + + 4(a − b + c) 4(a + b − c) GU (352)(4c)

 cosh ax cosh bx cosh cx dx =

4.

sinh(a + b + c)x sinh(−a + b + c)x − 4(a + b + c) 4(−a + b + c) sinh(a − b + c)x sinh(a + b − c)x + − 4(a − b + c) 4(a + b − c) GU (352)(4b)

 3.

121

sinh(a + b + c)x sinh(−a + b + c)x + 4(a + b + c) 4(−a + b + c) sinh(a − b + c)x sinh(a + b − c)x + + 4(a − b + c) 4(a + b − c) GU (352)(4d)

2.427 1.



12

 2.

 3.

1 sinh x sinh ax dx = p+a p

sinhp x sinh(2n + 1)x dx =

sinhp x sinh 2nx dx =

   p p−1 sinh x cosh ax − p sinh x cosh(a − 1)x dx Γ(p + 1)  Γ ⎡p+3 2 +n  n−1

Γ p+1 + n − 2k 2 sinhp−2k x cosh(2n − 2k + 1)x ×⎣ 22k+1 Γ (p − 2k + 1) k=0 ⎤  p−1 Γ 2 + n − 2k sinhp−2k−1 x sinh(2n − 2k)x⎦ − 2k+2 2 Γ(p − 2k)   Γ p+3 2 −n sinhp−2n x sinh x dx + 2n 2 Γ(p + 1 − 2n) [p is not a negative integer]

Γ(p + 1) p Γ 2 +⎡n + 1  n−1

Γ p2 + n − 2k ⎣ sinhp−2k x cosh(2n − 2k)x × 22k+1 Γ(p − 2k + 1) k=0 ⎤  Γ p2 + n − 2k − 1 sinhp−2k−1 x sinh(2n − 2k − 1)x⎦ − 22k+2 Γ(p − 2k) [p is not a negative integer]

2.428 1.



1 sinh x cosh ax dx = p+a p

   p p−1 sinh x sinh ax − p sinh x sinh(a − 1)x dx

GU (352)(5)a

122

Hyperbolic Functions

 2.

 3.

sinhp x cosh(2n + 1)x dx =

2.429

Γ(p + 1)  Γ ⎧p+3 2 +n ⎡  ⎨ n−1

Γ p+1 + n − 2k 2 ⎣ sinhp−2k x sinh(2n − 2k + 1)x × ⎩ 22k+1 Γ(p − 2k + 1) k=0 ⎤  p−1 Γ 2 + n − 2k sinhp−2k−1 x cosh(2n − 2k)x⎦ − 2k+2 2 Γ(p − 2k) ⎫   ⎬ Γ p+3 − n 2 sinhp−2n x cosh x dx + 2n ⎭ 2 Γ(p + 1 − 2n) [p is not a negative integer]

sinhp x cosh 2nx dx =

Γ(p + 1) p Γ ⎧2 + n⎡+ 1  ⎨n−1

Γ p2 + n − 2k ⎣ sinhp−2k x sinh(2n − 2k)x × ⎩ 22k+1 Γ(p − 2k + 1) k=0 ⎤  Γ p2 + n − 2k − 1 sinhp−2k−1 x cosh(2n − 2k − 1)x⎦ − 22k+2 Γ(p − 2k) ⎫ p  ⎬ Γ −n+1 sinhp−2n x dx + 2n 2 ⎭ 2 Γ(p + 1 − 2n) [p is not a negative integer]

2.429



1.  2.

coshp x sinh ax dx =

1 p+a

GU (352)(6)a

   coshp x cosh ax + p coshp−1 x sinh(a − 1)x dx ⎡

 p+1 n−1 Γ(p + 1) ⎣ Γ 2 + n − k coshp−k x cosh(2n − k + 1)x cosh x sinh(2n + 1)x dx =  p+3 2k+1 Γ(p − k + 1) Γ 2 +n k=0 ⎤  p+3  Γ 2 coshp−n x sinh(n + 1)x dx⎦ + n 2 Γ(p − n + 1) p

[p is not a negative integer] p  n−1 Γ(p + 1) ⎣ Γ 2 + n − k p  cosh x sinh 2nx dx = coshp−k x cosh(2n − k)x 2k+1 Γ(p − k + 1) Γ p2 + n + 1 k=0 ⎤ p  Γ 2 +1 coshp−n x sinh nx dx⎦ + n 2 Γ(p − n + 1) ⎡

3.

[p is not a negative integer]

GU (352)(7)a

2.433

2.431

Powers of hyperbolic functions



1.  2.

 3.

coshp x cosh ax dx =

1 p+a

coshp x cosh(2n + 1)x dx =



coshp x sinh ax + p



 coshp−1 x cosh(a − 1)x dx

⎡

 p+1 n−1 Γ(p + 1) ⎣ Γ 2 + n − k  p+3 coshp−k x sinh(2n − k + 1)x 2k+1 Γ(p − k + 1) Γ 2 +n k=0 ⎤  p+3  Γ 2 coshp−n x cosh(n + 1)x dx⎦ + n 2 Γ(p − n + 1)

[p is not a negative integer] n−1 Γ(p + 1) ⎣ Γ 2 + n − k p coshp−k x sinh(2n − k)x cosh x cosh 2nx dx =  p 2k+1 Γ(p − k + 1) Γ 2 +n+1 k=0 ⎤ p Γ 2 +1 coshp−n x cosh nx dx⎦ + n 2 Γ(p − n + 1) ⎡

p

[p is not a negative integer] 2.432 1. 2. 3. 4. 2.433



GU (352)(8)a

1 sinhn x sinh nx n  1 sinh(n + 1)x coshn−1 x dx = coshn x cosh nx n  1 cosh(n + 1)x sinhn−1 x dx = sinhn x cosh nx n  1 cosh(n + 1)x coshn−1 x dx = coshn x sinh nx n 

1.  2.  3.  4.  5.

123

sinh(n + 1)x sinhn−1 x dx =

n−1

sinh(2n − 2k)x sinh(2n + 1)x dx = 2 +x sinh x 2n − 2k k=0

sinh 2nx dx = 2 sinh x

n−1

k=0

sinh(2n − 2k − 1)x 2n − 2k − 1

GU (352)(5d)

n−1

cosh(2n − 2k)x cosh(2n + 1)x dx = 2 + ln sinh x sinh x 2n − 2k k=0

cosh 2nx dx = 2 sinh x

n−1

k=0

cosh(2n − 2k − 1)x x + ln tanh 2n − 2k − 1 2 n−1

sinh(2n + 1)x cosh(2n − 2k)x (−1)k dx = 2 + (−1)n ln cosh x cosh x 2n − 2k k=0

GU (352)(6d)

124

Hyperbolic Functions

 6.  7.  8.  9.

10. 11.

12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

23.

2.433

n−1

cosh(2n − 2k − 1)x sinh 2nx dx = 2 (−1)k cosh x 2n − 2k − 1

GU (352)(7d)

k=0

n−1

sinh(2n − 2k)x cosh(2n + 1)x dx = 2 + (−1)n x (−1)k cosh x 2n − 2k k=0

cosh 2nx dx = 2 cosh x

n−1

(−1)k

k=0

sinh(2n − 2k − 1)x + (−1)n arcsin (tanh x) 2n − 2k − 1

2 sinh 2x dx = − sinhn x (n − 2) sinhn−2 x

For n = 2:  sinh 2x dx = 2 ln sinh x sinh2 x  2 sinh 2x dx = coshn x (2 − n) coshn−2 x For n = 2:  sinh 2x dx = 2 ln cosh x cosh2 x  x cosh 2x dx = 2 cosh x + ln tanh sinh x 2  cosh 2x dx = − coth x + 2x sinh2 x  x cosh x 3 cosh 2x dx = − + ln tanh 2 sinh3 x 2 sinh2 x 2  cosh 2x dx = 2 sinh x − arcsin (tanh x) cosh x  cosh 2x dx = − tanh x + 2x cosh2 x  sinh x 3 cosh 2x dx = − + arcsin (tanh x) cosh3 x 2 cosh2 x 2  sinh 3x dx = x + sinh 2x sinh x  x sinh 3x dx = 3 ln tanh + 4 cosh x 2 2 sinh x  sinh 3x dx = −3 coth x + 4x sinh3 x  4 1 sinh 3x dx = − n−3 coshn x (3 − n) cosh x (1 − n) coshn−1 x For n = 1 and n = 3:  sinh 3x dx = 2 sinh2 x − ln cosh x cosh x

GU (352)(8d)

2.442

Rational functions of hyperbolic functions



24. 25.

26. 27. 28. 29. 30.

1 sinh 3x dx = + 4 ln cosh x cosh3 x 2 cosh2 x  4 1 cosh 3x dx = + n−3 sinhn x (3 − n) sinh x (1 − n) sinhn−1 x For n = 1 and n = 3:  cosh 3x dx = 2 sinh2 x + ln sinh x sinh x  1 cosh 3x dx = − + 4 ln sinh x 3 sinh x 2 sinh2 x  cosh 3x dx = sinh 2x − x cosh x  cosh 3x dx = 4 sinh x − 3 arcsin (tanh x) cosh2 x  cosh 3x dx = 4x − 3 tanh x cosh3 x

2.44–2.45 Rational functions of hyperbolic functions 2.441



1.

2. 3.

2.442 1.

2.

cosh x aB − bA A + B sinh x · n dx = (n − 1) (a2 + b2 ) (a + b sinh x)n−1 (a + b sinh x)  (n − 1)(aA + bB) + (n − 2)(aB − bA) sinh x 1 + dx n−1 (n − 1) (a2 + b2 ) (a + b sinh x)

For n = 1:   A + B sinh x dx B aB − bA dx = x − a + b sinh x b b a + b sinh x √  a tanh x2 − b + a2 + b2 1 dx √ = √ ln a + b sinh x a2 + b2 a tanh x2 − b − a2 + b2 a tanh x2 − b 2 = √ arctanh √ a2 + b 2 a2 + b 2 

(see 2.441 3)

 B dx A + B cosh x dx = − + A n n n−1 (a + b sinh x) (a + b sinh x) (n − 1)b (a + b sinh x)

For n = 1:   B dx A + B cosh x dx = ln (a + b sinh x) + A a + b sinh x b a + b sinh x (see 2.441 3)

125

126

2.443

Hyperbolic Functions



1.

2. 3.

2.444 1.

2.12 2.445 1.

12

2.12

2.443

sinh x aB − bA A + B cosh x · dx = (a + b cosh x)n (n − 1) (a2 − b2 ) (a + b cosh x)n−1  (n − 1)(aA − bB) + (n − 2)(aB − bA) cosh x 1 + dx (n − 1) (a2 − b2 ) (a + b cosh x)n−1

For n = 1:   A + B cosh x B aB − bA dx dx = x − a + b cosh x b b a + b cosh x  1 dx b + a cosh x =√ arcsin a + b cosh x a + b cosh x b 2 − a2 b + a cosh x 1 arcsin = −√ a + b cosh x b 2 − a2 √ x a2 − b2 tanh a + b + 1 2 =√ ln √ a2 − b2 a + b − a2 − b2 tanh x 2

(see 2.443 3)  2 b > a2 ,

x a2 ,

x>0



 2 a > b2

  x+a x−a dx = cosech a ln cosh − ln cosh cosh a + cosh x 2 2  a x = 2 cosech a arctanh tanh tanh 2 2    x a dx = 2 cosec a arctan tanh tan cosec a + cosh x 2 2





sinh x 1 dx = − n (a + b cosh x) (n − 1)b(a + b cosh x)n−1 For n = 1:  1 sinh x dx = ln (a + b cosh x) a + b cosh x b

[n = 1]

(see 2.443 3)

In evaluating definite integrals by use of formulas 2.441–2.443 and 2.445, one may not take the integral over points at which the integrand becomes infinite, that is, over the points  a x = arcsinh − b in formulas 2.441 or 2.442 or over the points  a x = arccosh − b in formulas 2.443 or 2.445. Formulas 2.443 are not applicable for a2 = b2 . Instead, we may use the following formulas in these cases:

2.449

2.446

Rational functions of hyperbolic functions



1.

A + B cosh x dx (ε + cosh x)n =

  n−1

(2n − 2k − 3)!! (n − 1)! n B sinh x B sinh x + εA + n n−1 (2n − 1)!! (n − k − 1)! (1 − n) (ε + cosh x)

×

2. 2.447 1.12

2.

3. 2.448

3. 2.449 1.6

k=0

εk (ε + cosh x)

n−k

[ε = ±1,

For n = 1:  cosh x − ε A + B cosh x dx = Bx + (εA − B) ε + cosh x sinh x   b sinh x dx − bx a ln cosh x + arctanh a a cosh x + b sinh x = a2− b2 a bx − a ln sinh x + arctanh b = b 2 − a2 For a = b = 1:  x 1 sinh x dx = + e−2x cosh x + sinh x 2 4 For a = −b = 1:  x 1 sinh x dx = − + e2x cosh x − sinh x 2 4

n > 1]

[ε = ±1]





1.

2.

127

 ax − b ln cosh x + arctanh ab cosh x dx = a cosh x + b sinh x a2 −b2 −ax + b ln sinh x + arctanh ab = b 2 − a2

[a > |b|] [b > |a|]

MZ 215

MZ 215

[a > |b|] [b > |a|]

For a = b = 1:  cosh x dx x 1 = − e−2x cosh x + sinh x 2 4 For a = −b = 1:  x 1 cosh x dx = + e2x cosh x − sinh x 2 4 

dx 1 n =  2 (a cosh x + b sinh x) (a − b2 )n

MZ 214, 215



dx   b sinhn x + arctanh a  1 dx  =  n a (b2 − a2 ) coshn x + arctanh b

[a > |b|] [b > |a|]

128

2.

3.12

4. 2.451

Hyperbolic Functions

For n = 1:      1 b  dx  =√ arctan sinh x + arctanh a cosh x + b sinh x a  a2 − b 2  a   x + arctanh 1 b  =√ ln tanh  2 b 2 − a2  For a = b = 1:  dx = −e−x = sinh x − cosh x cosh x + sinh x For a = −b = 1:  dx = ex = sinh x + cosh x cosh x − sinh x 

1.

 2.

3.

2.451

[a > |b|] [b > |a|]

MZ 214

A + B cosh x + C sinh x n dx (a + b cosh x + c sinh x) Bc − Cb + (Ac − Ca) cosh x + (Ab − Ba) sinh x 1 = n−1 + (n − 1) (a2 − b2 + c2 ) (1 − n) (a2 − b2 + c2 ) (a + b cosh x + c sinh x)  (n − 1)(Aa − Bb + Cc) − (n − 2)(Ab − Ba) cosh x − (n − 2)(Ac − Ca) sinh x dx × n−1 (a + b cosh x + c sinh x)  2 a + c2 = b2   (n − 1)! Bc − Cb − Ca cosh x − Ba sinh x A n(Bb − Cc) + (c cosh x + b sinh x) = n + 2 (n − 1)a (a + b cosh x + c sinh x) a (n − 1)a (2n − 1)!! n−1

(2n − 2k − 3)!! 1 × (n − k − 1)!ak (a + b cosh x + c sinh x)n−k k=0  2 a + c2 = b 2

Cb − Bc A + B cosh x + C sinh x dx = 2 ln (a + b cosh x + c sinh x) a + b cosh x + c sinh x b − c2   dx Bb − Cc Bb − Cc x + A − a + 2 2 2 2 b −c b − c a + b cosh x + c sinh x b2 = c2 (see 2.451 4)    C ∓B A (B ∓ C) b A + B cosh x + C sinh x dx = (cosh x ∓ sinh x) + − x a + b cosh x ± b sinh x a 2a2 2a  C ±B A (C ∓ B) b + ± − ln (a + b cosh x ± b sinh x) 2b a 2a2 [ab = 0]

2.452

Rational functions of hyperbolic functions

 4.

dx a + b cosh x + c sinh x (b − a) tanh x2 + c 2 arctan √ =√ b 2 − a2 − c2 b 2 − a2 − c2 √ x (a − b) tanh a 2 − b 2 + c2 1 2 −c+ √ =√ ln x a2 − b2 + c2 (a − b) tanh 2 − c − a2 − b2 + c2 x 1  = ln a + c tanh c 2 2 = (a − b) tanh x2 + c

129

 

b 2 > a 2 + c2 ,

a = b

b 2 < a 2 + c2 ,

a = b



[a = b, c = 0]  2 b = a 2 + c2 GU (351)(18)

2.452 1.



A + B cosh x + C sinh x dx (a1 + b1 cosh x + c1 sinh x) (a2 + b2 cosh  x + c2 sinh x)  dx dx a1 + b1 cosh x + c1 sinh x = A0 ln + A1 + A2 a2 + b2 cosh x + c2 sinh x a1 + b1 cosh x + c1 sinh x a2 + b2 cosh x + c2 sinh x where

GU (351)(19)

  a1 b1 c1    A B C   a2 b2 c2  2    b1 c1 2 c1 b1    + −  b2  b 2 c2  c2

A0 =  2 a1 a1   a2 a2      a1    b 1   c1    C B   C A   B A          c2 b 2   c2 a2   b 2 a2      a2 b2 c2  A2 =     ,   a1 b1 2 b1 c1 2 c1 a1 2   +  −  a2 b2  b2 c2   c2 a2   2.

    a1   b 1   c1    b1 c1  c1 a1  a1 b1         B C   C A   A B     a2 b2 c2  A1 =     ,   a1 b1 2 b1 c1 2 c1 a1 2   +  −   a2 b 2   b 2 c2   c2 a2 

  a1   a2

2  b b1  +  1  b2 b2

2  c c1  =  1  c2 c2

2  a1  . a2 

A cosh2 x + 2B sinh x cosh x + C sinh2 x dx a cosh2 x + 2b sinh x cosh x + c sinh2 x ⎧ ⎨ 1 = 2 [4Bb − (A + C)(a + c)]x 4b − (a + c)2 ⎩  + [(A + C)b − B(a + c)] ln a cosh2 x + 2b sinh x cosh x + c sinh2 x ⎫ ⎬  + 2(A − C)b2 − 2Bb(a − c) + (Ca − Ac)(a + c) f (x) ⎭

130

Hyperbolic Functions

2.453

where

√ 1 c tanh x + b − b2 − ac √ f (x) = √ ln 2 b2 − ac c tanh x + b + b2 − ac c tanh x + b 1 arctan √ = √ ac − b2 ac − b2 1 =− c tanh x + b 2.453



1.

 2.

GU (351)(24)

 2 b > ac  2 b < ac  2 b = ac

    1 x  dx (A + B sinh x) dx  = A ln tanh  + (aB − bA) sinh x (a + b sinh x) a 2 a + b sinh x (see 2.441 3)        a + b cosh x  dx A x (A + B sinh x) dx    +B = 2 a ln tanh  + b ln  sinh x (a + b cosh x) a − b2 2 sinh x  a + b cosh x (see 2.443 3)

For a = b = 1:     x  1 x (A + B sinh x) dx A  2 x ln tanh  − tanh + B tanh = sinh x (1 + cosh x) 2 2 2 2 2      A x 1 x (A + B sinh x) dx x  = − ln coth  + coth2 + B coth sinh x (1 − cosh x) 2 2 2 2 2 2

3. 4. 2.454 1. 2.

2

    a + b sinh x  1 (A + B sinh x) dx   = 2 (Aa + Bb) arctan (sinh x) + (Ab − Ba) ln  cosh x (a + b sinh x) a + b2 cosh x          sinh x  (A + B cosh x) dx 1 x  dx    = A ln tanh  + B ln  − Ab sinh x (a + b sinh x) a 2 a + b sinh x  a + b sinh x 

(see 2.441 3) 2.455 1.



     a + b cosh x  (A + B cosh x) dx 1 x     = 2 (Aa + Bb) ln tanh  + (Ab − Ba) ln  sinh x (a + b cosh x) a − b2 2 sinh x 

For a2 = b2 = 1:  x A + B  x  A − B (A + B cosh x) dx = ln tanh  − tanh2 2. sinh x (1 + cosh x) 2 2 4 2  x A−B A+B x (A + B cosh x) dx = coth2 − ln coth 3. sinh x (1 − cosh x) 4 2 2 2       a + b sinh x  dx A (A + B cosh x) dx  +B  = 2 a arctan (sinh x) + b ln 2.456  cosh x  cosh x (a + b sinh x) a + b2 a + b sinh x (see 2.441 3)

2.459

2.457

Rational functions of hyperbolic functions



1.

131

   1 dx (A + B cosh x) dx = A arctan sinh x − (Ab − Ba) cosh x (a + b cosh x) a a + b cosh x (see 2.443 3)

2.458



1.

dx a + b sinh2 x



 b = arctan − 1 tanh x a a(b − a)   b 1 1 − tanh x arctanh = a a(a − b)   1 b = 1 − tanh x arccoth a a(a − b) 1

 

b >1 a



b b a < 1 or < 0 and sinh2 x < − a a b   b a < 0 and sinh2 x > − a b

MZ 195

 2.



0
0 or − 1 < < 0 and cosh2 x < − a a b



MZ 202

For a = b = 1:  dx = tanh x 1 + sinh2 x  √  1 dx = √ arctanh 2 tanh x 2 1 − sinh x 2 √  1 2 tanh x = √ arccoth 2  √  dx 1 = √ arccoth 2 coth x 2 1 + cosh x 2  dx = coth x 1 − cosh2 x 2

3. 4.

5. 6. 2.459 1.



2

 sinh2 x < 1  sinh2 x > 1

   b sinh x cosh x dx 1 + (b − 2a) =  2 2a(b − a) a + b sinh2 x a + b sinh2 x a + b sinh2 x dx

(see 2.458 1)

MZ 196

132

Hyperbolic Functions

 2.

 3.

2.461

   b sinh x cosh x dx 1 − + (2a + b)  2 = 2a(a + b) a + b cosh2 x a + b cosh2 x a + b cosh2 x dx

⎡

(see 2.458 2)

MZ 203

    2 p tanh x 1 ⎣ 3 3 2 3 − 2 + 4 arctan (p tanh x) + 3 − 2 − 4  3 = 3 2 8pa p p p p 1 + p2 tanh2 x a + b sinh x ⎤   2 1 2p tanh x + 1 + 2 − 2 tanh2 x  2 ⎦ p p 1 + p2 tanh2 x   b 2 p = −1>0 a ⎡     2 1 ⎣ 3 3 2 q tanh x = 3 + 2 + 4 arctanh (q tanh x) + 3 + 2 − 4 8qa3 q q q q 1 − q 2 tanh2 x ⎤   2 1 2q tanh x + 1 − 2 + 2 tanh2 x  2 ⎦ q q 1 − q 2 tanh2 x   b 2 q =1− >0 a dx

MZ 196

⎡

4.

     2 p coth x dx 1 ⎣ 3 3 2 3 − 2 + 4 arctan (p coth x) + 3 − 2 − 4  3 = 3 2 8pa p p p p 1 + p2 coth2 x a + b cosh x ⎤   2 1 2p coth x + 1 + 2 − 2 coth2 x  2 ⎦ p p 1 + p2 coth2 x   b p2 = −1 − > 0 a ⎡     q coth x 1 ⎣ 3 2 3 2 = 3 + 2 + 4 ϕ(x)∗ + 3 + 2 − 4 3 8qa q q q q 1 − q 2 coth2 x ⎤   2 1 2q coth x + 1 − 2 + 2 coth2 x  2 ⎦ q q 1 − q 2 coth2 x   b 2 q =1+ >0 a

2.46 Algebraic functions of hyperbolic functions 2.461  √ √ √ 1. tanh x dx = arctanh tanh x − arctan tanh x ∗ In 2.459.4, if b < 0 and cosh2 x > − a , then ϕ(x) = arctanh (q coth x). If a b ϕ(x) = arccoth (q coth x)

MZ 221 b a

< 0, but cosh2 x < − ab , or if

b a

> 0, then

2.462

2. 2.462

Algebraic functions of hyperbolic functions

 √



1.

133

√ √ coth x dx = arccoth coth x − arctan coth x

   cosh x sinh x dx  = arcsinh √ = ln cosh x + a2 + sinh2 x a2 − 1 a2 + sinh2 x    cosh x = ln cosh x + a2 + sinh2 x = arccosh √ 1 − a2 = ln cosh x

MZ 222

  

a2 > 1 a2 < 1 a2 = 1



 2. 3.

4.12 5.12 6.12

7.12 8.12 9.12

10. 11.

 sinh x dx cosh x  = arcsin √ sinh2 x < a2 2 2 2 a +1 a − sinh x     cosh x sinh x dx  = arccosh √ = ln cosh x + sinh2 x − a2 a2 + 1 sinh2 x − a2  sinh2 x > a2       sinh x cosh x dx  = ln sinh x + a2 + sinh2 x = arcsinh |a| a2 + sinh2 x     sinh x cosh x dx  sinh2 x < a2 = arcsin 2 |a| 2 a − sinh x       cosh x dx sinh x  = ln sinh x + sinh2 x − a2 = arccosh sinh2 x − a2 |a|     sinh x = − arccosh − = − ln − sinh x + sinh2 x − a2 |a|       cosh x sinh x dx  = ln cosh x + a2 + cosh2 x = arcsinh |a| a2 + cosh2 x     cosh x sinh x dx  cosh2 x < a2 = arcsin |a| a2 − cosh2 x       sinh x dx cosh x  = ln cosh x + cosh2 x − a2 = arccosh |a| cosh2 x − a2  cosh2 x > a2     sinh x cosh x dx  = arcsinh √ = ln sinh x + a2 + cosh2 x a2 + 1 a2 + cosh2 x   sinh x cosh x dx  = arcsin √ cosh2 x < a2 2 2 a −1 a2 − cosh x

MZ 199

[sinh x > |a|] [sinh x < −|a|]

MZ 215, 216

134

Hyperbolic Functions

 12.

12



 sinh x  = arccosh √ 2 cosh2 x − a2  a − 1 sinh x = −arccosh − √ a2  −1  sinh x = arcsinh √ 1 − a2  cosh x dx

a2 > 1,

a2 > 1,

 = ln sinh x + cosh2 x − a2    = − ln − sinh x + cosh2 x − a2

= ln (sinh x) = − ln (− sinh x)

14.

sinh x >



sinh x < −

 2 a 1,

a2 > 1,  2 a = 1,  2 a = 1,

 a2 − 1



 a2 − 1

 a2 − 1   sinh x < − a2 − 1 x>0 x



MZ 206





13.

2.463

√ coth x dx b = 2 a arccoth 1 + sinh x a a + b sinh x  √ b = 2 a arctanh 1 + sinh x a )   √ b = 2 −a arctanh − 1 + sinh x a   √ tanh x dx b √ = 2 a arccoth 1 + cosh x a a + b cosh x  √ b = 2 a arctanh 1 + cosh x a )   √ b = 2 −a arctanh − 1 + cosh x a √

[b sinh x > 0,

a > 0]

[b sinh x < 0,

a > 0]

a 0,

a > 0]

[b cosh x < 0,

a > 0]

[a < 0] MZ 220, 221

2.463 1.



√ sinh x a + b cosh x dx p + q cosh x ) =2 ) =2 ) =2

aq − bp arccoth q aq − bp arctanh q bp − aq arctanh q

) ) )

q (a + b cosh x) aq − bp q (a + b cosh x) aq − bp q (a + b cosh x) bp − aq

 b cosh x > 0,

aq − bp >0 q

b cosh x < 0,

aq − bp >0 q

 

aq − bp 0,

aq − bp >0 q

b sinh x < 0,

aq − bp >0 q

 

aq − bp 0] BY (295.00)(295.10)       dx dx 1  ,k 2. =  = F arcsin 2 2 2 cosh x 2  cosh x − k sinh x + k [x > 0] BY (295.40)(295.30)        1 tanh x dx  ,k 0 < x < arccosh  = F arcsin 3. BY (295.20) 2 2 k k  1 − k cosh x 1 1 − sinh 2ax ,r= √ [ax > 0] Notation: In 2.464 4–2.464 8, we set α = arccos 1 + sinh 2ax 2  1 dx √ BY (296.50) = 4. F (α, r) 2a sinh 2ax (  2  √ 1 sinh 2ax 1 + sinh 2ax 1 [F (α, r) − 2 E (α, r)] + 5. sinh 2ax dx = BY (296.53) 2a a 1 + sinh 2ax  cosh2 2ax dx 1 E (α, r) 6. BY (296.51) = 2√ 2a (1 + sinh 2ax) sinh 2ax  2 (1 − sinh 2ax) dx 1 [2 E (α, r) − F (α, r)] 7. BY (296.55) = 2√ 2a (1 + sinh 2ax) sinh 2ax  √ sinh 2ax dx 1 8. BY (296.54) 2 = 4a [F (α, r) − E (α, r)] (1 + sinh 2ax)  cosh 2ax − 1 1 ,r=√ [x = 0]: Notation: In 2.464 9–2.464 15, we set α = arcsin cosh 2ax 2  1 dx √ BY (296.00) = √ F (α, r) 9. a 2 cosh 2ax  √ sinh 2ax 1 10. cosh 2ax dx = √ [F (α, r) − 2 E (α, r)] + √ BY (296.03) a 2 a cosh 2ax  dx 1 √ 11. BY (296.04) = √ [2 E (α, r) − F (α, r)] 3 a 2 cosh 2ax

136

Hyperbolic Functions

 12. 13. 14. 15.

tanh 2ax 1 = √ F (α, r) + √ 3 2a 3a cosh 2ax cosh 2ax √  2 √ 2 1 sinh 2ax dx √ F (α, r) + sinh 2ax cosh 2ax =− 3a 3a cosh 2ax √  2 2 tanh 2ax tanh 2ax dx √ F (α, r) − √ = 3a cosh 2ax 3a cosh 2ax √   cosh 2ax dx 1 = √ Π α, p2 , r 2 2 p + (1 − p ) cosh 2ax a 2 Notation: In 2.464 16–2.464 20, we set: √ a2 + b2 − a − b sinh x , α = arccos √ a2 + b2 + a + b sinh x ) √

a + a2 + b 2 √ r= a > 0, b > 0, 2 a2 + b 2 √

2.464

dx

BY (296.04)

5

BY (296.07)

BY (296.05)

BY (296.02)

x > − arcsinh

a b

 16. 17. 18.

dx 1 = √ F (α, r) 4 2 a + b sinh x a + b2 √   √ 2b cosh x a + b sinh x 4 a + b sinh x dx = a2 + b2 [F (α, r) − 2 E (α, r)] + √ a2 + b2 + a + b sinh x √  √  a + b sinh x a2 + b 2 − a 4 √ dx = a2 + b2 E (α, r) − F (α, r) 4 2 2 2 cosh x √ 2 a +b √ √ a2 + b2 − a − b sinh x a + b sinh x a + a2 + b 2 ·√ · − 2 2 b cosh x a + b + a + b sinh x √

BY (298.00)

BY (298.02)

BY (298.03)

 19.

20.

1 cosh2 x dx = √ E (α, r) √ 2 √ 4 2 2 2 b a2 + b 2 a + b + a + b sinh x a + b sinh x √  a + b sinh x dx 1 √ E (α, r) √ 2 = − √ 4 2 + b2 2 2 a a2 + b 2 − a a + b − a − b sinh x √ b cosh x a + b sinh x +√ · a2 + b2 − a a2 + b2 − (a + b sinh x)2 

21. 22. 23.

BY (298.01)

BY (298.04)

 a−b x ,r= [0 < b < a, x > 0]: Notation: In 2.464 21–2.464 31, we set α = arcsin tanh 2 a+b  dx 2 √ = √ F (α, r) BY (297.25) a + b cosh x a+b  √ √ x√ a + b cosh x dx = 2 a + b [F (α, r) − E (α, r)] + 2 tanh a + b cosh x BY (297.29) 2 √  2 x√ 2 2 a+b cosh x dx √ E (α, r) + tanh = √ F (α, r) − a + b cosh x BY (297.33) b b 2 a + b cosh x a+b

2.464

Algebraic functions of hyperbolic functions

√ 2 a+b √ [F (α, r) − E (α, r)] dx = BY (297.28) a−b a + b cosh x √ √  tanh4 x2 sinh x2 a + b cosh x 2 a+b 2 √ dx = [(3a + b) F (α, r) − 4a E (α, r)] + x 3(a − b)2 3(a − b) a + b cosh x cosh3 2

 24. 25.11

tanh2

27.

x 2

BY (297.28)

 26.

2 

 √ x √ tanh a + b cosh x − a + b E (α, r) 2

cosh x − 1 √ dx = b a + b cosh x √  2 4 a+b (cosh x − 1) √ dx = [(a + 3b) E (α, r) − b F (α, r)] 3b2 a + b cosh x  x x√ 4 a + b cosh x + 2 b cosh2 − (a + 3b) tanh 3b 2 2

29. 30.

√ a + b cosh x dx = a + b E (α, r) BY (297.26) cosh x + 1 √  a+b 2b dx √ √ E (α, r) − = F (α, r) BY (297.30) a−b (cosh x + 1) a + b cosh x (a − b) a + b   dx 1 √ b(5b − a) F (α, r) = √ 2 (cosh x + 1) a + b cosh x 3(a − b)2 a + b  sinh x2 √ 1 · a + b cosh x + (a − 3b)(a + b) E (α, r) + 6(a − b) cosh3 x2 297.30)

 31.

BY (297.31)

BY (297.31)

 √ 28.

137

 2 (1 + cosh x) dx √ = √ Π α, p2 , r 2 [1 + + (1 − p ) cosh x] a + b cosh x a+b Notation: In 2.464 32–2.464 40, we set:  a − b cosh x α = arcsin a−b 

a−b a 0 < b < a, 0 < x < arccosh r= a+b b p2

BY (297.27)

 32. 33.

dx 2 = √ F (α, r) a − b cosh x a+b  √ √ a − b cosh x dx = 2 a + b [F (α, r) − E (α, r)] √

√ 2 2 a+b cosh x dx √ E (α, r) − √ = F (α, r) b a − b cosh x a+b √  √ 2(b − 2a) 4a a + b cosh2 x dx 2 √ = √ F (α, r) + E (α, r) + sinh x a − b cosh x 2 3b 3b a − b cosh x 3b a + b √  2 a+b (1 + cosh x) dx √ E (α, r) = b a − b cosh x

BY (297.50) BY (297.54)

 34. 35. 36.

BY (297.56)

BY (297.56)

BY (297.51)

138

Hyperbolic Functions

2.464

  2b a−b dx √ ,r = √ Π α, BY (297.57) a cosh x a − b cosh x a a+b  x√ 1 1 dx √ tanh =√ E (α, r) − a − b cosh x BY (297.58) a+b 2 (1 + cosh x) a − b cosh x a+b  dx 1 [(a + 3b) E (α, r) − b F (α, r)] =  2√ (1 + cosh x) a − b cosh x 3 (a + b)3 √ tanh x2 a − b cosh x 1 [2a + 4b + (a + 3b) cosh x] − 3(a + b)2 cosh x + 1 

37. 38. 39.

BY (297.58)

 40.

 2 dx √ √ = Π α, p2 , r (a − b − + cosh x) a − b cosh x (a − b) a + b Notation: In 2.464 41 –2.464 47, we set:  b (cosh x − 1) , α = arcsin b cosh x − a  a+b [0 < a < b, x > 0] r= 2b ap2

 √

41. 42.

 √ 

43.  44.

46.

47.



2 F (α, r) b 

dx ( 3 (b cosh x − a)

BY (297.52)

BY (297.00)

√ 2b sinh x 2 F (α, r) − 2 2b E (α, r) + √ b b cosh x − a  2 1 [2b E (α, r) − (b − a) F (α, r)] = 2 · b − a2 b

b cosh x − a dx = (b − a)

BY (297.05)

BY (297.06)



2 ( [(b − 3a)(b − a) F (α, r) + 8ab E (α, r)] = 2 2 2 b 5 3 (b − a ) (b cosh x − a) sinh x 2b ( + 3 (b2 − a2 ) 3 (b cosh x − a) dx

 45.

dx = b cosh −a

bp2

1



BY (297.06)

2 2 sinh x [F (α, r) − 2 E (α, r)] + √ BY (297.03) b b cosh x − a   2 (cosh x + 1) dx 2 ( E (α, r) = BY (297.01) b−a b 3 (b cosh x − a)  √  b cosh x − a dx 2  = Π α, p2 , r BY (297.02) 2 2 p b − a + b (1 − p ) cosh x b )  b cosh x − a 2b Notation: In 2.464 48–2.464 55, we set α = arcsin and r = for b (cosh x − 1) a+b

a 0 < b < a, x > arccosh : b √

cosh x dx = b cosh x − a

2.464

Algebraic functions of hyperbolic functions



139

2 dx = √ F (α, r) BY (297.75) b cosh x − a a+b √ √ x√ b cosh x − a dx = −2 a + b E (α, r) + 2 coth b cosh x − a BY (297.79) 2 √ coth2 x2 dx 2 a+b √ E (α, r) = BY (297.76) a−b b cosh x − a √ √ b cosh x − a dx = a + b [F (α, r) − E (α, r)] BY (297.77) cosh x − 1 √ a+b 1 dx √ E (α, r) − √ = F (α, r) BY (297.78) a − b (cosh x − 1) b cosh x − a a+b  dx 1 √ (a − 2b)(a − b) F (α, r) = √ 2 (cosh x − 1) b cosh x − a 3(a − b)2 a + b  cosh x2 √ a+b · b cosh x − a + (3a − b)(a + b) E (α, r) + 6b(a − b) sinh3 x2 √

48.  49.  50.  51.  52.  53.

BY (297.78)

√ 1 2 b cosh x − a dx √ =√ [F (α, r) − E (α, r)] + BY (297.80) (a + b) sinh x (cosh x + 1) b cosh x − a a+b   dx 1 (a + 2b) F (α, r) =  2√ (cosh x + 1) b cosh x − a 3 (a + b)3 √    b cosh x − a a + 3b 2 x − (a + 3b) E (α, r) + 2 − tanh 3(a + b) sinh x a+b 2

 54. 55.12

BY (297.80)

Notation: In 2.464 56–2.464 60, we set √ 4 2 b − a2 , α = arccos √ a sinh x + b cosh x   1 a r= √ 0 < a < b, − arcsinh √ 2] GU (353)(6a)

 2.

 3.  4.

5.

q

q

q−1

x sinh x cosh x (p − 2) cosh x + qx cosh dx = − xp (p − 1)(p − 2)xp−1   coshq−2 x coshq x q(q − 1) q2 − dx + dx (p − 1)(p − 2) xp−2 (p − 1)(p − 2) xp−2 sinh x 1 dx = − x2n x(2n − 1)! sinh x 1 dx = − x2n+1 x(2n)!

n−2

(2k + 1)! k=0

n−1

(2k)!

x2k+1

cosh x +

k=0

 6.

10.

11.

+

1 chi(x) (2n − 1)!



n−1

GU (353)(6b)

cosh x 1 dx = − x2n+1 (2n)!x

n−1

(2k)! k=0

x2k

k=0

sinh x +

n−1

k=0



(2k + 1)! 1 chi(x) cosh x + x2k+1 (2n)!

k=0

GU (353)(7b) GU (353)(7b)

GU (353)(6c)

GU (353)(6d)

  m−1

 2m  1 1 2m cosh2m x dx = 2m−1 chi(2m − 2k)x + 2m ln x GU (353)(7c) x 2 2 k m k=0   m  1 2m + 1 cosh2m+1 x dx = 2m chi(2m − 2k + 1)x GU (353)(7c) x 2 k k=0    (−1)m−1 2m sinh2m x dx = x2 22m x m    m−1

2m cosh(2m − 2k)x 1 − (2m − 2k) shi(2m − 2k)x + 2m−1 (−1)k+1 2 x k 

9.

(2k)! sinh x x2k

    m−1

1 2m (−1)m 2m sinh2m x k dx = 2m−1 chi(2m − 2k)x + 2m ln x (−1) x 2 2 k m k=0    m 1

2m + 1 sinh2m+1 x k dx = 2m shi(2m − 2k + 1)x (−1) x 2 k 

8.

GU (353)(7a)



(2k + 1)! 1 shi(x) sinh x + GU (353)(6b) x2k+1 (2n)! k=0 k=0 n−2   n−1

(2k + 1)!

(2k)! cosh x 1 1 shi(x) dx = − sinh x + cosh x + 2n 2k+1 2k x x(2n − 1)! x x (2n − 1)! x2k

cosh x +

k=0

7.

n−1

[p > 2]

k=0

144

Hyperbolic Functions

 12.

 13.

  m 2m + 1 sinh2m+1 x 1

k+1 dx = (−1) x2 22m k   k=0 sinh(2m − 2k + 1)x − (2m − 2k + 1) chi(2m − 2k + 1)x × x cosh2m x dx x2 =−

 14.

2.476

2m+1

cosh x2

1 22m x

x



2m m

 −

1 22m−1

m−1

 k=0

2m k



 cosh(2m − 2k)x − (2m − 2k) shi(2m − 2k)x x

dx

  m  cosh(2m − 2k + 1)x 1 2m + 1 − (2m − 2k + 1) shi(2m − 2k + 1)x = − 2m 2 x k k=0

2.476



1.

 2.

 3.  4.  5.

 6.

 7.

 8.

  1 ka ka sinh kx dx = cosh shi(u) − sinh chi(u) a + bx b  b   b     ka k ka 1 exp − Ei(u) − exp Ei(−u) u = (a + bx) = 2b b b b   1 ka ka cosh kx dx = cosh chi(u) − sinh shi(u) a + bx b  b b        ka ka k 1 exp − Ei(u) + exp Ei(−u) u = (a + bx) = 2b b b b  sinh kx 1 sinh kx k cosh kx + dx (see 2.476 2) dx = − 2 (a + bx) b a + bx b a + bx  cosh kx 1 cosh kx k sinh kx + dx (see 2.476 1) dx = − (a + bx)2 b a + bx b a + bx  sinh kx k 2 sinh kx sinh kx k cosh kx + dx dx = − − 2 (a + bx)3 2b(a + bx)2 2b (a + bx) 2b2 a + bx cosh kx k cosh kx k sinh kx + dx = − − 2 (a + bx)3 2b(a + bx)2 2b (a + bx) 2b2 2



(see 2.476 1) cosh kx dx a + bx

(see 2.476 2)  sinh kx k 3 cosh kx sinh kx k cosh kx k sinh kx + dx dx = − − − (a + bx)4 3b(a + bx)3 6b2 (a + bx)2 6b3 (a + bx) 6b3 a + bx 2

(see 2.476 2)  cosh kx k 3 sinh kx cosh kx k sinh kx k cosh kx + dx dx = − − − (a + bx)4 3b(a + bx)3 6b2 (a + bx)2 6b3 (a + bx) 6b3 a + bx 2

(see 2.476 1)

2.477

Hyperbolic functions and powers

 9.

 10.

 11.

 12.

2.477 1.

2.

3.

sinh kx sinh kx k cosh kx k 2 sinh kx dx = − − − (a + bx)5 4b(a + bx)4 12b2 (a+ bx)3 24b3 (a + bx)2 k4 sinh kx k 3 cosh kx + dx − 24b4(a + bx) 24b4 a + bx (see 2.476 1) cosh kx k sinh kx k 2 cosh kx cosh kx dx = − − − 5 4 2 3 (a + bx) 4b(a + bx) 12b (a+ bx) 24b3 (a + bx)2 3 4 k cosh kx k sinh kx + dx − 24b4(a + bx) 24b4 a + bx (see 2.476 2) sinh kx k cosh kx k 2 sinh kx k 3 cosh kx sinh kx dx = − − − − (a + bx)6 5b(a + bx)5 20b2 (a +bx)4 60b3 (a + bx)3 120b4(a + bx)2 4 5 k cosh kx k sinh kx + dx − 120b5(a + bx) 120b5 a + bx (see 2.476 2) cosh kx k sinh kx k 2 cosh kx k 3 sinh kx cosh kx dx = − − − − (a + bx)6 5b(a + bx)5 20b2 (a +bx)4 60b3 (a + bx)3 120b4(a + bx)2 k5 sinh kx k 4 cosh kx + dx − 120b5(a + bx) 120b5 a + bx (see 2.476 1)

 −pxp−1 sinh x − (q − 2)xp cosh x xp−2 xp dx p(p − 1) = + dx sinhq x (q − 1)(q − 2) sinhq−2 x (q  − 1)(q − 2) sinhq−1 x q−2 xp dx − q − 1 sinhq−2 x [q > 2]   p p−1 p cosh x + (q − 2)x sinh x px xp−2 dx p(p − 1) x dx = − q cosh x (q − 1)(q − 2) coshq−2 x (q −1)(q − 2) coshq−1 x q−2 xp dx + q − 1 coshq−2 x [q > 2]   ∞ 2k

2−2 B2k n+2k xn dx = x [|x| < π, n > 0] sinh x (n + 2k)(2k)! 

 4.

GU (353)(8a)

GU (353)(10a) GU(353)(8b)

k=0 ∞

E2k xn+2k+1 xn dx = cosh x (n + 2k + 1)(2k)! k=0

 5.

145

xn

π |x| < , 2

n≥0

[|x| < π,

n ≥ 1]

 GU (353)(10b)

dx 2n−1 − 1 = − [1 + (−1)n ] Bn ln x sinh x n! ∞

2 − 22k B2k x2k−n + (2k − n)(2k)! k=0 k= n 2

GU (353)(9b)

146

Hyperbolic Functions

 6.12

∞

E2k dx 1 En−1 = x2k−n+1 + [1 + (−1)n ] ln x xn cosh x (2k − n + 1)(2k)! 2 (n − 1)! k=0 k= n−1 2

 7.

 8.

 9.12

π |x| < 2

10.12

∞

 2k

[n > 1,



11.

 12.

|x|
1,

k=1 k= n+1 2



GU (353)(11b)

∞

22k B2k xn n xn+2k−1 dx = −x coth x + n 2 (n + 2k − 1)(2k)! sinh x k=0

k=0 k= n+1 2



2.477

dx =

n−1

(2n − 3)(2n − 5) . . . (2n − 2k + 1) (2n − 2)(2n − 4) . . . (2n − 2k) k=1    x dx x cosh x 1 n−1 (2n − 3)!! × + + (−1) 2n−2k 2n−2k−1 (2n − 2)!! sinh x sinh x (2n − 2k − 1) sinh x GU (353)(8e) (see 2.477 15) (−1)k

n−1

(2n − 2)(2n − 4) . . . (2n − 2k + 2) x dx = 2n (2n − 1)(2n − 3) . . . (2n − 2k + 1) cosh x k=1    x dx x sinh x 1 (2n − 2)!! × + + 2n−2k+1 2n−2k (2n − 1)!! cosh2 x cosh x (2n − 2k) cosh x (see 2.477 18) GU (353)(10e)

2.478

Hyperbolic functions and powers

 14.

x 2n−1

cosh

 15.  16.  17.  18.  19.  20.  21.  22.  23.  24.

x

dx =

147

n−1

(2n − 3)(2n − 5) . . . (2n − 2k + 1) (2n − 2)(2n − 4) . . . (2n − 2k) k=1    x dx x sinh x 1 (2n − 3)!! × + + 2n−2k 2n−2k−1 (2n − 2)!! cosh x cosh x (2n − 2k − 1) cosh x GU (353)(10e) (see 2.477 16)

∞

2 − 22k x dx = B2k x2k+1 sinh x (2k + 1)(2k)!

[|x| < π]

GU (353)(8b)a

k=0 ∞

E2k x2k+2 x dx = cosh x (2k + 2)(2k)! k=0

π |x| < 2

GU (353)(10b)a

x dx = −x coth x + ln sinh x sinh2 x x dx = x tanh x − ln cosh x cosh2 x  1 x dx x cosh x 1 x dx − (see 2.477 15) = − − sinh3 x 2 sinh2 x 2 sinh x 2 sinh x  1 x dx x sinh x 1 x dx + (see 2.477 16) = + cosh3 x 2 cosh2 x 2 cosh x 2 cosh x 2 x cosh x 1 2 x dx =− − + x coth x − ln sinh x 3 sinh4 x 3 sinh3 x 6 sinh2 x 3 2 x sinh x 1 2 x dx = + + x tanh x − ln cosh x 3 cosh4 x 3 cosh3 x 6 cosh2 x 3  3 x dx x cosh x 1 3x cosh x 3 x dx + =− − + + 5 4 3 2 8 sinh x 8 sinh x sinh x 4 sinh x 12 sinh x 8 sinh x (see 2.477 15)  3 x dx x sinh x 1 3x sinh x 3 x dx + = + + + 5 4 3 2 cosh x 4 cosh x 12 cosh x 8 cosh x 8 cosh x 8 cosh x

MZ 257 MZ 262 MZ 257 MZ 262 MZ 258 MZ 262

MZ 258

(see 2.477 16) 2.478



1.  2.  3. 4.

xn cosh x dx xn n m = − m−1 + (m − 1)b (a + b sinh x) (m − 1)b (a + b sinh x) xn sinh x dx xn m = − m−1 (a + b cosh x) (m − 1)b (a + b cosh x)

x x x dx = x tanh − 2 ln cosh 1 + cosh x 2 2  x x x dx = x coth − 2 ln sinh 1 − cosh x 2 2



xn−1 dx (a + b sinh x)

MZ 262

m−1

[m = 1]  xn−1 dx n + (m − 1)b (a + b cosh x)m−1 [m = 1]

MZ 263

MZ 263

148

Hyperbolic Functions

 5.  6.  7.

 8.

2.479 1.

2.



3. 

5. 6.

=−

2

=

(1 + cosh x) x sinh x dx (1 − cosh x)

x x + tanh 1 + cosh x 2

x x − coth 1 − cosh x 2

MZ 262-264

1 x dx = [L(u + t) − L(u − t) − 2 L(t)] cosh 2x − cos 2t 2 sin 2t [u = arctan (tanh x cot t) ,        1 υ+t u+t u−t x cosh x dx = L −L +L π− cosh 2x − cos 2t 2 sin t 2 2   2   t π−t υ−t −2L −2L + L 2 2    2  x x t t u = 2 arctan tanh · cot , υ = 2 arctan coth · cot ; 2 2 2 2

m

x dx = x (−1)m+k n cosh x k m

t = ±nπ] LO III 402

 t = ±nπ

LO III 403

p sinh

2m



p

xp

sinh x x p dx = − + n−1 coshn x n − 1 (n − 1) cosh x

xp

cosh x dx = sinhn x 2m

m 



[n > 1] dx x n−1 cosh x [n > 1]

 m

(see 2.479 3)

p−1

(see 2.477 2)

GU (353)(12)

xp

dx (see 2.477 1) sinhn−2k x   p m 

cosh2m+1 x m x cosh x dx = xp dx (see 2.479 6) n sinh x k sinhn−2k x k=0   cosh x xp xp−1 dx p dx = − + xp n n−1 sinh x (n − 1) sinh x n − 1 sinhn−1 x [n > 1] (see 2.477 1) 

7. 

xp tanh x dx =



k=0

∞

2 k=1

p

x coth x dx =

8. 9.

2

xp dx (see 4.477 2) coshn−2k x k=0  m  

sinh2m+1 x sinh x m+k m dx = xp xp (−1) dx n−2k coshn x k cosh x k=0



4.12

x sinh x dx

2.479

∞

k=0

k

 2k



22k − 1 B2k p+2k x (2k + p)(2k)!

22k B2k xp+2k (p + 2k)(2k)!

x x cosh x x dx = ln tanh − 2 2 sinh x sinh x

GU (353)(13c)

p ≥ 1, [p ≥ +1,

π 2

GU (353)(12d)

|x| < π]

GU (353)(13d)

|x|
0]      ax + b ac − b2 ax + b √ √ − sin S a a a [a > 0]



5. 6.

171

x (sin ln x − cos ln x) 2  x cos ln x dx = (sin ln x + cos ln x) 2 sin ln x dx =

PE (444) PE (445)

2.55–2.56 Rational functions of the sine and cosine 2.551



1.

⎡ 1 A + B sin x ⎣ (Ab − aB) cos x n dx = (n − 1) (a2 − b2 ) (a + b sin x)n−1 (a + b sin x)  +

(Aa − Bb)(n − 1) + (aB − bA)(n − 2) sin x (a + b sin x)

n−1

⎤ dx⎦ TI (358)a

2. 3.

2.552 1.

2.

For n = 1:   B Ab − aB dx A + B sin x dx = x + a + b sin x b b a + b sin x  a tan x2 + b 2 dx = √ arctan √ a + b sin x a2 − b 2 a2 − b2√ x a tan 2 + b − b2 − a2 1 = √ ln  b2 − a2 a tan x + b + b2 − a2 2 

(see 2.551 3)

TI (342)

 2 a > b2  2 a < b2

 A + B cos x B dx n dx = − n n−1 + A (a + b sin x) (a + b sin x) (n − 1)b (a + b sin x) (see 2.552 3)

TI (361)

(see 2.551 3)

TI (344)

For n = 1:   B dx A + B cos x dx = ln (a + b sin x) + A a + b sin x b a + b sin x

172

Trigonometric Functions

 3.

2.553

⎡ b cos x dx 1 ⎣ n = (n − 1) (a2 − b2 ) (a + b sin x)n−1 (a + b sin x) ⎤  (n − 1)a − (n − 2)b sin x ⎦ + dx (a + b sin x)n−1

(see 2.551 1) TI (359)

2.553



1.

 B dx A + B sin x dx = + A n n n−1 (a + b cos x) (a + b cos x) (n − 1)b (a + b cos x) (see 2.554 3)

2.

3.

TI (355)

For n = 1:   B dx A + B sin x dx = − ln (a + b cos x) + A a + b cos x b a + b cos x (see 2.553 3)

√  a2 − b2 tan x2 2 dx = √ arctan a + b cos x a+b a2 − b 2 √ b2 − a2 tan x2 + a + b 1 = √ ln √ b 2 − a2 b2 − a2 tan x2 − a − b 

4.12

TI (343)

 2 a > b2  2 a < b2 TI II 93, 94, TI (305)

dx a + b cos x    (a − b) tan x2 2 √ arctan = √ a2 − b 2 a2 − b 2   x √   2  (b − a) tan 2 + b2 − a2  x √ ln  = √  b2 − a2  (b − a) tan 2 − b2 − a2     (a − b) tan x2 2 √ arctanh = √ b 2 − a2 b 2 − a2    (a − b) tan x2 2 √ arccoth = √ b 2 − a2 b 2 − a2

 2 b < a2  2 b > a2

b 2 > a2 ,

b 2 > a2 ,

   x      < b 2 − a2 (b − a) tan 2    x      > b 2 − a2 (b − a) tan 2

(compare with 2.551 3) 2.554 1.



⎡ 1 A + B cos x ⎣ (aB − Ab) sin x dx = (a + b cos x)n (n − 1) (a2 − b2 ) (a + b cos x)n−1  +

(Aa − bB)(n − 1) + (n − 2)(aB − bA) cos x (a + b cos x)

n−1

⎤ dx⎦ TI (353)

2.557

2.

3.

Rational functions of sine and cosine

173

For n = 1:   B Ab − aB dx A + B cos x dx = x + (see 2.553 3) a + b cos x b b a + b cos x ⎧  ⎨ b sin x dx 1 n = − 2 2 (n − 1) (a − b ) ⎩ (a + b cos x)n−1 (a + b cos x) ⎫  (n − 1)a − (n − 2)b cos x ⎬ − dx (see 2.554 1) ⎭ (a + b cos x)n−1

TI (341)

TI (354)

In integrating the functions in formulas 2.551 3 and 2.553 3, we may not take integration over points  the a in formula 2.551 3 or at which the integrand becomes infinite, that is, over the points x = arcsin − b  a over the points x = arccos − in formula 2.553 3. b 2.555 Formulas 2.551 3 and 2.553 3 are not applicable for a2 = b2 . Instead, we may use the following formulas in these cases:  n−2   

n − 2  tan2k+1 π ∓ x A + B sin x 1 4 2 1. dx = − n−1 2B (1 ± sin x)n 2 2k + 1 k k=0 π x   n−1 2k+1

n − 1 tan 4 ∓ 2 ± (A ∓ B) 2k + 1 k k=0

 2.



TI (361)a

  n − 2 tan2k+1 π4 ∓ π4 − x2 A + B cos x 1 dx = n−1 2B (1 ± cos x)n 2 2k + 1 k k=0      n−1

n − 1 tan2k+1 π ∓ π − x 4 4 2 ± (A ∓ B) 2k + 1 k n−2





k=0

TI (356)

3.12 4.

For n = 1 :  π x A + B sin x dx = ±Bx + (A ∓ B) tan ∓ 1 ± sin x 4 2 

 π π x  A + B cos x dx = ±Bx ± (A ∓ B) tan ∓ − 1 ± cos x 4 4 2

2.556  1. 2.

  1 − a2 dx x 1+a tan = 2 arctan 1 − 2a cos x + a2 1−a 2    x (1 − a cos x) dx x 1+a + arctan tan = 1 − 2a cos x + a2 2 1−a 2

2.557  1.

TI (250) TI (248)



1 dx n =  n 2 (a cos x + b sin x) (a + b2 )

 n

sin



dx

[0 < a < 1,

|x| < π]

FI II 93

[0 < a < 1,

|x| < π]

FI II 93

a b (see 2.515)

x + arctan

MZ 173a

174

Trigonometric Functions

2.558

 ax − b ln sin x + arctan ab sin x dx = a cos x + b sin x a2 + b 2   ax + b ln sin x + arctan ab cos x dx = a cos x + b sin x a2 + b 2 1   ln tan 2 x + arctan ab dx √ = a cos x + b sin x a2 + b 2     cot x + arctan ab a cos x − b sin x dx 1 = − = − 2 a2 + b 2 a2 + b2 a sin x + b cos x (a cos x + b sin x) 

2.

12

3. 4. 5.12

MZ 174a

MZ 174a

2.558  A + B cos x + C sin x 1. n dx (a + b cos x + c sin x) (Bc − Cb) + (Ac − Ca) cos x − (Ab − Ba) sin x 1 = n−1 + (n − 1) (a2 − b2 − c2 ) 2 2 2 (n − 1) (a − b − c ) (a + b cos x + c sin x)  (n − 1)(Aa − Bb − Cc) − (n − 2) [(Ab − Ba) cos x − (Ac − Ca) sin x] dx × n−1 (a + b cos x + c sin x)  2 2 2 n = 1, a = b + c  Cb − Bc + Ca cos x − Ba sin x A n(Bb + Cc) + (−c cos x + b sin x) = n + (n − 1)a (a + b cos x + c sin x) a (n − 1)a2 n−1 1 (n − 1)! (2n − 2k − 3)!! · × n−k k (2n − 1)!! (n − k − 1)!a (a + b cos x + c sin x) k=0  n = 1, a2 = b2 + c2

11

2.

For n = 1 :  Bc − Cb A + B cos x + C sin x Bb + Cc dx = 2 ln (a + b cos x + c sin x) + 2 x a + b cos x + c sin x b+ c2 b + c2  dx Bb + Cc a + A− 2 2 b +c a + b cos x + c sin x 

3.

4.

(see 2.558 4) GU (331)(18)



dx d(x − α) n = n, (a + b cos x + c sin x) [a + r cos(x − α)] where b = r cos α, c = r sin α (see 2.554 3)  dx a + b cos x + c sin x (a − b) tan x2 + c 2 = √ arctan √ a 2 − b 2 − c2 a2 − b2 − c2√ (a − b) tan x2 + c − b2 + c2 − a2 1 = √ ln  b2 + c2 − a2 (a − b) tan x + c + b2 + c2 − a2 2 1  x = ln a + c tan c 2 −2 = x c + (a − b) tan 2

 

a2 > b 2 + c2 a2 < b 2 + c2



TI (253), FI II 94 TI (253)a

[a = b] 

a2 = b 2 + c2



TI (253)a

2.561

Rational functions of sine and cosine

2.559  1. 2.12

175

   x c (a sin x − c cos x) 1 − a ln a + c tan = 3 c a (1 + cos x) + c sin x 2 [a (1 + cos x) + c sin x]2  A + B cos x + C sin x dx (a1 + b1 cos x + c1 sin x) (a2 + b2 cos x+ c2 sin x)  dx dx a1 + b1 cos x + c1 sin x = A0 ln + A1 + A2 a2 + b2 cos x + c2 sin x a1 + b1 cos x + c1 sin x a2 + b2 cos x + c2 sin x (see 2.558 4) GU (331)(19) dx

where   A B C    a1 b 1 c 1     a2 b 2 c2  A0 =     ,    a 1 b 1  2  b 1 c1  2  c1 a1  2   −  +   a2 b 2   b 2 c2   c2 a2         C B   C A   A B        c2 b2  c2 a2  a2 b2     a1 b1 c1    a2 b2 c2  A2 =  2  2  2 ,  a1 b 1        −  b 1 c1  +  c1 a1   a2 b 2   b 2 c2   c2 a2   3.

      B C   A C  B A        b1 c1  a1 c1  b1 a1     a1 b1 c1    a2 b2 c2  A1 =  2  2  2 ,  a1 b 1        − b1 c1  + c1 a1   a2 b 2   b 2 c2  c2 a2 

 a1  a2

2  c b1  +  1 b2  c2

2  b a1  =  1 a2  b2

2  c1  c2 

A cos2 x + 2B sin x cos x + C sin2 x dx a cos2 x + 2b sin x cos x + c sin2 x

⎧ ⎨ 1 = 2 [4Bb + (A − C)(a − c)]x + [(A − C)b − B(a − c)] 4b + (a − c)2 ⎩  × ln a cos2 x + 2b sin x cos x + c sin2 x ⎫ ⎬  + 2(A + C)b2 − 2Bb(a + c) + (aC − Ac)(a − c) f (x) ⎭

where

√ 1 c tan x + b − b2 − ac √ f (x) = √ ln 2 b2 − ac c tan x + b + b2 − ac c tan x + b 1 arctan √ = √ 2 ac − b ac − b2 1 =− c tan x + b

GU (331)(24)

 2 b > ac  2 b < ac  2 b = ac

2.561   A x Ba − Ab dx (A + B sin x) dx = ln tan + 1. sin x (a + b sin x) a 2 a a + b sin x (see 2.551 3)

TI (348)

176

Trigonometric Functions

 2.

2.561

  A a + b cos x x (A + B sin x) dx = a ln tan + b ln 2 sin x (a + b cos x) a2 − 2 sin x b dx +B a + b cos x

(see 2.553 3) TI (349)

For a = b (= 1) :    A x 1 x (A + B sin x) dx = ln tan + + B tan sin x (1 + cos x) 2 2 1 + cos x 2    A x 1 x (A + B sin x) dx = ln tan − − B cot sin x (1 − cos x) 2 2 1 − cos x 2      1 a + b sin x π x (A + B sin x) dx = 2 + − (Ab − aB) ln (Aa − Bb) ln tan cos x (a + b sin x) a − b2 4 2 cos x 2

3.12 4. 5.

2

TI (346)

For a = b (= 1):  π x A±B A∓B (A + B sin x) dx = ln tan + ∓ cos x (1 ± sin x) 2 4 2 2 (1 ± sin x)  π A x  B a + b cos x (A + B sin x) dx = ln tan + + ln cos x (a + b cos x) a  4 2 a cos x dx Ab − a a + b cos x 2

6. 7.

 8.  9.

10. 11.

2

A x B a + b sin x Ab (A + B cos x) dx = ln tan − ln − sin x (a + b sin x) a 2 a sin x a



(see 2.553 3) TI (351)a

dx a + b sin x

(see 2.551 3)   1 a + b cos x x (A + B cos x) dx = 2 + (Ab − Ba) ln (Aa − Bb) ln tan sin x (a + b cos x) a − b2 2 sin x

(see 2.551 3)

13.

TI (345)

For a2 = b2 (= 1) :  A∓B A±B x (A + B cos x) dx =± + ln tan sin x (1 ± cos x) 2 (1 ± cos x) 2 2       A a + b sin x dx π x (A + B cos x) dx = 2 + − b ln + B a ln tan 2 cos x (a + b sin x) a −b 4 2 cos x a + b sin x TI (350)

For a = b (= 1):  π x A±B A∓B (A + B sin x) dx = ln tan + ∓ cos x (1 ± sin x) 2 4 2 2 (1 ± sin x)   π A x  Ba − Ab dx (A + B cos x) dx = ln tan + + cos x (a + b cos x) a 4 2 a a + b cos x 2

12.

TI (352)

2

(see 2.553 3)

TI (347)

2.563

2.562

Rational functions of sine and cosine



1.

 2.

 a+b tan x a   sign a a+b tan x =  − arctanh a −a(a + b)   sign a a+b tan x =  − arccoth a −a(a + b)

dx sign a =  arctan a + b sin2 x a(a + b)







 

b > −1 a



b < −1, a

a sin x < − b

b < −1, a

a sin x > − b



4. 5. 6. 2.563

 

b > −1 a

b < −1, a

a cos x < − b

b < −1, a

cos2 x > −



2

a b



MZ 162

 √ 1 dx = √ arctan 2 tan x 2 1 + sin x 2  dx = tan x 1 − sin2 x   √ 1 dx √ = − 2 cot x arctan 1 + cos2 x 2  dx = − cot x 1 − cos2 x 

1.

 2.



2





3.



2

MZ 155

a+b cot x a   − sign a a+b cot x = − arctanh a −a(a + b)   − sign a a+b cot x = − arccoth a −a(a + b)

− sign a dx = arctan 2 a + b cos x a(a + b)



177

   dx b sin x cos x 1 (2a + b) +  2 = 2a(a + b) a + b sin2 x a + b sin2 x a + b sin2 x dx

dx (a + b cos2 x)2

=

1 2a(a + b)



 (2a + b)

(see 2.562 1)  dx b sin x cos x − a + b cos2 x a + b cos2 x

MZ 155

(see 2.562 2)

MZ 163

178

Trigonometric Functions

 3.



⎡

  1 ⎣ 3 2 arctan (p tan x) = + 3 +  3 8pa3 p2 p4 a + b sin2 x ⎤     2 2 3 1 p tan x 2p tan x + 3+ 2 − 4 + 1 − 2 − 2 tan2 x  2 ⎦ p p p p 1 + p2 tan2 x 1 + p2 tan2 x   b p2 = 1 + > 0 a ⎡   1 ⎣ 3 2 = 3 − 2 + 4 arctanh (q tan x) 3 8qa q q ⎤     2 2 q tan x 3 1 2q tan x + 3− 2 − 4 + 1 + 2 + 2 tan2 x  2 ⎦ q q q q 1 − q 2 tan2 x 1 − q 2 tan2 x dx

b q = −1 − > 0, a 2

dx 3

(a + b cos2 x)



a sin x < − ; b 2

a for sin x > − , change arctanh (q tan x) to arccoth (q tan x) b

=−

MZ 156

  1 ⎣ 3 2 arctan (p cot x) + 3 + 8pa3 p2 p4

⎤     2 2 p cot x 3 1 2p cot x + 3+ 2 − 4 + 1 − 2 − 2 cot2 x  2 ⎦ p p p p 1 + p2 cot2 x 1 + p2 cot2 x   b 2 p =1+ >0 a ⎡   1 ⎣ 3 2 =− 3 − 2 + 4 arctanh (q cot x) 8qa3 q q ⎤     2 2 3 1 q cot x 2p cot x + 3− 2 − 4 + 1 + 2 + 2 cot2 x  2 ⎦ q q q q 1 − q 2 cot2 x 1 − q 2 cot2 x

b q = −1 − > 0, a 2



2

⎡

 4.

2.564

a cos x < − ; b 2

a for cos x > − , change arctanh (q cot x) to arccoth (q cot x) b 2

MZ 163a

2.564   ln cos2 x + m2 sin2 x tan x dx = 1. 2 (m2 − 1) 1 + m2 tan2 x  tan α − tan x dx = sin 2α ln sin(x + α) − x cos 2α 2. tan α + tan x  1 tan x dx = 2 3. {bx − a ln (a cos x + b sin x)} a + b tan x a + b2      1 dx b b x− = 4. arctan tan x a−b a a a + b tan2 x

LA 210 (10) LA 210 (11)a PE (335)

PE (334)



2.571

Integrals with

2.57 Integrals containing Notation:

 α = arcsin  γ = arcsin

2.571

 √

1.

3.

√

√

a ± b cos x

179

√ √ a ± b sin x or a ± b cos x 

1 − sin x , 2

b (1 − sin x) , a+b ) (a + b) (1 − cos x) δ = arcsin , 2 (a − b cos x)

β = arcsin

b (1 − cos x) , a+b

dx −2 = √ F (α, r) a + b sin x a + b   1 2 =− F β, b r

a > b > 0,

0 < |a| < b,

 r=

2b a+b

π π ≤x< 2 2 a π − arcsin < x < b 2

−

sin x dx a + b sin x

√

π π 2a 2 a+b BY (288.03) a > b > 0, − ≤ x < = √ F (α, r) − E (α, r) 2 2 b b a + b    

a π 2 1 1 0 < |a| < b, − arcsin < x < BY (288.54) F β, − 2 E β, = b 2 b r r  √  √ 2 2a2 + b2 4a a + b 2 sin2 x dx √ √ cos x a + b sin x = F (α, r) − E (α, r) − 2 2 3b 3b a + b sin x 3b a + b π π a > b > 0, − ≤ x < 2 2       √ 1 2a + b 1 2 2 4a = E β, − F β, − cos x a + b sin x b 3b r 3b r

3b π a 0 < |a| < b, − arcsin < x < b 2

BY (288.03, 288.54)

 √

4.

x

2 dx ,r =√ F 2 a + b cos x a + b  2 1 = F γ, b r



[a > b > 0,

b ≥ |a| > 0,

0 ≤ x ≤ π]  a  0 ≤ x < arccos − b BY (289.00)

 5.

a ± b sin x or

BY (288.00, 288.50)

 2.

√

√

2 dx =√ F (δ, r) a − b cos x a+b

[a > b > 0,

0 ≤ x ≤ π]

BY (291.00)

180

Trigonometric Functions

 √

6.

2.572

 x , x  2 cos x dx ,r − aF ,r = √ (a + b) E 2 2 a + b cos x b a + b [a > b > 0,       2 1 1 2 E γ, − F γ, = b r r

0 ≤ x ≤ π] BY (289.03)

b > |a| > 0,

 a  0 ≤ x < arccos − b

BY (290.04)



7.

6

8.

   cos x dx 2 = √ (b − a) Π δ, r2 , r + a F (δ, r) a − b cos x b a+b [a > b > 0, 0 ≤ x ≤ π] BY (291.03)    ,  √  2 x 2 2 x cos2 x dx √ , r − 2a(a + b) E ,r + sin x a + b cos x = 2√ 2a + b2 F 2 2 3b a + b cos x 3b a + b [a > b > 0, 0 ≤ x ≤ π] √

1 = 3b

BY (289.03)       √ 1 1 2 2 (2a + b) F γ, − 4a E γ, + sin x a + b cos x b r 3b

r  a  b ≥ |a| > 0, 0 ≤ x < arccos − b BY (290.04)

 9.

12

 2.572

 2  2 cos2 x dx √ = 2√ 2a + b2 F (δ, r) − 2a(a + b) E (δ, r) a − b cos x 3b a + b a + b cos x 2 [a > b > 0, 0 ≤ x < π] + sin x √ 3b a − b cos x BY (291.04)a

1 a tan2 x dx √ √ = √ F (α, r) + E (α, r) a + b sin x a+b (a − b) a + b

π b − a sin x √ π < x < − 2 a + b sin x 0 < b < a, − 2 2 2 (a − b ) cos x     1 ab 1 2 2a + b = F β, + E β, b 2(a + b) r powera2 − b2 r

b − a sin x √ a π − 2 a + b sin x 0 < |a| < b, − arcsin < x < 2 (a − b ) cos x b 2 BY(288.08, 288.58)

2.573 1.

2.

⎫ ⎬ π dx x √ 1 − sin x 2 ,√ √ − = a + b E (α, r) − tan a + b sin x ⎭ 1 + sin x a + b sin x a − b 4 2

π π 0 < b < a, − ≤ x < 2 2 √  dx x√ 2 1 − cos x 2 a + b x  √ tan E ,r = a + b cos x − 1 + cos x a + b cos x a − b 2 a−b 2 [a > b > 0, 0 ≤ x < π]



BY (288.07)

BY (289.07)

2.575

2.574

Integrals with



1.

(2 −

 2.

p2

+

p2

4.

√

a ± b cos x

 1 dx √ Π α, p2 , r =− a+b sin x) a + b sin x

0 < b < a, 

181

−

π π ≤x< 2 2 BY (288.02)

  2 1 Π β, p2 , b r 0 < |a| < b,

− arcsin

π a b > 0, √   2 dx 1 √ √ Π γ, p2 , = r (a + b − p2 b + p2 b cos x) a + b cos x (a + b) b

b ≥ |a| > 0,

0 ≤ x < π]

BY (289.02)

 a  0 ≤ x < arccos − b

BY (290.02)

2.575 1.



2 2b cos x dx ( √ √ − E (α, r) = 2 2 ) a + b sin x 3 (a − b (a − b) a + b

(a + b sin x) 0 < b < a,

−

π π ≤x< 2 2

BY (288.05)       2 1 1 2b 2b 1 cos x − F β, + 2 = E β, ·√ b b 2 − a2 r a+b b − a2 a + b sin x

r π a 0 < |a| < b, − arcsin < x < b 2 BY (288.56)

182

Trigonometric Functions

 2.

2.575

  2 dx 2 ( = a − b2 F (α, r) − 4a(a + b) E (α, r) 2√ 2 2 5 3 (a − b ) a + b (a + b sin x)  2b 5a2 − b2 + 4ab sin x ( + cos x 2 3 3 (a2 − b2 ) (a + b sin x)

π π 0 < b < a, − ≤ x < 2 2 BY (288.05)       2 1 1 1 (3a − b)(a − b) F β, + 8ab E β, =− 2 2 2 b r r 3 (a − b )  2b a2 − b2 + 4a (a + b sin x) ( cos x + 3 (a2 − b2 )2 (a + b sin x)3

0 < |a| < b,

− arcsin

π a b > 0,

0 ≤ x ≤ π]

BY (289.05)       2 1 1 2b sin x (a − b) F γ, + 2b E γ, + 2 ·√ b r r b − a2 a + bcos x

a b ≥ |a| > 0, 0 ≤ x < arccos − b BY (290.06)

 4.

dx 2 ( √ E (δ, r) = 3 (a − b) a + b (a − b cos x)

[a > b > 0,

0 ≤ x ≤ π]

(291.01)

2.578

Integrals with



√ 2 a+b

dx

( = 2 5 3 (a2 − b2 ) (a + b cos x)

5.

−

√

,

2b 3 (a2 − b2 )2

4a E

·

a ± b sin x or

√

a ± b cos x

183

  x , r − (a − b) F ,r 2 2

x

5a2 − b2 + 4ab cos x ( sin x 3 (a + b cos x) [a > b > 0,

      2 1 1 1 (a − b)(3a − b) F γ, + 8ab E γ, = r r 3 (a2 − b2 )2 b  2b 5a2 − b2 + 4ab cos x sin x ( + 2 3 3 (a2 − b2 ) (a + b cos x)

0 ≤ x ≤ π] BY (289.05)

 a  0 ≤ x < arccos − b

b ≥ |a| > 0,

BY (290.06)

2.576 1.

 √

x  √ ,r a + b cos x dx = 2 a + b E 2

[a > b > 0,

0 ≤ x ≤ π] BY (289.01)

      2 1 1 (a − b) F γ, + 2b E γ, = b r r

 a  0 ≤ x < arccos − b

b ≥ |a| > 0,

BY (290.03)

 2. 2.577 1.

√

√ 2b sin x a − b cos x dx = 2 a + b E (δ, r) − √ a − b cos x

[a > b > 0,

0 ≤ x ≤ π]

  a − b cos x 2(a − b) 2ap √ dx = ,r Π δ, 1 + p cos x (a + b)(1 + p) (1 + p) a + b [a > b > 0,

0 ≤ x ≤ π,

 √

3

 ) 2.12

 2.578

BY (291.05)



a − b cos x 2(a − b) dx =  Π δ, r2 , 1 + p cos x (1 + p)(a + b)

)

2(ap + b) (1 + p)(a + b)

√  tan x dx b−a 1 √ √ √ arccos cos x = b−a b a + b tan2 x

[b > a,

BY (291.02)



[a > b > 0,

p = −1]

0 ≤ x ≤ π,

b > 0]

p = −1] PE (333)

184

Trigonometric Functions

2.580

2.58–2.62 Integrals reducible to elliptic and pseudo-elliptic integrals 2.580  √ 1. 2.12





 c ϕ = 2ψ + α, tan α = , p = b2 + c2 b   dx  =2 √ 2 2 A + Bx + Cx2 + Dx3 + Ex4 a + b cos ϕ + c sin ϕ + d cos ϕ + e sin ϕ cos ϕ + f sin ϕ 

ϕ tan = x, A = a + b + d, B = 2c + 2e, C = 2a − 2d + 4f, D = 2c − 2e, E = a − b + d 2 dϕ =2 a + b cos ϕ + c sin ϕ

Forms containing

dψ  a − p + 2p cos2 ψ dϕ

 1 − k2 sin2 x

ndexsquare roots √ Notation: Δ = 1 − k 2 sin2 x, k  = 1 − k 2 2.581  1.12 sinm x cosn xΔr dx ⎧ ⎨  1 = sinm−3 x cosn+1 xΔr+2 + m + n − 2(m + r − 1)k 2 2 (m + n + r)k ⎩ ⎫   ⎬ × sinm−2 x cosn xΔr dx − (m − 3) sinm−4 x cosn xΔr dx ⎭ ⎧ ⎨ 

1 m+1 n−3 r+2 2 2 = x cos xΔ + (n + r − 1)k − (m + n − 2)k sin (m + n + r)k 2 ⎩ ⎫   ⎬ 2 sinm x cosn−4 xΔr dx × sinm x cosn−2 xΔr dx + (n − 3)k  ⎭

2.

3.

4. 5.

[m + n + r = 0] For r = −3 and r = −5:  sinm x cosn x sinm−1 x cosn−1 x dx = Δ3 k 2Δ  n − 1 sinm x cosn−2 x m − 1 sinm−2 x cosn x dx + dx − 2 k Δ k2 Δ  sinm x cosn x sinm−1 x cosn−1 x dx = Δ5 3k 2Δ3  m − 1 sinm−2 x cosn x n − 1 sinm x cosn−2 x − dx + dx 3k 2 Δ3 3k 2 Δ3 For m = 1 or n = 1:  2  cosn−1 xΔr+2 (n − 1)k  sin x cosn xΔr dx = − cosn−2 x sin xΔr dx − (n + r + 1)k 2 (n + r + 1)k 2   sinm−1 xΔr+2 m−1 m r sin x cos xΔ dx = − sinm−2 x cos xΔr dx + (m + r + 1)k 2 (m + r + 1)k 2 For m = 3 or n = 3:

2.583

Elliptic and pseudo-elliptic integrals



 (n + r + 1)k 2 cos2 x − (r + 2)k 2 + n + 1 sin x cos xΔ dx = cosn−1 xΔr+2 4 (n + r + 1)(n + r + 3)k  2 (r + 2)k 2 + n + 1 (n − 1)k  cosn−2 x sin xΔr dx − (n + r + 1)(n + r + 3)k 4 3

6.

 7.12

185

n

r

sinm x cos3 xΔr dx



2 (m + r + 1)k 2 sin2 x − (r + 2)k 2 − (m + 1)k 

=

4 (m + r + 

1)(m + r + 3)k 2 (r + 2)k 2 − (m + 1)k  (m − 1)  sinm−2 x cos xΔr dx × sinm−1 xΔr+2 + (m + r + 1)(m + r + 3)k 4

2.582    n−1 n−2 2 − k2 1 − k2 Δn−2 dx − Δn−4 dx 1. Δn dx = n n k2 + sin x cos xΔn−2 n  2. 3.

dx n−3 1 − Δn−1 n − 1 k 2   sinn−3 x sinn x n − 2 1 + k 2 sinn−2 x dx = dx cos xΔ + Δ (n − 1)k 2 n − 1 k2 Δ  n−3 sinn−4 x dx − (n − 1)k 2 Δ 

4.

 5.

 6.

dx k 2 sin x cos x n−22−k =− + n+1 Δ n − 1 k 2 (n − 1)k  2 Δn−1

2

cosn−3 x cosn x n − 2 2k 2 − 1 dx = sin xΔ + 2 Δ (n − 1)k n − 1 k2 2  n−4 x n−3k cos + dx 2 n−1 k Δ 



(n − 2) 2 − k tann x tann−3 x Δ dx = 2 cos2 x −  Δ (n − 1)k (n − 1)k  2  tann−4 x n−3 dx − 2  Δ (n − 1)k n−1

LA (316)(1)a

dx Δn−3

LA 317(8)a

LA 316(1)a

cosn−2 x dx Δ

2 

x Δ cot n−2  cot x dx = − − 2 − k2 2 Δ n − 1 cos x n − 1 n − 3  2 cotn−4 x − k dx n−1 Δ n





LA 316(2)a

tann−2 x dx Δ

LA 317(3) n−2

cot

Δ

x

dx

LA 317(6)

2.583  1. Δ dx = E (x, k)

186

Trigonometric Functions



2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

12. 13.

14. 15.

16.

17.

Δ cos x k  − ln (k cos x + Δ) 2 2k  Δ sin x 1 Δ cos x dx = + arcsin (k sin x) 2 2k  2 k Δ 2k 2 − 1 E (x, k) Δ sin2 x dx = − sin x cos x + 2 F (x, k) + 3 3k 3k 2  Δ3 Δ sin x cos x dx = − 2 3k  2 k Δ k2 + 1 sin x cos x − 2 F (x, k) + Δ cos2 x dx = E (x, k) 3 3k 3k 2  2k 2 sin2 x + 3k 2 − 1 3k 4 − 2k 2 − 1 Δ sin3 x dx = − Δ cos x + ln (k cos x + Δ) 2 8k 8k 3  2k 2 sin2 x − 1 1 Δ sin2 x cos x dx = Δ sin x + 3 arcsin (k sin x) 2 8k 8k  4 2 2 2 2k cos x + k k Δ cos x + 3 ln (k cos x + Δ) Δ sin x cos2 x dx = − 2 8k 8k  2 2 2 2 2k cos x + 2k + 1 4k − 1 Δ sin x + arcsin (k sin x) Δ cos3 x dx = 2 8k 8k 3  3k 2 sin2 x + 4k 2 − 1 Δ sin4 x dx = − Δ sin x cos x  4 15k2 2 2 2k − k − 1 8k 4 − 3k 2 − 2 F (x, k) + E (x, k) − 4 15k 15k 4  3k 4 sin4 x − k 2 sin2 x − 2 Δ Δ sin3 x cos x dx = 15k 4  3k 2 cos2 x − 2k 2 + 1 Δ sin x cos x Δ sin2 x cos2 x dx = −  15k 2  2 2 k 1 + k 2 k4 − k2 + 1 − F (x, k) + E (x, k) 15k 4 15k 4   3k 4 sin4 x − k 2 5k 2 + 1 sin2 x + 5k 2 − 2 3 Δ sin x cos x dx = − Δ 15k 4  3k 2 cos2 x + 3k 2 + 1 Δ sin x cos x Δ cos4 x dx =  15k 2  2 2 2k  k  − 2k 2 3k 4 + 7k 2 − 2 + F (x, k) + E (x, k) 15k 4 15k 4   −8k 4 sin4 x − 2k 2 5k 2 − 1 sin2 x − 15k 4 + 4k 2 + 3 Δ sin5 x dx = Δ cos x 48k 4 6 4 2 5k − 3k − k − 1 ln (k cos x + Δ) + 16k 5  8k 4 sin4 x − 2k 2 sin2 x − 3 1 Δ sin x + arcsin (k sin x) Δ sin4 x cos x dx = 4 48k 16k 5 2

Δ sin x dx = −

2.583

2.583

Elliptic and pseudo-elliptic integrals



 8k 4 sin4 x − 2k 2 k 2 + 1 sin2 x − 3k 4 + 2k 2 − 3 Δ sin x cos x dx = Δ cos x 48k 4  4 k k2 + 1 + ln (k cos x + Δ) 16k 5  −8k 4 sin4 x + 2k 2 6k 2 + 1 sin2 x − 6k 2 + 3 2 3 Δ sin x cos x dx = Δ sin x 48k 4 2 2k − 1 arcsin (k sin x) + 16k 5  −8k 4 sin4 x + 2k 2 7k 2 + 1 sin2 x − 3k 4 − 8k 2 + 3 4 Δ sin x cos x dx = Δ cos x 48k 4 6 k − ln (k cos x + Δ) 16k 5  8k 4 sin4 x − 2k 2 12k 2 + 1 sin2 x + 24k 4 + 12k 2 − 3 5 Δ cos x dx = Δ sin x 48k 4 4 2 8k − 4k + 1 arcsin (k sin x) + 16k 5 2  k2 k 2 2 1 + k  E (x, k) − F (x, k) + Δ sin x cos x Δ3 dx = 3 3 3 3

18.

 19.

 20.

 21.

 22. 

2



24. 25.

26. 27.12

28.

29.

3k  2k 2 sin2 x + 3k 2 − 5 Δ cos x − ln (k cos x + Δ) 8 8k 4

Δ3 sin x dx =

23.

3 −2k 2 sin2 x + 5 Δ sin x + arcsin (k sin x) 8 8k  2 k  3 − 4k 2 3k 2 sin2 x + 4k 2 − 6 3 2 Δ sin x cos x + Δ sin x dx = F (x, k) 15 15k 2 4 2 8k − 13k + 3 E (x, k) − 15k 2  Δ5 Δ3 sin x cos x dx = − 2 5k  2 2 k k2 + 3 −3k sin2 x + k 2 + 6 3 2 Δ sin x cos x − Δ cos x dx = F (x, k) 15 15k 2 4 2 2k − 7k − 3 E (x, k) − 15k 2   8k 4 sin4 x + 2k 2 5k 2 − 7 sin2 x + 15k 4 − 22k 2 + 3 3 3 Δ sin x dx = Δ cos x 48k 2 6 4 2 5k − 9k + 3k + 1 ln (k cos x + Δ) − 16k 3  −8k 4 sin4 x + 14k 2 sin2 x − 3 Δ3 sin2 x cos x dx = Δ sin x 48k 2 1 arcsin (k sin x) + 16k 3 Δ3 cos x dx =

187

188

Trigonometric Functions

 −8k 4 sin4 x + 2k 2 k 2 + 7 sin2 x + 3k 4 − 8k 2 − 3 Δ sin x cos x dx = 48k 2 k 6 ×Δ cos x + ln (k cos x + Δ) 16k 3   8k 4 sin4 x − 2k 2 6k 2 + 7 sin2 x + 30k 2 + 3 3 3 Δ cos x dx = Δ sin x 48k 2 2 6k − 1 arcsin (k sin x) + 16k 3  1 Δ + cos x Δ dx = − ln + k ln (k cos x + Δ) sin x 2 Δ − cos x  k  Δ + k  sin x Δ dx = ln + k arcsin (k sin x) cos x 2 Δ − k  sin x  Δ dx 2 = k  F (x, k) − E (x, k) − Δ cot x sin2 x  1 1 − Δ k Δ + k Δ dx = ln + ln sin x cos x 2 1+Δ 2 Δ − k  Δ dx = F (x, k) − E (x, k) + Δ tan x cos2 x   k Δ + k sin x Δ dx = Δ tan x dx = −Δ + ln cos x 2 Δ − k   1 1−Δ cos x Δ dx = Δ cot x dx = Δ + ln sin x 2 1+Δ  2 Δ + cos x Δ cos x k Δ dx ln =− + 3 2 4 Δ − cos x sin x 2 sin x  1 + k 2 Δ − k  sin x −Δ Δ dx − = ln 2 sin x 2k  Δ + k  sin x sin x cos x  Δ 1 Δ + cos x Δ dx = + ln 2 sin x cos x cos x 2 Δ − cos x  Δ sin x 1 Δ dx Δ + k  sin x = +  ln 3 2 cos x 2 cos x 4k Δ − k  sin x  Δ Δ sin x dx = − k ln (k cos x + Δ) cos2 x cos x  Δ Δ cos x dx − k arcsin (k sin x) =− sin x sin2 x  Δ sin2 x dx Δ sin x 2k 2 − 1 k  Δ + k  sin x =− + arcsin (k sin x) + ln cos x 2 2k 2 Δ − k  sin x  Δ cos x k 2 + 1 1 Δ + cos x Δ cos2 x dx = + ln (k cos x + Δ) + ln sin x 2 2k 2 Δ − cos x    1, Δ dx 2 −Δ cot3 x + k 2 − 3 Δ cot x + 2k  F (x, k) + k 2 − 2 E (x, k) = 4 3 sin x 

30.

31.

32.12 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47.

3

2

2.583

2.583

Elliptic and pseudo-elliptic integrals



48. 49. 50. 51.12 52. 53. 54. 55. 56. 57. 58.12 59. 60. 61. 62. 63. 64. 65. 66.

Δ k Δ + k Δ dx k2 − 2 1 + Δ ln ln = − + + 2 Δ − k 4 1−Δ sin3 x cos x 2 sin2 x    2 1 1 + k Δ dx = tan x − cot x Δ + 2 F (x, k) − E (x, k) 2 2 2 sin x cos2 x k k  Δ 1 1 + Δ 2 − k2 Δ + k Δ dx = − ln + ln 3 2 sin x cos x 2 cos x 2 1 − Δ 4k  Δ − k  ,   2  2 1 Δ dx 2 3 2 = k tan x − 2k − 3 tan x Δ + 2k F (x, k) + k − 2 E (x, k) cos4 x 3k  2  Δ k2 sin x Δ + k Δ dx = + ln cos3 x 2 cos2 x 4k  Δ − k   Δ k2 1 + Δ cos x ln Δ dx = − + 4 1−Δ sin3 x 2 sin2 x   2 sin x Δ dx = tan2 xΔ dx = Δ tan x + F (x, k) − 2 E (x, k) cos2 x   cos2 x 2 cot2 xΔ dx = −Δ cot x + k  F (x, k) − 2 E (x, k) 2 Δ dx = sin x  k 2 sin2 x + 3k 2 − 1 sin3 x k Δ + k Δ dx = − Δ + ln 2 cos x 3k 2 Δ − k  k 2 sin2 x − 3k 2 − 1 cos3 x 1 1−Δ Δ dx = − Δ + ln sin x 3k 2 2 1+Δ  2  2  2 k − 3 sin x + 2 k k2 + 3 Δ + cos x Δ dx ln = Δ cos x + 16 Δ − cos x sin5 x 8 sin4 x   3 − k 2 sin2 x + 1 k  Δ − k  sin x Δ dx =− Δ − ln 4 3 2 Δ + k  sin x sin x cos x 3 sin x  3 sin2 x − 1 k 2 − 3 Δ − cos x Δ dx ln = Δ + 4 Δ + cos x sin3 x cos2 x 2 sin2 x cos x  2 2k 2 − 3 Δ + k  sin x 3 sin x − 2 Δ dx Δ− = ln 2 2 3 2 sin x cos x 4k  Δ − k  sin x sin x cos x  2 2  2k − 3 sin x − 3k 2 + 4 1 Δ + cos x Δ dx = Δ + ln 2 4  3 sin x cos x 2 Δ − cos x 3k cos x  2 2  2 2k − 3 sin x − 4k + 5 4k 2 − 3 Δ + k  sin x Δ dx = sin xΔ − ln 2 cos5 x Δ − k  sin x 8k  cos4 x 16k  3   2 2 2 4 2 − 2k + 1 k sin x + 3k − k + 1 sin x Δ dx = Δ 4 cos x 3k  2 cos3 x  Δ3 cos x 4 Δ dx = − sin x 3 sin3 x  sin x 2k 2 − 1 Δ + k  sin x sin2 x Δ dx = Δ + − k arcsin (k sin x) ln cos3 x 2 cos2 x 4k  Δ − k  sin x

189

190

Trigonometric Functions

 67.  68.  69.  70.

 71.

2.584 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.



cos x k 2 + 1 Δ + cos x cos2 x ln − k ln (k cos x + Δ) Δ dx = − Δ − 4 Δ − cos x sin3 x 2 sin2 x sin3 x sin2 x − 3 3k 2 − 1 Δ dx = − Δ− ln (k cos x + Δ) 2 cos x 2 cos x 2k cos3 x 2k 2 + 1 sin2 x + 2 Δ − arcsin (k sin x) Δ dx = − 2 sin x 2k sin2 x sin4 x 2k 2 sin2 x + 4k 2 − 1 Δ dx = − sin xΔ cos x 8k 2 4 2 k  Δ + k  sin x 8k − 4k − 1 ln arcsin (k sin x) + + 8k 3 2 Δ − k  sin x −2k 2 sin2 x + 5k 2 + 1 cos4 x Δ dx = cos xΔ sin x 8k 2 4 1 Δ + cos x 3k + 6k 2 − 1 + ln (k cos x + Δ) + ln 2 Δ − cos x 8k 3

dx = F (x, k) Δ  1 Δ − k cos x 1 sin x dx = ln = − ln (k cos x + Δ) Δ 2k Δ + k cos x k  1 1 k sin x cos x dx = arcsin (k sin x) = arctan Δ k k Δ  2 1 1 sin x dx = 2 F (x, k) − 2 E (x, k) Δ k k  Δ sin x cos x dx =− 2 Δ k  2 cos2 x dx k 1 = 2 E (x, k) − 2 F (x, k) Δ k k  3 cos xΔ 1 + k 2 sin x dx = − ln (k cos x + Δ) Δ 2k 2 2k 3  sin xΔ arcsin (k sin x) sin2 x cos x dx =− + Δ 2k 2 2k 3  2 cos xΔ sin x cos2 x dx k =− + ln (k cos x + Δ) Δ 2k 2 2k 3  sin xΔ 2k 2 − 1 cos3 x dx = + arcsin (k sin x) Δ 2k 2 2k 3   2 1 + k2 sin x cos xΔ 2 + k 2 sin4 x dx = + F (x, k) − E (x, k) Δ 3k 2 3k 4 3k 4  1  sin3 x cos x dx = − 4 2 + k 2 sin2 x Δ Δ 3k

2.584

2.584

Elliptic and pseudo-elliptic integrals



13. 14. 15. 16. 17. 18.12 19. 20. 21. 22.

23. 24.12

25. 26.

27. 28.

sin x cos xΔ 2 − k 2 sin2 x cos2 x dx 2k 2 − 2 =− + E (x, k) + F (x, k) Δ 3k 2 3k 4 3k 4   1  sin x cos3 x dx 2 = − 4 k 2 cos2 x − 2k  Δ Δ 3k  4 cos x dx 3k 4 − 5k 2 + 2 sin x cos xΔ 4k 2 − 2 + E (x, k) + F (x, k) = Δ 3k 2 3k 4 3k 4  2k 2 sin2 x + 3k 2 + 3 sin5 x dx 3 + 2k 2 + 3k 4 = cos xΔ − ln (k cos x + Δ) 4 Δ 8k 8k 5  2k 2 sin2 x + 3 sin4 x cos x dx 3 =− sin xΔ + 5 arcsin (k sin x) Δ 8k 4 8k  2k 2 cos2 x − k 2 − 3 sin3 x cos2 x dx k 4 + 2k 2 − 3 = cos xΔ − ln (k cos x + Δ) 4 Δ 8k 8k 5  2k 2 cos2 x + 2k 2 − 3 sin2 x cos3 x dx 4k 2 − 3 =− sin xΔ + arcsin (k sin x) Δ 8k 4 8k 5  3 − 5k 2 + 2k 2 sin2 x sin x cos4 x dx 3k 4 − 6k 2 + 3 = cos xΔ − ln (k cos x + Δ) 4 Δ 8k 8k 5  2k 2 cos2 x + 6k 2 − 3 cos5 x dx 8k 4 − 8k 2 + 3 = sin xΔ + arcsin (k sin x) Δ 8k 4 8k 5  sin6 x dx 3k 2 sin2 x + 4k 2 + 4 = sin x cos xΔ Δ 15k 4 4 2 8k 4 + 7k 2 + 8 4k + 3k + 8 F (x, k) − E (x, k) + 15k 6 15k 6  3k 4 sin4 x + 4k 2 sin2 x + 8 sin5 x cos x dx =− Δ Δ 15k 6  sin4 x cos x2 dx 3k 2 cos2 x − 2k 2 − 4 = sin x cos xΔ Δ 15k 4 4 2 2k 4 + 3k 2 − 8 k + 7k − 8 F (x, k) − E (x, k) + 6 15k 15k 6   3k 4 sin4 x − 5k 4 − 4k 2 sin2 x − 10k 2 + 8 sin3 x cos3 x dx = Δ Δ 15k 6  3k 2 cos2 x + 3k 2 − 4 sin2 x cos4 x dx =− sin x cos xΔ Δ 15k 4 4 2 3k 4 − 13k 2 + 8 9k − 17k + 8 F (x, k) − E (x, k) + 15k 6 15k 6  2 −3k 4 cos4 x + 4k 2 k  cos2 x − 8k 4 + 16k 2 − 8 sin x cos5 x dx = Δ Δ 15k 6  cos6 x dx 3k 2 cos2 x + 8k 2 − 4 = sin x cos xΔ Δ 15k 4 6 4 2 23k 4 − 23k 2 + 8 15k − 34k + 27k − 8 F (x, k) + E (x, k) + 15k 6 15k 6

191

192

Trigonometric Functions

 29.

 30.  31.

 32.

 33.12

 34.

 35.

 36.



37. 38. 39. 40.11 41. 42.

 sin7 x dx 8k 4 sin4 x + 10k 2 k 2 + 1 sin2 x + 15k 4 + 14k 2 + 15 = cos xΔ Δ  4  2 48k 6 2 5k − 2k + 5 k + 1 ln (k cos x + Δ) − 16k 7 8k 4 sin4 x + 10k 2 sin2 x + 15 sin6 x cos x dx 5 =− sin xΔ + arcsin (k sin x) Δ 48k 6 16k 7  sin5 x cos2 x dx −8k 4 sin4 x + 2k 2 k 2 − 5 sin2 x + 3k 4 + 4k 2 − 15 = cos xΔ Δ 48k 6 6 4 2 k + k + 3k − 5 ln (k cos x + Δ) − 16k 7  sin4 x cos3 x dx 8k 4 sin4 x − 2k 2 6k 2 − 5 sin2 x − 18k 2 + 15 = sin xΔ Δ 48k 6 2 6k − 5 arcsin (k sin x) + 16k 7  sin3 x cos4 x dx 8k 4 sin4 x − 2k 2 7k 2 − 5 sin2 x + 3k 4 − 22k 2 + 15 = cos xΔ Δ 48k 6 k 6 + 3k 4 − 9k 2 + 5 ln (k cos x + Δ) − 16k 7  sin2 x cos5 x dx −8k 4 sin4 x + 2k 2 12k 2 − 5 sin2 x − 24k 4 + 36k 2 − 15 = sin xΔ Δ 48k 6 4 2 8k − 12k + 5 arcsin (k sin x) + 16k 7  sin x cos6 x dx −8k 4 sin4 x + 2k 2 13k 2 − 5 sin2 x − 33k 4 + 40k 2 − 15 = cos xΔ Δ 48k 6 6 5k + ln (k cos x + Δ) 16k 7  cos7 x dx 8k 4 sin4 x − 2k 2 18k 2 − 5 sin2 x + 72k 4 − 54k 2 + 15 = sin xΔ Δ 48k 6 16k 6 − 24k 4 + 18k 2 − 5 arcsin (k sin x) + 16k 7

1 k 2 sin x cos x dx =  2 E (x, k) −  2 3 Δ Δ k k  sin x dx cos x =− 2 Δ3 k Δ  sin x cos x dx = Δ3 Δ  2 sin x dx 1 1 1 sin x cos x = 2 E (x, k) − 2 F (x, k) − 2 3  2 Δ k Δ k k k  sin x cos x dx 1 = 2 Δ3 k Δ  2 1 1 sin x cos x cos x dx = 2 F (x, k) − 2 E (x, k) + Δ3 k k Δ

2.584

2.584

Elliptic and pseudo-elliptic integrals

 43.  44.  45.  46.  47.  48.  49.  50.  51.  52.9  53.  54.  55.  56. 

57.

58.

59. 60.

1 sin3 x dx cos x + 3 ln (k cos x + Δ) =− 2 3 2  Δ k k Δ k 1 sin2 x cos x dx sin x = 2 − 3 arcsin (k sin x) Δ3 k Δ k 1 sin x cos2 x dx cos x = 2 − 3 ln (k cos x + Δ) 3 Δ k Δ k 1 k  sin x cos3 x dx + 3 arcsin (k sin x) = − Δ3 k2 Δ k 2

k + 1 2 sin x cos x sin4 x dx =  2 4 E (x, k) − 4 F (x, k) − 2  2 3 Δ k k k k k Δ 2

sin3 x cos x dx 2 − k 2 sin2 x = Δ3 k4 Δ 2 − k2 2 sin x cos x sin2 x cos2 x dx = F (x, k) − 4 E (x, k) + 3 Δ k4 k k2 Δ k 2 sin2 x + k 2 − 2 sin x cos3 x dx = Δ3 k4 Δ cos4 x dx k + 1 2k  k  sin x cos x = E (x, k) − 4 F (x, k) − 3 4 Δ k k k2 Δ 2

2

2

sin5 x dx k2 + 3 k 2 k  sin2 x + k 2 − 3 cos x + = ln (k cos x + Δ) 2 3 4  Δ 2k 5 2k k Δ 2

3 sin4 x cos x dx −k 2 sin2 x + 3 sin x − 5 arcsin (k sin x) = Δ3 2k 4 Δ 2k −k 2 sin2 x + 3 k2 − 3 sin3 x cos2 x dx = cos x + ln (k cos x + Δ) 4 Δ 2k Δ 2k 5 2k 2 − 3 sin2 x cos3 x dx k 2 sin2 x + 2k 2 − 3 sin x − = arcsin (k sin x) Δ3 2k 4 Δ 2k 5 3k  k 2 sin2 x + 2k 2 − 3 sin x cos4 x dx cos x + = ln (k cos x + Δ) 3 4 Δ 2k Δ 2k 5 2

4k 2 − 3 cos5 x dx −k 2 sin2 x + 2k 4 − 4k 2 + 3 sin x + = arcsin (k sin x) Δ3 2k 4 Δ 2k 5   2 2  1 dx −k 2 sin x cos x 2k k + 1 sin x cos x − = − F (x, k) 2 4 5  3  Δ 3k Δ 3k  2 3k Δ  2 2 k + 1 E (x, k) + 3k  4  sin x dx 2k 2 sin2 x + k 2 − 3 = cos x Δ5 3k  4 Δ3  cos x dx −2k 2 sin2 x + 3 = sin x Δ5 3Δ3

193

194

Trigonometric Functions



61.

62. 63.12 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79.

sin2 x dx k 2 + 1 1 = E (x, k) − F (x, k) 4 2 5  2  Δ 3k k 2 3k k 2 2 2 k k + 1 sin x − 2 + sin x cos x 3k  4 Δ3  1 sin x cos x dx = 2 3 Δ5 3k Δ   k 2 2k 2 − 1 sin2 x − 3k 2 + 2 cos2 x dx 1 2k 2 − 1 sin x cos x = 2 F (x, k) + E (x, k) + Δ5 3k 3k 2 k  2 3k  2 Δ  2  3k − 1 sin2 x − 2 sin3 x dx = cos x Δ5 3k  4 Δ3  sin2 x cos x sin3 x dx = Δ5 3Δ3  sin x cos2 x cos3 x dx = −  2 3 5 Δ 3k Δ  2  − 2k + 1 sin2 x + 3 cos3 x dx = sin x Δ5 3Δ3  1 Δ + cos x dx = − ln Δ sin x 2 Δ − cos x  1 dx Δ − k  sin x = −  ln Δ cos x 2k Δ + k  sin x   1 + cot2 x dx dx = F (x, k) − E (x, k) − Δ cot x = Δ Δ sin2 x   dx 1 1−Δ 1 dx Δ + k = (tan x + cot x) = ln +  ln Δ sin x cos x Δ 2 1 + Δ 2k Δ − k    dx 1 dx 1 = 1 + tan2 x = F (x, k) − 2 E (x, k) + 2 Δ tan x 2  Δ cos x Δ k k   dx 1 Δ + k sin x dx = tan x =  ln cos x Δ Δ 2k Δ − k   dx 1 1−Δ cos x dx = cot x = ln sin x Δ Δ 2 1+Δ  Δ cos x 1 + k 2 Δ + cos x dx ln =− − 3 4 Δ − cos x Δ sin x 2 sin2 x  1 dx Δ Δ − k  sin x − = − ln sin x 2k  Δ + k  sin x Δ sin2 x cos x  Δ 1 Δ − cos x dx = 2 + ln 2  Δ sin x cos x k cos x 2 Δ + cos x  Δ sin x 2k 2 − 1 Δ − k  sin x dx = + ln 2 3  2 Δ cos x Δ + k  sin x 2k cos x 4k  3  Δ sin x dx = 2 cos2 x Δ k  cos x

2.584

2.584

Elliptic and pseudo-elliptic integrals

 80.  81.  82.  83.  84.

 85.

 86.12

 87.

 88. 89. 90. 91. 92. 93. 94.12

Δ cos x dx =− sin x sin2 x Δ 1 Δ + k  sin x 1 sin2 x dx =  ln − arcsin (k sin x) cos x Δ 2k Δ − k  sin x k cos2 x dx 1 Δ + cos x 1 = ln + ln (k cos x + Δ) sin x Δ 2 Δ − cos x k     1 dx −Δ cot3 x − Δ 2k 2 + 3 cot x + k 2 + 2 F (x, k) − 2 k 2 + 1 E (x, k) = 4 3 Δ sin x   dx dx = tan x + 2 cot x + cot3 x 3 Δ Δ sin x cos x 1 Δ + k k2 + 2 1 + Δ Δ ln +  ln − =− 2 Δ − k 4 1−Δ 2 sin x 2k   2 dx dx = tan x + 2 + cot2 x 2 2 Δ Δ sin x cos x   k2 − 2 tan x − cot x Δ + E (x, k) + 2 F (x, k) = k 2 k 2   dx dx = cot x + 2 tan x + tan3 x 3 Δ sin x cos x Δ 1 1 + Δ 2 − 3k 2 Δ + k  Δ + − ln ln = Δ − k 2k  2 cos2 x 2 1 − Δ 4k  3 ⎧  1 ⎨ dx 5k 2 − 3 = Δ tan x − 3k 2 − 2 F (x, k) Δ tan3 x − 2 2 4   Δ cos x 3k ⎩ k ⎫  2 ⎬ 2 2k − 1 E (x, k) + 2 ⎭ k 

 dx Δ k2 Δ + k tan x 1 + tan2 x = 2 − ln Δ 2k cos2 x 4k  3 Δ − k   Δ k2 1 + Δ cos x dx =− ln − 3 2 4 1−Δ sin x Δ 2 sin x   2 2 tan x Δ 1 sin x dx = dx = 2 tan x − 2 E (x, k)  cos2 x Δ Δ k k   2 2 cot x cos x dx = dx = −Δ cot x − E (x, k) 2 Δ sin x Δ  sin3 x dx Δ 1 Δ + k = 2 +  ln cos x Δ k 2k Δ − k  Δ cos3 x dx 1 1+Δ = 2 − ln sin x Δ k 2 1−Δ    3 1 + k 2 sin2 x + 2 3k 4 + 2k 2 + 3 Δ + cos x dx ln = − Δ cos x + 5 4 16 Δ − cos x Δ sin x 8 sin x sin x dx = cos3 x Δ

195

196

Trigonometric Functions

 95.  96.  97.  98.  99.  100.12  101.  102.  12

103.



104. 105. 106. 107. 2.585 1.12

2.585

 3 + 2k 2 sin2 x + 1 1 dx Δ − k  sin x =− Δ −  ln 4 3 2k Δ + k  sin x Δ sin x cos x 3 sin x  2 3 − k 2 sin2 x − k  dx k 2 + 3 Δ − cos x ln = Δ + 4 Δ + cos x Δ sin3 x cos2 x 2k  2 sin2 x cos x  2 2 2 2 3 − 2k sin x − 2k dx 4k − 3 Δ + k  sin x = Δ − ln Δ − k  sin x Δ sin2 x cos3 x 2k  2 sin x cos2 x 4k  3  2 2 2 5k − 3 sin x − 6k + 4 1 Δ + cos x dx = Δ − ln 4 Δ sin x cos x 2 Δ − cos x 3k  4 cos3 x  2 2 2 3 2k − 1 sin x − 8k + 5 8k 4 − 8k 2 + 3 Δ + k  sin x dx = Δ sin x + ln Δ cos5 x Δ − k  sin x 8k  4 cos4 x 16k  5 2k 2 cos2 x − k  sin x dx =− Δ 4 cos x Δ 3k  4 cos3 x 2

2k 2 sin2 x + 1 cos x dx = − Δ sin4 x Δ 3 sin3 x Δ sin x sin2 x dx 1 Δ + k  sin x = 2 −  3 ln 3 2 cos x Δ Δ − k  sin x 2k cos x 4k Δ cos x k  Δ + cos x cos2 x dx = − ln + 3 2 Δ 4 Δ − cos x sin x 2 sin x 2

Δ 1 sin3 x dx = 2 + ln (k cos x + Δ) cos2 x Δ k  cos x k  −Δ 1 cos3 x dx = − arcsin (k sin x) sin x k sin2 x Δ  sin4 x dx 1 Δ + k  sin x 2k 2 + 1 Δ sin x + ln arcsin (k sin x) = − cos x Δ 2k 2 2k  Δ − k  sin x 2k 3  Δ cos x 1 Δ + cos x 3k 2 − 1 cos4 x dx = + + ln ln (k cos x + Δ) sin x Δ 2k 2 2 Δ − cos x 2k 3 

(a + sin x)p+3 dx Δ  1 p = (a + sin x) cos xΔ (p + 2)k 2   p+2  (a + sin x)p+1 dx dx (a + sin x) + (p + 1) 1 + k 2 − 6a2 k 2 +2(2p + 3)ak 2 Δ  Δ (a + sin x)p dx  2 2 2 −a(2p + 1) 1 + k − 2a k Δ   p−1   dx (a + sin x) 2 2 2 1−a k −p 1−a Δ   1 p = −2, a = ±1, a =  ± k For p = n a natural number, this integral can be reduced to the following three integrals:

2.586

Elliptic and pseudo-elliptic integrals

 2. 

3. 4.12

5. 2.586 1.

2.

3.

197

1 Δ − k cos x a + sin x dx = a F (x, k) + ln Δ 2k Δ + k cos x 2

1 + k 2 a2 1 a Δ − k cos x (a + sin x) dx = F (x, k) − 2 E (x, k) + ln Δ k2 k k Δ + k cos x     1 1 dx sin x dx , = Π x, − 2 , k −  2 (a + sin x) Δ a a a − sin2 x Δ where √ √  1 − a2 Δ − 1 − k 2 a2 cos x −1 sin x dx  =  √ ln √ a2 − sin2 x Δ 1 − a2 Δ + 1 − k 2 a2 cos x 2 (1 − a2 ) (1 − a2 k 2 )  cos xΔ 1 dx − = n (a + sin x) Δ (n − 1) (1 − a2 ) (1 − a2 k 2 ) (a + sin x)n−1   dx −(2n − 3) 1 + k 2 − 2a2 k 2 a (a + sin x)n−1 Δ   dx −(n − 2) 6a2 k 2 − k 2 − 1 (a + sin x)n−2 Δ    dx dx 2 2 − (10 − 4n)ak − (n − 3)k n−3 n−4 (a + sin x) Δ (a + sin x) Δ   1 n = 1, a = ±1, a = ± k This integral can be reduced to the integrals:     dx cos xΔ dx 1 2 2 2 − − a 1 + k − 2a k = 2 2 2 2 a + sin x (a + sin x) Δ (a + sin x) Δ (1 − a ) (1 − a k )    2 dx (a + sin x) dx (a + sin x) + k2 − 2ak 2 Δ Δ (see 2.585 2, 3, 4) ⎡    dx dx 1 ⎣ − cos xΔ − 3a 1 + k 2 − 2a2 k 2 = 3 2 2 (a + sin x) Δ 2 (1 − a2 ) (1 − a2 k 2 ) (a + sin x) (a + sin x) Δ ⎤   2 2 dx + 2ak 2 F (x, k)⎦ − 6a k − k 2 − 1 (a + sin x) Δ 

(see 2.585 4 and 2.586 2)

4.

For a = ±1, we have:     1 cos xΔ dx dx 2 = ∓ n n + (n − 1) 1 − 5k 2 n−1  (1 ± sin x) Δ (2n − 1)k (1 ± sin x) (1 ± sin x) Δ    dx dx + 2(2n − 3)k 2 − (n − 2)k 2 n−2 n−3 (1 ± sin x) Δ (1 ± sin x) Δ GU (241)(6a)

This integral can be reduced to the integrals

198

Trigonometric Functions

2.587



∓ cos xΔ 1 dx = 2 + F (x, k) −  2 E (x, k) (1 ± sin x) Δ k (1 ± sin x) k    2 1 − 5k 2 cos xΔ k  cos xΔ dx 1 ∓ = 4 2 2 ∓ 1 ± sin x (1 ± sin x) Δ 3k  (1 ± sin x)   2 + 1 − 3k 2 k  F (x, k) − 1 − 5k 2 E (x, k)

5. 6.

GU (241)(6c)

GU (241)(6b)

1 For a = ± , we have k     1 dx k cos xΔ dx 2 = ± + (n − 1) 5 − k n n n−1 (1 ± k sin x) Δ (2n − 1)k  2 (1 ± k sin x) (1 ± k sin x) Δ   dx dx + (n − 2) − 2(2n − 3) n−2 n−3 (1 ± k sin x) Δ (1 ± k sin x) Δ

7.

GU (241)(7a)

8. 9.

This integral can be reduced to the integrals  k cos xΔ 1 dx =± 2 + 2 E (x, k)  (1 ± k sin x) Δ k (1 ± k sin x) k     k 5 − k 2 cos xΔ kk  2 cos xΔ dx 1 ± = 4 2 2 ± 1 ± k sin x (1 ± k sin x) Δ 3k  (1 ± k sin x)   2 − 2k  F (x, k) + 5 − k 2 E (x, k)

GU (241)(7b)

GU(241)(7c)

2.587 1.

12

2. 3. 4.

  p+2 dx 1 (b + cos x) p 2 dx = (b + cos x) sin xΔ + 2(2p + 3)bk (p + 2)k 2 Δ   (b + cos x)p+1 dx  2 2 2 2 −(p + 1) k − k + 6b k Δ   (b + cos x)p dx  2 2 2 2 +(2p + 1)b k − k + b k Δ    (b + cos x)p−1 dx   2 + p 1 − b2 k  + k 2 b2 Δ   ik  p = −2, b = ±1, b = k For p = n a natural number, this integral can be reduced to the following three integrals:  1 b + cos x dx = b F (x, k) + arcsin (k sin x) Δ k  2 2 2 2 (b + cos x) b k −k 1 2b dx = arcsin (k sin x) F (x, k) + 2 E (x, k) + Δ k2 k k     b 1 cos x dx dx , = 2 Π x, 2 ,k +  (b + cos x) Δ b −1 b −1 1 − b2 − sin2 x Δ where 

p+3

(b + cos x) Δ

2.589

Elliptic and pseudo-elliptic integrals

 5. 2.588 1.

2.

 √ 1 − b2 Δ + k k  2 + k 2 b2 sin x 1 cos x dx  = (  ln √  1 − b2 − sin2 x Δ 1 − b2 Δ − k k  2 + k 2 b2 sin x 2 (1 − b2 ) k  2 + k 2 b2

 2 dx 1 −k  sin xΔ  2 = n −1 (b + cos x) Δ (n − 1) (1 − b2 ) k  + b2 k 2 (b + cos x)   dx −(2n − 3) 1 − 2k 2 + 2b2 k 2 b (b + cos x)n−1 Δ   dx −(n − 2) 2k 2 − 1 − 6b2 k 2 (b + cos x)n−2 Δ    dx dx 2 2 − (4n − 10)bk + (n − 3)k n−3 n−4 (b + cos x) Δ (b + cos x) Δ   ik  n = 1, b = ±1, b = ± k This integral can be reduced to the following integrals: ⎡   2  dx dx 1 ⎣ −k sin xΔ − 1 − 2k 2 + 2b2 k 2 b  = 2 2 2  2 2 b + cos x (b + cos x) Δ (b + cos x) Δ (1 − b ) k + b k ⎤   2 b + cos x (b + cos x) dx − k 2 dx⎦ + 2bk 2 Δ Δ



 3.

⎡

dx 3

(b + cos x) Δ

=

(see 2.587 2, 3, 4)

−k  sin xΔ 1 ⎣  2 2 2 (1 − b2 ) k  + b2 k 2 (b + cos x)   dx −3b 1 − 2k 2 + 2k 2 b2 2 (b + cos x) Δ ⎤   2 dx − 2bk 2 F (x, k)⎦ − 2k − 1 − 6b2 k 2 (b + cos x) Δ 2

(see 2.588 2 and 2.587 4) 2.589 1.



p+3

(c + tan x) Δ

⎡

dx

=

 p p+2 dx 1 ⎣ (c + tan x) Δ + 2(2n + 3)ck  2 (c + tan x) 2 2 cos x Δ (p + 2)k    (c + tan x)p+1 dx  2 2 2 −(p + 1) 1 + k + 6c k Δ   (c + tan x)p dx  2 2 +(2p + 1)c 1 + k  + 2c2 k  Δ ⎤  p−1    dx ⎦ (c + tan x) 2 − p 1 + c2 1 + k  c2 Δ [p = −2]

For p = n a natural number, this integral can be reduced to the following three integrals:

199

200

Trigonometric Functions

 2.  3. 4.

5. 2.591

where  √  1 + c2 k  2 + 1 + c2 Δ 1 sin x cos x dx  = ( √ ln   c2 − (1 + c2 ) sin2 x Δ 1 + c2 k  2 − 1 + c2 Δ 2 (1 + c2 ) 1 + c2 k  2 

⎡ 1 dx Δ ⎣−  = n n−1 (c + tan x) Δ (n − 1) (1 + c2 ) 1 + k  2 c2 (c + tan x) cos2 x   dx 2 2 +(2n − 3)c 1 + k  + 2c2 k  (c + tan x)n−1 Δ   dx 2 2 −(n − 2) 1 + k  + 6c2 k  (c + tan x)n−2 Δ ⎤   dx dx 2 2 ⎦ + (4n − 10)ck  − (n − 3)k  n−3 n−4 (c + tan x) Δ (c + tan x) Δ

This integral can be reduced to the integrals: ⎡  −Δ 1 dx ⎣  = 2 2 2 2  (c + tan x) cos2 x (c + tan x) Δ (1 + c ) 1 + k c    dx 2 2 +c 1 + k  + 2c2 k  (c + tan x) Δ ⎤   2 c + tan x (c + tan x) 2 2 dx + k  dx⎦ − 2ck  Δ Δ 

3.

1 c + tan x Δ + k dx = c F (x, k) +  ln Δ 2k Δ − k

1 (c + tan x)2 1 c Δ + k dx = 2 tan xΔ + c2 F (x, k) − 2 E (x, k) +  ln Δ k Δ − k k k    c 1 + c2 dx 1 = Π x, − F (x, k) + ,k (c + tan x) Δ 1 + c2 c (1 + c2 ) c2 sin x cos x dx , −  2 c − (1 + c2 ) sin2 x Δ

1.

2.

2.591

⎡ dx 3

(c + tan x) Δ

=

(see 2.589 2, 3, 4)

−Δ 1 ⎣  2 2 2  2 (1 + 1+k c (c + tan x) cos2 x    dx 2 2 +3c 1 + k  + 2c2 k  (c + tan x)2 Δ ⎤    dx 2 2 2 + 2ck  F (x, k)⎦ − 1 + k  + 6c2 k  (c + tan x) Δ c2 )

(see 2.591 2 and 2.589 4)

2.593

2.592 1.

Elliptic and pseudo-elliptic integrals

201

 

n a + sin2 x dx Pn = Δ The recursion formula Pn+2 =





n  a + sin2 x sin x cos xΔ + (2n + 2) 1 + k 2 + 3ak 2 Pn+1     − (2n + 1) 1 + 2a 1 + k 2 + 3a2 k 2 Pn + 2na(1 + a) 1 + k 2 a Pn−1

1 (2n + 3)k 2

reduces this integral (for n an integer) to the integrals 2.

P1

3.

P0

4.

P−1 =

5. 6.

(see 2.584 1 and 2.584 4) 

  1 1 dx  = Π x, , k a a a + sin2 x Δ

For a = 0  dx sin2 xΔ  dx Tn =  n h + g sin2 x Δ

(see 2.584 1)

(see 2.584 70)

can be calculated by means of the recursion formula:    −g 2 sin x cos xΔ 1 2 2 Tn−3 =  n−1 + 2(n − 2) g 1 + k + 3hk Tn−2 2 (2n − 5)k 2 h + g sin x    − (2n − 3) g 2 + 2hg 1 + k 2 + 3h2 k 2 Tn−1 + 2(n − 1)h(g + h) g + hk 2 Tn 2.593 1.12

H (124)a



 

n b + cos2 x dx Δ The recursion formula ⎧ ⎨ n  1 Qn+2 = b + cos2 x Δ sin x cos x − (2n + 2) 1 − 2k 2 − 3bk 2 Qn+1 2 (2n + 3)k ⎩

Qn =

⎫ ⎬  2  2  + (2n + 1) k + 2b k − k 2 − 3b2 k 2 Qn − 2nb(1 − b) k 2 − k 2 b Qn−1 ⎭ reduces this integral (for n an integer) to the integrals: 2.

Q1

(see 2.584 1 and 2.584 6)

3.

Q0

(see 2.584 1)

202

Trigonometric Functions

 Q−1 =

4.

5. 2.594 1.

12

2.594

  1 1 dx = Π x, − ,k (b + cos2 x) Δ b+1 b+1

For b = 0  dx cos2 xΔ

(see 2.584 72)

H (123)

 

n c + tan2 x dx Rn = Δ The recursion formula ⎧ ⎨ c + tan2 x n tan xΔ  1 − (2n + 2) 1 + k 2 − 3ck 2 Rn+1 Rn+2 = 2 2 (2n + 3)k ⎩ cos x

⎫ ⎬   + (2n + 1) 1 − 2c 1 + k 2 + 3c2 k 2 Rn + 2nc(1 − c) 1 − k 2 c Rn−1 ⎭ 

reduces this integral (for n an integer) to the integrals: 2.

R1

3.

R0

4.

R−1 =

(see 2.584 1 and 2.584 90) 

(see 2.584 1)   1 1−c 1 dx  F (x, k) + Π x, ,k = c−1 c(1 − c) c c + tan2 x Δ

For c = 0 see 2.582 5.



2.595 Integrals of the type

  ( 2 2 R sin x, cos x, 1 − p sin x dx for p2 > 1.

Notation: α = arcsin (p sin x). Basic formulas    2 1 dx 1  p >1 BY (283.00) 1. = F α, 2 2 p p 1 − p sin x      ( p2 − 1 1 1 2 2 − F α, 2. 1 − p sin x dx = p E α, p p p  2 p >1 BY (283.03)    2  2 r 1 dx 1 p >1 3. BY (283.02) = Π α, 2 ,   2 2 2 2 p p p 1 − r sin x 1 − p sin x    ( To evaluate integrals of the form R sin x, cos x, 1 − p2 sin2 x dx for p2 > 1, we may use formulas 

2.583 and 2.584, making the following modifications in them. We replace (1)

k with p;

(2)

k



2

with 1 − p2 ;

2.597

(3) (4)

Elliptic and pseudo-elliptic integrals

203

1  1 F α, p ; p     p2 − 1 1 1 − F α, . E (x, k) with p E α, p p p F (x, k) with

For example (see 2.584 15): 2.596 ⎡     2 4 2 2 cos x dx sin x cos x 1 − p sin x 4p − 2 ⎣ 1  + 1.10 = p E α, 3p2 3p4 p 1 − p2 sin2 x ⎤     1 ⎦ 2 − 5p2 + 3p4 1 1 p2 − 1 F α, − · F α, + p p 3p4 p p      4p2 − 2 1 1 sin x cos x 1 − p2 sin2 x p2 − 1 + − F α, E α, = 3p2 3p3 p 3p3 p

2.

3.



p2 > 1



For example (see 2.583 36):     ( 1 1 1 − p2 sin2 x 2 2 dx = tan x 1 − p sin x + F α, cos2 x p  p     p2 − 1 1 1 − − p E α, F α, p p p      ( 1 1 − E α, + tan x 1 − p2 sin2 x = p F α, p p  2 p >1 For example (see 2.584 37):       1 p2 − 1 1 −1 dx ( p E α, − F α, = 3 p2 − 1 p p p 1 − p2 sin2 x − sin x cos x p2 1 · + F 2 2 2 p −1 1 − p sin x p

    1 p 1 α, − 2 E α, p p −1 p



p2 > 1

   ( 2.597 Integrals of the form R sin x, cos x, 1 + p2 sin2 x dx   1 + p2 sin x Notation: α = arcsin  1 + p2 sin2 x Basic formulas    p dx 1  F α,  1. = 1 + p2 1 + p2 1 + p2 sin2 x    (  sin x cos x p 2 − p2  2. 1 + p2 sin x dx = 1 + p2 E α,  1 + p2 1 + p2 sin2 x



p2 sin x cos x · = 1 − p2 1 − p2 sin2 x

BY (282.00)

BY (282.03)

204

Trigonometric Functions

   1 + p2 sin2 x dx 1 p 2 =  Π α, r ,  1 + (p2 − r2 p2 − r2 ) sin2 x 1 + p2 1 + p2    p cos x sin x dx 1  = − arcsin  2 2 p 1 + p2 1 + p sin x    ( cos x dx 1 2 2  = ln p sin x + 1 + p sin x p 1 + p2 sin2 x   1 + p2 sin2 x − cos x dx 1   ln = 2 sin x 1 + p2 sin2 x 1 + p2 sin2 x + cos x    1 + p2 sin2 x + 1 + p2 sin x dx 1  ln  =   2 1 + p2 cos x 1 + p2 sin2 x 1 + p2 sin2 x − 1 + p2 sin x    1 + p2 sin2 x + 1 + p2 tan x dx 1    ln =  2 1 + p2 1 + p2 sin2 x 1 + p2 sin2 x − 1 + p2   cot x dx 1 1 − 1 + p2 sin2 x   = ln 2 1 + 1 + p2 sin2 x 1 + p2 sin2 x

3.

4. 5.

6.

7.

8.

9.

2.598

BY (282.02)

    2.598 To calculate integrals of the form R sin x, cos x, 1 + p2 sin2 x dx, we may use formulas 2.583 and 2.584, making the following modifications in them. We replace (1)

k 2 with −p2 ;

(2)

k  with 1 + p2 ; 2

√1 2 1+p

 F α, √ p

 ; 2

(3)

F (x, k) with

(4)

  E (x, k) with 1 + p2 E α, √ p

1+p

1+p2

(5)

1 k

(6)

1 k

ln (k cos x + Δ) with

1 p



− p2 √sin x2cos x2 ; 1+p sin x

p cos x √ ; 1+p2

arcsin    arcsin (k sin x) with 1p ln p sin x + 1 + p2 sin2 x .

For example (see 2.584 90):  1.

⎡ ( 1 ⎣ tan x 1 + p2 sin2 x  = 2 1 + p2 sin2 x (1 + p ) ⎤    p sin x cos x ⎦ + p2  − 1 + p2 E α,  1 + p2 1 + p2 sin2 x   tan x p 1 + E α,  = − 2 2 1+p 1+p 1 + p2 sin2 x tan2 x dx

For example (see 2.584 37):

2.611

Elliptic and pseudo-elliptic integrals

 2.

(

dx 1+

p2

1

3 = 1 + p2 E sin x 2



p

205



α,  1 + p2

    2.599 Integrals of the form R sin x, cos x, a2 sin2 x − 1 dx   a cos x . Notation: α = arcsin  a2 − 1



a2 > 1



Basic formulas:  √    2 dx a2 − 1 1  1. a >1 = − F α, 2 2 a a a sin x − 1   √   √   2−1 2−1 1 a a − a E α, 2. a2 sin2 x − 1 dx = F α, a a a  2 a >1    √  r 2 a2 − 1 a2 − 1 1 dx Π α, 2 2 , = 3.   a (r2 − 1) a (r − 1) a 1 − r2 sin2 x a2 sin2 x − 1  2 a > 1, r2 > 1   2 α sin x dx  a >1 =− 4. 2 a a2 sin x − 1     2 cos x dx 1   5. = ln a sin x + a2 sin2 x − 1 a >1 2 2 a a sin x − 1   2 dx cos x  6. a >1 = − arctan  sin x a2 sin2 x − 1 a2 sin2 x − 1  √  a2 − 1 sin x + a2 sin2 x − 1 1 dx   = √ ln √ 7. 2 a2 − 2 cos x a2 sin2 x − 1 a2 − 1 sin x − a2 sin2 x − 1  2 a >1  √  a2 − 1 + a2 sin2 x − 1 1 tan x dx   = √ 8. ln √ 2 a2 − 1 a2 sin2 x − 1 a2 − 1 − a2 sin2 x − 1  2 a >1     2 1 cot x dx  a >1 = − arcsin 9. a sin x a2 sin2 x − 1     2.611 To calculate integrals of the type R sin x, cos x, a2 sin2 x − 1 dx for a2 > 1,

BY (285.00)a

BY (285.06)a

BY (285.02)a

we may use

formulas 2.583 and 2.584. In doing so, we should follow the procedure outlined below: (1)

In the right members of these formulas, the following functions should be replaced with integrals equal to them:

206

Trigonometric Functions

2.611

 F (x, k)

should be replaced with

E (x, k)

should be replaced with

1 − ln (k cos x + Δ) k

should be replaced with

1 arcsin (k sin x) k

should be replaced with

1 Δ − cos x ln 2 Δ + cos x

should be replaced with

1 Δ + k  sin x ln 2k  Δ − k  sin x

should be replaced with

1 Δ + k ln 2k  Δ − k 

should be replaced with

1 1−Δ ln 2 1+Δ

should be replaced with

dx Δ

 Δdx      



sin x dx Δ cos x dx Δ dx Δ sin x dx Δ cos x tan x dx Δ cot x dx Δ

a2 sin2 x − 1, k with a and k 

2

(2)

Then, on both sides of the equations, we should replace Δ with i with 1 − a2 .

(3)

Both sides of the resulting equations should be multiplied by i, as a result of which only real functions a2 > 1 should appear on both sides of the equations.

(4)

The integrals on the right sides of the equations should be replaced with their values found from formulas 2.599.

Examples: 1.

We rewrite equation 2.584 4 in the form     1 1 sin2 x dx   dx = 2 − 2 i a2 sin2 x − 1 dx, a i a2 sin2 x − 1 i a2 sin2 x − 1 a from which we get  √       2−1 a dx 1 1 sin2 x dx   = 2 + a2 sin2 x − 1 dx = − E α, a a a a2 sin2 x − 1 a2 sin2 x − 1  2 a >1

2.

We rewrite equation 2.584 58 as follows:    2a4 a2 − 2 sin2 x − 3a2 − 5 a2 dx ( ( 5 = − 3 sin x cos x a2 sin2 x − 1 a2 sin2 x − 1 i5 3 (1 − a2 )2 i3    dx 1 2a2 − 4  − − i a2 sin2 x − 1 dx 3 (1 − a2 ) i a2 sin2 x − 1 3 (1 − a2 )2

2.613

Elliptic and pseudo-elliptic integrals

207

from which we obtain    2a4 a2 − 2 sin2 x − 3a2 − 5 a2 1 dx ( ( = sin x cos x + 5 3 2 3 (1 − a2 )2 a a2 sin2 x − 1 a2 sin2 x − 1 3 (1 − a2 )     √  √  2  2 a2 − 1 a2 − 1 2 − 2a a − 2 E α, × a − 3 F α, a a  2 a >1 3.

We rewrite equation 2.584 71 in the form    cot x dx tan x dx dx    = + , 2 2 2 2 sin x cos xi a sin x − 1 i a sin x − 1 i a2 sin2 x − 1 from which we obtain  √    a2 − 1 + a2 sin2 x − 1 dx 1 1   = √ − arcsin ln √ a sin x 2 a2 − 1 sin x cos x a2 sin2 x − 1 a2 − 1 − a2 sin2 x − 1  2 a >1

√   2.612 Integrals of the form R x, cos x, 1 − k 2 cos2 x dx.  sin    π To find integrals of the form R sin x, cos x, 1 − k 2 cos2 x dx we make the substitution x = −y, 2 which yields      (   R sin x, cos x, 1 − k 2 cos2 x dx = − R cos y, sin y, 1 − k 2 sin2 y dy    ( 2 2 The integrals R cos y, sin y, 1 − k sin y dy are found from formulas 2.583 and 2.584. As a result of the use of these formulas (where it is assumed that the original integral can be reduced only to integrals of the first and second Legendre forms), when we replace the functions F (x, k) and E (x, k) with the corresponding integrals, we obtain an expression of the form   ( dy −g (cos y, sin y) − A  1 − k 2 sin2 y dy − B 1 − k 2 sin2 y Returning now to the original variable x, we obtain        dx R sin x, cos x, 1 − k 2 cos2 x dx = −g (sin x, cos x) − A √ 1 − k 2 cos2 x dx −B 1 − k 2 cos2 x The integrals appearing in this expression are found from the formulas      dx sin x √ 1. = F arcsin √ ,k 1 − k 2 cos2 x 1 − k 2 cos2 x       sin x k 2 sin x cos x 2 2 1 − k cos x dx = E arcsin √ ,k − √ 2. 1 − k 2 cos2 x 1 − k 2 cos2 x 

   [p > 1]. R sin x, cos x, 1 − p2 cos2 x dx     To find integrals of the type R sin x, cos x, 1 − p2 cos2 x dx, where [p > 1], we proceed as in

2.613 Integrals of the form

section 2.612. Here, we use the formulas

208

Trigonometric Functions

  1 1  [p > 1] = − F arcsin (p cos x) , p p 1 − p2 cos2 x       1 1 p2 − 1 2 2 F arcsin (p cos x) , − p E arcsin (p cos x) , 1 − p cos x dx = p p p 

1. 2.

2.614

dx



   R sin x, cos x, 1 + p2 cos2 x dx.     To find integrals of the type R sin x, cos x, 1 + p2 cos2 x dx, we need to make the substitution π x = − y. This yields 2      (   R sin x, cos x, 1 + p2 cos2 x dx = − R cos y, sin y, 1 + p2 sin2 y dy    ( To calculate the integrals − R cos y, sin y, 1 + p2 sin2 y dy, we need to use first what was said 2.614 Integrals of the form

in 2.598 and 2.612 and then, after returning to the variable x, the formulas    dx p 1  1. F x,  = 1 + p2 cos2 x 1 + p2 1 + p2      p 1 + p2 cos2 x dx = 1 + p2 E x,  2. 1 + p2 

   R sin x, cos x, a2 cos2 x − 1 dx [a > 1] .     To find integrals of the type R sin x, cos x, a2 cos2 x − 1 dx, we need to make the substitution π x = − y. This yields 2      (   2 2 2 2 R sin x, cos x, a cos x − 1 dx = − R cos y, sin y, a sin y − 1 dy     To calculate the integrals − R cos y, sin y, a2 sin2 y − 1 dy, we use what was said in 2.611 and then, after returning to the variable x, we use the formulas     √ 2  dx a −1 a sin x 1 √ 1. = F arcsin √ , a a a2 cos2 x − 1 a2 − 1 2.615 Integrals of the form

2.

[a > 1]     √ 2   a −1 a sin x 2 2 a cos x − 1 dx = a E arcsin √ , a a2 − 1     √ 2 a −1 a sin x 1 [a > 1] − F arcsin √ , a a a2 − 1

2.616

Elliptic and pseudo-elliptic integrals

209

 ( ( 2 2 2 2 2.616 Integrals of the form R sin x, cos x, 1 − p sin x, 1 − q sin x dx.   1 − p2 sin x . Notation: α = arcsin  1 − p2 sin2 x   )  dx q 2 − p2 1 ( 1. F α,  =  1 − p2 1 − p2 1 − p2 sin2 x 1 − q 2 sin2 x

π 0 < p2 < q 2 < 1, 0 < x ≤ 2 



11

BY (284.00)

2.

3.12

4.

5.

  tan2 x dx tan x 1 − q 2 sin2 x (   = (1 − q 2 ) 1 − p2 sin2 x ) 1 − p2 sin2 x 1 − q 2 sin2 x   q 2 − p2 1  − E α, 1 − p2 (1 − q 2 ) 1 − p2

π BY (284.07) 0 < p2 < q 2 < 1, 0 < x ≤ 2  tan4 x dx (  1 − p2 sin2 x 1 − q 2 sin2 x    )  )  2 − p2 2 − p2   q q 1 − 1 − q 2 F α, 2 2 − p2 − q 2 E α, = 3 × 2 1 − p2 1 − p2 3 (1 − q 2 ) (1 − p2 ) 2 )  2p2 + q 2 − 3 + sin2 x 4 − 3p2 − 2q 2 + p2 q 2 sin x 1 − q 2 sin2 x + cos3 x 1 − p2 sin2 x 3 (1 − p2 ) (1 − q 2 )2

π BY (284.07) 0 < p2 < q 2 < 1, 0 < x ≤ 2  sin2 x dx (  3 1 − p2 sin2 x 1 − q 2 sin2 x  )    )  1 − p2 q 2 − p2 q 2 − p2 1  E α, − = F α, (1 − q 2 ) (q 2 − p2 ) 1 − p2 1 − p2 (q 2 − p2 ) 1 − p2 sin x cos x ( −  1 − p2 sin2 x 1 − q 2 sin2 x (1 − q 2 )

π BY (284.06) 0 < p2 < q 2 < 1, 0 < x ≤ 2  cos2 x dx ( 3  1 − p2 sin2 x 1 − q 2 sin2 x    )  )  1 − p2 q 2 − p2 q 2 − p2 1 − q2  − = 2 E α, F α, q − p2 1 − p2 1 − p2 (q 2 − p2 ) 1 − p2 

π BY (284.05) 0 < p2 < q 2 < 1, 0 < x ≤ 2

210

Trigonometric Functions

2.617

 6.

7.

cos4 x dx 5  1 − p2 sin2 x 1 − q 2 sin2 x   3 ⎡   ) 2 2 2 − p2 1−p 2 + p2 − 3q 2 1 − q 2 q ⎣ F α, = 2 2 1 − p2 3 (q 2 − p2 ) (1 − p2 ) ⎤   )  1 − p2 sin x cos x 1 − q 2 sin2 x q 2 − p2 ⎦ 2q 2 − p2 − 1 ( +2 E α, + 3 1 − p2 1 − p2 1 − p2 sin2 x 3 (q 2 − p2 ) 

π 0 < p2 < q 2 < 1, 0 < x ≤ BY (284.05) 2 )   )  1 − q 2 sin2 x q 2 − p2 1 dx = E α, 1 − p2 1 − p2 sin2 x 1 − p2 sin2 x 1 − p2

π 0 < p2 < q 2 < 1, 0 < x ≤ 2 (

BY (284.01)

8.

.   )   / 2 2 − p2 / 1 − p2 sin2 x 1 − p q sin x cos x q 2 − p2 0 ( − dx = E α,  3 2 2 2 2 2 1 − q 1 − p 1 − q 2 1 − q sin x 1 − p sin2 x 1 − q 2 sin2 x

π . 0 < p2 < q 2 < 1, 0 < x ≤ 2 

9.

) dx 2 2 1 + (p r − p2 − r2 ) sin2 x



)

BY (284.04)



1 − p2 sin2 x q 2 − p2 1 2  = Π α, r , 1 − p2 1 − q 2 sin2 x 1 − p2

π . 0 < p2 < q 2 < 1, 0 < x ≤ 2 BY (284.02)

)√ ) √ b2 + c2 − b sin x − c cos x 2 b 2 + c2 √ √ 2.617 Notation: α = arcsin , r= . 2 b 2 + c2 a + b 2 + c2  dx √ 1. a + b sin x + c cos x 2 F (α, r) = − √ a + b 2 + c2    b b 2 2 0 < b + c < a, arcsin √ − π ≤ x < arcsin √ b 2 + c2 b 2 + c2 BY (294.00) √ 2 = −√ F (α, r) 4 2 b + c2      b b a 2 2 0 < |a| < b + c , arcsin √ ≤ x < arcsin √ − arccos − √ b 2 + c2 b 2 + c2 b 2 + c2 BY (293.00)

2.618

Elliptic and pseudo-elliptic integrals

 2.

√ 2b sin x dx 2c √ √ = −( {2 E (α, r) − F (α, r)} + 2 a + b sin x + c cos x b + c2 4 a + b sin x + c cos x (b2 + c2 )3      b b a 0 < |a| < b2 + c2 , arcsin √ ≤ x < arcsin √ − arccos − √ b 2 + c2 b 2 + c2 b 2 + c2 BY (293.05)

 3. 4.

211

√ (b cos x − c sin x) dx √ = 2 a + b sin x + c cos x a + b sin x + c cos x  √ 2 b + c2 + b sin x + c cos x √ dx a + b sin x + c cos x √  ( 2 + c2  2 a − b = −2 a + b2 + c2 E (α, r) +  F (α, r) √ a + b 2 + c2    b b 2 2 0 < b + c < a, arcsin √ − π ≤ x < arcsin √ b 2 + c2 b 2 + c2

BY (294.04)

√  4 = −2 2 b2 + c2 E (α, r)      b b a 2 2 0 < |a| < b + c , arcsin √ ≤ x < arcsin √ − arccos − √ b 2 + c2 b 2 + c2 b 2 + c2

BY (293.01)

 5.

√

a + b sin x + c cos x dx (  = −2 a + b2 + c2 E (α, r)   0 < b2 + c2 < a,

 b b arcsin √ − π ≤ x < arcsin √ b 2 + c2 b 2 + c2 BY (294.01) √ √ 2 + c2 − a √  2 b 4 √ = −2 2 b2 + c2 E (α, r) + F (α, r) 4 2 b + c2      b b −a ≤ x < arcsin √ 0 < |a| < b2 + c2 , arcsin √ − arccos √ b 2 + c2 b 2 + c2 b 2 + c2 BY (293.03)

      √ 1 R sin t, cos t, 1 − 2 sin2 t dt R sin ax, cos ax, cos 2ax dx = a where the substitution √ t = ax has been used. Notation: α = arcsin   2 sin ax  √ The integrals R sin ax, cos ax, cos 2ax dx are special cases of the integrals 2.595. for (p = 2). 

2.618 Integrals of the form

We give some formulas:    dx 1 1 √ 1. = √ F α, √ cos 2ax a 2 2    2 cos ax 1 1 √ 2. dx = √ E α, √ cos 2ax a 2 2

π 0 < ax ≤ 4

π 0 < ax ≤ 4

212

Trigonometric Functions

 3.

√   2 1 dx tan x √ √ E α, √ cos 2ax = − 2 a a cos ax cos 2ax 2

2.619

π 0 < ax ≤ 4 √    √    2 6 cos ax + 1 sin ax √ 1 1 dx 2 2 2 √ E α, √ F α, √ 4. = − − cos 2ax a 3a 3acos3 ax cos4 ax cos 2ax 2 2 π 0 0, x > 0] ET I 317(3) 

1, −μ −μ (iβ) γ(μ, iβx) + (−iβ) γ (μ, −iβx) 2

[Re μ > 0, x > 0]   ,  π π 1 Γ(μ, −iax) + exp −iμ Γ(μ, iax) xμ−1 cos ax dx = − μ exp iμ 2a 2 2

ET I 319(22) ET I 319(23)

  1 k! cos ax + kπ x sin ax dx = − 2 k ak+1 k=0    n  n  xn−k

1 n x cos ax dx = k! sin ax + kπ 2 k ak+1 k=0   n  n−1 2n−2k



x2n−2k−1 2n k+1 x k cos x + sin x x sin x dx = (2n)! (−1) (−1) (2n − 2k)! (2n − 2k − 1)!



n

n

 n  xn−k

k=0

k=0

TI (487)

TI (486)

216

Trigonometric Functions

 n 2n−2k

x2n−2k+1 k x cos x + sin x x sin x dx = (2n + 1)! (−1) (−1) (2n − 2k + 1)! (2n − 2k)! k=0 k=0   n  n−1 2n−2k 2n−2k−1



x x sin x + cos x x2n cos x dx = (2n)! (−1)k (−1)k (2n − 2k)! (2n − 2k − 1)! k=0 k=0   n  n 2n−2k



x2n−2k+1 2n+1 k k x sin x + cos x x cos x dx = (2n + 1)! (−1) (−1) (2n − 2k + 1)! (2n − 2k)! 



4.

5.

6.

2.634

2n+1

n

k+1

k=0

k=0

2.634  1.

n/2 (n+1)/2 (2k) (2k−1) (x) sin mx

(x) cos mx

k Pn k−1 Pn (−1) + (−1) Pn (x) sin mx dx = − m m2k m m2k−1 k=0

 2.

k=1

n/2

Pn (x) cos mx dx =

sin mx

m k=0

(2k) Pn (x) (−1)k m2k

+

cos mx m (k)

(n+1)/2



k=1

(−1)k−1

(2k−1)

Pn (x) m2k−1

In formulas 2.634, Pn (x) is any nth -degree polynomial and Pn (x) is its k th derivative with respect to x. 2.635 Notation: z1 = a + bx.  1 b 1. z1 sin kx dx = − z1 cos kx + 2 sin kx k k  1 b 2. z1 cos kx dx = z1 sin kx + 2 cos kx k k    2bz1 1 2b2 2 2 3. z1 sin kx dx = − z1 cos kx + 2 sin kx k k2 k    2bz1 1 2b2 z12 − 2 sin kx + 2 cos kx 4. z12 cos kx dx = k k k  2     3b z1 6b 2b2 3 2 2 5. z1 sin kx dx = − z1 cos kx + 2 z1 − 2 sin kx k k2 k k      2 3b z1 6b 2b2 6. z13 cos kx dx = z12 − 2 sin kx + 2 z12 − 2 cos kx k k k k      2 4 4bz1 1 12b 2 24b 6b2 4 4 2 7. z1 sin kx dx = − z1 − 2 z1 + 4 cos kx + 2 z1 − 2 sin kx k k k k k      2 4 2 4bz1 1 12b 2 24b 6b 4 4 2 8. z1 cos kx dx = z1 − 2 z1 + 4 sin kx + 2 z1 − 2 cos kx k k k k k      2 4 z1 5b 12b 24b 20b2 120b4 9. z15 sin kx dx = 2 z14 − 2 z12 + 4 z14 − 2 z12 + sin kx − cos kx k k k k k k4      z1 5b 12b2 2 24b4 20b2 2 120b4 5 4 4 10. z1 cos kx dx = 2 z1 − 2 z1 + 4 z1 − 2 z1 + cos kx + sin kx k k k k k k4

2.637

Trigonometric functions and powers

217

  6bz1 20b2 2 120b4 4 sin kx sin kx dx = 2 z1 − 2 z1 + k  k k4  30b2 360b4 2 720b6 1 z16 − 2 z14 + cos kx z − − k k k4 1 k6    6bz1 20b2 2 120b4 6 4 cos kx z1 cos kx dx = 2 z1 − 2 z1 + k  k k4  30b2 360b4 2 720b6 1 z16 − 2 z14 + sin kx z − + k k k4 1 k6

 11.

12.

2.636

z16



1.

xn sin2 x dx =

xn+1 2(n + ⎫ ⎧1) n/2 (n−1)/2 ⎬

(−1)k+1 xn−2k−1 n! ⎨ (−1)k+1 xn−2k sin 2x + cos 2x + ⎭ 4 ⎩ 22k (n − 2k)! 22k+1 (n − 2k − 1)! k=0

 2.

k=0

GU (333)(2e)

xn cos2 x dx =

xn+1 2(n + ⎫ ⎧1) n/2 (n−1)/2 ⎬

(−1)k+1 xn−2k−1 n! ⎨ (−1)k+1 xn−2k sin 2x + cos 2x − ⎭ 4 ⎩ 22k (n − 2k)! 22k+1 (n − 2k − 1)! k=0

k=0

GU (333)(3e)

 3. 4. 5. 6. 2.637 1.11

x 1 x2 − sin 2x − cos 2x 4 4 8    3 x 1 x 1 x2 sin2 x dx = − cos 2x − x2 − sin 2x 6 4 4 2  x 1 x2 + sin 2x + cos 2x x cos2 x dx = 4 4 8    3 x 1 x 1 x2 cos2 x dx = + cos 2x + x2 − sin 2x 6 4 4 2 x sin2 x dx =



MZ 241

MZ 245

⎧  ⎨n/2

(−1)k xn−2k  cos 3x n! 3 n x sin x dx = − 3 cos x 4 ⎩ (n − 2k)! 32k+1 k=0

(n−1)/2

−



k=0

xn−2k−1 (−1)k (n − 2k − 1)!



⎫ ⎬ sin 3x − 3 sin x ⎭ 32k+2 GU(333)(2f)

218

Trigonometric Functions

 2.

⎧   n/2 n! ⎨ (−1)k xn−2k sin 3x xn cos3 x dx = + 3 sin x 4 ⎩ (n − 2k)! 32k+1 k=0



[(n−1)/2]

+

k=0

(−1)k

n−2k−1

x (n − 2k − 1)!



⎫ ⎬ cos 3x + 3 cos x ⎭ 32k+2 GU(333)(3f)

 3. 4. 5. 6. 2.638

1 3 x 3 sin x − sin 3x − x cos x + cos 3x 4 36 4 12   2    x 1 3 x 3 2 3 2 3 x sin x dx = − x + cos x + + cos 3x + x sin x − sin 3x 4 2 12 54 2 18  1 3 x 3 cos 3x + x sin x + sin 3x x cos3 x dx = cos x + 4 36 4 12   2    3 1 x x 3 2 3 x2 cos3 x dx = sin x + sin 3x + x cos x + x − − cos 3x 4 2 12 54 2 18 x sin3 x dx =



1.

 2.

3.6

4.6

2.638

sinq x sinq−1 x [(p − 2) sin x + qx cos x] dx = − xp (p − 1)(p − 2)xp−1   sinq x dx sinq−2 x dx q2 q(q − 1) − + p−2 (p − 1)(p − 2) x (p − 1)(p − 2) xp−2 [p = 1, p = 2]

cosq x cosq−1 x [(p − 2) cos x − qx sin x] dx = − p p−1 x (p − 1)(p  − 2)x  2 q cos x dx cosq−2 x dx q q(q − 1) − + p−2 (p − 1)(p − 2) x (p − 1)(p − 2) xp−2 [p = 1, p = 2]   cos x dx sin x dx sin x 1 =− + p p−1 x (p − 1)x p−1 xp−1  sin x dx sin x cos x 1 =− − − p−1 p−2 (p − 1)x (p − 1)(p − 2)x (p − 1)(p − 2) xp−2 (p > 2)   sin x dx cos x dx cos x 1 =− − xp (p − 1)xp−1 p−1 xp−1  cos x dx cos x sin x 1 =− + − (p − 1)xp−1 (p − 1)(p − 2)xp−2 (p − 1)(p − 2) xp−2 (p > 2)

MZ 241

MZ 245, 246

TI (496)

TI (495)

TI (492)

TI (491)

2.641

2.639

Trigonometric functions and powers



1.

219

⎧ n−2 (−1)n+1 ⎨ (−1)k (2k + 1)! sin x dx = cos x x2n x(2n − 1)! ⎩ x2k+1 k=0 ⎫ n−1 ⎬ (−1)n+1

(−1)k+1 (2k)! Ci(x) sin x + + ⎭ (2n − 1)! x2k k=0

 2.

⎧ ⎨ n−1

(−1)k+1 (2k)!

sin x (−1)n+1 dx = x2n+1 x(2n)! ⎩ +

n−1

k=0

 3.

cos x

⎫ ⎬ (−1)n (−1)k+1 (2k + 1)! Si(x) sin x + ⎭ x2k+1 (2n)! ⎧ ⎨ n−1

(−1)k+1 (2k)!

cos x dx (−1)n+1 dx = x2n x(2n − 1)! ⎩ −

n−2

k=0



x2k

k=0

GU (333)(6b)a

k=0

x2k

GU (333)(6b)a

cos x

⎫ ⎬ (−1)k (2k + 1)! (−1)n Si(x) sin x + ⎭ (2n − 1)! x2k+1 GU (333)(7b)

⎧ ⎨ n−1

(−1)k+1 (2k + 1)!

cos x dx (−1)n+1 = x2n+1 x(2n)! ⎩

cos x x2k+1 ⎫ n−1 ⎬ (−1)n

(−1)k+1 (2k)! Ci(x) sin x + − ⎭ x2k (2n)!

4.

k=0

k=0

GU (333)(7b)

2.641 1. 2. 3. 4. 5.

  1 ka ka sin kx dx = cos si(u) − sin ci(u) a + bx b b b    cos kx 1 ka ka dx = cos ci(u) + sin si(u) a + bx b b b   k cos kx sin kx 1 sin kx + dx dx = − (a + bx)2 b a + bx b a + bx   k sin kx cos kx 1 cos kx − dx dx = − (a + bx)2 b a + bx b a + bx   sin kx k2 sin kx k cos kx − dx = − − (a + bx)3 2b(a + bx)2 2b2 (a + bx) 2b2





 k u = (a + bx) b   k u = (a + bx) b (see 2.641 2) (see 2.641 1)

sin kx dx a + bx (see 2.641 1)

220

Trigonometric Functions

 6.  7.12

 8.  12

9.

 10.

 11.

 12.

2.642 1.

2.

4.



cos kx dx a + bx (see 2.641 2)

sin kx sin kx k cos kx dx = − − 2 (a + bx)4 3b(a + bx)3 6b (a + bx)2 2 3  k cos kx k sin kx − dx + 3 6b (a + bx) 6b3 a + bx

(see 2.641 2)  k3 sin kx cos kx k sin kx k cos kx cos kx + 3 dx dx = − + 2 + 3 4 3 2 (a + bx) 3b(a + bx) 6b (a + bx) 6b (a + bx) 6b a + bx (see 2.641 1) 2

sin kx sin kx k cos kx dx = − − (a + bx)5 4b(a + bx)4 12b2 (a + bx)3  k4 sin kx k 2 sin kx k 3 cos kx + dx + + 3 2 4 4 24b (a + bx) 24b (a + bx) 24b a + bx (see 2.641 1) cos kx cos kx k sin kx dx = − + 5 4 (a + bx) 4b(a + bx) 12b2 (a + bx)3  2 k4 cos kx k cos kx k 3 sin kx + dx + − 24b3(a + bx)2 24b4 (a + bx) 24b4 a + bx (see 2.641 2) sin kx k cos kx k 2 sin kx k 3 cos kx sin kx dx = − − + + (a + bx)6 5b(a + bx)5 20b2 (a +bx)4 60b3 (a + bx)3 120b4(a + bx)2 k5 cos kx k 4 sin kx + dx − 120b5(a + bx) 120b5 a + bx (see 2.641 2) cos kx k sin kx k 2 cos kx cos kx dx = − + + 6 5 2 4 (a + bx) 5b(a + bx) 20b (a + bx) 60b3 (a + bx)3 3 4 k5 sin kx k sin kx k cos kx − dx − − 4 2 5 120b (a + bx) 120b (a + bx) 120b5 a + bx (see 2.641 1) 



  m−1 ln x 2m (−1)m

k ci[(2m − 2k)x] + (−1) 22m 22m−1 k k=0    m sin2m+1 x (−1)m

2m + 1 k dx = 2m si[(2m − 2k + 1)x] (−1) x 2 k 

sin2m x dx = x



2m m



k=0

m−1

 2m  ln x 1 ci[(2m − 2k)x] + 22m 22m−1 k k=0   m  cos2m+1 x 1 2m + 1 dx = 2m ci[(2m − 2k + 1)x] x 2 k



3.

k2 cos kx cos kx k sin kx − dx = − + (a + bx)3 2b(a + bx)2 2b2 (a + bx) 2b2

2.642

cos2m x dx = x

2m m

k=0

2.643

Trigonometric functions and powers

 5.

7.

8.

2.643



1.  2.

3.4

 2m 1 2m m 2 x    m−1 2m cos(2m − 2k)x (−1)m

+ (2m − 2k) si[(2m − 2k)x] + 2m−1 (−1)k+1 2 x k

sin2m x dx = − x2



k=0

  m 2m + 1 (−1)m

k+1 (−1) x2 22m k k=0   sin(2m − 2k + 1)x − (2m − 2k + 1) ci[(2m − 2k + 1)x] × x    2m cos x 2m 1 dx = − 2 2m x m 2 x  m−1

 2m   cos(2m − 2k)x 1 + (2m − 2k) si[(2m − 2k)x] − 2m−1 2 x k k=0 ⎧   m  cos2m+1 x 1 2m + 1 ⎨ cos(2m − 2k + 1)x = − ⎩ x2 22m x k k=0 ⎫ ⎬ + (2m − 2k + 1) si[(2m − 2k + 1)x] ⎭

 6.

221

sin

2m+1

x

dx =

xp dx xp−1 [p sin x + (q − 2)x cos x] q − 2 =− + q sin x q−1 (q − 1)(q − 2) sinq−1 x



xp dx p(p − 1) + sinq−2 x (q − 1)(q − 2)



xp−2 dx sinq−2 x

xp−1 [p cos x − (q − 2)x sin x] xp dx =− q cos x (q − 1)(q − 2) cosq−1 x  p−2 dx xp dx p(p − 1) x q−2 + + q − 1 cosq−2 x (q − 1)(q − 2) cosq−2 x  2k−1  ∞ −1 xn

xn k+1 2 2 dx = + B2k xn+2k (−1) sin x n (n + 2k)(2k)! k=1

[|x| < π,

 4.12

xn 

5.8  6.

n−1 n 2 −1 1 dx = − n − [1 + (−1)n ] (−1) 2 Bn ln x − sin x nx n!

xn dx = cos x

∞

k=0

∞

k=1 k= n 2

n > 0] TU (333)(8b)  2n 2 2 −1 B2k x2k−n (−1)k (2k − n) · (2k)!

[n > 1, |x| < π]

 π |x| < , n > 0 2

|E2k |xn+2k+1 (n + 2k + 1)(2k)!

GU (333)(9b) GU (333)(10b)

∞

|E2k |x2k−n+1 1 dx n |En−1 | = [1 − (−1) ln x + ] n x cos x 2 (n − 1)! (2k − n + 1) · (2k)! k=0 k= n−1 2

π |x| < 2

GU (333)(11b)

222

Trigonometric Functions

 7.  8.

∞

22k xn+2k−1 xn dx n n n−1 x B2k = −x cot x + + n (−1)k 2 n−1 (n + 2k − 1)(2k)! sin x k=1 [|x| < π, n > 1] n+1 2 n cot x dx n Bn+1 ln x =− n + − [1 − (−1)n ] (−1) 2 2 n+1 x (n + 1)x (n + 1)! sin x ∞ (−1)k (2x)2k n

B2k − n+1 2 (2k − n − 1)(2k)! k=1 k= n+1 2

9.

xn dx = xn tan x + n cos2 x

 10.

xn

∞

(−1)k

k=1

2

 2k





1.

 2.

 3.

[|x| < π]

22k − 1 xn+2k−1 B2k (n + 2k − 1) · (2k)!

n > 1,

GU (333)(9c)

|x|
1]

(see 2.643 2)

GU (333)(12)

(see 2.643 1) (see 2.645 6) [n > 1]

(see 2.643 1)

GU (333)(13)

224

Trigonometric Functions

2.646

 7. 8. 2.646

x x x cos x + ln tan dx = − sin x 2 sin2 x   x x π x sin x dx = − ln tan + 2 cos x cos x 2 4



p

x tan x dx =

1.

∞

(−1)

k=1

 2.

xp cot x dx =

∞

(−1)k

k=0

 

22k B2k xp+2k (p + 2k)(2k)!



1.  2.  3.  4.  5.  6.12  7.  8.  9. 

p ≥ −1, [p ≥ 1,

|x|
0]

TI (603)

238

Logarithms and Inverse-Hyperbolic Functions

2.721

2.72–2.73 Combinations of logarithms and algebraic functions 2.721

 m xn+1 lnm x − xn lnm−1 x dx (see 2.722) 1. x ln x dx = n+1 n+1 For n = −1 and m = −1  m lnm+1 x ln x dx = 2. x m+1 For n = −1 and m = −1  dx = ln (ln x) 3. x ln x  dx 4.∗ = Ei(ln x) ln x  m xn+1

lnm−k x (−1)k (m + 1)m(m − 1) · · · (m − k + 1) 2.722 xn lnm x dx = m+1 (n + 1)k+1 

12

2.723 1. 2. 3. 2.724

2.725 1. 2.

m

TI (604)

k=0

 1 ln x − n + 1 (n + 1)2    2 2 ln x 2 2 n n+1 ln x − x ln x dx = x + n + 1 (n + 1)2 (n + 1)3  3   3 ln2 x 6 ln x 6 3 n n+1 ln x − x ln x dx = x + − n + 1 (n + 1)2 (n + 1)3 (n + 1)4





1.

2.

n

xn ln x dx = xn+1



xn dx xn+1 n+1 m = − m−1 + m − 1 (ln x) (m − 1) (ln x)



TI 375

TI 375

xn dx

m−1

(ln x)

For m = 1  n  x dx = li xn+1 ln x    1 (a + bx)m+1 dx m+1 (a + bx) ln x − (a + bx) ln x dx = (m + 1)b x  m  m m−k k k+1

 b x 1 m m+1 m+1 k a (a + bx) ln x − (a + bx) ln x dx = −a (m + 1)b (k + 1)2



m

k=0

For m = −1 see 2.727 2. 2.726      a2 1 (a + bx)2 − ln x − ax + bx2 1. (a + bx) ln x dx = 2b 2b 4    b2 x3 1  abx2 2 3 3 2 (a + bx) − a ln x − a x + + 2. (a + bx) ln x dx = 3b 2 9

TI 374

2.727

Combinations of logarithms and algebraic functions



239

  1  3 2 2 1 2 3 1 3 4 4 4 3 (a + bx) − a ln x − a x + a bx + ab x + b x (a + bx) ln x dx = 4b 4 3 16 3

3. 2.727 8

1.

2.8 3. 4. 5.

6.∗

7.∗

8.∗

   ln x ln x dx 1 dx − = + TI 376 (a + bx)m b(m − 1) (a + bx)m−1 x(a + bx)m−1 For m = 1   1 1 ln(a + bx) dx ln x dx = ln x ln(a + bx) − (see 2.728 2) a + bx b b x  1 x ln x dx ln x + ln =− 2 (a + bx) b(a + bx) ab a + bx  1 x ln x dx ln x 1 + ln =− + (a + bx)3 2b(a + bx)2 2ab(a + bx) 2a2 b a + bx     √ √ 2 ln x dx (a + bx)1/2 − a1/2 √ (ln x − 2) a + bx − 2 a ln [a > 0] = x1/2  a + bx b   √ √ 2 a + bx = (ln x − 2) a + bx + 2 −a arctan [a < 0] b −a  ln |x − b| √ x dx = x2 + a2       x a2 + bx 2 2 2 2 √ + a + b arctanh √ x + a (ln |x − b| − 1) − b arcsinh |a| a2 + b2 x2 + a2  ln |x − b| √ x dx a2 − x2    x 2 2 = a − x (1 − ln |x − b|) − b arcsinh |a|  √ √  a2 − b2 a2 − x2 + a2 − bx 2 2 [|b| ≤ |a|] − a − b ln |a(x − b)|    x = a2 − x2 (1 − ln |x − b|) − b arcsin |a|    a2 − bx √ [|b| ≥ |a|] − b2 − a2 arctan √ b2 − a2 a2 − x2   ln |x − b| √ x dx = x2 − a2 (ln |x − b| − 1) + I1 + I2 x2 − a2 where  √ x2 − a2 + x [x ≥ |a|] I1 = −b ln |a|  √ x2 − a2 − x [x ≤ −|a|] = b ln |a| 

240

Logarithms and Inverse-Hyperbolic Functions

2.728

   a2 − bx 2 2 √ [|b| ≤ |a|] I2 = a − b arccot √ a2√− b2 x2 − a2  √  b2 − a2 x2 − a2 − a2 + bx = b2 − a2 ln |a(x − b)|  = − b2 − a2 ln

[either b ≥ |a|, x ≥ |a| or b ≤ −|a|, x ≤ −|a|]  √ √ b2 − a2 x2 − a2 + a2 − bx |a(x − b)| [either b ≥ |a|, x ≤ −|a| or b ≤ −|a|, x ≥ −|a|]

2.728

  m+1  dx 1 x xm+1 ln(a + bx) − b m+1 a + bx    bx bx ln(a + bx) = ln a ln x + Φ − , 2, 1 [a > 0] x a a 

1. 2.9 2.729



1.  2.  12

3.

 4.  2.731

 2.732

7

xm ln(a + bx) dx =

  m+1 1 (−1)k xm−k+2 ak−1 1 (−a)m+1 m+1 x ln(a + bx) dx = x ln(a + bx) + − m+1 bm+1 m+1 (m − k + 2)bk−1 k=1     ax 1 x2 1 2 a2 x − 2 ln(a + bx) − − x ln(a + bx) dx = 2 b 2 2 b   3   3 ax2 a2 x 1 x 1 3 a 2 x − 3 ln(a + bx) − − + 2 x ln(a + bx) dx = 3 b 3 3 2b b     ax3 a2 x2 1 x4 1 4 a4 a3 x x − 4 ln(a + bx) − − + − x3 ln(a + bx) dx = 4 b 4 4 3b 2b2 b3 ⎡   1 ⎣x2n+1 ln x2 + a2 + (−1)n 2a2n+1 arctan x x2n ln x2 + a2 dx = 2n + 1 a ⎤ n

(−1)n−k 2n−2k 2k+1 ⎦ a −2 x 2k + 1 k=0 ⎡    1 ⎣ x2n+2 + (−1)n a2n+2 ln x2 + a2 x2n+1 ln x2 + a2 dx = 2n + 2 ⎤ n+1

(−1)n−k a2n−2k+2 x2k ⎦ + k m

k=1

2.733 1. 2.



  x ln x2 + a2 dx = x ln x2 + a2 − 2x + 2a arctan a     2 1 x2 + a2 ln x2 + a2 − x2 x ln x + a2 dx = 2

DW DW

2.737*

Combinations of logarithms and algebraic functions

241

  2 3 1 3  2 x 2 2 3 x ln x + a − x + 2a x − 2a arctan 3. x ln x + a dx = DW 3 3 a     x4  1  4 x − a4 ln x2 + a2 − + a2 x2 4. x3 ln x2 + a2 dx = DW 4 2    2 5 2 2 3  2 1 5  2 x 4 2 2 4 5 x ln x + a − x + a x − 2a x + 2a arctan 5. x ln x + a dx = DW 5 5 3 a    2.734 x2n ln x2 − a2  dx     n

    x + a 1 1 −2 x2n+1 ln x2 − a2  + a2n+1 ln  a2n−2k x2k+1 = 2n + 1 x − a 2k + 1 k=0    n+1

1      1 2n+1 2 2 2n+2 2n+2 2 2 2n−2k+2 2k   x a ln x − a − ln x − a dx = −a x 2.735 x 2n + 2 k 

2

2.736



2

2



k=1



1. 

    2  x + a  2 2 2     ln x − a dx = x ln x − a − 2x + a ln  x − a

DW

     1  2 x − a2 ln x2 − a2  − x2 x ln x2 − a2  dx = 2       2  x + a 1 3  2  x ln x − a2  − x3 − 2a2 x + a3 ln  x2 ln x2 − a2  dx = 3 3 x − a      x4  2  2 1  4 3 2 4 2 2 2   x − a ln x − a − −a x x ln x − a dx = 4 2       2 5 2 2 3  2 x + a 1 5  2 4 2 2 4 5    x ln x − a − x − a x − 2a x + a ln  x ln x − a dx = 5 5 3 x − a

2. 3. 4. 5. 2.737* 1.∗



1 x

t −1 ln t

k + n(x + 1) k+n

DW



[n = 2, 4, 6, . . . ,

x > −1]

[n = 1, 3, 5, . . . , m = 1, 2, . . . , n−1  n−1 1



1 tx (ln t)m (−1)k tk/n dt = (−1)m nm+1 m! [k + n(x + 1)]m+1 0

x > −1]



1 x

t −1 ln t

0

 0

4.∗

n−1 3

DW

x > −1]

2.

3.∗

 k/n

DW

[n = 1, 2, . . . ,

0 ∗

n−1

DW

1

t

dt = ln

k=0

k=0

n−1  k n−1

3  k + n(x + 1) (−1) k k/n dt = ln (−1) t k+n

tx (ln t)m

k=0

n−1

k=0

k=0

 tk/n

dt = (−1)m nm+1 m!

n−1

k=0

1 [k + n(x + 1)]m+1



k=0

k=0

[n = 2, 4, 6, . . . ,

m = 1, 2, . . . ,

x > −1]

242

Inverse Trigonometric Functions

2.741

2.74 Inverse hyperbolic functions Assume a > 0 2.741  x  x 1. arcsinh dx = x arcsinh − x2 + a2 a a  x  x 2. arccosh dx = x arccosh − x2 − a2 a a x  2 = x arccosh + x − a2 a  x a  x 3. arctanh dx = x arctanh + ln a2 − x2 a a 2  x a  x 4. arccoth dx = x arccoth + ln x2 − a2 a a 2 2.742 1. 2.





x2 a2 + 2 4

 x arccosh > 0 a 

x arccosh < 0 a

DW DW

DW DW



x x 2 − x + a2 a 4   2  x x 2 x a2 x arccosh − x arccosh dx = x − a2 − a 4 a 4  22 a2 x x 2 x − arccosh + x − a2 = 2 4 a 4 x x arcsinh dx = a

DW

arcsinh

DW

 x arccosh > 0 a 

x arccosh < 0 a DW

2.75 Logarithms and exponential functions 2.751*

 e

1.  2.

ax

   ax p−1 x 1 ax p e ln e ln x − p dx ln x dx = a x p

eax ln x dx =

1 ax [e ln x − Ei(ax)] a

[a = 0] [a = 0]

2.8 Inverse Trigonometric Functions 2.81 Arcsines and arccosines Assume a > 0   n/2 n 

x n−2k x n · (2k)! arcsin dx = x (−1)k 2.811 arcsin a a 2k k=0   (n+1)/2  

n x n−2k+1 k−1 2 2 · (2k − 1)! arcsin + a −x (−1) a 2k − 1 k=1

2.822

2.812

The arcsecant, the arccosecant, the arctangent and the arccotangent

  arccos

243

n/2 n 

x n−2k x n · (2k)! arccos dx = x (−1)k a a 2k k=0   (n+1)/2  

n x n−2k+1 · (2k − 1)! arccos + a2 − x2 (−1)k a 2k − 1 k=1

2.813 1.11 2.9

3.

   x x dx = sign(a) x arcsin + a2 − x2 a |a| 2     x 2 x x − 2x arcsin dx = x arcsin + 2 a2 − x2 arcsin a |a| |a| ⎡ 3 2       3 x x x dx = sign(a) ⎣x arcsin + 3 a2 − x2 arcsin arcsin a |a| |a| ⎤  x − 6 a2 − x2 ⎦ − 6x arcsin |a| 

arcsin

2.814  x  x 1. arccos dx = x arccos − a2 − x2 a a     x 2 x 2 x 2. arccos dx = x arccos − 2 a2 − x2 arccos − 2x a a a       x 3 x 2 x 3 x 3. arccos dx = x arccos − 3 a2 − x2 arccos − 6x arccos + 6 a2 − x2 a a a a

2.82 The arcsecant, the arccosecant, the arctangent and the arccotangent Assume a > 0 2.821      a a x arccosec dx = arcsin dx = x arcsin + a ln x + x2 − a2 1.12 a x x    a = x arcsin − a ln −x + x2 − a2 x  2.12

2.822 1.

8



   a a x arcsec dx = arccos dx = x arccos + a ln x + x2 − a2 a x x    a = x arccos − a ln −x + x2 − a2 x

 arctan

x a  x dx = x arctan − ln a2 + x2 a a 2

π a 0 < arcsin < x 2 

π a − < arcsin < 0 2 x

DW

π a 0 < arccos < x 2

π a < arccos < π 2 x

DW

DW

244

Inverse Trigonometric Functions

 2.

12

x a  x dx = x arccot + ln a2 + x2 a a 2 1 2 x x ax x + a2 arctan − x arctan dx = a 2 a 2 2 ax πx 1 2 x x + − x + a2 arctan x arccot dx = a 2 4 2 a ax2  1 x x 1 x2 arctan dx = x3 arctan + a3 ln x2 + a2 − a 3 a 6 6 πx3  1 ax2 x x 1 + x2 arccot dx = − x3 arctan − a3 ln x2 + a2 + a 3 a 6 6 6 arccot

 3.9  9

4.

 5.9  9

6.

2.831

DW

2.83 Combinations of arcsine or arccosine and algebraic functions Assume a > 0

 n+1 dx xn+1 x 1 x x √ dx = arcsin − (see 2.263 1, 2.264, 2.27) 2 − x2 a n + 1 a n + 1 a   xn+1 x 1 xn+1 dx x √ arccos + 2.832 xn arccos dx = (see 2.263 1, 2.264, 2.27) a n+1 a n+1 a2 − x2   arccos x arcsin x dx and dx) cannot be expressed as a 1. For n = −1, these integrals (that is, x x finite combination of elementary functions.   arccos x π 1 arcsin x 2. dx = − ln − dx x 2 x x xn arcsin

2.831

2.8339



1.  2.  3.  4.  5.  6. 2.834 12

1.



  x2 a2 x x 2 2 − arcsin + a −x 2 4 |a| 4   πx2 1 2 x 2 x x − sign(a) 2x − a2 arcsin + a − x2 x arccos dx = a 4 4 |a| 4  3   x x 1 2 x 2 2 2 2 arcsin + x + 2a a −x x arcsin dx = sign(a) a 3 |a| 9  3   πx3 x x 1 2 x − sign(a) arcsin + x + 2a2 a2 − x2 x2 arccos dx = a 6 3 |a| 9  4   4  3a x 1  2 x x 3 2 2 2 − arcsin + x 2x + 3a a −x x arcsin dx = sign(a) a 4 32 |a| 32    8x4 − 3a4 x πx4 x 1  2 3 2 2 2 a −x x arccos dx = − sign(a) arcsin + x 2x + 3a a 8 32 |a| 32 x x arcsin dx = sign(a) a



1 x 1 1 x arcsin dx = − arcsin − ln 2 x a x a a



a+

 √ a2 − x2 |x|

Arcsecant and arccosecant with powers of x

2.841

 2.12

1 x 1 1 x arccos dx = − arccos + ln x2 a x a a



a+

√

a2 − x2 |x|

245



  1 arcsin x arcsin x b + ax √ √ √ + dx = − arctan 2 (a + bx)2 b(a + bx) b a2 − b2 a2 −√b2 1 − x   √ 2 2 arcsin x 1 b + ax − b − a 1 − x2 √ √ ln =− + √ b(a + bx) 2b b2 − a2 b + ax + b2 − a2 1 − x2   1 + x arcsin x 1 + =− 2 b 1−x x−1   1 − x arcsin x 1 + =− 2 b 1+x x+1 √  c + 1x 1 x arcsin x arcsin x √ √ 2.83612 + arctan dx = 2 2 2c (1 + cx2 ) 2c c + 1 1 (1 + cx2 )  √−x 1 − x2 + x −(c + 1) 1 arcsin x  +  =− ln √ 2c (1 + cx2 ) 4c −(c + 1) 1 − x2 − x −(c + 1) 

2.83512

 

a2 > b 2 a2 < b 2



[a = −b] [a = b] [c > −1] [c < −1]

2.837   x arcsin x √ 1. dx = x − 1 − x2 arcsin x 1 − x2  2 x x arcsin x x2 1 2 √ − 2.12 dx = 1 − x2 arcsin x + (arcsin x) 2 4 2 4 1−x  3  2x 1  2 x arcsin x x3 √ + − x +2 3. dx = 1 − x2 arcsin x 9 3 3 1 − x2 2.838



1.  2.

arcsin x x arcsin x 1  ( dx = √ + ln 1 − x2 2 3 1 − x2 (1 − x2 ) arcsin x 1 1−x x arcsin x ( dx = √ + ln 2 2 1+x 3 1−x (1 − x2 )

2.84 Combinations of the arcsecant and arccosecant with powers of x Assume a > 0 2.841     a 1 2 x a x arccos − a x2 − a2 x arcsec dx = x arccos dx = 1.12 a x 2  x  a 1 2 2 2 x arccos + a x − a = 2 x

π a 0 < arccos < x 2

π a < arccos < π 2 x

DW

246

Inverse Trigonometric Functions

 2.

 3.12

   1 3 a a a  2 a3  2 2 2 x arccos dx = x arccos − x x − a − ln x + x − a x 3 x 2 2 π a 0 < arccos < x 2    a a  2 a3  1 3 2 2 2 x arccos + x x − a − ln −x + x − a = 3 x 2 2 

π a < arccos < π 2 x

x x arcsec dx = a 2

12

2.851



2



DW

1

  a x a x2 arcsin + a x2 − a2 x arccosec dx = x arcsin dx = a x 2  x  a 1 2 2 2 x arcsin − a x − a = 2 x

π a 0 < arcsin < x 2 

π a − < arcsin < 0 2 x

DW

2.85 Combinations of the arctangent and arccotangent with algebraic functions Assume a > 0

xn+1 x a x arctan − x arctan dx = a n+1 a n+1 n

2.851 2.852



1.

2. 3. 2.853

xn arccot

xn+1 x a x dx = arccot + a n+1 a n+1





xn+1 dx a2 + x2 xn+1 dx a2 + x2

For n = −1  arctan x dx cannot be expressed as a finite combination of elementary functions. x   arccot x π arctan x dx = ln x − dx x 2 x 

x 1 2 x ax dx = x + a2 arctan − a 2 a 2  1 2 ax x x x + a2 arccot + 2. x arccot dx = a 2 a 2  3 3  ax2 x x a x arctan + ln x2 + a2 − x2 arctan dx = 3.9 a 3 a 6 6  3 3  πx3 x x a ax2 x ln x2 + a2 + + x2 arccot dx = − arctan − 4.9 a 3 a 6 6 6  1 x 1 a2 + x2 x 1 dx = − arctan − ln arctan 2.854 2 a x  a 2a x2   x a + bx arctan x 1 b − ax ln √ 2.855 dx = 2 − arctan x (a + bx)2 a + b2 a + bx 1 + x2 2.856     1 ln 1 + x2 dx 1 x arctan x 2 dx = arctan x ln 1 + x − 1. 1 + x2 2 2 1 + x2 1.

x arctan

TI (689)

2.859

Arctangent and arccotangent with algebraic functions

247

 2. 3.

1 x2 arctan x 1  2 dx = x arctan x − ln 1 + x2 − (arctan x) 2 1+x 2 2   3 1 x arctan x x arctan x 1 2 x + 1 + x arctan x − dx = − dx 1 + x2 2 2 1 + x2 



(see 2.8511)  1 x − x arctan x + (arctan x)2 3 2

TI (405)

x arctan x 1 2  dx = − x2 + ln 1 + x2 + 2 1+x 6 3    n

(2n − 2k)!!(2n − 1)!! x arctan x dx 1 (2n − 1)!! arctan x arctan x 2.857 = + (2n)!!(2n − 2k + 1)!! (1 + x2 )n−k+1 2 (2)!! (1 + x2 )n+1 k=1 n 1

(2n − 1)!!(2n − 2k)!! 1 + 2 (2n)!!(2n − 2k + 1)!!(n − k + 1) (1 + x2 )n−k+1 k=1 √   √ x arctan x x 2 √ 2.858 dx = − 1 − x2 arctan x + 2 arctan √ − arcsin x 1 − x2 1 − x2   a + bx2 arctan x x arctan x 1 ( [a < b] arctan 2.859 dx = √ − √ 2 b − a√ 3 a b−a a a + bx √ (a + bx2 ) a + bx2 − a − b x arctan x 1 √ = √ ln √ [a > b] + √ 2a a − b a a + bx2 a + bx2 + a − b 4.

4

3

Table of Integrals, Series, and Products. http://dx.doi.org/10.1016/B978-0-12-384933-5.00003-5 c 2015 Elsevier Inc. All rights reserved. Copyright 

3–4 Definite Integrals of Elementary Functions 3.0 Introduction 3.01 Theorems of a general nature 3.011 Suppose that f (x) is integrable† over the largest of the intervals (p, q), (p, r), (r, q). Then (depending on the relative positions of the points p, q, and r) it is also integrable over the other two intervals and we have    q

p

f (x) dx =

r

p

f (x) dx +

q

r

f (x) dx

FI II 126

3.012 The first mean-value theorem. Suppose (1) that f (x) is continuous and that g(x) is integrable over the interval (p, q), (2) that m ≤ f (x) ≤M and (3) that g(x) does not change sign anywhere in the interval (p, q). Then, there exists at least one point ξ (with p ≤ ξ ≤ q) such that  q  q f (x)g(x) dx = f (ξ) g(x) dx FI II 132 p

p

3.013 The second mean-value theorem. If f (x) is monotonic and non-negative throughout the interval (p, q), where p 1 p q c ≥ u]

BY (231.00)

[a > b > c > u]

BY (232.00)

[a > b ≥ u > c]

BY (233.00)

3.133

Square roots of polynomials



b

4. u



u

5. b



a

6. u



u

7. a



u

3.132



c

1. u

 c

b

u

[a ≥ u > b > c]

BY (235.00)

2 F (λ, p) = √ a −c (a − x)(x − b)(x − c)

[a > u ≥ b > c]

BY (236.00)

dx 2  F (μ, q) =√ a−c (x − a)(x − b)(x − c)

[u > a > b > c]

BY (237.00)

[u ≥ a > b > c]

BY (238.00)



dx

dx 2  F (ν, q) =√ a−c (x − a)(x − b)(x − c) x dx

2  [c F (β, p) + (a − c) E (β, p)] − 2 = √ a−c (a − x)(b − x)(c − x)



(a − u)(c − u) b−u BY (232.19)

√ x dx 2a  F (γ, q) − 2 a − c E (γ, q) =√ a−c (a − x)(b − x)(x − c) BY (233.17)



2 x dx  (b − a) Π δ, q 2 , q + a F (δ, q) =√ a − c (a − x)(b − x)(x − c) [a > b > u ≥ c] x dx 2  =√ a −c (a − x)(x − b)(x − c)

a

x dx a√ 2c  F (λ, p) + 2 a − c E (λ, p) = √ b a−c (a − x)(x − b)(x − c)

b

BY (234.16)



(b − c) Π κ, p2 , p + c F (κ, p) [a ≥ u > b > c]

5.12 u



2 dx  F (κ, p) =√ a−c (a − x)(x − b)(x − c)

u

4. 

BY (234.00)

[a > b ≥ u > c]

3. 

[a > b > u ≥ c]

[a > b > c > u] u

2. 

dx 2  F (δ, q) =√ a−c (a − x)(b − x)(x − c)

∞

8.

257

BY (235.16)

[a > u ≥ b > c] u

6.12 a

BY (236.16)

x dx 2  [(a − b) Π(μ, 1, q) + b F (μ, q)] =√ a−c (x − a)(x − b)(x − c) [u > a > b > c]

3.133  1.

u



c

BY (237.16)

dx 2 √  [F (α, p) − E (α, p)] = 3 (a − b) a − c (a − x) (b − x)(c − x) −∞ [a > b > c ≥ u]

2. u

2 2 dx √  [F (β, p) − E (β, p)] + = 3 a−c (a − b) a − c (a − x) (b − x)(c − x) [a > b > c > u]



BY (231.08)

c−u (a − u)(b − u) BY (232.13)

258

Power and Algebraic Functions



u

3. c

 4.

b

12 u



u

5. b

6.

3.133

 dx (b − u)(u − c) 2 2 √  E (γ, q) − = (a − b)(a − c) a−u (a − x)3 (b − x)(x − c) (a − b) a − c [a > b ≥ u > c] BY (233.09) dx 2 √  E (δ, q) = 3 (a − b) a − c (a − x) (b − x)(x − c)

[a > b > u ≥ c]

2 2 √  [F (κ, p) − E (κ, p)] + = 3 a − b (a − b) a − c (a − x) (x − b)(x − c) dx

BY (234.05)



u−b (a − u)(u − c)

[a > u > b > c]   ∞ dx u−b 2 2 √  E (ν, q) + = 3 a − b (u − a)(u − c) (b − a) a − c (x − a) (x − b)(x − c) u

BY (235.04)

[u > a > b > c] √ u dx 2 2 a−c √  E (α, p) − F (α, p) = 3 (a − b)(b − c) (a − b) a−c (a − x)(b − x) (c − x) −∞  2 c−u − b − c (a − u)(b − u) [a > b > c ≥ u] √  c dx 2 2 a−c √  E (β, p) − F (β, p) = 3 (a − b)(b − c) (a − b) a − c (a − x)(b − x) (c − x) u

BY (238.05)

[a > b > c > u] √ dx 2 a−c 2 √  E (γ, q) F (γ, q) − = (a − b)(b − c) (a − x)(b − x)3 (x − c) (b − c) a − c  c 2 (a − u)(u − c) + (a − b)(b − c) b−u [a > b > u > c] √  a 2 a−c 2 dx √  E (λ, p) F (λ, p) − = (a − b)(b − c) (a − x)(x − b)3 (x − c) (a − b) a − c  u 2 (a − u)(u − c) + (a − b)(b − c) u−b [a > u > b > c] √  u dx 2 2 a−c √  E (μ, q) − F (μ, q) = 3 (x − c) (a − b)(b − c) (b − c) a−c (x − a)(x − b) a

BY (232.14)

[u > a > b > c] √ ∞ dx 2 2 a−c √  E (ν, q) − F (ν, q) = 3 (a − b)(b − c) (b − c) a−c (x − a)(x − b) (x − c) u  2 u−a − a − b (u − b)(u − c) [u ≥ a > b > c]

BY (237.12)

 7.

8.

 9.

10.

11.

 12.

BY (231.09)

u

BY (233.10)

BY (236.09)

BY (238.04)

3.134

Square roots of polynomials

 13.

259



u

2 2 √  E (α, p) + = 3 b − c (c − b) a − c (a − x)(b − x)(c − x) −∞ dx

b−u (a − u)(c − u)

[a > b > c > u] 

b

14. u



dx 2 2 √  [F (δ, q) − E (δ, q)] + = 3 b−c (b − c) a − c (a − x)(b − x)(x − c)

b

BY (234.04)

dx 2 √  E (κ, p) = (b − c) a − c (a − x)(x − b)(x − c)3

[a > u ≥ b > c] u

17.12 a



dx 2 2 √  [F (μ, q) − E (μ, q)] + = a−c (b − c) a − c (x − a)(x − b)(x − c)3 [u > a > b > c]

∞

18. u





BY (236.10)

u−a (u − b)(u − c) BY (237.13)

dx 2 √  [F (ν, q) − E (ν, q)] = (b − c) a − c (x − a)(x − b)(x − c)3 [u ≥ a > b > c]

BY (238.03)

u

dx  5 (a − x) (b − x)(c − x) −∞

1.

=

3(a − b)2 +



c

2. u

 3.

b−u (a − u)(u − c)

[a ≥ u > b > c] BY (235.01)   a dx (a − u)(u − b) 2 2 √  E (λ, p) − = 3 (b − c)(a − c) u−c (b − c) a − c (a − x)(x − b)(x − c) u 

3.134

BY (231.10)

[a > b > u > c] u

15.

16.



12 c

u

2  [(3a − b − 2c) F (α, p) − 2(2a − b − c) E (α, p)] 3 (a − c) 

2 3(a − c)(a − b)

(c − u)(b − u) (a − u)3 [a > b > c ≥ u]

BY (231.08)

2 dx   [(3a − b − 2c) F (β, p) − 2(2a − b − c) E (β, p)] = 5 2 (a − x) (b − x)(c − x) 3(a − b) (a − c)3

 2 4a2 − 3ab − 2ac + bc − u(3a − 2b − c) c−u + 3(a − b)(a − c)2 (a − u)3 (b − u) [a > b > c > u] BY (232.13) dx 2   [2(2a − b − c) E (γ, q) − (a − b) F (γ, q)] = 5 2 (a − x) (b − x)(x − c) 3(a − b) (a − c)3

 2 5a2 − 3ab − 3ac + bc − 2u(2a − b − c) (b − u)(u − c) − 3(a − b)2 (a − c)2 (a − u)3 [a > b ≥ u > c] BY (233.09)

260

Power and Algebraic Functions



b

4.12 u

dx 2   [2(2a − b − c) E (δ, q) − (a − b) F (δ, q)] = 5 2 3 (a − x) (b − x)(x − c) 3(a − b) (a − c)  +



u

5.12 b



∞

u

 7.

2 3(a − b)(a − c)

(b − u)(u − c) (a − u)3 [a > b > u ≥ c]

BY (234.05)

dx  5 (a − x) (x − b)(x − c) =

6.

3.134

2  [(3a − b − 2c) F (κ, p) − 2(2a − b − c) E (κ, p)] 3(a − b)2 (a − c)3

 2 4a2 − 2ab − 3ac + bc − u(3a − b − 2c) u−b + 2 3(a − b) (a − c) (a − u)3 (u − c) [a > u > b > c] BY (235.04)

dx 2   [2(2a − b − c) E (ν, q) − (a − b) F (ν, q)] = (x − a)5 (x − b)(x − c) 3(a − b)2 (a − c)3

 2 4a2 − 2ab − 3ac + bc + u(b + 2c − 3a) u−b + 2 3(a − b) (a − c) (u − a)3 (u − c) [u > a > b > c] BY (238.05)

u

dx 2  √ = 2 5 (a − x)(b − x) (c − x) 3(a − b) (b − c)2 a − c −∞



× [2(a − c)(a + c − 2b) E (α, p) + (b − c)(3b − a − 2c) F (α, p)]

 2 3ab − ac + 2bc − 4b2 − u(2a − 3b + c) c−u − 2 3(a − b)(b − c) (a − u)(b − u)3 [a > b > c ≥ u] BY (231.09) c

8. u

 9. c

dx 2  √ = 2 (b − c)2 a − c 5 3(a − b) (a − x)(b − x) (c − x) × [(b − c)(3b − a − 2c) F (β, p) + 2(a − c)(a − 2b + c) E (β, p)]  2 (a − u)(c − u) + 3(a − b)(b − c) (b − u)3 [a > b > c > u] BY (232.14)

u

dx 2  √ = (a − x)(b − x)5 (x − c) 3(a − b)2 (b − c)2 a − c × [(a − b)(2a − 3b + c) F (γ, q) + 2(a − c)(2b − a − c) E (γ, q)]

 2 3ab + 3bc − ac − 5b2 − 2u(a − 2b + c) (a − u)(u − c) + 2 2 3(a − b) (b − c) (b − u)3 [a > b > u > c] BY (233.10)

3.134

Square roots of polynomials



a

10. u



u

a

14.

15.

16.

− c)

=

2 √ 3(a − b)2 (b − c)2 a − c

dx 2  √ = (x − a)(x − b)5 (x − c) 3(a − b)2 (b − c)2 a − c

∞

u



(a − x)(x −

b)5 (x

× [(a − b)(2a + c − 3b) F (μ, q) + 2(a − c)(2b − a − c) E (μ, q)]  (u − a)(u − c) 2 + 3(a − b)(b − c) (u − b)3 [u > a > b > c] BY (237.12)

12.

13.

dx

× [(b − c)(3b − 2c − a) F (λ, p) + 2(a − c)(a + c − 2b) E (λ, p)]

 2 3ab + 3bc − ac − 5b2 + 2u(2b − a − c) (a − u)(u − c) + 3(a − b)2 (b − c)2 (u − b)3 [a > u > b > c] BY (236.09)

11.





261

dx 2  √ = 2 5 (x − a)(x − b) (x − c) 3(a − b) (b − c)2 a − c × [(a − b)(2a + c − 3b) F (ν, q) + 2(a − c)(2b − c − a) E (ν, q)]

 2 3bc + 2ab − ac − 4b2 + u(3b − a − 2c) u−a − 3(a − b)2 (b − c) (u − b)3 (u − c) [u ≥ a > b > c] BY (238.04)

u

dx 2   = [2(a + b − 2c) E (α, p) − (b − c) F (α, p)] 5 2 (a − x)(b − x)(c − x) 3(b − c) (a − c)3 −∞

 2 ab − 3ac − 2bc + 4c2 + u(2a + b − 3c) b−u + 2 3(a − c)(b − c) (a − u)(c − u)3 [a > b > c > u] By (231.10)  b dx 2   = [(2a + b − 3c) F (δ, q) − 2(a + b − 2c) E (δ, q)] 5 2 (a − x)(b − x)(x − c) 3(b − c) (a − c)3 u

 2 ab − 3ac − 2bc + 4c2 + u(2a + b − 3c) b−u + 2 3(b − c) (a − c) (a − u)(u − c)3 [a > b > u > c] BY (234.04)  u dx 2   = [2(a + b − 2c) E (κ, p) − (b − c) F (κ, p)] 5 2 3 (a − x)(x − b)(x − c) 3(b − c) (a − c) b  (a − u)(u − b) 2 + 3(a − c)(b − c) (u − c)3 [a ≥ u > b > c] BY (235.20)  a dx 2   = [2(a + b − 2c) E (λ, p) − (b − c) F (λ, p)] 5 2 (a − x)(x − b)(x − c) 3(b − c) (a − c)3 u

 2 ab − 3ac − 3bc + 5c2 + 2u(a + b − 2c) (a − u)(u − b) − 3(b − c)2 (a − c)2 (u − c)3 [a > u ≥ b > c] BY (236.10)

262

Power and Algebraic Functions

 17.

u

12 a



dx 2   = [(2a + b − 3c) F (μ, q) − 2(a + b − 2c) E (μ, q)] 5 2 (x − a)(x − b)(x − c) 3(b − c) (a − c)3

 2 ab − 2ac − 3bc + 4c2 + u(a + 2b − 3c) u−a + 3(b − c)(a − c)2 (u − b)(u − c)3 [u > a > b > c] BY (237.13)

∞

18. u

dx 2   = [(2a + b − 3c) F (ν, q) − 2(a + b − 2c) E (ν, q)] 5 2 3 (x − a)(x − b)(x − c) 3(b − c) (a − c)  +

3.135  1.12

2.

3.

4.

5.

6.

u

3.135

2 3(a − c)(b − c)

(u − a)(u − b) (u − c)3 [u ≥ a > b > c]

BY (238.03)

dx 2  √ [(b − c) F (α, p) − (2a + b − c) E (α, p)] = 3 3 (a − b)(b − c)2 a − c (a − x)(b − x) (c − x) −∞ 2(b + c − 2u)  + (b − c)2 (a − u)(b − u)(c − u) BY (231.13) [a > b > c > u]  a dx 2  √ [(b − c) F (λ, p) − 2(2a − b − c) E (λ, p)] = 3 (x − c)3 (a − b)(b − c)2 a − c  (a − x)(x − b) u 2(a − b − c + u) a−u + (a − b)(b − c)(a − c) (u − b)(u − c) [a > u > b > c] BY (236.15)  u dx 2  √ [(2a − b − c) E (μ, q) − 2(a − b) F (μ, q)] = 3 3 (a − b)(b − c)2  a−c (x − a)(x − b) (x − c) a u−a 2 + (a − c)(b − c) (u − b)(u − c) [u > a > b > c] BY (236.14)  ∞ dx 2  √ [(2a − b − c) E (ν, q) − 2(a − b) F (ν, q)] = 3 3 (a − b)(b − c)2  a−c (x − a)(x − b) (x − c) u u−a 2 − (a − b)(b − c) (u − b)(u − c) [u ≥ a > b > c] BY (238.13)  u dx 2   = [(2b − a − c) E (α, p) − (b − c) F (α, p)] 3 3 (a − x) (b − x)(c − x) (a − b)(b − c)  (a − c)3 −∞ b−u 2 + (b − c)(a − c) (a − u)(c − u) [a > b > c > u] BY(231.12)  b dx 2   = [(a − b) F (δ, q) + (2b − a − c) E (δ, q)] 3 3 (a − x) (b − x)(x − c) (b − c)(a − b)  (a − c)3 u b−u 2 + (b − c)(a − c) (a − u)(u − c) [a > b > u > c] BY (234.03)

3.136

Square roots of polynomials



u

7. b

dx 2   = [(b − c) F (κ, p) − (2b − a − c) E (κ, p)] 3 3 (a − x) (x − b)(x − c) (a − b)(b − c)  (a − c)3 +



∞

8. u

2 (a − b)(a − c)

9.

10.

11.

12.12

3.136 1.

u



u

u−b (a − u)(u − c) [a > u > b > c]

BY (235.15)

dx 2   = [(a + c − 2b) E (ν, q) − (a − b) F (ν, q)] 3 3 (x − a) (x − b)(x − c) (a − b)(b − c)  (a − c)3 +



263

2 (a − b)(a − c)

u−b (u − a)(u − c) [u > a > b > c]

BY (238.14)

dx 2  √ [(a + b − 2c) E (α, p) − 2(b − c) F (α, p)] = 3 (b − x)3 (c − x) (b − c)(a − b)2  a−c (a − x) −∞ c−u 2 − (a − b)(b − c) (a − u)(b − u) [a > b > c ≥ u] BY (231.11)  c dx 2 √  [(a + b − 2c) E (β, p) − 2(b − c) F (β, p)] = 2 (b − c) a − c 3 (b − x)3 (c − x) (a − b) (a − x) u  c−u 2 + (a − b)(a − c) (a − u)(b − u) [a > b > c > u] BY (232.15)  u dx 2 √  [(a − b) F (γ, q) − (a + b − 2c) E (γ, q)] = 2 (b − c) a − c 3 3 (a − b) (a − x) (b − x) (x − c) c 2

 2 a + b2 − ac − bc − u(a + b − 2c) u−c + 2 (a − b) (b − c)(a − c) (a − u)(b − u) [a > b > u > c] BY (233.11)  ∞ dx 2 √  [(a − b) F (ν, q) − (a + b − 2c) E (ν, q)] = 2 3 3 (x − a) (x − b) (x − c) (a − b) (b − c) a − c u 2(2u − a − b)  + 2 (a − b) (u − a)(u − b)(u − c) BY (238.15) [u > a > b > c] dx  3 (a − x) (b − x)3 (c − x)3 −∞ =

2

 (a − − c)2 (a − c)3 

× (b − c)(a + b − 2c) F (α, p) − 2 c2 + a2 + b2 − ab − ac − bc E (α, p) b)2 (b

+

2[c(a − c) + b(a − b) − u(2a − c − b)]  (a − b)(a − c)(b − c)2 (a − u)(b − u)(c − u) [a > b > c > u]

BY (231.14)

264

Power and Algebraic Functions



∞

2. u

dx  3 (x − a) (x − b)3 (x − c)3 =

2

 (a − − c)2 (a − c)3 

× (a − b)(2a − b − c) F (ν, q) − 2 a2 + b2 + c2 − ab − ac − bc E (ν, q) b)2 (b

+

3.137 1.6

2.

3.

u



c

BY (238.16)

[a > b > c ≥ u]

BY (231.15)

2(c − b) dx √  = (r −b)(r − c) a− c u (r − x) (a − x)(b − x)(c − x) 2 r−b √ ,p + F (β, p) × Π β, r−c (r − b) a − c [a > b > c > u, r = 0]    u dx b−c 2 √  ,q Π γ, = r−c (r − c) a − c c (r − x) (a − x)(b − x)(x − c) [a > b ≥ u > c, b

4. u

6.8

2[u(a + b − 2c) − a(a − c) − b(b − c)]  (a − b)2 (a − c)(b − c) (u − a)(u − b)(u − c) [u > a > b > c]

    dx a−r 2 √  , p − F (α, p) Π α, = a−c (a − r) a − c −∞ (r − x) (a − x)(b − x)(c − x)





5.

3.137



u



u

r = c]

dx 2 √  = (r − x) (a − x)(b − x)(x − c) (r− a)(r − b)  a−c   r−a × (b − a) Π δ, q 2 , q + (r − b) F (δ, q) r−b [a > b > u ≥ c, r = b]

BY (232.17)

BY (233.02)

BY (234.18)

2 dx √  = (c − r)(b − r) b (x − r) (a − x)(x − b)(x − c)     a−c 2c − r , p + (b − r) F (κ, p) × (c − b) Π κ, p b−r [a ≥ u > b > c, r = b] BY (235.17)    a dx a−b 2 √  ,p Π λ, = a −r (a − r) a − c u (x − r) (a − x)(x − b)(x − c)

7. a

[a > u ≥ b > c,

r = a]

2 dx √  = (x − r) (x − a)(x − b)(x − c) (b− r)(a − r)    a−c b−r , q + (a − p) F (μ, q) × (b − a) Π μ, a−b [u > a > b > c, r = a]

BY (236.02)

BY (237.17)

3.139

Square roots of polynomials



∞

8. u

    r−c 2 √  , q − F (ν, q) Π ν, = a−c (r − c) a − c (x − r) (x − a)(x − b)(x − c) dx

[u ≥ a > b > c] 3.138



u

1. 0

 2.

3.

4.

5.

 √ dx  = 2 F arcsin u, k x(1 − x) (1 − k 2 x)

[0 < u < 1]

 √ dx  [0 < u < 1]  2 = 2 F arccos u, k u x(1 − x) k  + k 2 x √    1 dx 1−u  , k [0 < u < 1] = 2 F arcsin  k u x(1 − x) x − k  2  u  √ dx  [0 < u < 1]  = 2 F arctan u, k 2 0 x(1 + x) 1 + k  x  u  √ dx  = F 2 arctan u, k  2 0 x 1 + x2 + 2 k  − k 2 x 

1

u

dx 

=F x k  2 (1 + x2 ) + 2 (1 + k 2 ) x

π 2

[0 < u < 1]  √ − 2 arctan u, k

u

[0 < u < 1]     dx u−α p+m−α 1  , = √ F 2 arctan p p 2p (x − α) [(x − m)2 + n2 ]

a

[α < u]     dx α−u p−m+α 1  , = √ F 2 arccot p p 2p (α − x) [(x − m)2 + n2 ]

7. a



BY (238.06)

PE (532), JA

1

6.



265

8. u

PE(533)

PE (534)

PE (535)

JA

JA

[u < α]  where p = (m − α)2 + n2 .

√ 1− 3−u √ , 3.139 Notation α = arccos 1+ 3−u √ 3+1−u , γ = arccos √ 3−1+u  u dx 1 √ = √ F (α, sin 75◦ ) 1. 4 3 3 −∞ 1 − x  1 dx 1 √ F (β, sin 75◦ ) 2. = √ 4 3 1 − x 3 u

√ 3−1+u β = arccos √ , 3+1−u √ u−1− 3 √ . δ = arccos u−1+ 3 H 66 (285)

H 65 (284)

266

3. 4. 5. 6.12 7. 8. 9. 10. 11.

Power and Algebraic Functions



u



∞

3.139

dx 1 √ H 65 (283) = √ F (γ, sin 15◦ ) 4 3 x −1 3 1  ∞ dx 1 √ F (δ, sin 15◦ ) H 65 (282) = √ 4 3 3 x −1 u   3  1 dx 1 1 √ √ √ Γ = MO 9 3 3 3 2π 3 2 1−x 0 √   2  1 x dx 2 1 3 √ Γ = √ MO 9 3 3 π 4 3 1−x 0  1   1 √ 4 1 − x3 dx = 27 F (β, sin 75◦ ) − 2u 1 − u3 BY (244.01) 5 u √  1   1 √ x dx 1 2 1 − u3 4 √ = 3− 4 − 3 4 F (β, sin 75◦ ) + 2 3 E (β, sin 75◦ ) − √ BY (244.05) 1 − x3 3+1−u u √   1 m x dx 2(m − 2) 1 xm−3 dx 2um−2 1 − u3 √ √ + = BY (244.07) 2m − 1 2m − 1 u 1 − x3 1 − x3 u √  u   1 √ x dx 1 2 u3 − 1 4 −4 ◦ ◦ 4 √ BY (240.05) = 3 + 3 F (γ, sin 15 ) − 2 3 E (γ, sin 15 ) + √ x3 − 1 3−1+u 1 √  u dx 1 + u + u2 1 2 ◦ ◦ √ √ √  √ = √ ) − 2 E (α, sin 75 )] + [F (α, sin 75 4 3 27 3 1+ 3−u 1−u −∞ (1 − x) 1 − x [u = 1]

12. u

√

BY (246.06)

1 2 dx 1 + u + u2 √ √ √ = √ [F (δ, sin 15◦ ) − 2 E (δ, sin 15◦ )] + √  4 (x − 1) x3 − 1 27 3 u−1+ 3 u−1

[u = 1] √ u (1 − x) dx 2− 3 [F (α, sin 75◦ ) − E (α, sin 75◦ )] = √ √  2 √ 4 27 −∞ 1 + 3−x 1 − x3 √  1 (1 − x) dx 2− 3 √ [F (β, sin 75◦ ) − E (β, sin 75◦ )] = 4 √  2 √ 27 u 1+ 3−x 1 − x3 √ √ √  u 2 3−2 (x − 1) dx u3 − 1 2− 3 √ √ − 4 = E (γ, sin 15◦ ) √  2 √ 2 − 2u − 2 u 3 3 27 1 1+ 3−x x −1 √ √  √  ∞ 2 2− 3 (x − 1) dx u3 − 1 2− 3 √ √ − = E (δ, sin 15◦ ) √  2 √ 4 2 − 2u − 2 u 3 3 27 u 1+ 3−x x −1  √  √ √  u (1 − x) dx 2 + 3 2 4 3 1 − u3 − E (α, sin 75◦ ) = √ √  2 √ 4 u2 − 2u − 2 27 −∞ 1 − 3−x 1 − x3 √  u (x − 1) dx 2+ 3 √ = 4 [F (γ, sin 15◦ ) − E (γ, sin 15◦ )] √  2 √ 3 27 1 1− 3−x x −1

BY (242.03)

 13.

14.

15.

16.

17.

18.

BY (246.07)

BY (244.04)

BY (240.08)

BY (242.07)

BY (246.08)

BY (240.04)

3.141

Square roots of polynomials

√ (x − 1) dx 2+ 3 = √ [F (δ, sin 15◦ ) − E (δ, sin 15◦ )] √  2 √ 4 3 27 u 1− 3−x x −1  2  u x + x + 1 dx 1 = √ E (α, sin 75◦ ) √  2 √ 4 3 −∞ 1 + 3−x 1 − x3  2  1 x + x + 1 dx 1 = √ E (β, sin 75◦ ) √ 2 √  4 3 3 u x−1+ 3 1−x  2  u x + x + 1 dx 1 = √ E (γ, sin 15◦ ) √ 2 √ 4 3 3 1 3+x−1 x −1  2  ∞ x + x + 1 dx 1 = √ E (δ, sin 15◦ ) √ 2 √  4 3 3 u x−1+ 3 x −1 √  u (x − 1) dx 2+ 3 4 ◦ √ E (γ, sin 15 ) − √ F (γ, sin 15◦ ) = √ 4 4 2 3 27 √ 27 1 (x + x + 1) x − 1 √ 2 − 3 2(u − 1) 3 + 1 − u − √ √ √ 3 3 − 1 + u u3 − 1 

19.

20.

21.

22.

23.

24.

267

∞

BY (242.05)

BY (246.01)

BY (244.02)

BY (240.01)

BY (242.01)

BY (240.09)



25.

2  u  1 + 3 − x dx 1  √ Π α, p2 , sin 75◦ = √ √ √ 2 4 3 −∞ 1 + 3 − x − 4 3p2 (1 − x) 1 − x3 

1

26. u



u

27. 1



u

BY (246.02)

√ 2  1 + 3 − x dx 1  √ Π β, p2 , sin 75◦ = √ √ √ 2 4 3 1 + 3 − x − 4 3p2 (1 − x) 1 − x3

BY (244.03)

√ 2  1 − 3 − x dx 1 √ = √ Π γ, p2 , sin 15◦ √ √ 2 4 3 1 − 3 − x − 4 3p2 (x − 1) x3 − 1

BY (240.02)

√ 2  1 − 3 − x dx 1  √ = √ Π δ, p2 , sin 15◦ √ √ 2 4 3 1 − 3 − x − 4 3p2 (x − 1) x3 − 1

BY (242.02)







∞

28.

3.141

√



Notation: In 3.141 and 3.142 we set:    a−c c−u u−c , β = arcsin , γ = arcsin , α = arcsin a−u b−u b−c    (a − c)(b − u) (a − c)(u − b) a−u , κ = arcsin , λ = arcsin , δ = arcsin (b − c)(a − u) (a − b)(u − c) a−b     u−a a−c a−b b−c , ν = arcsin , p= , q= . μ = arcsin u−b u−c a−c a−c

268

Power and Algebraic Functions

 c 1. u

√ a−x dx = 2 a − c [F (β, p) − E (β, p)] + 2 (b − x)(c − x)

 u c

√ a−x dx = 2 a − c E (γ, q) (b − x)(x − c)

u

√ a−x dx = 2 a − c E (δ, q) − 2 (b − x)(x − c)

2.  b 3.

 u 4. b

 a u

 u 6. a

7. u

8. c

9. u

[a > b ≥ u > c]

BY (233.01)

(b − u)(u − c) a−u

[a > b > u ≥ c]  √ a−x (a − u)(u − b) dx = 2 a − c [F (κ, p) − E (κ, p)] + 2 (x − b)(x − c) u−c

BY (234.06)

[a ≥ u > b > c]

BY (235.07)

√ x−a dx = −2 a − c E (μ, q) + 2 (x − b)(x − c)



[a > u ≥ b > c]

BY (236.04)

(u − a)(u − c) u−b

[a > b > c > u]

BY (237.03)

BY (232.07)

√ b−x 2(a − b) dx = 2 a − c E (γ, q) − √ F (γ, q) (a − x)(x − c) a−c BY (233.04)

[a > b > u ≥ c]  √ x−b (a − u)(u − b) 2(b − c) dx = 2 a − c E (κ, p) − √ F (κ, p) − 2 (a − x)(x − c) u−c a−c

BY (234.07)

10. b

 a 11. u

 u a

BY (232.06)

[a > b ≥ u > c]  √ b−x (b − u)(u − c) 2(a − b) dx = 2 a − c E (δ, q) − √ F (δ, q) − 2 (a − x)(x − c) a−u a−c

 u

12.

[a > b > c > u]

[u > a > b > c]  √ b−x (a − u)(c − u) 2(b − c) dx = √ F (β, p) − 2 a − c E (β, p) + 2 (a − x)(c − x) b−u a−c

 u

 b

(a − u)(c − u) b−u

√ a−x dx = 2 a − c [F (λ, p) − E (λ, p)] (x − b)(x − c)

5.

 c





3.141

[a ≥ u > b > c]

BY (235.06)

√ x−b 2(b − c) dx = 2 a − c E (λ, p) − √ F (λ, p) (a − x)(x − c) a−c [a > u ≥ b > c]  √ x−b (u − a)(u − c) 2(a − b) dx = √ F (μ, q) − 2 a − c E (μ, q) + 2 (x − a)(x − c) u−b a−c [u > a > b > c]

BY (236.03)

BY (237.04)

3.141

Square roots of polynomials

 c 13. u

√ c−x dx = −2 a − c E (β, p) + 2 (a − x)(b − x)

 u 14. c

 b 15. u

16. b

 a

[a > b > u ≥ c]

BY (233.03)

BY (234.08)

(a − u)(u − b) u−c [a ≥ u > b > c]

BY (235.07)

[a > u ≥ b > c]

BY (236.01)

a

 c

22.

√ x−c dx = 2 a − c E (κ, p) − 2 (a − x)(x − b)



√ x−c dx = 2 a − c [F (μ, q) − E (μ, q)] + 2 (x − a)(x − b)

18.

BY (232.08)

√ x−c dx = 2 a − c [F (γ, q) − E (γ, q)] (a − x)(b − x)

u

 u

21.11

[a > b > c > u]

√ x−c dx = 2 a − c E (λ, p) (a − x)(x − b)

17.

20.

(a − u)(c − u) b−u

[a > b ≥ u > c]  √ x−c (b − u)(u − c) dx = 2 a − c [F (δ, q) − E (δ, q)] + 2 (a − x)(b − x) a−u

 u

19.



269



(u − a)(u − c) u−b

[u > a > b > c]

(b − x)(c − x) 2√ dx = a − c [(2a − b − c) E (β, p) − (b − c) F (β, p)] a − x 3  u (a − u)(c − u) 2 + (2b − 2a + c − u) 3 b−u [a > b > c > u]  u (x − c)(b − x) 2√ dx = a − c [(2a − b − c) E (γ, q) − 2(a − b) F (γ, q)] a−x 3 c  2 (a − u)(b − u)(u − c) − 3 [a > b ≥ u > c]  b (x − c)(b − x) 2√ dx = a − c [2(b − a) F (δ, q) + (2a − b − c) E (δ, q)] a−x 3  u (b − u)(u − c) 2 + (b + c − a − u) 3 a−u [a > b > u ≥ c]  u (x − b)(x − c) 2√ dx = a − c [(2a − b − c) E (κ, p) − (b − c) F (κ, p)] a−x 3  b (a − u)(u − b) 2 + (b + 2c − 2a − u) 3 u−c [a ≥ u > b > c]

BY (237.05)

BY (232.11)

BY (233.06)

BY (234.11)

BY (235.10)

270

Power and Algebraic Functions

 a 23.

11 u

 u 24. a

 c 25.

26.

27.

28.

29.

30.11

3.141

(x − b)(x − c) 2√ dx = a − c [(2a − b − c) E (λ, p) − (b − c) F (λ, p)] a−x 3  2 (a − u)(u − b)(u − c) + 3 [a > u ≥ b > c]

BY (236.07)

(x − b)(x − c) 2√ dx = a − c [2(a − b) F (μ, q) + (b + c − 2a) E (μ, q)] x−a 3  (u − a)(u − b) 2 + (u + 2a − 2b − c) 3 u−c [u > a > b > c]

BY (237.08)

2√ (a − x)(c − x) dx = a − c [(2b − a − c) E (β, p) − (b − c) F (β, p)] b−x 3  u (a − u)(c − u) 2 + (a + c − b − u) 3 b−u [a > b > c > u]  u (a − x)(x − c) 2√ dx = a − c [(2b − a − c) E (γ, q) + (a − b) F (γ, q)] b−x 3 c  2 (a − u)(b − u)(u − c) − 3 [a > b ≥ u > c]   b (a − x)(x − c) 2√ dx = a − c [(a − b) F (δ, q) + (2b − a − c) E (δ, q)] b−x 3  u (b − u)(u − c) 2 + (2a + c − 2b − u) 3 a−u [a > b > u ≥ c]  u (a − x)(x − c) 2√ dx = a − c [(b − c) F (κ, p) + (a + c − 2b) E (κ, p)] x−b 3  b 2 (a − u)(u − b) + (2b − a − 2c + u) 3 u−c [a ≥ u > b > c]  a (a − x)(x − c) 2√ dx = a − c [(a + c − 2b) E (λ, p) + (b − c) F (λ, p)] x−b 3 u 2 (a − u)(u − b)(u − c) − 3 [a > u ≥ b > c]  u (x − a)(x − c) 2√ dx = a − c [(a + c − 2b) E (μ, q) − (a − b) F (μ, q)] x − b 3  a (u − a)(u − c) 2 + (u + b − a − c) 3 u−b [u > a > b > c]

BY (232.10)

BY (233.05)

BY (234.10)

BY (235.11)

BY (236.06)

BY (237.06)

3.142

Square roots of polynomials

271

 c 31.

32.

33.

34.

35.

36.

3.142

(a − x)(b − x) 2√ dx = a − c [2(b − c) F (β, p) + (2c − a − b) E (β, p)] c−x 3  u (a − u)(c − u) 2 + (a + 2b − 2c − u) 3 b−u [a > b > c > u]  u (a − x)(b − x) 2√ a − c [(a + b − 2c) E (γ, q) − (a − b) F (γ, q)] dx = x−c 3 c  2 (a − u)(b − u)(u − c) + 3 [a > b ≥ u > c]  b (a − x)(b − x) 2√ dx = a − c [(a + b − 2c) E (δ, q) − (a − b) F (δ, q)] x − c 3  u (b − u)(u − c) 2 + (2c − 2a − b + u) 3 a−u [a > b > u ≥ c]  u (a − x)(x − b) 2√ dx = a − c [(a + b − 2c) E (κ, p) − 2(b − c) F (κ, p)] x−c 3  b (a − u)(u − b) 2 + (u + c − a − b) 3 u−c [a ≥ u > b > c]  a 2√ (a − x)(x − b) dx = a − c [(a + b − 2c) E (λ, p) − 2(b − c) F (λ, p)] x−c 3 u  2 (a − u)(u − b)(u − c) − 3 [a > u ≥ b > c]  u (x − a)(x − b) 2√ dx = a − c [(a + b − 2c) E (μ, q) − (a − b) F (μ, q)] x−c 3  a (u − a)(u − c) 2 + (u + 2c − a − 2b) 3 u−b [u > a > b > c] 

u



1. −∞

 b 2. u

+2

3. b

BY (233.07)

BY (234.09)

BY (235.09)

BY (236.05)

BY (237.07)

 √ a−x b−u 2(a − c) 2 a−c 2 E (α, p) + F (α, p) − dx = √ (b − x)(c − x)3 b−c b−c (a − u)(c − u) a−c

[a > b > c > u] √ a−x 2 a−c a−b √ E (δ, q) F (δ, q) − dx = 2 3 (b − x)(x − c) b−c (b − c) a − c

 u

BY (232.09)

a−c b−c

BY (231.05)

b−u (a − u)(u − c)

[a > b > u > c] √ a−x 2 2 a−c E (κ, p) − √ F (κ, p) dx = (x − b)(x − c)3 b−c a−c [a ≥ u > b > c]

BY (234.13)

BY (235.12)

272

Power and Algebraic Functions

 √ a−x (a − u)(u − b) 2 2 2 a−c E (λ, p) − √ F (λ, p) − dx = (x − b)(x − c)3 b−c b−c u−c a−c

 a 4. u

 u 5. a

BY (236.12) [a > u ≥ b > c] √  x−a u−a 2(a − b) 2 a−c √ E (μ, q) − F (μ, q) − 2 dx = 3 (x − b)(x − c) b−c (u − b)(u − c) (b − c) a − c

[u > a > b > c] √ x−a 2(a − b) 2 a−c √ E (ν, q) − F (ν, q) dx = 3 (x − b)(x − c) b−c (b − c) a − c

BY (237.10)

[u ≥ a > b > c] √  a−x c−u 2 a−c a−b dx = E (α, p) − 2 3 (b − x) (c − x) b−c b − c (a − u)(b − u)

BY (238.09)

[a > b > c ≥ u] √ a−x 2 a−c dx = E (β, p) [a > b > c > u] 3 (b − x) (c − x) b−c u  √  u a−x (a − u)(u − c) 2 a−c 2 dx = [F (γ, q) − E (γ, q)] + 3 (b − x) (x − c) b−c b−c b−u c

BY (231.03)



∞

6. u



u



7. −∞

 c 8. 9.

3.142

 a 10. u

 u 11. a



u



u

−∞

 c 14. u

[a > u > b > c]

BY (236.11)

√ x−a 2 a−c dx = [F (μ, q) − E (μ, q)] (x − b)3 (x − c) b−c

[u > a > b > c] √  x−a u−a 2 a−c dx = [F (ν, q) − E (ν, q)] + 2 3 (x − b) (x − c) b−c (u − b)(u − c) b−x 2 dx = √ E (α, p) (a − x)3 (c − x) a−c

b−x 2 2(a − b) dx = √ E (β, p) − (a − x)3 (c − x) a−c a−c

 u c

BY (233.15)



13.

15.

[a > b > u > c]  √ a−x (a − u)(u − c) 2 a−c 2 dx = E (λ, p) + 3 (x − b) (x − c) c−b b−c u−b

∞

12.

BY (232.01)

BY (237.09)

[u ≥ a > b > c]

BY (238.10)

[a > b > c ≥ u]

BY (231.01)



c−u (a − u)(b − u)

[a > b > c > u]  b−x (b − u)(u − c) 2 2 dx = √ [F (γ, q) − E (γ, q)] + 3 (a − x) (x − c) a−c a−u a−c [a > b ≥ u > c]

BY (232.05)

BY (233.13)

3.142

Square roots of polynomials

 b 16. u

2 b−x dx = √ [F (δ, q) − E (δ, q)] (a − x)3 (x − c) a−c

 u 17.

18.

x−b 2 dx = − √ E (κ, p) + 2 3 (a − x) (x − c) a−c



u

19. −∞

 b 20. u

21. b

22. u

 u a



25.

∞



[a > b > c > u]

[a > b > u > c]

[a > u ≥ b > c] x−b b−c 2 E (μ, q) − 2 dx = √ 3 (x − a)(x − c) a−c a−c



u

BY (238.07)

BY (231.04)

BY (234.14)

BY (235.03)

BY (236.14)

u−a (u − b)(u − c) [u > a > b > c]

x−b 2 E (ν, q) [u ≥ a > b > c] dx = √ 3 (x − a)(x − c) a−c u √  u  c−x 2 a−c 2(b − c) √ dx = E (α, p) − F (α, p) 3 (b − x) (a − x) a − b (a − b) a − c −∞

26.

BY (235.08)

x−b 2 [F (κ, p) − E (κ, p)] dx = √ (a − x)(x − c)3 a−c



 c

BY (234.15)

b−u (a − u)(u − c)

[a ≥ u > b > c]  x−b (a − u)(u − b) 2 2 √ [F (λ, p) − E (λ, p)] + dx = (a − x)(x − c)3 a−c u−c a−c

 a

23.

[u > a > b > c]  b−x b−u 2 [F (α, p) − E (α, p)] + 2 dx = √ (a − x)(c − x)3 (a − u)(c − u) a−c

b−x 2 E (δ, q) + 2 dx = − √ 3 (a − x)(x − c) a−c

 u

24.

[a > b > u ≥ c]



u−b (a − u)(u − c) b [a > u > b > c]    ∞ x−b u−b 2 dx = √ [F (ν, q) − E (ν, q)] + 2 3 (x − c) (x − a) (u − a)(u − c) a − c u 

273

BY (237.11) BY (238.01)

[a > b > c ≥ u] BY (231.07) √  c−x c−u 2 a−c 2(b − c) √ dx = E (β, p) − F (β, p) − 2 (a − x)3 (b − x) a−b (a − u)(b − u) (a − b) a − c [a > b > c > u]

BY (232.03)

274

Power and Algebraic Functions

 u 27. c

 b 28. u

3.143

 √ x−c (b − u)(u − c) 2 a−c 2 2 dx = E (γ, q) − √ F (γ, q) − (a − x)3 (b − x) a−b a−b a−u a−c

[a > b ≥ u > c] √ x−c 2 a−c 2 dx = E (δ, q) − √ F (δ, q) (a − x)3 (b − x) a−b a−c

BY (233.14)

BY (234.20) [a > b > u ≥ c] √ x−c 2(b − c) 2 a−c √ dx = E (κ, p) F (κ, p) − 3 (a − x) (x − b) a−b (a − b)  a−c b u−b a−c +2 a − b (a − u)(u − c) [a > u > b > c] BY (235.13)  √  ∞ x−c u−b 2 2(a − c) 2 a−c dx = √ E (ν, q) + F (ν, q) − 3 (x − b) (x − a) a − b a − b (u − a)(u − c) a−c u

 u 29.

30.





u

31. −∞

 c 32. u

[u > a > b > c] √  c−x c−u 2 a−c [F (α, p) − E (α, p)] + 2 dx = (a − x)(b − x)3 a−b (a − u)(b − u) [a > b > c ≥ u]

BY (231.06)

[a > b > c > u]  √ 2 a−c x−c (a − u)(u − c) 2 dx = − E (γ, q) + (a − x)(b − x)3 a−b a−b b−u

BY (232.04)

[a > b > u > c]  √ x−c (a − u)(u − c) 2 2 a−c [F (λ, p) − E (λ, p)] + dx = (a − x)(x − b)3 a−b a−b u−b

BY (233.16)

√ c−x 2 a−c [F (β, p) − E (β, p)] dx = (a − x)(b − x)3 a−b

 u 33. c

 a 34. u

[a > u > b > c] √ x−c 2 a−c E (μ, q) [u > a > b > c] dx = 3 (x − a)(x − b) a−b a √   ∞ x−c u−a b−c 2 a−c E (ν, q) − 2 dx = 3 (x − a)(x − b) a−b a − b (u − b)(u − c) u

BY (236.13)

[u ≥ a > b > c]

BY (238.11)

 u 35. 36.

3.143

BY (238.08)



1

1.6 u

dx 1 √ = F 4 2 1+x



 √  √  1 + 2 (1 − u) √ 4 arctan ,2 2 2−1 (1 + u)

BY (237.01)

H 66 (286)

3.144

Square roots of polynomials



∞

2. u

3.144

1.

2.



∞

u



∞

4. u



6.



√  2 u2 − 1 arccos 2 , u +1 2

H 66 (287)

1 Notation: α = arcsin √ . u2 − u + 1  √   ∞ dx 3  [u ≥ 1] = F α, 2 2 x(x − 1) (x − x + 1) u  √   ∞ 2(2u − 1) dx 3  = − 4 E α, 3 3 2 2 2 x (x − 1) (x − x + 1) u(u − 1) (u − u + 1) u

3.12

5.

dx 1 √ = F 2 1 + x4

 

(2x − 1)2 dx  =4 F x3 (x − 1)3 (x2 − x + 1)

 

4 dx  F = 3 x(x − 1) (x2 − x + 1)3

[u > 1]  √  √  3 3 α, − E α, 2 2

√  3 α, 2

∞

[u ≥ 1]

(2x − 1)2 dx  [u > 1] = 4E u x(x − 1) (x2 − x + 1)3  √   √   ∞ x(x − 1) 3 3 1 4 α, − F α, 3 dx = 3 E 2 2 3 2 (x − x + 1) u 

∞

7. u



∞

8. u



dx (2x − 1)2

 dx (2x − 1)2

∞

9. u

(2x −

1)2

 

x(x − 1) 1 = F 2 x −x+1 3

x2 − x + 1 =E x(x − 1)

BY (261.50)

[u > 1] BY (261.54)  √   √  2u − 1 3 3 − E α, +  α, 2 2 2 u(u − 1) (u2 − u + 1)





275

BY (261.56)

BY (261.52) BY (261.51)

[u > 1] BY (261.53)  √   √  3 3 u(u − 1) 1 α, − E α, + 2 2 2(2u − 1) u2 − u + 1

[u > 1]   √ 3 u(u − 1) 3 α, − 2 2(2u − 1) u2 − u + 1



dx 4  = E 2 3 x(x − 1) (x − x + 1)



BY (261.57)

[u > 1] BY (261.58)  √   √  3 3 u(u − 1) 1 2 α, − F α, − 2 3 2 2u − 1 u2 − u + 1 [u > 1]

BY (261.55)

276

Power and Algebraic Functions



∞

10. u



∞

11. u



∞

12. u

3.145

dx

40  E = 5 5 2 3 x (x − 1) (x − x + 1)

44 dx  F = 27 x(x − 1) (x2 − x + 1)5





3.145

 √   √  2(2u − 1) 9u2 − 9u − 1 3 3 4 α, − F α, −  2 3 2 3 u3 (u − 1)3 (u2 − u + 1)

[u > 1] BY (261.54)  √   √  2(2u − 1) u(u − 1) 56 3 3  α, − E α, + 2 27 2 9 (u2 − u + 1)3

[u > 1]  √  √  dx 3 3 1 16  E α, − F α, = 4 2 2 27 2 (2x − 1) x(x − 1) (x − x + 1) 27   2 8 5u − 5u + 2 u(u − 1) − 9(2u − 1)3 u2 − u + 1 [u > 1] 

BY (261.52)

BY (261.55)

For the following, define (m − α)2 + n2 = p2 , and (m − β)2 + n2 = q 2 .∗ u dx  1.12 (x − α)(x − β) [(x − m)2 + n2 ] α  φ dθ 1  = √ pq 0 1 − k 2 sin2 θ 1 [2m > α + β, u < u∗ ] or [2m < α + β] = √ F (φ, k) pq 1 [2m > α + β, u > u∗ ] = √ [2K(k) − F (π − φ, k)] pq   q(u − α) 1 (p + q)2 − (α − β)2 pβ − qα , k= , u∗ = for β < α < u, φ = 2 arctan q(u − β) 2 pq p−q  u dx  2.12 (α − x)(x − β) [(x − m)2 + n2 ] β  ψ dθ 1  = √ pq 0 1 − κ2 sin2 θ 1 [β < u < u ¯ < α] = √ F (ψ, κ) pq 1 = √ [2K(κ) − F (π − ψ, κ)] [β < u ¯ < u < α] pq   p(u − β) 1 (α − β)2 − (p − q)2 pβ − qα for ψ = 2 arctan , κ= , u ¯= q(α − u) 2 pq p+q ∗ Formulas

3.145 are not valid for α + β = 2m. In this case, we make the substitution x − m = z, which leads to one of the formulas 3.152.

3.146

Square roots of polynomials



3.

β

dx  2 2 u (xω − α)(x − β) [(x − m) + n ] dθ 1  = √ pq 0 1 − λ2 sin2 θ 1 = √ F (ω, λ) pq 1 = √ [2K(λ) − F (π − ω, λ)] pq  q(α − u) for u < β < α, ω = 2 arctan , p(β − u)

12



[2m < α + β,

3(2).

u ˆ < u]

4.

Set

[2m > α + β]

u ˆ=

pβ − qα p−q

a

[β < u ˜ < u < α] [β < u < u ˜ < α]  1 Λ= 2

(α − β)2 − (p − q)2 , pq 

12

or

[2m < α + β, u < u ˆ]  1 (p + q)2 − (α − β)2 λ= , 2 pq

dx  (α − x)(x − β)[(x − m)2 + n2 ] u  Ω dθ 1  = √ pq 0 1 − Λ2 sin2 θ 1 = √ F (Ω, Λ) pq 1 = √ [2K (Λ) − F (π − Ω, Λ)] pq  q(α − u) for β < u < α, Ω = 2 arctan , p(u − β)

∗

277

2

2

2

2

(m1 − m) + (n1 + n) = p , (m1 − m) + (n1 − n) = 2

p21 , cot α

=

u ˜=

pβ + qα p+q

2

(p + p1 ) − 4n2 4n2 − (p − p1 )

2;

(Note that m − n tan α and m + n cot α are the points of intersection in the complex plane of the real line and the circle passing through the 4 points {m ± in, m1 ± in1 }) then    u √ dx u − m 2 pp1 2  , F α + arctan =  p+p  n p + p1 1 m−n tan α 2 [(x − m)2 + n2 ] (x − m1 ) + n21 [m − n tan α < u < m + n cot α, 3.146



1

1. 0



1

2. 0

 3. 0

1

1 dx π 1√ √ = + 2K 4 1+x 8 4 1 − x4

√  2 2

x2 dx π √ = 1 + x4 1 − x4 8 x4 dx π 1√ √ + = − 2K 1 + x4 1 − x4 8 4

m1 > 1,

n > 0]

BI (13)(6)

BI (13)(7)

√  2 2

BI (13)(8)

278

Power and Algebraic Functions

3.147



3.147

(a − c)(d − u) , (a − d)(c − u)   (a − c)(u − d) (b − d)(c − u) , γ = arcsin , β = arcsin (c − d)(a − u) (c − d)(b − u)   (b − d)(u − c) (a − c)(b − u) , κ = arcsin , δ = arcsin (b − c)(u − d) (b − c)(a − u)   (a − c)(u − b) (b − d)(a − u) , μ = arcsin , λ = arcsin (a − b)(u − c) (a − b)(u − d)    (b − d)(u − a) (b − c)(a − d) (a − b)(c − d) , q= , r= . ν = arcsin (a − d)(u − b) (a − c)(b − d) (a − c)(b − d)

Notation: In 3.147–3.151 we set: α = arcsin



d

1. u



[a > b > c > d > u] u

2. d



u

4. c

 5.

u

 6. b

 7.11

u

 8. a



BY (256.00)

dx 2 =  F (μ, r) (a − x)(x − b)(x − c)(x − d) (a − c)(b − d) [a > u ≥ b > c > d]

u

BY (255.00)

2 dx  =  F (λ, r) (a − x)(x − b)(x − c)(x − d) (a − c)(b − d) [a ≥ u > b > c > d]

a

BY (254.00)

dx 2  =  F (κ, q) (a − x)(b − x)(x − c)(x − d) (a − c)(b − d) [a > b > u ≥ c > d]

u

BY (253.00)

dx 2  =  F (δ, q) (a − x)(b − x)(x − c)(x − d) (a − c)(b − d) [a > b ≥ u > c > d]

b

BY (254.00)

dx 2  = F (γ, r) (a − x)(b − x)(c − x)(x − d) (a − c)(b − d) [a > b > c > u ≥ d]

u

BY (251.00)

dx 2  =  F (β, r) (a − x)(b − x)(c − x)(x − d) (a − c)(b − d) [a > b > c ≥ u > d]

c

3. 

dx 2  = F (α, q) (a − x)(b − x)(c − x)(d − x) (a − c)(b − d)

BY (257.00)

dx 2  =  F (ν, q) (x − a)(x − b)(x − c)(x − d) (a − c)(b − d) [u > a > b > c > d]

BY (258.00)

3.149

3.148

Square roots of polynomials



d

1.8 u

x dx 2  = (a − x)(b − x)(c − x)(d − x) (a − c)(b − d)

279

    a−d , q + c F (α, q) (d − c) Π α, a−c

[a > b > c > d > u] BY (251.03)     u x dx 2 d−c   , r + a F (β, r) = (d − a) Π β, a−c (a − x)(b − x)(c − x)(x − d) (a − c)(b − d) d

 2.



c

3. u



u

4. c



b

5. u

[a > b > c ≥ u > d] BY (252.11)     x dx 2 c−d  , r + b F (γ, r) = (c − b) Π γ, b−d (a − x)(b − x)(c − x)(x − d) (a − c)(b − d) [a > b > c > u ≥ d] BY (253.11)     x dx 2 b−c  , q + d F (δ, q) =  (c − d) Π δ, b−d (a − x)(b − x)(x − c)(x − d) (a − c)(b − d) x dx 2  =  (a − x)(b − x)(x − c)(x − d) (a − c)(b − d)



[a > b ≥ u > c > d] BY (254.10)    b−c , q + a F (κ, q) (b − a) Π κ, a−c

[a > b > u ≥ c > d] BY (255.17)     u x dx 2 a−b  , r + c F (λ, r) =  (b − c) Π λ, a−c (a − x)(x − b)(x − c)(x − d) (a − c)(b − d) b

 6.8



a

7. u



u

8. a

[a ≥ u > b > c > d] BY (256.11)     x dx 2 b−a  , r + d F (μ, r) =  (a − d) Π μ, b−d (a − x)(x − b)(x − c)(x − d) (a − c)(b − d) [a > u ≥ b > c > d] BY (257.11)     x dx 2 a−d  , q + b F (ν, q) (a − b) Π ν, =  b−d (x − a)(x − b)(x − c)(x − d) (a − c)(b − d) [u > a > b > c > d]

3.149



d

1. u

dx  x (a − x)(b − x)(c − x)(d − x) =

 2. d

u

2  cd (a − c)(b − d)

dx  x (a − x)(b − x)(c − x)(x − d)

BY (258.11)

    c(a − d) , q + d F (α, q) (c − d) Π α, d(a − c) [a > b > c > d > u] BY (251.04)

    a(d − c) , r + d F (β, r) =  (a − d) Π β, d(a − c) ad (a − c)(b − d) [a > b > c ≥ u > d] BY (252.12) 2

280

Power and Algebraic Functions



c

3. u

dx  x (a − x)(b − x)(c − x)(x − d) =



u

4. c



b

5. u



u

b



a

u

    d(b − c) , q + c F (δ, q) (d − c) Π δ, c(b − d) [a > b ≥ u > c > d] BY (254.11)

    2 a(b − c)  , q + b F (κ, q) × (a − b) Π κ, b(a − c) ab (a − c)(b − d) [a > b > u ≥ c > d] BY (255.18)

dx  x (a − x)(x − b)(x − c)(x − d) =

7.

2  cd (a − c)(b − d)

dx  x (a − x)(b − x)(x − c)(x − d) =

6.

   b(c − d) , r + c F (γ, r) (b − c) Π γ, c(b − d) [a > b > c > u ≥ d] BY (253.12)



2  bc (a − c)(b − d)

dx  x (a − x)(b − x)(x − c)(x − d) =

3.151

    2 c(a − b)  , r + b F (λ, r) × (c − b) Π λ, b(a − c) bc (a − c)(b − d) [a ≥ u > b > c > d] BY (256.12)

dx  x (a − x)(x − b)(x − c)(x − d)

    d(b − a) , r + a F (μ, r) =  × (d − a) Π μ, a(b − d) ad (a − c)(b − d) [a > u ≥ b > c > d] BY (257.12) 2



u

8. a

dx  x (x − a)(x − b)(x − c)(x − d) 2 =  ab (a − c)(b − d)

3.151



d

1. u

    b(a − d) , q + a F (ν, q) (b − a) Π ν, a(b − d) [u > a > b > c > d] BY (258.12)

dx  (p − x) (a − x)(b − x)(c − x)(d − x) =

2  (p− c)(p − d)   (a − c)(b − d)  (a − d)(p − c) , q + (p − d) F (α, q) × (d − c) Π α, (a − c)(p − d) [a > b > c > d > u, p = d] BY (251.39)

3.151

Square roots of polynomials



u

2. d

dx  (p − x) (a − x)(b − x)(c − x)(x − d) =



c

3. u



u

4. c



b

5. u



u

b

2  (p− a)(p − d)   (a − c)(b − d)  (d − c)(p − a) , r + (p − d) F (β, r) × (d − a) Π β, (a − c)(p − d) [a > b > c ≥ u > d, p = d] BY (252.39)

2 dx   = (p − x) (a − x)(b − x)(c − x)(x − d) (p− b)(p − c)   (a − c)(b − d)  (c − d)(p − b) , r + (p − c) F (γ, r) × (c − b) Π γ, (b − d)(p − c) [a > b > c > u ≥ d, p = c] BY (253.39) dx 2   = (p − x) (a − x)(b − x)(x − c)(x − d) (p− c)(p − d)   (a − c)(b − d)  (b − c)(p − d) , q + (p − c) F (δ, q) × (c − d) Π δ, (b − d)(p − c) [a > b ≥ u > c > d, p = c] BY (254.39) dx  (p − x) (a − x)(b − x)(x − c)(x − d) =

6.

2  (p− a)(p − b)   (a − c)(b − d)  (b − c)(p − a) , q + (p − b) F (κ, q) × (b − a) Π κ, (a − c)(p − b) [a > b > u ≥ c > d, p = b] BY (255.38)

dx  (x − p) (a − x)(x − b)(x − c)(x − d) =



a

7. u

281

2  (b− p)(p − c)   (a − c)(b − d)  (a − b)(p − c) , r + (p − b) F (λ, r) × (b − c) Π λ, (a − c)(p − b) [a ≥ u > b > c > d, p = b] BY (256.39)

dx  (p − x) (a − x)(x − b)(x − c)(x − d) =

2  (p− a)(p − d)   (a − c)(b − d)  (b − a)(p − d) , r + (p − a) F (μ, r) × (a − d) Π μ, (b − d)(p − a) [a > u ≥ b > c > d, p = a] BY (257.39)

282

Power and Algebraic Functions



u

8. a

dx  (p − x) (x − a)(x − b)(x − c)(x − d) =

3.152

3.152

Notation: In 3.152–3.163 we set: 

2  (p− a)(p − b)   (a − c)(b − d)  (a − d)(p − b) , q + (p − a) F (ν, q) × (a − b) Π ν, (b − d)(p − a) [u > a > b > c > d, p = a] BY (258.39) u α = arctan , b

u β = arccot , a

 a2 + b 2 b u ε = arccos , ξ = arcsin δ = arccos , , b u a2 + u 2   a b2 − u2 a u 2 − b2 ζ = arcsin , κ = arcsin , b a2 − u 2 u a2 − b 2  √ a u 2 − a2 a2 − b 2 , q = , μ = arcsin , ν = arcsin u 2 − b2 u a a b b , s= √ , t= . r=√ 2 2 2 2 a a +b a +b

a2 + b 2 , a2 + u 2 u η = arcsin , b  2 a − u2 , λ = arcsin a2 − b 2

u γ = arcsin b



u

1. 0



1 dx  = F (α, q) a (x2 + a2 ) (x2 + b2 )

[a > b > 0]

H 62(258), BY (221.00)

[a > b > 0]

H 63 (259), BY (222.00)

dx 1  F (γ, r) = √ 2 2 2 2 2 a + b2 (x + a ) (b − x )

[b ≥ u > 0]

H 63 (260)

dx 1  F (δ, r) = √ a2 + b 2 (x2 + a2 ) (b2 − x2 )

[b > u ≥ 0]

H 63 (261), BY (213.00)

dx 1  F (ε, s) = √ 2 2 2 2 2 a + b2 (x + a ) (x − b )

[u > b > 0]

H 63 (262), BY (211.00)

[u > b > 0]

H 63 (263), BY (212.00)

1 dx  = F (η, t) 2 2 2 2 a (a − x ) (b − x )

[a > b ≥ u > 0]

H 63 (264), BY (219.00)

dx 1  = F (ζ, t) 2 2 2 2 a (a − x ) (b − x )

[a > b > u ≥ 0]

H 63 (265), BY (220.00)

dx 1  = F (κ, q) 2 2 2 2 a (a − x ) (x − b )

[a ≥ u > b > 0]

H 63 (266), BY (217.00)

∞

2. u



u

3. 0



b

4. u



u

5. b



∞

6. u



u

7. 0



b

8. u

 9.

b

u

1 dx  = F (β, q) 2 2 2 2 a (x + a ) (x + b )

dx 1  F (ξ, s) = √ 2 2 2 2 2 a + b2 (x + a ) (x − b )

3.153

Square roots of polynomials



a

10. u



u

11. a



u

3.153



u

1. 0



u

u

5. 0 b

6. u

b

a

8. u

9.

u

6 a

 10. 0

H 64(269), BY (215.00)

x2 dx  =u 2 (x + a2 ) (x2 + b2 )



a2 + u 2 − a E (α, q) b2 + u2

[u > 0,

a > b] 

2

1

BY (221.09)

b2 − u2 a2 + u 2 BY (214.05)

2

BY (213.06)

 x2 dx 1 2 b2  F (ε, s) − a2 + b2 E (ε, s) + (u + a2 ) (u2 − b2 ) = √ u a2 + b 2 (a2 + x2 ) (x2 − b2 ) x2 dx  = a {F (η, t) − E (η, t)} (a2 − x2 ) (b2 − x2 ) x2 dx



 = a {F (ζ, t) − E (ζ, t)} + u (a2 − x2 ) (b2 − x2 )

[u > b > 0]

BY (211.09)

[a > b ≥ u > 0]

BY (219.05)

b2 − u2 a2 − u 2 [a > b > u ≥ 0]

u

7.



[u ≥ a > b > 0]

dx 1  = F (ν, t) a (x2 − a2 ) (x2 − b2 )

[b > u ≥ 0]

b



H 63 (268), BY (216.00)

 x dx a  F (δ, r) = a2 + b2 E (δ, r) − √ 2 2 2 2 2 a + b2 (a + x ) (b − x )

4.



[u > a > b > 0]

=

[b ≥ u > 0]

u



1 dx  = F (μ, t) 2 2 2 2 a (x − a ) (x − b )

−

b2 )

b

3.



H 63 (257), BY (218.00)

−

x2 ) (x2

 x2 dx a2  F (γ, r) − u = a2 + b2 E (γ, r) − √ a2 + b 2 (a2 + x2 ) (b2 − x2 )

0



[a > u ≥ b > 0]

(a2

u

2.



1 F (λ, q) a



∞

12.

dx

283

BY (220.06)

1 2 x2 dx  (a − u2 ) (u2 − b2 ) = a E (κ, q) − u (a2 − x2 ) (x2 − b2 )



x2 dx (a2 − x2 ) (x2 − b2 )

= a E (λ, q) 

x2 dx

 = a {F (μ, t) − E (μ, t)} + u (x2 − a2 ) (x2 − b2 ) x2 dx 1  = 2 2 2 2 k (1 + x ) (1 + k x )



[a ≥ u > b > 0]

BY (217.05)

[a > u ≥ b > 0]

BY (218.06)

u 2 − a2 u 2 − b2

[u > a > b > 0] ! π   1 + k2 2 −E , 1−k 2 4

BY (216.06) BI (14)(9)

284

3.154

Power and Algebraic Functions



u

1. 0



BY (213.06)

u

 2   x4 dx 1  2b − a2 b2 F (ε, s) + 2 a4 − b4 E (ε, s) = √ 2 2 2 2 2 2 (a + x ) (x − b ) 3 a + b 2b2 − 2a2 + u2  2 (u + a2 ) (u2 − b2 ) + 3u [u > b > 0]

BY (211.09)

u

  u x dx a  2  2a + b2 F (η, t) − 2 a2 + b2 E (η, t) + (a2 − u2 ) (b2 − u2 ) = 2 2 2 2 3 3 (a − x ) (b − x )

u

4. b

5. 0



b

6. u



u

7. b



a

8. u

 9. a

BY (221.09)

 2   x4 dx 1  2a − b2 a2 F (δ, r) − 2 a4 − b4 E (δ, r) = √ 2 2 2 2 2 2 (a + x ) (b − x ) 3 a + b u 2 (a + u2 ) (b2 − u2 ) + 3 [b > u ≥ 0]

b

3.



a2 + u 2 b2 + u2

BY (214.05)

0



u > 0]



 2   1 x4 dx  2a − b2 a2 F (γ, r) − 2 a4 − b4 E (γ, r) = √ (a2 + x2 ) (b2 − x2 ) 3 a2 + b2  b2 − u2 u 2 2 2 − 2b − a + u 3 a2 + u 2 [a ≥ u > 0]

2.



 u 2 a  2  2 a + b2 E (α, q) − b2 F (α, q) + u − 2a2 − b2 = 2 2 2 2 3 3 (x + a ) (x + b ) x4 dx

[a > b, u

3.154

u

4

[a > b ≥ u > 0]

BY (219.05)

  x4 dx a  2  2a + b2 F (ζ, t) − 2 a2 + b2 E (ζ, t) = 2 2 2 2 3 (a − x ) (b − x )  b2 − u2 u 2 2 2 + u + a + 2b 3 a2 − u 2 [a > b > u ≥ 0]

BY (220.06)

 x4 dx a  2  2 a + b2 E (κ, q) − b2 F (κ, q) = 2 2 2 2 (a − x ) (x − b ) 3 2 u + 2a2 + 2b2  2 (a − u2 ) (u2 − b2 ) − 3u [a ≥ u > b > 0]

BY (217.05)



 u x dx a  2 2 a + b2 E (λ, q) − b2 F (λ, q) + (a2 − u2 ) (u2 − b2 ) = 2 2 2 3 3 − x ) (x − b ) 4

(a2

[a > u ≥ b > 0]

BY (218.06)

  x4 dx a  2  2a + b2 F (μ, t) − 2 a2 + b2 E (μ, t) = (x2 − a2 ) (x2 − b2 ) 3  u 2 − a2 u 2 2 2 + u + 2a + b 3 u 2 − b2 [u > a > b > 0]

BY (216.06)

3.155

Square roots of polynomials

285

3.155  a  u a  2 a + b2 E (λ, q) − 2b2 F (λ, q) − (a2 − x2 ) (x2 − b2 ) dx = (a2 − u2 ) (u2 − b2 ) 1. 3 3 u

2.

3.

4.

5.9

6.

7.

8.

9.

[a > u ≥ b > 0]  u   a  2 a + b2 E (μ, t) − a2 − b2 F (μ, t) (x2 − a2 ) (x2 − b2 ) dx = 3  a u 2 − a2 u 2 2 2 + u − a − 2b 3 u 2 − b2 [u > a > b > 0]  u   a 2 2b F (α, q) − a2 + b2 E (α, q) (x2 + a2 ) (x2 + b2 ) dx = 3  0 a2 + u 2 u 2 2 2 + u + a + 2b 3 b2 + u2 [a > b, u > 0]  u

  1 2 (a2 + x2 ) (b2 − x2 ) dx = a + b2 a2 F (γ, r) − a2 − b2 E (γ, r) 3  0 b2 − u2 u 2 2 2 + u + 2a − b 3 a2 + u 2 [a ≥ u > 0]  b 

  1 (a2 + x2 ) (b2 − x2 ) dx = a2 + b2 a2 F (δ, r) + b2 − a2 E (δ, r) 3 u u 2 (a + u2 ) (b2 − u2 ) + 3 [b > u ≥ 0]  u

  1 2 (a2 + x2 ) (x2 − b2 ) dx = a + b2 b2 − a2 E (ε, s) − b2 F (ε, s) 3 b u 2 + a2 − b 2  2 (a + u2 ) (u2 − b2 ) + 3u [u > b > 0]  u   a  2 a + b2 E (η, t) − a2 − b2 F (η, t) (a2 − x2 ) (b2 − x2 ) dx = 3 0 u 2 (a − u2 ) (b2 − u2 ) + 3 [a > b ≥ u > 0]  b   a  2 a + b2 E (ζ, t) − a2 − b2 F (ζ, t) (a2 − x2 ) (b2 − x2 ) dx = 3  u b2 − u2 u 2 2 2 + u − 2a − b 3 a2 − u 2 [a > b > u ≥ 0]  u  a  2 a + b2 E (κ, q) − 2b2 F (κ, q) (a2 − x2 ) (x2 − b2 ) dx = 3 b u 2 − a2 − b 2  2 (a − u2 ) (u2 − b2 ) + 3u [a ≥ u > b > 0]

BY (218.11)

BY (216.10)

BY (221.08)

BY (214.12)

BY (213.13)

BY (211.08)

BY (219.11)

BY (220.05)

BY (217.09)

286

3.156 1.

Power and Algebraic Functions



∞

6 u



u

5.

BY (222.04)

BY (213.09)

 2  dx 1  √ a + b2 E (ε, s) − b2 F (ε, s) = 2 2 2 2 2 2 2 2 a b a +b (x + a ) (x − b ) [u > b > 0]

BY (211.11)

∞

 2  dx 1  √ a + b2 E (γ, s) − b2 F (γ, s) = 2 2 2 2 2 2 2 2 a +b (x + a ) (x − b ) a b  u u 2 − b2 1 − 2 b u a2 + u 2 [u ≥ b > 0]   b dx b2 − u2 1 1  = 2 {F (ζ, t) − E (ζ, t)} + 2 2 (a2 − x2 ) (b2 − x2 ) ab b u a2 − u 2 u x u

6.

x2

b



a

7. u

dx 1  = 2 E (κ, q) 2 2 2 2 ab (a − x ) (x − b )

BY (212.06)

[a > b > u > 0]

BY (220.09)

[a ≥ u > b > 0]

BY (217.01)

dx 1  2 1  (a − u2 ) (u2 − b2 ) = 2 E (λ, q) − 2 2 ab a b u x2 (a2 − x2 ) (x2 − b2 )

[a > u ≥ b > 0]   u dx u 2 − a2 1 1  = 2 {F (μ, t) − E (μ, t)} + 2 2 2 2 2 2 ab a u u 2 − b2 (x − a ) (x − b ) a x

BY (218.12)

[u > a > b > 0]

BY (216.09)

[u ≥ a > b > 0]

BY (215.07)



∞

9.

x2

u

3.157

u > 0]

x2



8.

1 b2 + u2 − 2 E (β, q) a2 + u 2 ab

2   dx 1  √ a F (δ, r) − a2 + b2 E (δ, r) = 2 2 2 2 2 2 2 2 (x + a ) (b − x ) a b a + b 1  2 (a + u2 ) (b2 − u2 ) + 2 2 a b u [b > u > 0]

x2

b

4.

x2

u

3. 



[a ≥ b, b

2.



1 dx  = 2 2 2 2 2 2 ub x (x + a ) (x + b )

3.156



u

1. 0



dx 1  = a (p + b2 ) (p − x2 ) (x2 + a2 ) (x2 + b2 )

∞

2. u

1 dx  = 2 {F (ν, t) − E (ν, t)} 2 2 2 2 ab (x − a ) (x − b )



   b2 p + b2 Π α, , q + F (α, q) p p

[p = 0]     dx a2 + p 1  Π β, , q − F (β, q) =− a (a2 + p) a2 (p − x2 ) (x2 + a2 ) (x2 + b2 )

BY (221.13) BY (222.11)

3.157

Square roots of polynomials



u

3. 0

dx

1 √  = 2 2 2 2 2 2 p (p + a ) a2 + b 2 (p − x ) (a + x ) (b − x )

287

!   b 2 p + a2 , r + p F (γ, r) a Π γ, p (a2 + b2 ) 

2

[b ≥ u > 0, p = 0] BY (214.13)a   b dx b2 1 √  ,r Π δ, 2 = 2 b −p (p − b2 ) a2 + b2 (a2 + x2 ) (b2 − x2 ) u (p − x )

BY (213.02) b > u ≥ 0, p = b2      u  dx p 1 2 2 √  b Π ε, , s + p − b F (ε, s) = 2 p − b2 (a2 + x2 ) (x2 − b2 ) p (p − b2 ) a2 + b2 b (p − x )

BY (211.14) u > b > 0, p = b2      ∞ 2 dx a +p 1 √  Π ξ, 2 , s − F (ξ, s) = 2 2 2 2 2 2 2 2 a + b2 (a + p) a + b (a + x ) (x − b ) u (x − p) 

4.

5.

6.



u

7. 0

 8.

9.

10.

11.

  dx b2 1  Π η, , t = ap p (p − x2 ) (a2 − x2 ) (b2 − x2 )

[u ≥ b > 0] [a > b ≥ u > 0,

BY (212.12)

p = b]

BY (219.02)

b

dx 1  = 2 ) (p − b2 ) 2 − x2 ) (b2 − x2 ) a (p − a (a u  !     2 b 2 p − a2 2 2 , t + p − b F (ζ, t) × b − a Π ζ, 2 a (p − b2 )

BY (220.13) a > b > u ≥ 0, p = b2  !   2  u 2  p a − b dx 1  b2 Π κ, 2 , q + p − b2 F (κ, q) = 2 ap (p − b2 ) a (p − b2 ) (a2 − x2 ) (x2 − b2 ) b (p − x )

BY (217.12) a ≥ u > b > 0, p = b2    a dx a2 − b 2 1  Π λ, 2 ,q = 2 2 a (a − p) a −p (a2 − x2 ) (x2 − b2 ) u (x − p)

BY (218.02) a > u ≥ b > 0, p = a2  u dx  2) 2 − a2 ) (x2 − b2 ) (p − x (x a      2  1 p − b2 2 2 a = Π μ, F (μ, t) − b , t + p − a a (p − a2 ) (p − b2 ) p − a2

u > a > b > 0, p = a2 , p = b2 BY (216.12) (p − x2 )



∞

12. u

 dx 1   p   Π ν, 2 , t − F (ν, t) = ap a (x2 − p) (x2 − a2 ) (x2 − b2 ) [u ≥ a > b > 0,

p = 0]

BY (215.12)

288

Power and Algebraic Functions

3.158  1.

u

0



2  1 dx  a E (α, q) − b2 F (α, q) = 2 2 2 ab (a − b ) (x2 + a2 ) (x2 + b2 )3 [a > b,

∞

2. u

 0

[a > b, ∞

u

0 b

6. u



u

b



dx 1  = √ E (γ, r) 2 3 a a2 + b 2 (a2 + x2 ) (b2 − x2 )

1 u  = √ E (δ, r) − 2 2 2 2 2 a (a + b2 ) 3 a a +b (a2 + x2 ) (b2 − x2 )

b2 − u2 a2 + u 2 

u

u 2 − b2 u 2 + a2 BY (211.05)

dx 1  {F (ξ, s) − E (ξ, s)} = √ 2 a2 + b 2 3 a (a2 + x2 ) (x2 − b2 )

[b > u > 0]

u

BY (213.08)

BY (212.03)

1 u dx  = √ {F (γ, r) − E (γ, r)} +  2 2 2 2 2 3 b a +b b (a + u2 ) (b2 − u2 ) (a2 + x2 ) (b2 − x2 )

∞

10.

0



BY (214.01)a

[u ≥ b > 0] u

0

11.

BY (222.03)

[u > b > 0] ∞

9.



u ≥ 0]

dx 1 1  = √ {F (ε, s) − E (ε, s)} + 2 2 a2 + b 2 (a + b2 ) u 3 a (a2 + x2 ) (x2 − b2 )

u



BY (221.06)

[b ≥ u > 0]

dx

8.



u > 0]

[b > u ≥ 0]

7.

BY (222.05)

dx 1  {F (β, q) − E (β, q)} = 2 a (a − b2 ) 3 (a2 + x2 ) (x2 + b2 ) [a > b,

u

5. 

u ≥ 0]

dx u 1  {F (α, q) − E (α, q)} +  = 2 2 2 2 a (a − b ) a (u + a2 ) (u2 + b2 ) (x2 + a2 )3 (x2 + b2 )

4.



u > 0] BY (221.05)

2  1 u dx  a E (β, q) − b2 F (β, q) −  = 2 2 2) 2 2 ab (a − b 3 b (a + u2 ) (b2 + u2 ) (x2 + a2 ) (x2 + b2 ) [a > b,

u

3.



3.158

BY (214.10)

dx u 1  =  E (ξ, s) − √ 2 2 2 2 2 2 3 b (a + u ) (u − b ) b a2 + b2 (a2 + x2 ) (x2 − b2 )

dx 1  = 2 2 a (a − b2 ) 3 (a2 − x2 ) (b2 − x2 )

[u ≥ b > 0] !  b2 − u2 a E (η, t) − u a2 − u 2 [a > b ≥ u > 0]

BY (212.04)

BY (219.07)

3.159

Square roots of polynomials



b

12. u



u

13. b



dx 1  E (ζ, t) = 2 a (a − b2 ) 3 (a2 − x2 ) (b2 − x2 ) dx

 (a2 − x2 )3 (x2 − b2 )

∞

14. u



u

15. 0

1 = a (a2 − b2 )

dx 1  = 2 − a2 ) a (b 3 (x2 − a2 ) (x2 − b2 )

289

[a > b > u ≥ 0]

a F (κ, q) − E (κ, q) + u



u 2 − b2 a2 − u 2

a

16.

dx 1 1  = 2 F (η, t) − 2 2 ab b (a − b2 ) 3 (a2 − x2 ) (b2 − x2 )

u



[u > a > b > 0]  a E (η, t) − u

a

∞

u



u

1. 0



0

−

b2 )

b2 F (λ, q) − a2 E (λ, q) + au

a2 − u 2 u 2 − b2

!

BY (218.04)

dx 1  = 2 2 b (a − b2 ) 3 (x2 − a2 ) (x2 − b2 )

BY (216.11)

[u ≥ a > b > 0]

BY (215.06)

u > 0]

BY (221.12)

x dx a u  = 2 {F (β, q) − E (β, q)} +  2 2 2 a −b 3 (a + u ) (b2 + u2 ) (x2 + a2 ) (x2 + b2 ) 2

[a > b, u

BY (219.06)

[u > a > b > 0] !  b 2 u 2 − a2 1 a E (ν, t) − − 2 F (ν, t) u u 2 − b2 ab

[a > b,

u

3.

3

(a2 − x2 ) (x2 − b2 )

1 ab2 (a2



x2 dx a  = 2 {F (α, q) − E (α, q)} a − b2 3 (x2 + a2 ) (x2 + b2 )

∞

2.



=

a2 − u 2 b2 − u2

BY (215.04)

!

dx 1 a  E (μ, t) − 2 F (μ, t) = 2 2 2 b (a − b ) ab 3 (x2 − a2 ) (x2 − b2 )

18.

3.159

dx

BY (217.10)

[a > u > b > 0] u

17.





!

[a > u > b > 0] !  a u 2 − b2 E (ν, t) − u u 2 − a2

[a > b > u > 0] 

BY (220.10)

u ≥ 0]

BY (222.10)

2  x dx 1 u  a E (α, q) − b2 F (α, q) −  = 2 2 2 2 a (a − b ) 3 (a + u ) (b2 + u2 ) (x2 + a2 ) (x2 + b2 ) 2

[a > b,

u > 0]

BY (221.11)

290

Power and Algebraic Functions



∞

4. u



2  1 x2 dx  a E (β, q) − b2 F (β, q) = 2 2 a (a − b ) 3 (x2 + a2 ) (x2 + b2 ) [a > b,

u ≥ 0]

u

x2 dx 1  = √ {F (γ, r) − E (γ, r)} 2 3 a + b2 (a2 + x2 ) (b2 − x2 )

b

x dx 1 u  = √ {F (δ, r) − E (δ, r)} + 2 2 2 a + b2 3 a +b (a2 + x2 ) (b2 − x2 )

5. 0



3.159

[b ≥ u > 0]

6. u

2



BY (222.07)

BY (214.04)

b2

u2

− a2 + u 2

[b > u ≥ 0]  u x2 dx u 2 − b2 1 a2  = √ E (ε, s) − 2 2 u (a + b ) u2 + a2 a2 + b 2 b (a2 + x2 )3 (x2 − b2 )

BY (213.07)

 7.



∞

8. u



u

9. 0



∞



BY (212.01)

[u > b > 0] ! b2 − u2 1 − F (η, t) a2 − u 2 a

x2 dx 1  = 2 a − b2 3 (a2 − x2 ) (b2 − x2 )

b

x dx a 1  = 2 E (ζ, t) − F (ζ, t) 2 a − b a 3 (a2 − x2 ) (b2 − x2 )

0

a E (η, t) − u

[a > b ≥ u > 0]

12. u

u

13. b



[u ≥ b > 0]

u

BY (212.10)

BY (219.04)

2

[a > b > u ≥ 0] !  3 2 − b2 a u b2 F (κ, q) − a2 E (κ, q) + u a2 − u 2

x2 dx 1  = 2 − b2 ) a (a 3 (a2 − x2 ) (x2 − b2 )

∞

14.

BY (214.07)

x2 dx 1 u  = √ {F (ξ, s) − E (ξ, s)} +  2 + b2 2 + u2 ) (u2 − b2 ) 3 a (a (a2 + x2 ) (x2 − b2 )

u

11.



BY (211.13)

[b > u > 0]

u



[u > b > 0]

x2 dx u 1  =  E (γ, r) −√ 2 2 2 2 2 3 a + b2 (a + u ) (b − u ) (a2 + x2 ) (b2 − x2 )

10.



x dx 1  = √ E (ξ, s) 2 3 a + b2 (a2 + x2 ) (x2 − b2 ) 2

x2 dx a  = 2 a − b2 3 (x2 − a2 ) (x2 − b2 )

a u



[a > u > b > 0] ! 1 u 2 − b2 − E (ν, t) + F (ν, t) u 2 − a2 a [u > a > b > 0]

BY (220.08)

BY (217.06)

BY (215.09)

3.161

Square roots of polynomials



u

15. 0



a

16. u



u

17. a



x2 dx

 3 (a2 − x2 ) (b2 − x2 )



x2 dx

1 = 2 a − b2

= 3

(a2 − x2 ) (x2 − b2 )

a2

1 − b2

 u

! a2 − u 2 − a E (η, t) b2 − u2

[a > b > u > 0] !  a2 − u 2 a F (λ, q) − a E (λ, q) + u u 2 − b2

x dx a  = 2 E (μ, t) a − b2 3 (x2 − a2 ) (x2 − b2 )

∞

18. u

2

x2 dx

 3 (x2 − a2 ) (x2 − b2 )

1 = 2 a − b2

291

b2 a E (ν, t) − u



[a > u > b > 0]

BY (218.07)

[u > a > b > 0]

BY (216.01)

u 2 − a2 u 2 − b2

!

[u ≥ a > b > 0] 3.161



∞

1. u

 2.

3.

4.

5.

BY (219.12)

BY (215.11)

  a2 b2 − u2 2a2 + b2 1  2 2 2  = 3 4 2 a + b E (β, q) − b F (β, q) + 3a b 3a2 b4 u3 x4 (x2 + a2 ) (x2 + b2 ) dx

[a > b,

u > 0]

BY (222.04)

b

2 2   dx 1  √ a 2a − b2 F (δ, r) − 2 a4 − b4 E (δ, r) = 4 (x2 + a2 ) (b2 − x2 ) 3a4 b4 a2 + b2 u x a2 b2 + 2u2 a2 − b2  2 (b − u2 ) (a2 + u2 ) + 3a4 b4 u3 [b > u > 0] BY (213.09) √  2  u a2 + b 2 dx 2 a − b2 2b2 − a2  √ F (ε, s) + E (ε, s) = 4 4 4 (x2 + a2 ) (x2 − b2 ) 3 a b 3a4 b2 a2 + b2 b x  1 + 2 2 3 (u2 + a2 ) (u2 − b2 ) 3a b u [u > b > 0] BY (211.11)  ∞

 4   dx 1  √ 2 a − b4 E (ξ, s) + b2 2b2 − a2 F (ξ, s) = 4 (x2 + a2 ) (x2 − b2 ) 3a4 b4 a2 +b2 u x  a2 b2 + u2 2a2 − b2 u 2 − b2 − 2 4 3 3a b u u 2 + a2 [u ≥ b > 0] BY (212.06) ⎧  b  dx 1 ⎨ 2  = 3 4 2a + b2 F (ζ, t) − 2 a2 + b2 E (ζ, t) 4 2 2 2 2 ⎩ (a − x ) (b − x ) 3a b u x ⎫  2

 2a + b2 u2 + a2 b2 a b2 − u2 ⎬ + u3 a2 − u 2 ⎭ [a > b > u > 0]

BY (220.09)

292

Power and Algebraic Functions

 6.

7.

3.162

u

 dx 1   = 3 4 2 a2 + b2 E (κ, q) − b2 F (κ, q) 2 2 2 2 3a b (a − x ) (x − b ) b  1 + 2 2 3 (a2 − u2 ) (u2 − b2 ) 3a b u [a ≥ u > b > 0] ⎧  a 1 ⎨  dx  = 3 4 2 a2 + b2 E (λ, q) − b2 F (λ, q) 4 (a2 − x2 ) (x2 − b2 ) 3a b ⎩ u x ⎫  ⎬ 2 a 2 + b 2 u 2 + a2 b 2  2 2 ) (u2 − b2 ) (a − u − ⎭ au3 x4



u

8. a



⎧ ⎨

[a > u ≥ b > 0]

 1 dx  = 3 4 2a2 + b2 F (μ, t) − 2 a2 + b2 E (μ, t) x4 (x2 − a2 ) (x2 − b2 ) 3a b ⎩ ⎫  2 2

2 2 2 2 2 2 a + 2b u + a b b u −a ⎬ + au3 u 2 − b2 ⎭

∞

9. u

⎧  dx 1 ⎨ 2  = 3 4 2a + b2 F (ν, t) − 2 a2 + b2 E (ν, t) x4 (x2 − a2 ) (x2 − b2 ) 3a b ⎩ ⎫ ⎬ 2 ab + 3 (u2 − a2 ) (u2 − b2 ) ⎭ u [u ≥ a > b > 0]

BY (217.14)

BY (218.12)

[u > a > b > 0]

BY (216.09)

BY (215.07)

3.162  1. 0

 2.

3.

u

 2   dx 1  = 3a − b2 F (α, q) − 2 2a2 − b2 E (α, q) 2 3 2 2 (x2 + a2 )5 (x2 + b2 ) 3a (a2 − b2 )  u a 4a − 3b2 + u2 3a2 − 2b2  + 3a4 (a2 − b2 ) (u2 + a2 )3 (u2 + b2 ) [a > b, u > 0] BY (221.06)

∞

 2   dx 1  3a − b2 F (β, q) − 2 2a2 − b2 E (β, q) = 2 3 2 2 u (x2 + a2 )5 (x2 + b2 ) 3a (a − b )  u 2 + b2 u + 2 2 2 3a (a − b ) (a2 + u2 )3 [a > b, u ≥ 0] BY (222.03)  2  u 2 a 2a − 4b dx 3b2 − a2  = 2 F (α, q) + 2 E (α, q) 2 2 2 5 3ab (a − b )  3b4 (a2 − b2 ) 0 (x2 + a2 ) (x2 + b2 ) u 2 + a2 u + 2 2 3b (a − b2 ) (u2 + b2 )3 [a > b, u > 0] BY (221.05)

3.162

Square roots of polynomials



∞

4. u



u

5. 0

 6.

u



u

7. b



 2  dx 1   = 2 b + 2a2 E (γ, r) − a2 F (γ, r) (a2 + x2 )5 (b2 − x2 ) 3a4 (a2 + b2 )3  b2 − u2 u + 2 2 2 3a (a + b ) (a2 + u2 )3

 2  dx 1   = 4a + 2b2 E (δ, r) − a2 F (δ, r) 5 3 (a2 + x2 ) (b2 − x2 ) 3a4 (a2 + b2 )   2 2 u a 5a + 3b2 + u2 4a2 + 2b2 b2 − u2 − 2 4 2 2 3a (a + b ) (a2 + u2 )3 [b > u > 0]

∞

u

9.12 0



u

BY (212.03)

 2   dx 1   2a + 3b2 F (γ, r) − 2a2 + 4b2 E (γ, r) = 5 3 (a2 + x2 ) (b2 − x2 ) 3b4 (a2 + b2 )  2  u 3a + 4b2 b2 − 2a2 + 3b2 u2  + 3 3b4 (a2 + b2 ) (a2 + u2 ) (b2 − u2 ) [b > u > 0] BY (214.10)

∞

10.

BY (213.08)

 2   dx 1   = 3a + 2b2 F (ξ, s) − 4a2 + 2b2 E (ξ, s) (a2 + x2 )5 (x2 − b2 ) 3a4 (a2 + b2 )3  u 2 − b2 u + 2 2 3a (a + b2 ) (a2 + u2 )3 [u > b > 0]

u

BY (214.15)

 2   dx 1   = 3a + 2b2 F (ε, s) − 4a2 + 2b2 E (ε, s) 5 3 (a2 + x3 ) (x2 − b2 ) 3a4 (a2 + b2 )    2 3a + b2 u2 + 2 2a2 + b2 a2 u 2 − b2 + 2 3 3a2 (a2 + b2 ) u (u2 + a2 ) [u > b > 0] BY (211.05)

8.



2 2   dx 1  a − 2b2 E (β, q) + b2 3b2 − a2 F (β, q) = 2 2a 4 2 2 5 3ab (a − b ) (x2 + a2 ) (x2 + b2 )  u b2 3a2 − 4b2 + u2 2a2 − 3b2  − 3b4 (a2 − b2 ) (u2 + a2 ) (u2 + b2 )3 [a > b, u ≥ 0] BY (222.05)

[b ≥ u > 0] b

293

 2  dx 1   2a + 4b2 E (ξ, s) − b2 F (ξ, s) = 5 3 (a2 + x2 ) (x2 − b2 ) 3b4 (a2 + b2 )  2  u 3a + 4b2 b2 − 2a2 + 3b2 u2  + 3 3b4 (a2 + b2 ) (a2 + u2 ) (u2 − b2 ) [u > b > 0]

BY (212.04)

294

Power and Algebraic Functions

 2a 2b2 − a2 2a2 − 3b2  F (η, t) + = 2 E (η, t) 3ab4 (a2 − b2 ) 5 3b4 (a2 − b2 )  0 (a2 − x2 ) (b2 − x2 )   2 u 3a − 5b2 b2 − 2 a2 − 2b2 u2 a2 − u 2 + 2 b2 − u2 3b4 (a2 − b2 ) (b2 − u2 ) [a > b, a > 0]   a 2a a2 − 2b2 dx 3b2 − a2  = F (λ, q) + E (λ, q) 4 (a2 − b2 )2 5 3ab2 (a2 − b2 )2 3b u (a2 − x2 ) (x2 − b2 )   u 2 2b2 − a2 u2 + 3a2 − 5b2 b2 a2 − u 2 + 2 u 2 − b2 3b4 (a2 − b2 ) (u2 − b2 ) [a > u > b > 0]   u 2 2 2a 2b2 − a2 dx 2a − 3b  F (μ, t) + = 2 E (μ, t) 3ab4 (a2 − b2 ) 5 3b4 (a2 − b2 ) a (x2 − a2 ) (x2 − b2 )  u u 2 − a2 + 2 2 3b (a − b2 ) (u2 − b2 ) u2 − b2 [u > a > b > 0]  2  ∞ 4b − 2a2 a dx 2a2 − 3b2  F (ν, t) = E (ν, t) + 2 3ab4 (a2 − b2 ) 5 3b4 (a2 − b2 ) u (x2 − a2 ) (x2 − b2 )   3b2 − a2 u2 − 4b2 − 2a2 b2 u2 − a2 − 2 u 2 − b2 3b2 u (a2 − b2 ) (u2 − b2 ) [u ≥ a > b > 0]  u

  dx 1  = 4a2 − 2b2 E (η, t) − a2 − b2 F (η, t) 2 3 2 − b2 ) 5 0 (a2 − x2 ) (b2 − x2 ) 3a (a !  2   u 5a − 3b2 a2 − 4a2 − 2b2 u2 b2 − u2 − a (a2 − u2 ) a2 − u 2 

11.

12.

13.

14.

15.

 16.

17.

18.

u

b

3.162

dx



2

[a > b ≥ u > 0]

BY (219.06)

BY (218.04)

BY (216.11)

BY (215.06)

BY (219.07)

2 2a2 − b dx 1  = 2 E (ζ, r) − 3a3 (a2 − b2 ) F (ζ, t) 3 2 2 5 u (a2 − x2 ) (b2 − x2 ) 3a (a − b )  b2 − u2 u + 2 2 3a (a − b2 ) (a2 − u2 ) a2 − u2 [a > b > u ≥ 0] BY (220.10)  u

 2   dx 1  = 3a − b2 F (κ, q) − 4a2 − 2b2 E (κ, q) 2 3 2 2 5 b (a2 − x2 ) (x2 − b2 ) 3a (a − b )   2 2a2 − b2 a2 + b2 − 3a2 u2 u2 − b2 , + 2 a2 − u 2 3a2 u (a2 − b2 ) (a2 − u2 ) BY (217.10) [a > u > b > 0]  ∞

 2   dx 1  = 4a − 2b2 E (ν, t) − a2 − b2 F (ν, t) 2 3 2 2 5 u (x2 − a2 ) (x2 − b2 ) 3a (a − b )   4a2 − 2b2 a2 + b2 − 3a2 u2 u2 − b2 + 2 u 2 − a2 3a2 u (a2 − b2 ) (u2 − a2 ) [u > a > b > 0] BY (215.04)

3.164

3.163

Square roots of polynomials



u

1. 0



 2  dx 1  a + b2 E (α, q) − 2b2 F (α, q) = 2 3 3 ab2 (a2 − b2 ) (x2 + a2 ) (x2 + b2 ) u  − 2 2 2 a (a − b ) (a2 + u2 ) (b2 + u2 ) [a > b, u > 0]

∞

BY (222.12)

2   dx 1   a F (γ, r) − a2 − b2 E (γ, r) = 3 3 3 (x2 + a2 ) (b2 − x2 ) a2 b2 (a2 + b2 ) u  + 2 2 2 b (a + b ) (a2 + u2 ) (b2 − u2 ) [b > u > 0]

BY (214.15)

u

u

3.12 0



∞

4. u



u

5. 0



dx b −a 1   = E (ξ, s) −  F (ξ, s) 3 3 3 3 (x2 + a2 ) (x2 − b2 ) a2 b2 (a2 + b2 ) a2 (a2 + b2 ) u  + 2 2 2 b (a + b ) (u2 + a2 ) (u2 − b2 ) [u > b > 0] 2

∞

u

2

∞

1. u



u

−

x2 dx

(x2

 = − ρρ) (x2 + ρ2 ) (x2 + ρ2 ) 2

BY (279.08)

BY (215.10)

2

(ρ − ρ) . ρρ

dx 1  = √ F (α, r) ρρ (x2 + ρ2 ) (x2 + ρ2 )

∞

2.

 1 r= 2

BY (212.05)

2

dx a2 + b2 1  F (ν, t) − = 2 2 2 E (ν, t) 2 ab (a − b ) ab2 (a2 − b2 ) (x2 − a2 )3 (x2 − b2 )3 1  + 2 2 u (a − b ) (u2 − a2 ) (u2 − b2 ) [u > a > b > 0]

u2 − ρρ , Notation: α = arccos 2 u + ρρ 

2

dx a +b 1  F (η, t) − = 2 2 2 E (η, t) 2 2 ab (a − b ) ab (a2 − b2 ) (a2 − x2 )3 (b2 − x2 )3 4  a + b 4 − a2 + b 2 u 2 u + 2 2 a2 b2 (a2 − b2 ) (a − u2 ) (b2 − u2 ) [a > b > u > 0]

6.

3.164

BY (221.07)

 2  dx 1  a + b2 E (β, q) − 2b2 F (β, q) = 2 2 2 2 3 3 ab (a − b ) (x2 + a2 ) (x2 + b2 ) u  − 2 2 2 b (a − b ) (a2 + u2 ) (b2 + u2 ) [a > b, u ≥ 0]

2.



295

BY (225.00)

 2u (u2 + ρ2 ) (u2 + ρ2 ) 2

(ρ + ρ)

(u4

−

ρ2 ρ 2 )

−

1 (ρ + ρ)

2

√ E (α, r) ρρ BY (225.03)

296

Power and Algebraic Functions

 3. 4.

3.165

∞

1 x2 dx  =− [F (α, r) − E (α, r)] 2 2√ 2 2 2 2 2 (x + ρ ) (x + ρ ) (ρ − ρ) ρρ u (x + ρρ) √  ∞ 4 ρρ 1 x2 dx  =− F (α, r) 2 E (α, r) + 2√ 2 2 (ρ − ρ ) u (x2 + ρ2 )3 (x2 + ρ2 )3  2 (ρ − ρ) ρρ 2u u − ρρ  − 2 2 (ρ + ρ) (u + ρρ) (u2 + ρ2 ) (u2 + ρ2 )

BY (225.07)

BY (225.05)



∞

5. u

 6.

7.

8.



2

√ 4 ρρ

x2 − ρρ dx  =− 2 [F (α, r) − E (α, r)] 3 3 (ρ − ρ)  (x2 + ρ2 ) (x2 + ρ2 ) 2u u2 − ρρ  + (u2 + ρρ) (u2 + ρ2 ) (u2 + ρ2 ) BY (225.06)

 ∞ (x2 + ρ2 ) (x2 + ρ2 )

1 dx = √ E (α, r) ρρ + ρρ) u  √  ∞ 2 2 x2 −  dx ( + ) 4   E (α, r) + = − F (α, r) √ 2 2 (x2 + 2 ) (x2 + 2 ) ( − )2 ( − )2  u (x + )  2 2  ∞  x +  dx 1   = √ Π α, p2 , r 2  u (x2 + ) − 4p2 x2 (x2 + 2 ) (x2 + 2 )

BY(225.01)

2

(x2

BY (225.08)

BY (225.02)

√ a2 − b 2 u 2 − a2 √ . 3.165 Notation: α = arccos 2 , r = u + a2 a 2 √  a dx 2 √ √ = √ 1. 4 2 2 4 x + 2b x + a a 2⎡+ a2 + b2 u ⎤     √ √ 2 2 2 a 2 (a2 − b2 ) 2 + a − b a − u a ⎦ √ √ , √ × F ⎣arctan a+u a2 + b 2 a 2 + a2 − b 2 

[a > b, ∞

2.12 u



∞

3. u



dx 1 √ F (α, r) = 4 2 2 4 2a x + 2b x + a

u

a2 > 0,

√ dx u4 + 2b2 u2 + a4 1 √ = 3 [F (α, r) − 2 E (α, r)] + 2a a2 u (u2 + a2 ) x2 x4 + 2b2 x2 + a4 [a > b > 0, u > 0]

∞

4.12

2 a > b2 ,

a > u ≥ 0]

x dx 1 [F (α, r) − E (α, r)] = √ 2 − b2 ) 4 2 2 4 4a (a x + 2b x + a 2 a > b2 , a2 > 0,

BY (264.00)

u≥0

BY (263.00, 266.00)

BY (263.06)

2

(x2

+

2 a2 )

u≥0

BY (263.03, 266.05)

3.166

Square roots of polynomials



∞

5. u



∞

u

8.

9.12

√ x2 dx 1 u u4 + 2b2 u2 + a4 − E (α, r) = 2√ 4 2 2 4 4 2 2 2 2 2 4 2 (a + b ) (u − a ) 4a (a + b2 ) (x − a ) x + 2b x + a 2

a > b2 > −∞, u2 > a2 > 0 BY (263.05, 266.02)

6.12

7.

x2 dx 1 a  E (α, r) − F (α, r) = 4 − b4 ) 2 − b2 ) 2 (a 4a (a 3  2 (x4 + 2b2 x2 + a4 ) u u − a2 √ − 2 (a2 + b2 ) (u2 + a 2 ) u4 + 2b2 u2 + a4

a2 > b2 , a2 > 0, u ≥ 0 

∞



√ ∞ x4 + 2b2 x2 + a4

x2 − a2

dx u a u 2 − a2  √ = 2 [F (α, r) − E (α, r)] + 2 2 2 4 a −b u +a 3 u + 2b2 u2 + a4 u (x4 + 2b2 x2 + a4 )  2 

 b  < a2 , u ≥ 0 BY (266.08)  2  ∞ 2 x + a2 dx a a 2 − b 2 u 2 − a2 u  = 2 E (α, r) − 2 · ·√ 2 2 u 2 + a2 4 + 2b2 u2 + a4 a + b a + b 3 u u (x2 + 2b2 x2 + a4 )  2 

 b  < a2 , u ≥ 0 BY (266.06)a   ∞ 2 x2 − a2 dx a a2 + b 2 F (α, r) = E (α, r) − √ 2 a2 − b 2 2a (a2 − b2 ) x4 + 2b2 x2 + a4 u (x2 + a2 ) 2

a > b2 , a2 > 0, u ≥ 0

u

 11.

(x2 +

2 a2 )

√ ∞ x4 + 2b2 x2 + a4

u

(x2 − a2 )2

BY (263.04, 266.07)

dx =

1 E (α, r) 2a

2 a > b2 ,

a2 > 0,

u≥0

BY (263.01, 266.01)

dx =

u  4 1 [F (α, r) − E (α, r)] + 4 u + 2b2 u2 + a4 2a u − a4

[a > b > 0, 2 ∞ x2 + a2 dx 1   √ Π α, p2 , r = 2 2a u (x2 + a2 ) − 4a2 p2 x2 x4 + 2b2 x2 + a4

u > a]

BY (263)

[a > b > 0,

u ≥ 0]

BY (263.02)



3.166

BY (263.08, 266.03)

2



10.

12.

297

Notation:



α = arccos

u2 − 1 , u2 + 1

β = arctan

  √ 1−u , 1+ 2 1+u

1 1 − u2 γ = arccos u, δ = arccos , ε = arccos , u 1 + u2 √  √  √ √ 2 4 , q =2 3 2−4=2 2 r= 2 − 1 ≈ 0.985171 2

298

Power and Algebraic Functions



∞

1. u



∞

2. u

dx 1 √ = F (α, r) 4 2 x +1

[u ≥ 0]

3.166

H (287), BY (263.50)

√ dx u4 + 1 1 √ = [F (α, r) − 2 E (α, r)] + 2 u (u2 + 1) x2 x4 + 1

[u > 0]  ∞ u u2 − 1 x2 dx 1 1 √ √ = E (α, r) − F (α, r) − 4 4 2 4 2 (u2 + 1) u4 + 1 u (x + 1) x + 1

BY (263.57)

 3.

 4. 5. 6.

x2 dx 1 = [F (α, r) − E (α, r)] 2√ 4 2 4 x +1 u (x + 1) √  ∞ x2 dx u u4 + 1 1 − E (α, r) = 2√ 4 2 (u4 − 1) 4 x +1 u (x2 − 1) √  ∞√ 4 x +1 u u4 + 1 1 2 dx = 2 [F (α, r) − E (α, r)] + u4 − 1 u (x2 − 1) 

7. 8.

9.

10. 11. 12. 13.

∞

∞



2

x2 − 1 dx 1 = E (α, r) − F (α, r) 2√ 4 2 2 x +1 u (x + 1) √  ∞ x4 + 1 dx 1 2 = 2 E (α, r) 2 (x + 1) u  2 2  ∞ x + 1 dx 1   √ = Π α, p2 , r 2 2 u (x2 + 1) − 4p2 x2 x4 + 1  u dx 1 √ = F (ε, r) 4 2 x +1 0  1  √  dx √ = 2 − 2 F (β, q) x4 + 1 u √  1  2  √  x + x 2 + 1 dx √  √ = 2 + 2 E (β, q) 2 x4 + 1 u x −x 2+1  1 (1 − x)2 dx 1 √  √ = √ [F (β, q) − E (β, q)] 2 4 2 x +1 u x −x 2+1

[u ≥ 0]

BY (263.59)

[u ≥ 0]

BY (263.53)

[u > 1]

BY (263.55)

[u > 1]

BY (263.58)

[u ≥ 0]

BY (263.54)

[u ≥ 0]

BY (263.51)

[u ≥ 0]

BY (263.52)

H 66(288)

[0 ≤ u < 1]

BY (264.50)

[0 ≤ u < 1]

BY (264.51)

[0 ≤ u < 1] √ √ 1 (1 + x)2 dx 3 2−4 3 2+4 √ E (β, q) − F (β, q)  √ = 2−x 2+1 4+1 2 2 x x u

BY (264.55)

[0 ≤ u < 1]

BY (264.56)

[0 < u < 1]

H 66 (290), BY (259.75)

 14.



1

15.12 u

dx 1 √ = √ F (γ, r) 4 2 1−x

3.167

Square roots of polynomials

  2 dx 1 1 √ Γ = √ 4 4 4 2π 1 − x 0  u dx 1 √ = √ F (δ, r) 4 2 x −1 1  1 2 √ x dx 1 √ = 2 E (γ, r) − √ F (γ, r) 4 1 − x u   22 1 3 = √ Γ 4 2π 

16. 17. 18.12

 19. 20.12 21.3 22.

299

1

[u > 1]

H 66 (289), BY (260.75)

[0 < u < 1] [u = 0] BY (259.76)

1

√ x dx 1 √ u4 − 1 [u > 1] = √ F (δ, r) − 2 E (δ, r) + 4 u 2 x −1 1  1 4 x dx u 1 √ = √ F (γ, r) + 1 − u4 [0 < u < 1] 3 1 − x4 3 2 u  u 4 x dx 1 1  √ [u > 1] = √ F (δ, r) + u u4 − 1 3 x4 − 1 3 2 1  √   √   u 1+ 1− 3 u 1 dx 2+ 3  √ ,  F arccos = √ 4 2 3 1+ 1+ 3 u x (1 + x3 ) 0 

u

u

23.12 0

2

dx 1  F = √ 4 3 3 x (1 − x )

[u > 0] √  √  1− 1+ 3 u 2− 3 √ , arccos 2 1+ 3−1 u



 Notation: In 3.167 and 3.168 we set:  β = arcsin  δ = arcsin

BY (259.76) BY (260.77)

BY (260.50)



[0 < u ≤ 1] 3.167

BY (260.77)

α = arcsin

(a − c)(d − u) , (a − d)(c − u) 

(a − c)(u − d) , (c − d)(a − u)

γ = arcsin

(b − d)(u − c) , (b − c)(u − d)

κ = arcsin

BY (259.50)



(b − d)(c − u) , (c − d)(b − u) (a − c)(b − u) , (b − c)(a − u)

 (a − c)(u − b) (b − d)(a − u) , μ = arcsin , λ = arcsin (a − b)(u − c) (a − b)(u − d)    (b − d)(u − a) (b − c)(a − d) (a − b)(c − d) , q= , r= . ν = arcsin (a − d)(u − b) (a − c)(b − d) (a − c)(b − d) 

 d 1. u

2(c − d) d−x dx =  (a − x)(b − x)(c − x) (a − c)(b − d)

    a−d , q − F (α, q) Π α, a−c [a > b > c > d > u]

BY (251.05)

300

Power and Algebraic Functions

 u 2. d

 c 3. u

4. c

5. u

6. b

7. u

 u 8. a

9. u

 u 10. d

c

2 x−d dx =  (x − a)(x − b)(x − c) (a − c)(b − d)



BY (256.13)

BY (257.02) [a > u ≥ b > c > d]    a−d , q + (b − d) F (ν, q) (a − b) Π ν, b−d BY (258.14)

[a > b > c > d > u] BY (251.02)     c−x 2 d−c dx =  , r − (a − c) F (β, r) (a − d) Π β, (a − x)(b − x)(x − d) a−c (a − c)(b − d)

[a > b > c ≥ u > d]     c−x 2(b − c) c−d dx =  , r − F (γ, r) Π γ, (a − x)(b − x)(x − d) b−d (a − c)(b − d)

BY (252.13)

[a > b > c > u ≥ d]     x−c 2(c − d) b−c dx =  , q − F (δ, q) Π δ, (a − x)(b − x)(x − d) b−d (a − c)(b − d)

BY (253.13)

 u 12.

BY (253.14)

BY (255.20) [a > b > u ≥ c > d]    a−b , r + (c − d) F (λ, r) (b − c) Π λ, a−c

[u > a > b > c > d]   c−x 2(c − d) a−d dx =  ,q Π α, (a − x)(b − x)(d − x) a−c (a − c)(b − d)

 d

u

x−d 2 dx =  (a − x)(x − b)(x − c) (a − c)(b − d)



[a ≥ u > b > c > d]   x−d 2(a − d) b−a dx =  ,r Π μ, (a − x)(x − b)(x − c) b−d (a − c)(b − d)

 a

11.

[a > b > c > u ≥ d]   x−d 2(c − d) b−c dx =  ,q Π δ, (a − x)(b − x)(x − c) b−d (a − c)(b − d)

BY (254.02) [a > b ≥ u > c > d]     2 b−c x−d dx =  , q + (a − d) F (κ, q) (b − a) Π κ, (a − x)(b − x)(x − c) a−c (a − c)(b − d)

 u

 c

    d−c , r − F (β, r) Π β, a−c

BY (252.14) [a > b > c ≥ u > d]     x−d 2 c−d dx =  , r + (b − d) F (γ, r) (c − b) Π γ, (a − x)(b − x)(c − x) b−d (a − c)(b − d)

 u

 b

2(d − a) x−d dx =  (a − x)(b − x)(c − x) (a − c)(b − d)

3.167

[a > b ≥ u > c > d]

BY (254.12)

3.167

Square roots of polynomials

 b 13. u

    x−c 2 b−c dx =  , q + (a − c) F (κ, q) (b − a) Π κ, (a − x)(b − x)(x − d) a−c (a − c)(b − d)

 u 14. b

u

 u 16. a

17. u

 u 18. d

20. c

21. u

b

 a 23.8 u

BY (252.15)

BY (253.02) [a > b > c > u ≥ d]    b−x 2 b−c dx =  , q + (b − d) F (δ, q) (d − c) Π δ, (a − x)(x − c)(x − d) b−d (a − c)(b − d)



[a > b ≥ u > c > d]     b−x 2(a − b) b−c dx =  , q − F (κ, q) Π κ, (a − x)(x − c)(x − d) a−c (a − c)(b − d)

BY (254.14)

[a > b > u ≥ c > d]     x−b 2(b − c) a−b  dx = , r − F (λ, r) Π λ, (a − x)(x − c)(x − d) a−c (a − c)(b − d)

BY (255.21)

 u 22.

BY (251.07) [a > b > c > d > u]     b−x 2 d−c dx =  , r − (a − b) F (β, r) (a − d) Π β, (a − x)(c − x)(x − d) a−c (a − c)(b − d)

[a > b > c ≥ u > d]   b−x 2(b − c) c−d dx =  ,r Π γ, (a − x)(c − x)(x − d) b−d (a − c)(b − d)

 u

 b

[a > u ≥ b > c > d] BY (257.13)     2 a−d x−c dx =  , q + (b − c) F (ν, q) (a − b) Π ν, (x − a)(x − b)(x − d) b−d (a − c)(b − d) [u > a > b > c > d] BY (258.13)     b−x 2 a−d dx =  , q + (b − c) F (α, q) (c − d) Π α, (a − x)(c − x)(d − x) a−c (a − c)(b − d)

 d

u

BY (259.19)



15.

19.

[a > b > u ≥ c > d]   x−c 2(b − c) a−b dx =  ,r Π λ, (a − x)(x − b)(x − d) a−c (a − c)(b − d)

[a ≥ u > b > c > d] BY (256.02)    2 b−a x−c dx =  , r + (d − c) F (μ, r) (a − d) Π μ, (a − x)(x − b)(x − d) b−d (a − c)(b − d)

 a

 c

301

BY (256.15) [a ≥ u > b > c > d]     x−b 2 b−a dx =  , r − (b − d) F (μ, r) (a − d) Π μ, (a − x)(x − c)(x − d) b−d (a − c)(b − d)

[a > u ≥ b > c > d]

BY (257.15)

302

Power and Algebraic Functions

 u 24. a



25. u

 u 26. d

27. u

c

u

BY (251.06)

[a > b > c ≥ u > d] BY (252.02)    a−x 2 c−d dx =  , r + (a − b) F (γ, r) (b − c) Π γ, (b − x)(c − x)(x − d) b−d (a − c)(b − d)

28.

29.

[a > b > c > d > u]   a−x 2(a − d) d−c dx =  ,r Π β, (b − x)(c − x)(x − d) a−c (a − c)(b − d) 

 u

 b

  2(a − b) a−d x−b dx =  ,q Π ν, (x − a)(x − c)(x − d) b−d (a − c)(b − d) BY (258.02) [u > a > b > c > d]    a−x 2 a−d dx =  , q + (a − c) F (α, q) (c − d) Π α, (b − x)(c − x)(d − x) a−c (a − c)(b − d)

 d

 c

3.168

[a > b > c > u ≥ d] BY (253.15)     a−x 2 b−c dx =  , q + (a − d) F (δ, q) (d − c) Π δ, (b − x)(x − c)(x − d) b−d (a − c)(b − d)

[a > b ≥ u > c > d]   b−c a−x 2(a − b) Π κ, dx =  ,q (b − x)(x − c)(x − d) a−c (a − c)(b − d)

 u 30. b

 a 31. u

 u 32. a

BY (254.13)

BY (255.02) [a > b > u ≥ c > d]     a−x 2 a−b dx =  , r + (a − c) F (λ, r) (c − b) Π λ, (x − b)(x − c)(x − d) a−c (a − c)(b − d)

[a ≥ u > b > c > d]     a−x 2(d − a) b−a dx =  , r − F (μ, r) Π μ, (x − b)(x − c)(x − d) b−d (a − c)(b − d)

BY (256.14)

[a > u ≥ b > c > d]     x−a 2(a − b) a−d  dx = , q − F (ν, q) Π ν, (x − b)(x − c)(x − d) b−d (a − c)(b − d)

BY (257.14)

[u > a > b > c > d] 3.168

 c

1. u

2 c−x dx = 3 (a − x)(b − x)(x − d) d−a

 u 2. c

x−c 2 dx = (a − x)(b − x)(x − d)3 a−d





a−c E (γ, r) − b−d



(a − u)(c − u) (b − u)(u − d)

[a > b > c > u > d]

BY (258.15)



BY (253.06)

a−c [F (δ, q) − E (δ, q)] b−d [a > b ≥ u > c > d]

BY (254.04)

3.168

Square roots of polynomials

 a−c (b − u)(u − c) 2 [F (κ, q) − E (κ, q)] + b−d b − d (a − u)(u − d) u [a > b > u ≥ c > d] BY (255.09)      u x−c a−c c − d (a − u)(u − b) 2 E (λ, r) − dx = 3 (a − x)(x − b)(x − d) a−d b−d b − d (u − c)(u − d) b  b

3.

4.

303

x−c 2 dx = 3 (a − x)(b − x)(x − d) a−d

 a u

 u 6. a

7. u

c

BY (256.06)

[a > u ≥ b > c > d]

BY (257.01)

x−c a−c 2 [F (ν, q) − E (ν, q)] dx = 3 (x − a)(x − b)(x − d) a−d  b−d (u − a)(u − c) 2 + a − d (u − b)(u − d) [u > a > b > c > d]

BY (258.10)





a−c E (μ, r) b−d

b−x 2  dx = (a − x)(c − x)(x − d)3 (a − d)(c − d) (a − c)(b − d)

 u 8.

[a ≥ u > b > c > d]

x−c 2 dx = 3 (a − x)(x − b)(x − d) a−d

5.

 c



× [(b − c)(a − d) F (γ, r) − (a − c)(b − d) E (γ, r)]  (a − u)(c − u) 2(b − d) + (a − d)(c − d) (b − u)(u − d) [a > b > c > u > d]

BY (253.03)

b−x 2  dx = (a − x)(x − c)(x − d)3 (a − d)(c − d) (a − c)(b − d) × [(a − c)(b − d) E (δ, q) − (a − b)(c − d) F (δ, q)]

 b 9. u

[a > b ≥ u > c > d]

BY (254.15)

b−x 2  dx = 3 (a − x)(x − c)(x − d) (a − d)(c − d) (a − c)(b − d) × [(a − c)(b − d) E (κ, q) − (a − b)(c − d) F (κ, q)]  (b − u)(u − c) 2 − c − d (a − u)(u − d) [a > b > u ≥ c > d]

BY (255.06)

304

Power and Algebraic Functions

 u 10. b

11.

12.

13.

14.

15.

16.

17.

3.168

2 x−b  dx = (a − x)(x − c)(x − d)3 (a − d)(c − d) (a − c)(b − d)

× [(a − c)(b − d) E (λ, r) − (a − d)(b − c) F (λ, r)]  (a − u)(u − b) 2 − a − d (u − c)(u − d) [a ≥ u > b > c > d] BY (256.03)    a (a − c)(b − d) x−b E (μ, r) dx = 2 3 (a − x)(x − c)(x − d) (a − d)(c − d) u 2(b − c)  − F (μ, r) (c − d) (a − c)(b − d) BY (257.09) [a > u ≥ b > c > d]   u x−b dx (x − a)(x − c)(x − d)3 a  (u − a)(u − c) 2(a − b) 2(b − d)  + F (ν, q) = (a − d)(c − d) (u − b)(u − d) (a − d) (a − c)(b − d)  (a − c)(b − d) E (ν, q) +2 (a − d)(c − d) [u > a > b > c > d] BY (258.09)     c a−x a−c (a − u)(c − u) 2 2 [F (γ, r) − E (γ, r)] + dx = 3 (b − x)(c − x)(x − d) c−d b−d c − d (b − u)(u − d) u [a > b > c > u > d] BY (253.04)   u a−x a−c 2 E (δ, q) dx = 3 (b − x)(x − c)(x − d) c − d b−d c BY (254.01) [a > b ≥ u > c > d]    b a−x a−c (b − u)(u − c) 2(a − d) 2 E (κ, q) − dx = 3 (b − x)(x − c)(x − d) c − d b − d (b − d)(c − d) (a − u)(u − d) u BY (255.08) [a > b > u ≥ c > d]    u a−x a−c (a − u)(u − b) 2 2 [F (λ, r) − E (λ, r)] + dx = 3 (x − b)(x − c)(x − d) c − d b − d b − d (u − c)(u − d) b [a ≥ u > b > c > d] BY (256.05)   a a−x a−c 2 [F (μ, r) − E (μ, r)] dx = 3 (x − b)(x − c)(x − d) c−d b−d u

 u 18. a

x−a −2 dx = (x − b)(x − c)(x − d)3 c−d



[a > u ≥ b > c > d]  a−c (u − a)(u − c) 2 E (ν, q) + b−d c − d (u − b)(u − d) [u > a > b > c > d]

BY (257.06)

BY (258.05)

3.168

Square roots of polynomials

 d 19. u

 u 20. d

2 d−x dx = (a − x)(b − x)(c − x)3 b−c

x−d −2 dx = 3 (a − x)(b − x)(c − x) b−c



305

b−d [F (α, q) − E (α, q)] a−c



[a > b > c > d > u]  b−d (b − u)(u − d) 2 E (β, r) + a−c b − c (a − u)(c − u)

BY (251.01)

BY (252.06) [a > b > c ≥ u > d]   x−d b−d (b − u)(u − d) 2 2 [F (κ, q) − E (κ, q)] + dx = 3 (a − x)(b − x)(x − c) b − c a − c b − c (a − u)(u − c) u [a > b > u > c > d] BY (255.05)   u x−d b−d 2 E (λ, r) dx = 3 (a − x)(x − b)(x − c) b − c a−c b

 b 21.

22.

 a 23. u

x−d 2 dx = 3 (a − x)(x − b)(x − c) b−c



[a ≥ u > b > c > d] BY (256.01)  b−d (a − u)(u − b) 2(c − d) E (μ, r) − a−c (a − c)(b − c) (u − c)(u − d)

BY (257.06) [a > u ≥ b > c > d]   x−d b−d (u − a)(u − d) 2 2 [F (ν, q) − E (ν, q)] + dx = 3 (x − a)(x − b)(x − c) b − c a − c a − c (u − b)(u − c) a [u > a > b > c > d] BY (258.06)   a b−x b−d 2 dx = E (α, q) 3 (a − x)(c − x) (d − x) c−d a−c u

 u 24.

25.

26.

27.

BY (251.01) [a > b > c > d > u]  b−x b−d (b − u)(u − d) 2 2 dx = [F (β, r) − E (β, r)] + (a − x)(c − x)3 (x − d) c−d a−c c − d (a − u)(c − u) d [a > b > c > u > d] BY (252.03)     b b−x b−d (b − u)(u − d) 2 2 dx = E (κ, q) + 3 (x − d) (a − x)(x − c) d − c a − c c − d (a − u)(u − c) u

 u



 u



28. b

 a 29. u

x−b 2 dx = 3 (a − x)(x − c) (x − d) c−d

x−b 2 dx = (a − x)(x − c)3 (x − d) c−d



[a > b > u > c > d]

BY (255.03)

b−d [F (λ, r) − E (λ, r)] a−c

[a ≥ u > b > c > d] BY (256.08)  b−d (a − u)(u − b) 2 [F (μ, r) − E (μ, r)] + a−c a − c (u − c)(u − d) [a > u ≥ b > c > d] BY (257.03)

306

Power and Algebraic Functions

 u 30. a

 d 31.

32.

33.

34.

35.

36.

37.

x−b 2 dx = 3 (x − a)(x − c) (x − d) c−d 



b−d 2(b − c) E (ν, q) − a−c (a − c)(c − d)

3.168



(u − a)(u − d) (u − b)(u − c)

[u > a > b > c > d]

BY (258.03)

2 (a − c)(b − d) a−x a−b 2  dx = E (α, q) − F (α, q) 3 (d − x) (b − x)(c − x) (b − c)(c − d) b − c (a − c)(b − d) u BY (251.08) [a > b > c > d > u]   u (a − c)(b − d) a−x 2(a − d)  dx = E (β, r) F (β, r) − 2 3 (b − x)(c − x) (x − d) (b − c)(c − d) (c − d) (a − c)(b d − d) (b − u)(u − d) a−c +2 (b − c)(c − d) (a − u)(c − u) [a > b > c > u > d] BY (252.04)   b a−x (a − c)(b − d) 2(a − b)  dx = E (κ, q) F (κ, q) − 2 (b − x)(x − c)3 (x − d) (b − c)(c − d) (b − c) (a − c)(b − d) u  (b − u)(u − d) 2(a − c) + (b − c)(c − d) (a − u)(u − c) [a > b > u > c > d] BY (255.04)   u 2 (a − c)(b − d) a−x 2(a − d)  dx = E (λ, r) − F (λ, r) 3 (x − d) (x − b)(x − c) (b − c)(c − d) (c − d) (a − c)(b − d) b BY (256.09) [a ≥ u > b > c > d]   a 2 (a − c)(b − d) a−x 2(a − d)  dx = E (μ, r) − F (μ, r) 3 (x − d) (x − b)(x − c) (b − c)(c − d) (c − d) (a − c)(b − d) u  (a − u)(u − b) 2 − b − c (u − c)(u − d) [a > u ≥ b > c > d] BY (257.04)   u 2 (a − c)(b − d) 2(a − b) x−a  dx = E (ν, q) − F (ν, q) 3 (x − d) (x − b)(x − c) (b − c)(c − d) (b − c) (a − c)(b − d) a  (u − a)(u − d) 2 − c − d (u − b)(u − c) [u > a > b > c > d] BY (258.04)   d 2 (a − c)(b − d) 2(c − d) d−x  dx = E (α, q) − F (α, q) 3 (c − x) (a − x)(b − x) (a − b)(b − c) (b − c) (a − c)(b − d) u  (a − u)(d − u) 2 − a − b (b − u)(c − u) [a > b > c > d > u] BY (251.11)

3.168

Square roots of polynomials

307

 u

38.

39.

40.

41.

42.

43.

44.

45.

 2 (a − c)(b − d) 2(a − d) x−d  dx = E (β, r) − F (β, r) 3 (c − x) (a − x)(b − x) (a − b)(b − c) (a − b) (a − c)(b − d) d  (c − u)(u − d) 2 + b − c (a − u)(b − u) [a > b > c ≥ u > d] BY (252.07)   c 2 (a − c)(b − d) x−d 2(a − d)  F (γ, r) dx = E (γ, r) − 3 (c − x) (a − x)(b − x) (a − b)(b − c) (a − b) (a − c)(b − d) u BY (253.07) [a > b > c > u ≥ d]   u 2 (a − c)(b − d) 2(c − d) x−d  dx = E (δ, q) F (δ, q) − 3 (a − x)(b − x) (x − c) (a − b)(b − c) (b − c) (a − c)(b c  − d) (a − u)(u − c) 2(b − d) + (a − b)(b − c) (b − u)(u − d) [a > b > u > c > d] BY (254.05)   a 2 (a − c)(b − d) x−d 2(a − d)  F (μ, r) − dx = E (μ, r) 3 (x − c) (a − x)(x − b) (a − b)(b − c) (a − b) (a − c)(b u  − d) (a − u)(u − c) 2(b − d) + (a − b)(b − c) (u − b)(u − d) [a > u > b > c > d] BY (257.07)   u 2 (a − c)(b − d) x−d 2(c − d)  dx = E (ν, q) − F (ν, q) 3 (x − c) (x − a)(x − b) (a − b)(b − c) (b − c) (a − c)(b − d) a BY (258.07) [u > a > b > c > d]     d c−x a−c (a − u)(d − u) 2 2(b − c) dx = E (α, q) − 3 (a − x)(b − x) (d − x) a−b b−d (a − b)(b − d) (b − u)(c − u) u [a > b > c > d > u]    u c−x a−c (c − u)(u − d) 2 2 dx = [F (β, r) − E (β, r)] + 3 (x − d) (a − x)(b − x) a − b b − d b − d (a − u)(b − u) d [a > b > c ≥ u > d] BY (252.10)   c c−x a−c 2 dx = [F (γ, r) − E (γ, r)] 3 (a − x)(b − x) (x − d) a−b b−d u  u

46. c

x−c 2 dx = (a − x)(b − x)3 (x − d) b−a



[a > b > c > u ≥ d]  a−c (a − u)(u − c) 2 E (δ, q) + b−d a − b (b − u)(u − d) [a > b ≥ u > c > d]

 a 47. u

x−c 2 dx = (a − x)(x − b)3 (x − d) a−b



a−c 2 [F (μ, r) − E (μ, r)] + b−d a−b [a > u ≥ b > c > d]



BY (254.08)

BY (254.08)

(a − u)(u − c) (u − b)(u − d) BY (257.10)

308

Power and Algebraic Functions

 u 48. a

49.

50.

 c



a−x 2 dx = (b − x)3 (c − x)(x − d) b−c

54. a

BY (252.09)

a−c E (γ, r) b−d

55. u

 u 56. d



x−a 2 dx = (x − b)3 (x − c)(x − d) b−c

 d (a −

d−x 2 dx = − x)(c − x) b−a



x)3 (b

x−d 2 dx = (a − x)3 (b − x)(c − x) a−b



[a > u > b > c > d]

BY (257.08)

a−c [F (ν, q) − E (ν, q)] b−d [u > a > b > c > d]  b−d (b − u)(d − u) 2 E (α, q) + a−c a − b (a − u)(c − u) [a > b > c > d > u]

BY (258.08)

BY (251.09)

b−d [F (β, q) − E (β, q)] a−c

BY (252.05) [a > b > c ≥ u > d]  x−d b−d (c − u)(u − d) 2 2 dx = [F (γ, r) − E (γ, r)] + 3 (b − x)(c − x) (a − x) a − b a − c a − c (a − u)(b − u) u [a > b > c > u ≥ d] BY (253.05)     u 2 2(a − d) x−d b−d (b − u)(u − c) dx = E (δ, q) − 3 (a − x) (b − x)(x − c) a−b a−c (a − b)(a − c) (a − u)(u − d) c [a > b ≥ u > c > d] BY (254.03)

 c

58.

[a > b > c ≥ u > d]

BY (253.01) [a > b > c > u ≥ d]    u a−x a−c (a − u)(u − c) 2 2 dx = [F (δ, q) − E (δ, q)] + 3 (x − c)(x − d) (b − x) b − c b − d b − c (b − u)(u − d) c [a > b > u > c > d] BY (254.06)    a a−x a−c (a − u)(u − c) 2 2 dx = E (μ, r) + 3 (x − c)(x − d) (x − b) c − b b − d b − c (u − b)(u − d) u

 u

57.

a−c E (ν, q) b−d

BY (258.01) [u > a > b > c > d]  a−x a−c (a − u)(d − u) 2 2 dx = [F (α, q) − E (α, q)] + 3 (b − x) (c − x)(d − x) b−c b−d b − d (b − u)(c − u) u [a > b > c > d > u] BY (251.12)    u a−x a−c (u − d)(c − u) 2 2(a − b) dx = E (β, r) − 3 (b − x) (c − x)(x − d) b−c b−d (b − c)(b − d) (a − u)(b − u) d



u

53.



 d

51.

52.

x−c 2 dx = 3 (x − a)(x − b) (x − d) a−b

3.168



3.168

Square roots of polynomials

 b 59. u

2 x−d dx = (a − x)3 (b − x)(x − c) a−b

61.

62.

63.

64.

65.

66.

67.

b−d E (κ, q) a−c

[a > b > u ≥ c > d] BY (255.01)  x−d b−d (u − b)(u − d) 2 2 dx = [F (λ, r) − E (λ, r)] + 3 (a − x) (x − b)(x − c) a−b a−c a − b (a − u)(u − c) b [a > u > b > c > d] BY (256.10)   d 2 (a − c)(b − d) c−x 2(c − d)  dx = E (α, q) F (α, q) − 3 (a − x) (b − x)(d − x) (a − b)(a − d) (a − d) (a − c)(b u  − d) (b − u)(d − u) 2(a − c) + (a − b)(a − d) (a − u)(c − u) [a > b > c > d > u] BY (251.15)   u 2 (a − c)(b − d) 2(b − c) c−x  dx = E (β, r) − F (β, r) 3 (b − x)(x − d) (a − x) (a − b)(a − d) (a − b) (a − c)(b − d) d BY (252.08) [a > b > c ≥ u > d]   c 2 (a − c)(b − d) c−x 2(b − c)  dx = E (γ, r) − F (γ, r) 3 (a − x) (b − x)(x − d) (a − b)(a − d) (a − b) (a − c)(b − d) u  (c − u)(u − d) 2 − a − d (a − u)(b − u) [a > b > c > u ≥ d] BY (253.10)   u 2 (a − c)(b − d) x−c 2(c − d)  dx = E (δ, q) − F (δ, q) 3 (b − x)(x − d) (a − x) (a − b)(a − d) (a − d) (a − c)(b − d) c  (b − u)(u − c) 2 − a − b (a − u)(u − d) [a > b ≥ u > c > d] BY (254.09)   b 2 (a − c)(b − d) 2(c − d) x−c  dx = E (κ, q) − F (κ, q) 3 (b − x)(x − d) (a − x) (a − b)(a − d) (a − d) (a − c)(b − d) u BY (255.10) [a > b > u ≥ c > d]   u 2 (a − c)(b − d) x−c 2(b − c)  dx = E (λ, r) F (λ, r) − 3 (x − b)(x − d) (a − x) (a − b)(a − d) (a − b) (a − c)(b − d) b  2(a − c) (u − b)(u − d) + (a − b)(a − d) (a − u)(u − c) [a > u > b > c > d] BY (256.07)     d b−x b−d (b − u)(d − u) 2 2 dx = [F (α, q) − E (α, q)] + 3 (c − x)(d − x) (a − x) a − d a − c a − d (a − u)(c − u) u [a > b > c > d > u] BY (251.13)  u

60.



309



310

Power and Algebraic Functions

 u 68. d

2 b−x dx = (a − x)3 (c − x)(x − d) a−d

 c 69. u

(a −

 u 70. c

 b 71. u

b−x 2 dx = − x)(x − d) a−d

(a −



BY (253.08) [a > b > c > u ≥ d]  b−d (b − u)(u − c) 2 [F (δ, q) − E (δ, q)] + a−c a − c (a − u)(u − d) [a > b ≥ u > c > d] BY (254.07)

x−b −2 dx = − c)(x − d) a−d

b−d [F (κ, q) − E (κ, q)] a−c



x)3 (x

b−d E (β, r) a−c

[a > b > c ≥ u > d] BY (252.01)  b−d (c − u)(u − d) 2(a − b) E (γ, r) − a−c (a − c)(a − d) (a − u)(b − u)



b−x 2 dx = (a − x)3 (x − c)(x − d) a−d

 u b



x)3 (c

b−x 2 dx = (a − x)3 (x − c)(x − d) a−d

72.



3.169

[a > b > u ≥ c > d]  b−d (u − b)(u − d) 2 E (λ, r) + a−c a − d (a − u)(u − c) [a ≥ u > b > c > d]

3.169

Notation: In 3.169–3.172, we set:

u α = arctan , b

BY (255.07)

BY (256.04)

a β = arctan , u

 a2 + b 2 b u ε = arccos , ξ = arcsin δ = arccos , , b u a2 + u 2   a b2 − u2 a u 2 − b2 ζ = arcsin , κ = arcsin , 2 2 b a −u u a2 − b 2  √ u 2 − a2 a2 − b 2 a , q = , μ = arcsin , ν = arcsin 2 2 u −b u a a b b , s= √ t= . r= √ 2 2 2 2 a a +b a +b



a2 + b 2 , a2 + u 2 u η = arcsin , b  2 a − u2 , λ = arcsin a2 − b 2

u γ = arcsin b

 u 1. 0

 u 2.6 0

 u 3. 4.

x2 + a2 dx = a {F (α, q) − E (α, q)} + u x2 + b2



x2 + b2 b F (α, q) − a E (α, q) + u dx = 2 2 x +a a 2

 x2 + a2 dx = a2 + b2 E (γ, r) − u 2 2 b −x 0  b 2  a + x2 dx = a2 + b2 E (δ, r) 2 2 b −x u



a2 + u 2 b2 + u2



[a > b,

u > 0]

BY (221.03)

[a > b,

u > 0]

BY (221.04)

a2 + u 2 b2 + u2

b2 − u2 a2 + u 2

[b ≥ u > 0]

BY (214.11)

[b > u ≥ 0]

BY (213.01), ZH 64 (273)

3.169

Square roots of polynomials

 u 5. b

 u 6. 0

 1 2 a2 + x2 dx = a2 + b2 {F (ε, s) − E (ε, s)} + (u + a2 ) (u2 − b2 ) 2 2 x −b u  b2 − x2 dx = a2 + b2 {F (γ, r) − E (γ, r)} + u 2 2 a +x

8.

0

b2 − x2 a2 − b 2 F (η, t) dx = a E (η, t) − a2 − x2 a

u

b2 − x2 a2 − b 2 F (ζ, t) − u dx = a E (ζ, t) − 2 2 a −x a

9.  b 10.

 u 11. b

 a

a

 u 14. 0

 a

18.

BY (213.03)

[u > b > 0]

BY (211.04)

[a > b ≥ u > 0]

BY (219.03)

b2 − u2 a2 − u 2 BY (220.04)

a2 − x2 dx = a E (η, t) b2 − x2



[a ≥ u > b > 0]

BY (217.04)

[a > u ≥ b > 0]

BY (218.03)

u 2 − a2 u 2 − b2 [u > a > b > 0]

BY (216.03)

[a > b ≥ u > 0]

H 64 (276), BY (219.01)

!  u b2 − u2 a2 − x2 [a > b > u ≥ 0] dx = a E (ζ, t) − b2 − x2 a a2 − u 2 u  u 2 a − x2 1 2 dx = a {F (κ, q) − E (κ, q)} + (a − u2 ) (u2 − b2 ) x2 − b2 u b  b

BY (214.03)

x2 − b2 1 2 b2 F (κ, q) − dx = a E (κ, q) − (a − u2 ) (u2 − b2 ) a2 − x2 a u

x2 − b2 a2 − b 2 F (μ, t) − a E (μ, t) + u dx = x2 − a2 a

13.

BY (211.03)

b2 − u2 a2 + u 2

[a > b > u ≥ 0]

u

 u

17.



b2 x2 − b2 F (λ, q) dx = a E (λ, q) − a2 − x2 a

12.

16.

[u > b > 0]

 b2 − x2 dx = a2 + b2 {F (δ, r) − E (δ, r)} [b > u ≥ 0] 2 2 a +x u   u  x2 − b2 1 2 dx = (a + u2 ) (u2 − b2 ) − a2 + b2 E (ε, s) 2 2 a +x u b  u

15.



[b ≥ u > 0]

 b 7.

311

a2 − x2 dx = a {F (λ, q) − E (λ, q)} x2 − b2 u   u 2 x − a2 u 2 − a2 dx = u − a E (μ, t) x2 − b2 u 2 − b2 a

BY (220.03)

[a ≥ u > b > 0]

BY (217.03)

[a > u ≥ b > 0]

BY (218.09)

[u > a > b > 0]

BY (216.04)

312

3.171

Power and Algebraic Functions



u

1. b



∞

u



b

3. u



u

4. b



a

5. u



u

6. a



dx x2

dx x2 dx x2

∞

u



∞

8. u



b

9. u

11.

 

dx x2

a2 − x2 a 1 = 2 E (κ, q) − F (κ, q) x2 − b2 b a a2 − x2 a 1 = 2 E (λ, q) − F (λ, q) − x2 − b2 b a



dx x2

dx x2

[u ≥ b > 0]  a2 − x2 b2 − u2 a2 − b 2 a a2 = F (ζ, t) − 2 E (ζ, t) + 2 2 2 2 b −x ab b b u a2 − u 2



a 1 a2 x2 + a2 F (β, q) − = E (β, q) + x2 + b2 a b2 b2 u x2 + b2 1 1 = {F (β, q) − E (β, q)} + 2 2 x +a a u

b2 − x2 = a2 + x2



BY (220.12)

[a ≥ u > b > 0]

BY (217.11)







[a > u ≥ b > 0]



b

12. u



dx x2

∞

13. u



[u > a > b > 0]

dx x2

[a > b,

u > 0]

BY (222.08)

u > 0]

BY (222.09)

b2 + u2 a2 + u 2

[a > b, √ (b2 − u2 ) (a2 + u2 ) a2 + b 2 − E (δ, r) 2 a u a2

a a2 − b 2 x2 − a2 = 2 E (ν, t) − F (ν, t) 2 2 x −b b ab2

BY (216.08)

b2 + u2 a2 + u 2

[u ≥ b > 0]  √ (b2 − u2 ) (a2 + u2 ) a2 + x2 a2 + b 2 = {F (δ, r) − E (δ, r)} + 2 2 2 b −x b b2 u



BY (218.10)

u 2 − a2 u 2 − b2



u

BY (212.09)

(a2 − u2 ) (u2 − b2 ) b2 u

x2 − a2 a a2 − b 2 1 = E (μ, t) − F (μ, t) − x2 − b2 b2 ab2 u







BY (211.01), ZH 64 (274)

[a > b > u > 0]

[b > u > 0] √ dx x2 − b2 a2 + b 2 = {F (ε, s) − E (ε, s)} [u > b > 0] 2 2 2 a +x a2 b x   √  ∞ dx x2 − b2 a2 + b 2 1 u 2 − b2 = {F (ξ, s) − E (ξ, s)} + x2 a2 + x2 a2 u a2 + u 2 u 

10.



dx x2

7.



√ a2 + x2 a2 + b 2 = E (ε, s) [u > b > 0] 2 2 x −b b2   √ dx a2 + x2 a2 + b 2 u 2 − b2 a2 = E (ξ, s) − 2 2 2 2 2 x x −b b b u a2 + u 2

dx x2

2.

3.171

BY (213.10) BY (211.07)

BY (212.11)

[b > u > 0]

BY (213.05)

[u ≥ a > b > 0]

BY (215.08)

3.172

Square roots of polynomials

15. 16.



 1 b2 − u2 1 b2 − x2 = − E (ζ, t) [a > b > u > 0] 2 − x2 2 − u2 a u a a u   u dx x2 − b2 1 [a ≥ u > b > 0] = {F (κ, q) − E (κ, q)} 2 2 2 a −x a b x    a (a2 − u2 ) (u2 − b2 ) dx x2 − b2 1 {F (λ, q) − E (λ, q)} + = 2 u2 − x2 a a2 u u x 

14.

313



b

u

17. a



dx x2

1 x2 − b2 1 = E (μ, t) − x2 − a2 a u  dx x2 − b2 1 = E (ν, t) x2 x2 − a2 a

dx x2

∞

18. u





u 2 − a2 u 2 − b2

BY (220.11)

BY (217.08)

[a > u ≥ b > 0]

BY (218.08)

[u > a > b > 0]

BY (216.07)

[u ≥ a > b > 0] BY (215.01), ZH 65 (281)

3.172

 u

1.

(x2

0



x2 + b2

∞



2.  u

∞



4.

5. 0

 b

b



∞

8. u

[a > b,

u > 0]

BY (221.10)

1 E (β, q) a

[a > b,

u ≥ 0]

H 64 (271)

a E (α, q) b2

[a > b,

u > 0]

H 64 (270)

u ≥ 0]

BY (222.06)

dx =

dx =

dx =

a a2 − b 2 u  E (β, q) − 2 2 2 2 b b (a + u ) (b2 + u2 )

[a > b, √ b2 − x2 a2 + b 2 1 dx = E (γ, r) − √ F (γ, r) 3 2 2 2 2 a a + b2 (a + x )

[b ≥ u > 0]  √ b2 − x2 a2 + b 2 b2 − u2 1 u √ dx = E (δ, r) − F (δ, r) − 3 a2 a2 a2 + u 2 a2 + b 2 (a2 + x2 )

BY (214.08)

[b > u ≥ 0]  √ x2 − b2 a2 + b 2 b2 1 u 2 − b2 E (ε, s) − √ F (ε, s) − 3 dx = a2 u u 2 + a2 a2 a2 + b 2 (a2 + x2 )

BY (213.04)

 u 7.

x2 + a2 3

 u

u

+

3 b2 )

(x2 + b2 )

u

6.

+

a2 ) 3

x2 + a2 (x2

0



x2 + b2 (x2

u

3.

+

u a2 − b 2 1  E (α, q) − 2 2 2 a a (a + u ) (b2 + u2 )

dx =

3 a2 )



[u > b > 0] √ x2 − b2 a2 + b 2 b2 √ dx = E (ξ, s) − F (ξ, s) a2 a2 a2 + b 2 (a2 + x2 )3

BY (211.06)

[u ≥ b > 0]

BY (212.08)

314

Power and Algebraic Functions

 u 9. 0



∞

x2 + a2

a2 dx = √ F (γ, r) − 3 b 2 a2 + b 2 (b2 − x2 )



10. u

 u 11.

b2 − x2

13.

14.

−

3 x2 )

dx =

1 a

F (η, t) − E (η, t) +

u a



[u > b > 0] ! b2 − u2 a2 − u 2 [a > b ≥ u > 0]

 b 12.

 2 a + b2 u a2 + b 2 E (γ, r) +  b2 b2 (a2 + u2 ) (b2 − u2 )

[b > u > 0] BY (214.09)  2 √ a + b2 u x2 + a2 a2 + b 2 1 √  dx = F (ξ, s) − E (ξ, s) + 3 b2 a2 + b 2 b2 (a2 + u2 ) (u2 − b2 ) (x2 − b2 )

(a2

0

√

b2 − x2

BY (212.07)

BY (219.09)

1 {F (ζ, t) − E (ζ, t)} [a > b > u ≥ 0] a  1 u 2 − b2 1 [a > u > b > 0] − E (κ, q) 3 dx = u 2 − u2 2 2 a a (a − x ) b   ∞ 2 x − b2 1 u 2 − b2 1 3 dx = a [F (ν, t) − E (ν, t)] + u u 2 − a2 (x2 − a2 ) u

BY (217.07)

[u > a > b > 0]

BY (215.05)

(a2 − x2 ) u   u x2 − b2

 u 15.  a 16.

x2 )3

a2 − x2 3

u

 u 17.

(x2 − b2 ) x2 − a2 (x2

a



3

a2 − x2 (b2 −

0

∞



18. u

−

3 b2 )

dx =

dx =

a u [F (η, t) − E (η, t)] + 2 b2 b 



a2 − u 2 a − 2 E (λ, q) u 2 − b2 b

dx =

u b2

dx =

a [F (μ, t) − E (μ, t)] b2

x2 − a2

a 1 3 dx = b2 [F (ν, t) − E (ν, t)] + u 2 2 (x − b )





1

1. u

 2. 1

u

dx x2

dx x2





BY (220.07)

a2 − u 2 b2 − u2 [a > b > u > 0]

BY (219.10)

[a > u > b > 0]

BY (218.05)

[u > a > b > 0]

BY (216.05)

u 2 − a2 u 2 − b2 [u ≥ a > b > 0]

3.173

3.173

BY (215.03)

   √  √ √  x2 + 1 √ 2 2 1 − u4 − E arccos u, + = 2 F arccos u, 2 1−x 2 2 u x2 + 1 √ = 2E x2 − 1



√  2 1 arccos , u 2

[u < 1]

BY (259.77)

[u > 1]

BY (260.76)

3.181

3.174

Fourth roots of polynomials

Notation: In 3.174 and 3.175, we set: √  1− 1+ 3 u √ , β = arccos 1+ 3−1 u 

u

1. 0

2.

3.

4.

3.175

 dx √ 2  1+ 1+ 3 x

315

√  1+ 1− 3 u √ ,  α = arccos 1+ 1+ 3 u  √ 2+ 3 , p= 2

1 − x + x2 1 = √ E (α, p) 4 x(1 + x) 3

 q=

2− 2

√ 3

.

[u > 0]

BY (260.51)

 1 dx 1 + x + x2 = √ E (β, q) [1 ≥ u > 0] BY (259.51) √ 2 4 x(1 − x) 3 0 1+ 3−1 x  √ √   √  u 2 2+ 3 1+ 1− 3 u 2− 3 1 dx x(1 + x) √ √  E (α, p) − √ F (α, p) − =√ 4 4 2 1 − x + x2 27 27 3 1+ 1+ 3 u 0 1−x+x  u(1 + u) × 1 − u + u2 [u > 0] BY (260.54)  √ √   √  u 2 2− 3 1− 1+ 3 u dx x(1 − x) 4 2+ 3 √ √ √ E (β, q) − √ F (β, q) − 4 4 2 2 1 + x + x 1 + x + x 27 27 3 1+ 3−1 u 0  u(1 − u) × 1 + u + u2 [1 ≥ u > 0] BY (259.55) 

u



u

1. 0



u

2. 0

dx 1+x

dx 1−x





 u (1 − u + u2 ) x 2 1 √ √ √  [F (α, p) − 2 E (α, p)] + = √ 4 1 + x3 27 3 1+u 1+ 1+ 3 u [u > 0]  u (1 + u + u2 ) 2 1 x √ [F (β, q) − 2 E (β, q)] + √ √ = √ 4 3 1−x 27 3 1−u 1+ 3−1 u [0 < u < 1]

BY (260.55)

BY (259.52)

3.18 Expressions that can be reduced to fourth roots of second-degree polynomials and their products with rational functions 3.181



1. b

u

      √ dx 1 1 4 4(a − u)(u − b)  + E arccos ,√ = a−b 2 E √ 4 (a − b)2 2 2 (a − x)(x − b)     !  4(a − u)(u − b) 1 1 − K √ + F arccos 4 ,√ (a − b)2 2 2 BY (271.05) [a ≥ u > b]

316

Power and Algebraic Functions



u

2. a

3.182



u

1. b

dx

 4 (x − a)(x − b)

3.182

   a − b − 2 (u − a)(u − b) 1 a−b  F arccos ,√ 2 2 a − b + 2 (u − a)(u − b)    a − b − 2 (u − a)(u − b) 1  ,√ − 2 E arccos 2 a − b + 2 (u − a)(u − b)  4 2(2u − a − b) (u − a)(u − b)  + a − b + 2 (u − a)(u − b) [u > a > b]



BY (272.05)

      4(a − u)(u − b) 1 1 2  K √ + F arccos =√ ,√ 4 (a − b)2 a−b 2 2 [(a − x)(x − b)]3 dx

[a ≥ u > b]    √ u a − b − 2 (u − a)(u − b) 1 dx 2   F arccos =√ ,√ 4 a−b 2 [(x − a)(x − b)]3 a − b + 2 (u − a)(u − b) a

BY (271.01)

 2.

[u > a > b] 3.183

Notation: In 3.183–3.186 we set: 1 α = arccos √ , 4 2 u +1

2. 3.

3.184 1.

2.

√ 1 − u2 − 1 √ . γ = arccos 1 + u2 − 1

     √ 1 2u 1 dx √ = 2 F α, √ − 2 E α, √ + √ 4 4 2 x +1 2 2 u2 + 1 0 [u > 0]       u √ dx 1 1 √ − F β, √ [0 < u ≤ 1] = 2 2 E β, √ 4 2 1 − x 2 2 0 √      u dx 1 1 2u 4 u2 − 1 √ √ = F γ, √ − 2 E γ, √ + 4 x2 − 1 2 2 1 + u2 − 1 1 [u > 1] 

1.

 4 β = arccos 1 − u2 ,

u

√       x2 dx 1 1 2u 4 2 2 3 √ √ √ 2 E β, − F β, − = (1 − u2 ) 4 2 5 5 2 2 1 − x 0 [0 < u ≤ 1] √ √      u dx u2 − 1 1 1 1 − u2 − 1 1 √ √ √ √ F γ, − − = E γ, · 4 2 x2 − 1 2 u 2 2 1 + u2 − 1 1 x



BY (272.00)

BY (273.55) BY (271.55)

BY (272.55)

u

[u > 1]

BY (271.59)

BY (272.54)

3.186

3.185

Fourth roots of polynomials



u

1. 0

3.

4.

5.

 4 (x2 + 1)3

√ = 2F

  1 α, √ 2

  √ dx 1  = 2 F β, √ 2 0 4 (1 − x2 )3    u dx 1  = F γ, √ 2 1 4 (x2 − 1)3 √    u x2 dx 1 2  2 2 4  √ F β, − u 1 − u2 = 3 3 4 3 2 0 (1 − x2 )      u √ √ dx 1 1  − 2 F α, √ = 2 2 E α, √ 2 2 0 4 (x2 + 1)5 

2.

dx



u

u

6. 0

3.186 1. 2.

[u > 0]

BY (273.50)

[0 < u ≤ 1]

BY (271.51)

[u > 1]

BY (272.50)

[0 < u ≤ 1]

BY (271.54)

[u > 0]      √ x dx 1 2u 1  − 2 E α, √ + √ = 2 2 F α, √ 4 4 5 2 2 u2 + 1 (x2 + 1)

  u x2 dx 1 1 u  = √ F α, √ −  4 7 4 3 2 2 0 (x2 + 1) 6 (u2 + 1)3

[u > 0]

BY (273.56)

[u > 0]

BY (273.53)

√   √ 1 + x2 + 1 1 √ √ 2 E α, dx = 2 [u > 0] 4 2 x2 + 1 2 0 (x + 1) √       u √ 1 u 4 1 − u2 1 dx √ √  √ − E β, √ + = 2 F β, √ 1 + 1 − x2 4 1 − x2 2 2 1 + 1 − u2 0



u

[0 < u ≤ 1]      u dx 1 1 1 √ √ √  √ F γ, = − E γ, 2 x2 + 2 x2 − 1 4 x2 − 1 2 2 1

 3.



u

4. 0

 5. 1

BY (273.54)

2

 7.

317

u

[u > 1] √ √      √ 1 − 1 − x2 dx 1 1 2u 4 1 − u2 √ √ · = 2 2 E β, √ − F β, √ − 1 + 1 − x2 4 (1 − x2 )3 2 2 1 + 1 − u2   x dx 1 = E γ, √ √   4 3 2 x2 + 2 x2 − 1 (x2 − 1) 2

BY (273.51)

BY (271.58)

BY (272.53)

[0 < u ≤ 1]

BY (271.57)

[u > 1]

BY (272.51)

318

Power and Algebraic Functions

3.191

3.19–3.23 Combinations of powers of x and powers of binomials of the form (α+βx) 3.191



a

xν−1 (a − x)μ−1 dx = aμ+ν−1 B(μ, ν)

1. 0



∞

2. u



1

3.

[Re μ > 0,

x−ν (x − u)μ−1 dx = uμ−ν B(ν − μ, μ)

xν−1 (1 − x)μ−1 dx =

0



1

[Re ν > Re μ > 0]



1

1. 0



1

2. 0



1

3. 0



xμ−1 (1 − x)ν−1 dx = B(μ, ν)

3.194

xa dx = −π cosec aπ (1 − x)a+1

[−1 < a < 0]

BI (3)(5)

(1 − x)a dx = −π cosec aπ xa+1

[−1 < a < 0]

BI (4)(6)

1

(x − 1)a− 2



u

0



∞

u

∞

0 ∞

4.11 0



a−ν

x dx u = ν (1 + bx)ν b (ν − a)

 2 F 1 ν, ν − a;

[Re ν > 0]

[|arg(1 + bu)| < π,

ν − a + 1;

−

1 bu

xa−1 dx = b−a B(a, ν − a) (1 + bx)ν xa−1 dx π = (−1)n a (1 + bx)n+1 b



a−1 n

Re a > 0] ET I 310(20)



[|arg b| < π,

EH I 2

ET I 310(21)

Re ν > Re a > 0] FI II 775a, ET I 310(19)

 cosec(aπ)

[|arg b| < π,

0 < Re a < n + 1] ET I 308(6)

u

5. 0



a−1

BI (23)(7)

[Re ν > Re a]

3. 

nν+n n! n!nν+n = (ν)n+1 ν(ν + 1)(ν + 2) . . . (ν + n)

xa−1 dx ua = 2 F 1 (ν, a; 1 + a; −bu) (1 + bx)ν a

2.6 

1

− 2 < a < 12

dx = π sec aπ x

xν−1 (n − x)n dx =

0

1.

FI II 774(1)

BI (3)(4)

∞

n

Re ν > 0]

2

a 0, 3.192

Re ν > 0]

xa−1 dx ua = 2 F 1 (1, a; 1 + a; −bu) 1 + bx a

[|arg(1 + ub)| < π,

Re a > 0] ET I 308(5)

∞

6.12 u

a−1

(1 − a)π x dx = 2 (1 + x) sin πa

[0 < Re a < 2]

BI (16)(4)

Powers of x and binomials

3.197



∞

7.

1

xm dx 1

(a + bx)n+ 2

0

 0

 3.19511

= 2m+1 m!

xn−1 dx = 2−n (1 + x)m

1

8.

∞

0

319

∞   k=0

(2n − 2m − 3)!! am−n+ 2 (2n − 1)!! bm+1

m < n − 12 ,

a > 0,

b>0

BI (21)(2)



m − n − 1 (−2)−k n+k k

(1 + x)p−1 1 − a−p dx = (a + x)p+1 p(a − 1) ln a = a−1 =1

BI (3)(1)

[p = 0,

a > 0,

a = 1]

[p = 0,

a > 0,

a = 1]

[a = 1] LI (19)(6)

3.196



a

1.

(x + b)ν (a − x)μ−1 dx =

0



∞

2. a



b

3. a

 4.12 5.12 3.197

(x + b)−ν (x − a)μ−1 dx = (a + b)μ−ν B(ν − μ, μ)

a−1

(x − a)

ν−1

(b − x)

a+ν−1

dx = (b − a)

B(a, ν)

 a −ν π dx 1− =− ν b sin πν b 1 (a − bx)(x − 1)  1  −ν dx a π − 1 = ν b sin πν b −∞ (a − bx)(1 − x) 

∞

∞

0



∞

12 a

 3. 0

 a   arg  < π, b [b > a,

ET II 185(8)

Re ν > Re μ > 0



ET II 201(7)

Re a > 0,

Re ν > 0] EH I 10(13)

1.

2.

 a b ν aμ 2 F 1 1, −ν; 1 + μ; − μ b   a   arg  < π b

1

[a < b,

b > 0,

[a > b > 0,

0 < ν < 1]

0 < ν < 1]

LI (23)(5)

LI (24)(10)

 γ xν−1 (b + x)−μ (x + γ)− dx = b−μ γ ν− B(ν, μ − ν + ) 2 F 1 μ, ν; μ + ; 1 − b [|arg b| < π, |arg γ| < π, Re ν > 0, Re μ > Re(ν − )] ET II 233(9) −λ

x

ν

μ−1

(x + b) (x − a)

  b dx = a (b + a) B(λ − μ − ν, μ) 2 F 1 λ, μ; λ − ν; −     a   b b arg  < π or   < 1, 0 < Re μ < Re(λ − ν) ET II 201(8)  a a −λ

μ+ν

xλ−1 (1 − x)μ−1 (1 − bx)−ν dx = B(λ, μ) 2 F 1 (ν, λ; λ + μ; b) [Re λ > 0,

Re μ > 0,

|b| < 1]

WH

320

Power and Algebraic Functions



1

4.

xμ−1 (1 − x)ν−1 (1 + ax)−μ−ν dx = (1 + a)−μ B(μ, ν)

0

[Re μ > 0,

Re ν > 0,

a > −1] BI(5)4, EH I 10(11)



∞

5. 0

xλ−1 (1 + x)ν (1 + ax)μ dx = B(λ, −μ − ν − λ) 2 F 1 (−μ, λ; −μ − ν; 1 − a) [|arg a| < π,

− Re(μ + ν) > Re λ > 0] EH I 60(12), ET I 310(23)



∞

6. 1





xλ−ν (x − 1)ν−μ−1 (ax − 1)−λ dx = a−λ B(μ, ν − μ) 2 F 1 ν, μ; λ; a [1 + Re ν > Re λ > Re μ,

EH I 115(6)

[Re μ > 0]  u xν−1 (x + a)λ (u − x)μ−1 dx = aλ uμ+ν−1 B(μ, ν) 2 F 1 −λ, ν; μ + ν; − a   u    arg  < π, Re μ > 0, a

BI 19(5)

∞

μ− 12

x

0



u

8. 0



−1

|arg(a − 1)| < π]

7.

∞

9. 0



3.198

−μ

(x + a)

−μ

(x + b)

 √ 1−2μ Γ μ − 1 √ √ 2 dx = π a+ b Γ(μ)

Re ν > 0



ET II 186(9)

xλ−1 (1 + x)−μ+ν (x + β)−ν dx = B(μ − λ, λ) 2 F 1 (ν, μ − λ; μ; 1 − β) [Re μ > Re λ > 0]

EH I 205

xq−1 dx π = cosec qπ [0 < q < 1, p > −1] BI (5)(1) q (1 + px) (1 − x) (1 + p)q 0  √  1 1 2 Γ p + 12 Γ(1 − p) xp− 2 dx √ sin (2p − 1) arctan q 2p √ √ cos (arctan q) 11. = p p π (2p − 1) sin arctan q 0 (1 − x) (1 + qx) 1

− 2 < p < 1, q > 0 BI (11)(1)     1 √ √ 1−2p 1−2p 1 Γ p + 12 Γ(1 − p) 1 − q − 1+ q xp− 2 dx √ 12. = √ p p π (2p − 1) q 0 (1 − x) (1 − qx) 1

− 2 < p < 1, 0 < q < 1 BI (11)(2)  1 3.198 xμ−1 (1 − x)ν−1 [ax + b(1 − x) + c]−(μ+ν) dx = (a + c)−μ (b + c)−ν B(μ, ν) 1

10.

0

[a ≥ 0,

 3.199

b

a

 3.211 0

b ≥ 0,

c > 0,

Re μ > 0,

Re ν > 0]

FI II 787

(x − a)μ−1 (b − x)ν−1 (x − c)−μ−ν dx = (b − a)μ+ν−1 (b − c)−μ (a − c)−ν B(μ, ν) [Re μ > 0,

Re ν > 0,

c < a < b] EH I 10(14)

1

xλ−1 (1 − x)μ−1 (1 − ux)− (1 − vx)−σ dx = B(μ, λ) F 1 ((λ, , σ, λ + μ; u, v)) [Re λ > 0,

Re μ > 0]

EH I 231(5)

Powers of x and binomials

3.223

321

  

q−1 a+b −p −p − q2 (1 + ax) + (1 + bx) x dx = 2(ab) B(q, p − q) cos q arccos √ 2 ab 0 BI (19)(9) [p > q > 0]     ∞

q a+b (1 + ax)−p − (1 + bx)−p xq−1 dx = −2i(ab)− 2 B(q, p − q) sin q arccos √ 2 ab 0 [p > q > 0] BI (19)(10)  1

(1 + x)μ−1 (1 − x)ν−1 + (1 + x)ν−1 (1 − x)μ−1 dx = 2μ+ν−1 B(μ, ν) 

3.212

3.213

3.214

∞

0

[Re μ > 0,

Re ν > 0] LI(1)(15), EH I 10(10)



1

3.215



 aμ xμ−1 (1 − ax)ν−1 + (1 − a)ν xν−1 [1 − (1 − a)x]μ−1 dx = B(μ, ν)

0

[Re μ > 0,

Re ν > 0,

|a| < 1] BI (1)(16)

3.216  1.

1

0

 2. 3.217 3.218 3.219

[Re μ > 0,

Re ν > 0]

FI II 775

∞

xμ−1 + xν−1 dx = B(μ, ν) [Re μ > 0, Re ν > 0] (1 + x)μ+ν 1   ∞  p p−1 b x (1 + bx)p−1 dx = π cot pπ [0 < p < 1, b > 0] − (1 + bx)p bp−1 xp 0 ∞ 2p−1 2p−1 x − (a + x) dx = π cot pπ [p < 1] (cf. 3.217) x)p xp  0 ∞  (a + xν xμ dx = ψ(μ + 1) − ψ(ν + 1) − ν+1 (x + 1) (x + 1)μ+1 0 [Re μ > −1, Re ν > −1]

3.221  1.

∞

FI II 775 BI(18)(13) BI (18)(7)

BI (19)(13)

(x − a)p−1 dx = π(a − b)p−1 cosec pπ x−b

[a > b,

0 < p < 1]

LI (24)(8)

(a − x)p−1 dx = −π(b − a)p−1 cosec pπ x−b −∞

[a < b,

0 < p < 1]

LI (24)(8)

a

 2.

xμ−1 + xν−1 dx = B(μ, ν) (1 + x)μ+ν

3.222  1. 0



a

1

xμ−1 dx = β(μ) 1+x

∞

2. 0

xμ−1 dx = π cosec(μπ)aμ−1 x+a = −π cot(μπ)(−a)μ−1

[Re μ > 0]

WH

for a > 0

FI II 718, FI II 737

for a < 0

BI(18)(2), ET II 249(28)

[0 < Re μ < 1] 3.223  1. 0

∞

xμ−1 dx π  μ−1 = b − cμ−1 cosec(μπ) (b + x)(c + x) c−b [|arg b| < π,

|arg c| < π,

0 < Re μ < 2]

ET I 309(7)

322

Power and Algebraic Functions



∞

2. 0

3. 3.224

3.225

∞



∞



∞

2. 1



∞

3. 0



1

1. 0



1

2. 0

0 < Re μ < 2]



∞

0



1. a

b

BI (23)(8)

(x − 1)1−p 1 dx = p(1 − p)π cosec pπ 3 x 2

[0 < p < 1]

BI (23)(1)

xp dx π = p(1 − p) cosec pπ (1 + x)3 2

[−1 < p < 2]

BI (16)(5)

BI (8)(1)

1

0



[−1 < p < 1]

(2n − 1)!! xn− 2 dx √ π. = (2n)!! 1−x

∞

2.

(x − 1)p−1 dx = (1 − p)π cosec pπ x2

xn dx (2n)!! √ =2 (2n + 1)!! 1−x

1.

3.228

a > 0,

xμ−1 dx aμ−1 − bμ−1 = π cot(μπ) [a > b > 0, 0 < Re μ < 2] ET I 309(9) b−a 0 (a − x)(b − x)    ∞ (x + b)xμ−1 dx c − b μ−1 d − b μ−1 = π cosec(μπ) c d + (x + c)(x + d) c−d d−c 0 [|arg c| < π, |arg d| < π, 0 < Re μ < 1] ET I 309(10)

1

3.227

π μ−1 xμ−1 dx = b cosec(μπ) + aμ−1 cot(μπ) (b + x)(a − x) a+b [|arg b| < π,

ET I 309(8)



1.

3.226

3.224

BI (8)(2)

 xν−1 (b + x)1−μ c dx = b1−μ cν−1 B(ν, μ − ν) 2 F 1 μ − 1, ν; μ; 1 − c+x b [|arg b| < π, |arg c| < π, 0 < Re ν < Re μ] x− (b − x)−σ dx = πc− (b − c)−σ cosec(π) I 1−c/b (σ, ) c+x [|arg b| < π, |arg c| < π,

− Re σ < Re  < 1]

  ν  (x − a)ν (b − x)−ν a−c dx = π cosec(νπ) 1 − for c < a x−c b−c  ν   c−a for a < c < b = π cosec(νπ) 1 − cos(νπ) ν b − c   c−a for c > b = π cosec(νπ) 1 − c−b [|Re ν| < 1]

ET II 217(9)

ET II 217(10)

ET II 250(31)

Powers of x and binomials

3.231



b

2. a



b

3.12 a

 ν−1 (x − a)ν−1 (b − x)−ν π cosec(νπ)  a − c  dx = b − c x−c b−c ν−1 π(c − a) cot(νπ) =− (b − c)ν

323

for c < a or c > b; for a < c < b [0 < Re ν < 1]

(x − a)ν−1 (b − x)μ−1 dx x−c   (b − a)μ+ν−1 b−a = B(μ, ν) 2 F 1 1, μ; μ + ν; b−c b−c

ET II 250(32)

for c < a or c > b;

= (c − a)ν−1 (b − c)μ−1 cot μπ − (b − a)μ+ν−2 B(μ − 1, ν)   b−c for a < c < b × 2 F 1 2 − μ − ν, 1; 2 − μ; b−a [Re μ > 0, Re ν > 0, μ + ν = 1, μ = 1, 2, . . .] ET II 250(33) 

1

4. 0

 5.

(1 − x)ν−1 x−ν π(a − b)ν−1 dx = cosec(νπ) a − bx aν

∞

12 0



∞

6.12 0



3.231



1

1. 0



1

2.11 0



1

3. 0

 4.

0

1

BI (5)(8)

for c < 0;

for c > 0 ET II 251(34)

⎡ ⎤  ν−1 n−1 −n ν−1  (1 − ν)j  γ − β j π (γ + x) β γ ⎣1 − ⎦ dx = xν−1 x+β sin πν (γ − β)n β j! γ j=0

1

0

0 < b < a]

 xν−1 (x + a)1−μ c dx = a1−μ (−c)ν−1 B(μ − ν, ν) 2 F 1 μ − 1, ν; μ; 1 + x−c a a1−μ−ν ν−1 1−μ B(μ − ν − 1, ν) =c (a + c) cot[(μ − ν)π] − π (a + c)  a × 2 F 1 2 − μ, 1; 2 − μ + ν; a+c [a > 0, 0 < Re ν < Re μ]

[|arg β| < π, 3.229

[0 < Re ν < 1,

|arg γ| < π,

0 < Re ν < n]

  π cosec μπ b xμ−1 dx a = − (1 − x)μ (1 + ax)(1 + bx) a−b (1 + a)μ (1 + b)μ [0 < Re μ < 1]

AS 256 (6.1.22)

BI (5)(7)

xp−1 − x−p dx = π cot pπ 1−x

2

p −1,

FI II 718

a > 0,

b > 0,



c > 0] BI (18)(14)

dx = ψ(ν) + C x

[Re ν > 0]

EH I 17, WH

dx = πa−q cot qπ

[0 < q < 1,

a > 0]

BI (5)(11)

dx = πa−q cosec qπ

[0 < q < 1,

a > 0]

BI (5)(10)

∞

(1 + x)μ − 1 dx = ψ(ν) − ψ(ν − μ) [Re ν > Re μ > 0] (1 + x)ν x   0 1 xμ/2 dx μ 1−μ (1 − a)−μ − (1 + a)−μ  √ Γ 1 + Γ = (μ+1)/2 2aμ π 2 2 0 [(1 − x) (1 − a2 x)] [−2 < μ < 1, |a| < 1]  u 2  n+1 ∞  u Γ dx = ln   2 2 (−1)n+1 [|arg u| < π] x + u u+1 n n=0 2 Γ 2

3.235

3.238

q−1

−x 1−x

0

1.

[Re μ > 0,

FI II 815, BI(4)(5)



3.234

3.232

∞

ν−1

|x| νπ ν−1 dx = −π cot |u| sign u x − u 2 −∞

[0 < Re ν < 1,

u real,

BI (18)(5)

BI (12)(32) ET II 216(1)

u = 0] ET II 249(29)

 2.

|x|ν−1 νπ ν−1 sign x dx = π tan |u| x − u 2 −∞ ∞

[0 < Re ν < 1,

u real,

u = 0] ET II 249(30)



b

3.

(b − x)μ−1 (x − a)ν−1 μ+ν

|x − u|

a

dx =

(b − a)μ+ν−1 Γ(μ) Γ(ν) |a − u|μ |b − u|ν Γ(μ + ν) [Re μ > 0, Re ν > 0, 0 < u < a < b or 0 < a < b < u]

MO 7

3.24–3.27 Powers of x, of binomials of the form α + βxp and of polynomials in x 3.241



1. 0

1

xμ−1 dx 1 = β 1 + xp p

  μ p

[Re μ > 0,

p > 0]

WH, BI (2)(13)

Powers of x and binomials and polynomials

3.244



∞

2. 0

μπ 1 π xμ−1 dx = B = cosec ν 1+x ν ν ν



3.11 4.12

6.10

3.242

μ ν −μ , ν ν

 [Re ν ≥ Re μ ≥ 0] ET I 309(15)a, BI (17)(10)

∞

p−1

x dx pπ π [p < q] = cot q 1 − x q q 0  μ/ν   ∞ xμ−1 dx p 1 μ μ , = B 1 + n − ν n+1 νpn+1 q ν ν 0 (p + qx )  μ 0 < < n + 1, ν PV

 5.



325

∞

p−1

x

(p − q)π (p − q)π [p < 2q] cosec 2 q q 0 (1 +  p       b b−u x 1/p x − du = (b − a)F G(x) = sign c b−a c a where ⎧ ⎪ x≤0  1 ⎨−1 F (x) = sign(x − t) dt = 2x − 1 0 < x < 1 ⎪ 0 ⎩ 1 x≥1 dx

2 xq )

=

BI (17)(11)

p = 0,

q = 0



BI (17)(22)a BI (17)(18)

  x2m dx π (2n − 2m − 1) (2m + 1)π 1. = sin t cosec t cosec 4n + 2x2n cos t + 1 x n 2n 2n −∞

2 m < n, t < π 2 FI II 642       ∞ c 1 1 x2 x2 + 1 dx 2.11 = 2−1/2−c (1 + a)1/2−c B c − , 4 + 2ax2 + 1 b+1 2 x x x 2 2 0  ∞ xμ−1 dx 3.24311 2ν 3ν 0 (1 + x ) (1 + x )      4π 2π π 8 cosec(2ρ) + 12 cosec(3ρ) − 8 cosec 2ρ − + 8 cosec 2ρ − = 48ν 3 3    π π cosec ρ + sec(ρ) −3 cosec ρ − 6 6 μπ where ρ = , [0 < Re μ < 5 Re ν] ET I 312(34) 6ν 3.244



∞



1

1. 0

 2. 3.

xp−1 + xq−p−1 pπ π dx = cosec 1 + xq q q

xp−1 − xq−p−1 pπ π dx = cot q 1 − x q q 0  1 ν−1   μ  μ−1 x −x 1 C + ψ dx = 1 − xν ν ν 0 1

[q > p > 0]

BI (2)(14)

[q > p > 0]

BI (2)(16)

[Re μ > Re ν > 0]

BI (2)(17)

326

Power and Algebraic Functions

     π x2m − x2n 2m + 1 2n + 1 cot π − cot π dx = 2l l 2l 2l −∞ 1 − x

 4.

3.245

∞

[m < l, 

∞

n < l]

FI II 640

ν−μ

x − xν (1 + x)−μ dx =

ν B(ν, μ − ν) ν − μ+1 0 [Re μ > Re ν > 0]  ∞ 1 − xq p−1 qπ pπ (p + q)π π cosec cosec x dx = sin r r r r r 0 1−x [p + q < r, p > 0]

3.245 3.246



BI (16)(13)

ET I 331(33), BI (17)(12)



dx can be transformed by the substitution x = et Integrals of the form f xp ± x−p , xq ± x−q , . . . x  1  ∞  1+p 1−p −1 −t x or x = e . For example, instead of +x dx, we should seek to evaluate sech px dx 0 0   1 n−m−1 ∞ x + xn+m−1 dx, we should seek to evaluate cosh mx (cosh nx − cos a)−1 dx and, instead of n 2n 1 + 2x cos a + x 0 0 (see 3.514 2). 3.247  1 α−1 ∞  x (1 − x)n−1 ξk 11 dx = (n − 1)! 1. 1 − ξxb (α + kb)(α + kb + 1) . . . (α + kb + n − 1) 0 k=0



[b > 0, ∞

2. 0

|ξ| < 1]

π  (p + ν)π π (1 − xp ) xν−1 πν cosec dx = sin cosec np 1−x np n np np [0 < Re ν < (n − 1)p]

3.248 1. 2.12 3.12 4.3

6.12

  1 xμ−1 dx μ 1 μ √ , − [Re ν > Re 2μ > 0] = B ν ν 2 ν 1 + xν 0 √  1 2n+1 x dx πn! (2n)!!  √ = 3 = (2n + 1)!! 2 2Γ n + 1 − x 0 2   √  1 2n x dx π 1 (2n − 1)!! π √ Γ n+ = = 2 2n! 2 (2n)!! 2 1−x 0  ∞ dx π √ = 2 2 3 −∞ (1 + x ) 4 + 3x ⎧   ⎪ 2 b ⎪ ⎪ √ −1 if a < b arctan ⎪ ⎪ ⎪ a b − a ⎪  ∞ ⎨ dx 2  = √  if a = b 2 ⎪ 2 −∞ 1 + x ⎪ b + ax √ √  ⎪ a ⎪ ⎪ 1 a+ a−b ⎪ ⎪ √ ln √ if a > b ⎩√ a−b a− a−b



AD (6704)

ET I 311(33)

∞

BI (21)(9)

BI (8)(14)

BI (8)(13)

Powers of x and binomials and polynomials

3.251

3.249 1.0 2.9 3. 4. 5.



327

∞

dx π (2n − 3)!! = 2n−1 2 + a2 ) n 2 · (2n − 2)!! a (x 0  a  2 n− 12 (2n − 1)!! a − x2 dx = a2n π. 2(2n)!! 0 n  1 1 − x2 dx = 2n+1 Q n (a) n+1 −1 (a − x)    1 μ x dx μ+1 1 = β 2 2 2 0 1+x  1  μ−1  1 − x2 dx = 22μ−2 B(μ, μ) = 12 B 12 , μ

FI II 743 FI II 156

EH II 181(31)

[Re μ > −1]

BI (2)(7)

[Re μ > 0]

FI II 784

[p > 0]

BI (7)(7)

0

 6. 7. 8.11 3.251

1



1

0



3.

1−

 ν−1 1 μ  ,ν xμ−1 1 − xλ dx = B λ λ

x2m dx (2m − 1)!!(2n − 2m − 3)!!π √ n = 2(2n − 2)!!am cn−m−1 ac (ax2 + c)

∞

x dx m!(n − m − 2)! n = 2 (ax + c) 2(n − 1)!am+1 cn−m−1

∞

12

1

12

0

x2a+1 dx

1

2 x2 )

x2a+1 dx (1 +

0

8.

2m+1

(1 +

0



Re ν > 0,

λ > 0]

Re μ > 0, [p > 0,



Re ν + 12 μ < 1

Re ν > 0,

Re μ < p − p Re ν]

[a > 0,

c > 0,

n > m + 1] GU (141)(8a)

0



[n > 1]

ET I 311(32)

5.

7.

μ

∞

0



|ν| > 1]

FI II 787

μ

∞

4. 

[Re μ > 0,

[Re μ > 0,

 ν−1 1 ,1 − ν − xμ−1 1 + x2 dx = B 2 2 2 0    ∞ μ 1 ν−1 xμ−1 (xp − 1) dx = B 1 − ν − , ν p p 1



6.

√ p−1 x dx =

2 p(p + 1) 0    1 1 1 1 1 μ −ν ,1 − (1 − x ) dx = B μ μ ν 0  −n/2    ∞  π(n − 1) x2 n−1 n Γ 1+ dx = n−1 2 Γ 2 −∞

1.

2.



x2 )2

= =

πa 2 sin πa

−p q

dx =

n > m + 1 ≥ 1]

[| Re a| < 1]

2aβ(a) − 1 4

xq+p−1 (1 − xq )

[ac > 0,

pπ pπ cosec q2 q

GU (141)(8b) WH

[Re a > 0]

LI (3)(11)

[q > p]

BI (9)(22)

328

Power and Algebraic Functions



1

9.

1 −p

q

x p −1 (1 − xq )

dx =

0

 10.

0



1

p

∞

−ν

xμ−1 (1 + bxp )

0

∞

0

 2. 3.

[p > 1,

pπ π cosec q q   μ 1 − μp μ ,ν − dx = b B p p p [|arg b| < π, p > 0,

xp−1 (1 − xq )− q dx =

11.

3.252  1.

π π cosec q p

n−1

(ax2

x dx n + 2bx + c) (−1)n ∂ n−2 = (n − 1)! ∂cn−2

ac > b2

GW (131)(4)

c > 0,

GW (131)(5)

√ b > − ac

! b 1 b − arccot √ 2 (ac − b2 ) 2 (ac − b2 ) 32 ac − b2 ! √ b 1 b + b2 − ac √ + ln 2 (ac − b2 ) 4 (b2 − ac) 32 b − b2 − ac

(−1)n ∂ n−2 (n − 1)! ∂cn−2 an−2 = 2(n − 1)(2n − 1)b2n−2

=

∞

5. −∞

(ax2

for ac > b2 ; for b2 > ac > 0; for ac = b2

[a > 0,

x dx (2n − 3)!!πban−2 n = − (2n−1) + 2bx + c) (2n − 2)!! (ac − b2 ) 2

ac > b2 ,

b > 0, a > 0,

n ≥ 2] n≥2

GW (141)(5)

GW (141)(6)

∞

m

6. −∞

(ax2

∞

m

(ax2

n−m−1 m

x dx b (−1) πa n = 1 + 2bx + c) (2n − 2)!! (ac − b2 )n− 2 k  m/2   m ac − b2 (2k − 1)!!(2n − 2k − 3)!! × b2 2k k=0

ac > b2 , 0 ≤ m ≤ 2n − 2 xn dx

7. 0

BI (17)(20), EH I 10(16)

GW (213)(4) ∞

0



BI (9)(20)

  (−1)n−1 ∂ n−1 b dx 1 √ √ = arccot (ax2 + 2bx + c)n (n − 1)! ∂cn−1 ac − b2 ac − b2

a > 0, ac > b2

∞

4.



BI (9)(23)a

0 < Re μ < p Re ν]

dx (2n − 3)!!πa a > 0, n = 1 n− + 2bx + c) −∞ (2n − 2)!! (ac − b2 ) 2    ∞ dx 1 (−2)n ∂ n √ √ = n+ 32 (2n + 1)!! ∂cn c ( ac + b) 0 (ax2 + 2bx + c) a ≥ 0,



q > 0]

[q > p > 0]

(ax2



3.252

n+ 32

+ 2bx + c)

=

n! √ √ n+1 (2n + 1)!! c ( ac + b) a ≥ 0,

c > 0,

GW (141)(17)

√ b > − ac GW (213)(5a)

Powers of x and binomials and polynomials

3.254



∞

8.

xn+1 dx n+ 32

(ax2 + 2bx + c)

0

=

n! √ √ n+1 (2n + 1)!! a ( ac + b) a > 0,

329

c ≥ 0,

√ b > − ac GW (213)(5b)



n+ 12

∞

9.

x (ax2

0

dx n+1

+ 2bx + c)

=

1 22n+ 2

(2n − 1)!!π √ n+ 1 √ (b + ac) 2 n! a

a > 0,

c > 0,

b+

√ ac > 0 LI (21)(19)



∞

xμ−1 dx ν− 12 ν = 2 (1 + 2x cos t + x2 )

∞



10.12 0

  1 1 1 −ν 2 −ν B(μ, 2ν − μ) P μ−ν− (sin t) 2 Γ ν + 1 (cos t) 2 2 [0 < t < π, 0 < Re μ < Re 2ν]

ET I 310(22)

 11.

1 + 2βx + x2

μ− 12

0

 x−ν−1 dx = 2−μ β 2 − 1 Γ(1 − μ) B(ν − 2μ + 1, −ν) P μν−μ (β) μ 2

[Re ν < 0,

Re(2μ − ν) < 1,

|arg (β ± 1)| < π] EH I 160(33)

1 2 −μ

= −π cosec νπ C ν (β)  −2 < Re 12 − μ < Re ν < 0, 

∞

12. 0

|arg (β ± 1)| < π

EH I 178(24)

xμ−1 dx = −πaμ−2 cosec t cosec(μπ) sin[(μ − 1)t] x2 + 2ax cos t + a2 [a > 0, 0 < |t| < π,

0 < Re μ < 2] FI II 738, BI(20)(3)



∞

13.

xμ−1 dx (x2 + 2ax cos t +

0

2 a2 )

=

πaμ−4 cosec μπ cosec3 t 2 × {(μ − 1) sin t cos [(μ − 2)t] − sin[(μ − 1)t]} [a > 0,



∞

14. 0

0 < |t| < π,

xμ−1 dx √ = π cosec(μπ) P μ−1 (cos t) 1 + 2x cos t + x2

0 < Re μ < 4] [−π < t < π,

LI(20)(8)a, ET I 309(13)

0 < Re μ < 1] ET I 310(17)



(1 + x)2μ−1 (1 − x)2ν−1 dx = 2μ+ν−2 B(μ, ν) (1 + x2 )μ+ν −1

3.253 3.254



1.12 0

u

1

 ν xλ−1 (u − x)μ−1 x2 + β 2 dx

[Re μ > 0,

Re ν > 0]

FI II 787

  λ λ + 1 λ + μ λ + μ + 1 −u2 ; , ; 2 = β 2ν uλ+μ−1 B(λ, μ) 3 F 2 −ν, , 2 2 2 β    2 u Re > 0, Re λ > 0, Re μ > 0 ET II 186(10) β

330

Power and Algebraic Functions

 2.

∞

12 u

3.255

 ν x−λ (x − u)μ−1 x2 + β 2 dx



1

3.255

1

xμ+ 2 (1 − x)μ− 2

1

μ+1

(c + 2bx − ax2 )

0

= uμ−λ+2ν B (λ − 2ν − μ, μ)   1+λ−μ λ 1+λ β2 λ−μ − ν, − ν; − ν, − ν; − 2 × 3 F 2 −ν, 2 2 2 u  2  β |u| > |β| and Re > 0, 0 < Re μ < Re(λ − 2ν) ET II 202(9) u  √ Γ μ + 12 π dx =  √ √ 2 μ+ 12 √ Γ(μ + 1) a+ c + 2b − a + c c + 2b − a   √ √ 2 1 c + 2b − a + c > 0, c + 2b − a > 0, Re μ > − 2 a+ BI (14)(2)

3.256



1

1.

(1 − x2 )

0

 2. 0

xp−1 + xq−1

1

p−1

x

p+q 2

q−1

−x

(1 − x2 )

dx =

p+q 2

dx =

1 cos 2

1 sin 2





    q−p q+p p q π sec π B , 4 4 2 2 

q−p π cosec 4



−p−1 2  ∞  b ax + 3.2579 +c dx x 0  √ π Γ p + 12 = 1 2acp+2 Γ(p + 1) 1 B p + 12 , 12 = 1 2 a (4ab + c)p+ 2 3.258



∞

1. b



∞

2. b



∞

3. 0

 4.

12 0

∞

 n  x − x2 − a2 dx =

 n x2 + 1 − x dx =



p + q < 2]

BI (8)(25)

BI (8)(26)

[p > 0,

q > 0,

p + q < 2]

a > 0,

b < 0,

c > 0,

p > − 12

a > 0,

b > 0,

c > −4ab,

BI (20)(4)

p > − 12

  n−1 n+1   a2 1 b − b 2 − a2 b − b 2 − a2 − 2(n − 1) 2(n + 1)

√

n−1

b2 + 1 − b 2(n − 1)

 n nan+1 x2 + a2 − x dx = 2 n −1 dx n √ n = n−1 2 2 2 a (n − 1) x+ x +a



[p > 0, q > 0, p q  q+p π B , 4 2 2 

[0 < a ≤ b, √ n+1 b2 + 1 − b + 2(n + 1)

n ≥ 2]

GW (215)(5)

[n ≥ 2]

GW (214)(7)

[n ≥ 2]

GW (214)(6a)

[n ≥ 2]

GW (214)(5a)

Powers of x and binomials and polynomials

3.261



∞

5.

xm



n x2 + a2 − x dx =

0

 6.

∞

7. a

1.



1

6

(x − a)m

p−1

x

0



[a > 0, 0 ≤ m ≤ n − 2] n  n · (n − m − 2)!(2m + 1)!am+n+1 x − x2 − a2 dx = 2m (n + m + 1)!

n−1

(1 − x)



GH (215)(6)

∞   k  l b Γ(p + km) (1 + bx ) dx = (n − 1)! k Γ(p + n + km) m l

k=0

u

GW (214)(5)

n ≥ m + 2]

[|b| < 1 unless l = 0, 1, 2, . . . ; 2.11

GW (214)(6)

x dx n · m! √ n = 2 2 (n − m − 1)(n − m + 1) . . . (m + n + 1)an−m−1 x+ x +a

[a > 0, 3.259

0 ≤ m ≤ n − 2]

m



0



n · m!am+n+1 (n − m − 1)(n − m + 1) . . . (m + n + 1) [a > 0,

∞

331

p, n, p + ml > 0]

BI (1)(14)

xν−1 (u − x)μ−1 (xm + bm )λ dx

0

3.11

= bmλ uμ+ν−1 B (μ, ν)   ν +m−1 μ+ν μ+ν +1 μ + ν + m − 1 −um ν ν+1 ,..., ; , ,..., ; m × m+1 F m −λ, , m m m m  b  m  u  m  π   Re μ > 0, Re ν > 0, arg ET II 186(11) < b m      ∞ λ b λ 1 λ −μ −ν ,μ + ν − ; μ + ν; 1 − ν, xλ−1 (1 + axp ) (1 + bxp ) dx = a−λ/p B F 2 1 p p p p a 0 [|arg a| < π, |arg b| < π, p > 0, 0 < Re λ < 2 Re(μ + ν)] ET I 312(35)

3.261 1.11

 PV 0

 2. 3.

4.12

1

∞

(1 − x cos t) xμ−1 dx  cos kt = 1 − 2x cos t + x2 μ+k

[Re μ > 0,

t = 2nπ]

BI (6)(9)

k=0

2

π sin νt (xν + x−ν ) dx ν < 1, t = (2n + 1)π = BI (6)(8) 2 1 + 2x cos t + x sin t sin νπ 0   1 1+p x + x1−p dx π (p sin t cos pt − cos t sin pt) = 2 )2 2 sin3 t sin pπ (1 + 2x cos t + x 0 2

p < 1, t = (2n + 1)π BI (6)(18)    1 xμ−1 dx π cosec t cosec μπ a sin t · = sin t − μ arctan 2 x2 (1 − x)μ μ/2 2 1 + 2ax cos t + a 1 + a cos t 0 (1 + 2a cos t + a ) 1

[a > 0,

0 < Re μ < 1]

BI (6)(21)

332

Power and Algebraic Functions



∞

x−p dx (1 − p)π π [−2 < p < 1] = cosec 3 1 + x 3 3 0 ∞   ν π νπ νπ x dx ν−1 ν ν = cb + b − 2c sec cosec cosec(νπ) 2 2 2 (b2 + c2 ) 2 2 0 (x + c) (x + b ) [Re b > 0, |arg c| < π, −1 < Re ν < 2, ν = 0]

3.262 3.263

3.264

3.262



∞

1. 0

π ap−2 + bp−2 cos pπ pπ xp−1 dx 2 = cosec (a2 + x2 ) (b2 − x2 ) 2 a2 + b 2 2

[0 < p < 4,

a > 0,

LI (18)(3)

ET II 216(7)

b > 0] BI (19)(14)



∞

2. 0

μ−1

x dx πc = 2 2 (b + x ) (c + x ) 2 =



∞

3. 0



∞

4. 0



∞

5. 0



∞

6.12 0

7. 0

−b b −c

π 2(c − b)

π 2 = 2 2 2 (b + x ) (a + b + x ) dx

π 3 = 4 (b + x2 ) (a + b + x2 ) dx

 

μ 2 −1

μπ cosec 2 1 1 √ −√ c b [|arg b| < π,



μ = 12 |arg c| < π,

0 < Re μ < 4]

1 1 1 − − 3/2 1/2 2 a2 b1/2 2a (a + b) a (a + b)

ET I 309(4)

 MC

3 1 2 2 − − − 5/2 3/2 1/2 2 3 a3 b1/2 4a (a + b) a (a + b) a (a + b)



⎛ π⎝ 2 5 3 − − 4 = 4 4 1/2 7/2 5/2 2 2 a b (b + x ) (a + b + x ) 8a (a + b) 4a2 (a + b) ⎞ 1 2 ⎠ − − 3/2 1/2 3 4 a (a + b) a (a + b) dx

dx (b + x2 ) (a + b + x2 )n     a+b 3 π 1 1 1 1 − n; = − B n− , 2 F 1 1 − n, 1; n−1/2 2 an b1/2 2 2 2 a 2a (a + b)

=



μ 2 −1

π 1 π − n−1/2 n 2 an b1/2 2a (a + b)



∞

(x2

+

n−1  j=0

 1 j  2

j!

[n > 0,

AS 263 (6.6.3.2)

a a+b



j

a + b > 0]

x2 dx π b c π = − = + b2 ) (x2 + c2 ) 2a (b2 − c2 ) b + a c + a 2(a + b)(a + c)(b + c)

a2 ) (x2

Powers of x and binomials and polynomials

3.271

 3.265

3.266

333

1 − xμ−1 dx = ψ(μ) + C [Re μ > 0] FI II 796, WH, ET I 16(13) 1−x 0 = ψ(1 − μ) + C − π cot(μπ) [Re μ > 0] EH I 16(15)a    ∞ ν (x − aν ) dx π b aν = bν cosec(νπ) − aν cot(νπ) − ln a+b π a 0 (x − a)(b + x) [|arg b| < π, |Re ν| < 1, ν = 0] 1

ET II 216(8)

3.267 1. 2. 3. 3.268

 Γ n + 13 2π x3n dx  √ √ = 3 1 − x3 3 3 Γ 13 Γ(n + 1) 0   1 3n−1 (n − 1)! Γ 23 x dx  √ = 3 3 Γ n + 23 1 − x3 0    1 3n−2 Γ n − 13 Γ 23 x dx  √ = 3 3 Γ n + 13 1 − x3 0





1

1



1. 0

 2. 3.

pxp−1 1 − 1 − x 1 − xp

BI (9)(6)

BI (9)(7)

 dx = ln p

BI (5)(14)

1 − xμ ν−1 dx = ψ(μ + ν) − ψ(ν) [Re ν > 0, x 0 1−x     1 n  xμ−1 n n−k √ − dx = nC + ψ μ + 1−x 1− nx n 0 1

Re μ > 0]

BI (2)(3)

k=1

[Re μ > 0] 3.269



1

1. 0

 2. 3. 3.271

xp − x−p pπ 1 π − x dx = cot 2 1−x 2 2 p

xp − x−p pπ 1 π x dx = − cosec 2 1 + x p 2 2 0      1 μ x − xν ν +1 1 μ+1 1 ψ − ψ dx = 2 2 2 2 2 0 1−x 

1

∞

1. 0

 2. 0

∞

xp − xq dx π = x−1 x+a 1+a p

p

p



x −a x −1 π dx = x−a x−1 a−1

a −1 1 − ap ln a sin(2pπ) π 2p

2

p −1]

BI (2)(9)

 2 p < 1,

2 p < 14

q 2 < 1,

a>0

BI (19)(2)

BI (19)(3)

334

3.

4.

Power and Algebraic Functions

3.272

  1 p p 2 (a − 1) cot pπ − (a + 1) ln a π 0

2 BI (18)(9) p 0]

3.

3.274

BI (16)(3)

[p > 0]

0



0 < p2 < 14

k=1

2. 

2 xp − x−p dx = 2 (1 − 2pπ cot 2pπ) 1−x

1

n  k=1

2k k

xq − 1 dx π qπ = tan xp − x−p x 2p 2p



xn−1 pxnp−1 − 1−x 1 − x1/p

BI (20)(13) BI (5)(3)

[p > q]

BI (18)(6)

[p > 0]

BI (13)(9)

 dx = p ln p

Powers of x and binomials and polynomials

3.277

   n 1 nxn−1 xmn−1 n−k dx = C + − ψ m + 1 − xn 1−x n n 0 k=1   1  p−1 qxpq−1 x − dx = ln q [q > 0] 1−x 1 − xq 0   ∞ 1 dx 1 = 0. − n m 2 2 1+x 1+x x 0 

2.

3. 4. 3.276

1.

335

10

1



 −p−1 2 b +c dx  ∞ ax + x x2

0



∞

10

2.

0

3.277



∞

12

1.

0

1 = 2|b|

BI (5)(12) BI (18)(17)

 B p + 12 , 12 1 p+ 2

(2a (b + |b|) + c)

a > 0, 1

−p−1  2    B p + 12 , 2 b b a+ 2 ax + +c dx = 1 x x (4ab + c)p+ 2 a > 0, xμ−1

BI (5)(13)

c > −4ac,

b > 0,

p > − 12

c > −4ac,

p > − 12

√

ν ν+μ  ν +μ μ 1 + x2 + b μ √ Γ(1 − μ − ν) P μ 2−1 (b) dx = 2 2 −1 b2 − 1 2 4 Γ 2 2 1 + x2 [Re b > −1, 0 < Re μ < 1 − Re ν]

√

ν   ∞ μ−1 x b2 + x2 + x bμ+ν−1 1−μ−ν √ dx = B μ, 2μ 2 b2 + x2 0 [Re b > 0,

ET I 310(25)

 2.

0 < Re μ < 1 − Re ν] ET I 311(28)

√ 

ν ∞ μ−1 Γ μ2 Γ(1 − μ − ν) cos t ± i sin t 1 + x2 x μ−1 1−μ 2 2 √ dx = 2 sin t Γ(−ν) 1 + x2 0   μ−1 1 i 1 μ−1 −2 2 2 2 × π Q − μ+1 −ν (cos t) ∓ π P − μ+1 −ν (cos t) 2 2 2 [Re μ > 0] ET I 311 (27)  ν  ∞ xμ−1 (b2 − 1) (x2 + 1) + b √ dx x2 + 1 0 μ (1−μ)/4 μ−1 2(μ−1)/2 − 12 iπ(μ−1) Γ 2 Γ(1 − μ − ν)  2 e b −1 Q −2μ+1 −ν (b) = √ π Γ(−ν) 2 [Re b > 1, Re ν < 0, Re μ < 1 − Re ν] ET I 311(26) 

3.12

4.12



∞

5. u

(x − u)μ−1

√ 2ν √ 1  2μ−1 12 −μ x+1− x−1 1 2ν+ 2 √ dx = √ e(μ− 2 )πi u2 − 1 4 Q ν− 1 (u) 2 π x2 − 1 [|arg(u − 1)| < π, 0 < Re μ < 1 + Re ν]

ET II 202(10)

336

Exponential Functions

3.278

√ √ ν  −ν    x2 − 1 + x − x2 − 1 1−μ+ν 1−μ−ν −μ √ , dx = 2 B 2 2 x2 − 1 1 ET I 311(29) [Re μ < 1 + Re ν] √  √ 2ν √ √ 2ν  u (u − x)μ−1 x+2+ x + x+2− x  dx x(x + 2) 0  1 1 (2μ+1)/2 2 −μ π[u(u + 2)]μ− 2 P ν− =2 1 (u + 1) 

6.

7.

∞ xμ−1



x−

[|arg u| < π, 3.2788





∞

1. 0

xp 1 + x2p

q

dx =0 1 − x2

2

Re μ > 0]

ET II 186(12)

[pq > 1]

3.3–3.4 Exponential Functions 3.31 Exponential functions 

3.310 3.311

∞

11 0



∞

1. 0



∞

2. 0



3.12 4. 5. 6. 7.

e−μx dx = β(μ) 1 + e−x

∞

e−ax dx π = −x sin πa −∞ 1 + e  ∞ −qx ∞  e dx ak = −px q + kp 0 1 − ae k=0  ∞ 1 − eνx dx = ψ(ν) + C + π cot(πν) ex − 1 0  ∞ −x e − e−νx dx = ψ(ν) + C 1 − e−x 0  ∞ −μx e − e−νx dx = ψ(ν) − ψ(μ) 1 − e−x 0 PV 

9.

1 p

∞

[Re p > 0]

dx ln 2 = 1 + epx p

 8.12

e−px dx =

LO III 284a

[Re μ > 0] [Re a > 0]

EH I 20(3), ET I 144(7)

(cf. 3.241 2)

[0 < a < 1] [Re ν < 1]

BI (27)(7)

(cf. 3.265)

[Re ν > 0] [Re μ > 0,

BI (28)(7)

EH I 16(16)

WH, EH I 16(14)

Re ν > 0]

(cf. 3.231 5) BI (27)(8)

∞

−μx

e dx = πbμ−1 cot(μπ) −x −∞ b − e

e−μx dx = πbμ−1 cosec(μπ) −x −∞ b + e

[b > 0,

0 < Re μ < 1]

[|arg b| < π,

ET I 120(14)a

0 < Re μ < 1] ET I 120(15)a

3.315

Exponential functions



10.

11

11.

∞

e−px − e−qx pπ π cot dx = −(p+q)x p + q p +q 0 1−e       ∞ px e − eqx r−q r−p 1 ψ − ψ dx = rx − esx r−s r−s r−s 0 e 

∞

12. 0





e +e π cosec dx = −(p+q)x p+q 1+e

∞

 ν−1 x 1 − e− β e−μx dx = β B(βμ, ν)

0



x

∞

13. 3.312

  c  c  ln b ln a a −b 1 −ψ dx = c ψ c x x c −d ln d ln d ln dc x

1.

−px

−qx

πp p+q

337

[p > 0,

q > 0]

GW (311)(16c)

[r > s, r > p, r > q] [c > a > 0,

GW (311)(16)

b > 0,

d > 0] GW (311)(16a)



[Re β > 0,

Re ν > 0,

Re μ > 0]

0



LI(25)(13), EH I 11(24) ∞

2.



1 − e−x

0

 3.

−1 

1 − e−αx

[Re p > 0, ∞

11



1 − e−x

 1 − e−βx e−px dx = ψ(p + α) + ψ(p + β) − ψ(p + α + β) − ψ(p) Re p > − Re α,

ν−1 

1 − βe−x

0

 PV 

7

2.

3.314

3.315 1.

e−μx dx = π cot πμ −x −∞ 1 − e

Re ν > 0,

|arg(1 − β)| < π]



ET I 145(15)

EH I 116(15)

[0 < Re μ < 1]

∞

e−μx dx = B(μ, ν − μ) [0 < Re μ < Re ν] −x )ν −∞ (1 + e     ∞ ν e−μx dx  = γ exp β μ − B(γμ, ν − γμ) β/γ + e−x/γ ν γ −∞ e     ν Re > Re μ > 0, |Im β| < π Re γ γ

ET I 120(21)

∞

 e−μx dx ν−β ν = exp[γ(μ − ) − βν] B(μ, ν +  − μ) 2 F 1 ν, μ; ν + ; 1 − e β −x γ −x ) (e + e ) −∞ (e + e [|Im β| < π, |Im γ| < π, 0 < Re μ < Re(ν + )] ET I 121(22)  π β μ−1 − γ μ−1 e−μx dx = cosec(μπ) −x ) (γ + e−x ) γ−β −∞ (β + e [|arg β| < π, |arg γ| < π,

 2.

∞

Re p > − Re(α + β)]

e−μx dx = B(μ, ν) 2 F 1 (, μ; μ + ν; β) [Re μ > 0,

3.313 1.7

−

Re p > − Re β,

∞

β = γ,

0 < Re μ < 2]

ET I 120(18)

338

Exponential Functions



ν

∞

(1 + e−x ) − 1 dx = ψ(μ) − ψ(μ − ν) −x )μ −∞ (1 + e

3.316

3.316

[Re μ > Re ν > 0]

(cf. 3.235) BI (28)(8)

3.317





∞

1. −∞  ∞



2. −∞

1 1 − −x 1+e (1 + e−x )μ

 dx = C + ψ(μ)

1 1 ν − μ (1 + e−x ) (1 + e−x )

[Re μ > 0]

(cf. 3.233)

BI (28)(10)

 dx = ψ(μ) − ψ(ν)

[Re μ > 0,

Re ν > 0]

(cf. 3.219) BI (28)(11)

3.318



∞

1. 0



∞

2.7 u

√ √

−ν

−ν b + 1 − e−x + b − 1 − e−x √ e−μx dx 1 − e−x

1 √ 1 − e−2x

 (μ−ν)/2 2μ+1 e(μ−ν)πi b2 − 1 Γ(μ) Q ν−μ μ−1 (b) = Γ(ν) [Re μ > 0] ET I 145(18)  ν   e−u 1 − e−2x − e−x 1 − e−2u e−μx dx √ 1 u − 1 (μ+ν) √ 2− 2 (μ+ν) πe− 2 (μ+ν) Γ(μ) Γ(ν + 1) P − 21 (μ−ν) 1 − e−2u 2 = Γ[(μ + ν + 1)/2] [u > 0, Re μ > 0, Re ν > −1] ET I 145(19)

3.32–3.34 Exponentials of more complicated arguments 3.321 1.11

√ √  u ∞  2 π π (−1)k u2k+1 Φ(u) = erf(u) = e−x dx = 2 2 k!(2k + 1) 0 k=0 ∞ k 2k+1 2  2 u [u > 0] = e−u (2k + 1)!! k=0



2. 3. 4. 5. 6.

u

√

π Φ(qu) 2q 0 √  ∞ 2 2 π e−q x dx = 2q 0  u  2 2 2 2 1  xe−q x dx = 2 1 − e−q u 2q 0 √   u π 1 2 −q2 x2 −q2 u2 Φ(qu) − que x e dx = 3 2q 2 0  u   2 2 2 2 1  x3 e−q x dx = 4 1 − 1 + q 2 u2 e−q u 2q 0 e−q

2

x2

dx =

(see 8.25)

AD 6.700

[q > 0] [q > 0]

FI II 624

3.326

Exponentials of more complicated arguments



u

7.

4 −q2 x2

x e

0

3.322  1.12 2.12 3.11

3.323  1.12 2.10 3.12 4.∗ 3.324

 √    1 3 π 3 2 2 −q2 u2 dx = 5 Φ(qu) − + q u que 2q 4 2

√ π a2 e [1 − erf(a + u)] e dx = 2 u √  ∞ 2 π a2 e [1 − erf(a)] e−x −2ax dx = 2 0   ∞ 1 π ±πi/4 ±iλx2 e PV e dx = 2 λ 0 ∞

339

−x2 −2ax

[Reβ > 0,

u > 0]

ET I 146(21) NT 27(1)a

[λ > 0]

PBM 343 (2.3.15(2))

√ π a2 e [1 − erf (a + 1)] 2 1  2 √  ∞  2 2

π q Re p2 > 0 exp −p x ± qx dx = exp 2 4p p −∞ √  ∞  4 a 2a2 exp −x − 4ax2 dx = e K1/4 (2a2 ) 2 0  4   4   4   ∞ γ γ γ πγ 2 4 2 2 exp I−1/4 ± I1/4 exp (−β x ± 2γ x ) dx = 2 2 2 4 β 2β 2β 2β 0 ∞

e−x

2

−2ax

dx =

BI (29)(4) BI (28)(1) ET I 147(34)a

     β β − γx dx = K1 1. exp − βγ [Re β ≥ 0, Re γ > 0] ET I 146(25) 4x γ 0   2n     ∞ 1 1 b 11 [b ≥ 0] dx = Γ 2. exp − x − x n 2n −∞     ∞  √  1 π b 2 3.325 exp −2 ab exp −ax − 2 dx = [a > 0, b > 0] FI II 644 x 2 a 0 3.326    ∞ 1 1 [Re μ > 0] exp (−xμ ) dx = Γ BI (26)(4) 1.8 μ μ 0  ∞ Γ(γ) m+1 10 [Re β > 0, Re m > 0, Re n > 0] 2. xm exp (−βxn ) dx = γ= γ nβ n 0    ∞ Γ n2 , β(−b)n Γ n1 , β(−b)n 3. (x − a) exp (−β(x − b)n ) dx = − (a − b) nβ 2/n nβ 1/n 0 

∞



u

4. 0

(x − a) exp (−β(x − b)n ) dx =

2

n



2

[Re n > 0, n



Re β > 0,

|arg b| < π]

− Γ n , β(u − b) 2/n  1 nβ  Γ n , β(−b)n − Γ n1 , β(u − b)n −(a − b) nβ 1/n [Re n > 0, Re β > 0, |arg b| < π, |arg(u − b)| < π] Γ

n , β(−b)

340

Exponential Functions



∞

5. u

n

(x − a) exp (−β(x − b) ) dx =

Γ

2

n n , β(−b) nβ 2/n

3.327

− (a − b)

Γ

1

n , β(u − nβ 1/n

[Re n > 0, Exponentials of exponentials  ∞ 1 exp (−aenx ) dx = − Ei(−a) 3.327 n 0 ∞ 3.328 exp (−ex ) eμx dx = Γ(μ) −∞    ∞ a exp (−ceax ) b exp −cebx b dx = e−c ln 3.329 − −ax −bx 1 − e 1 − e a 0 3.331



∞

1. 0



∞

2. 0

 3.

∞

11

[n ≥ 1,

Re β > 0,

|arg(u − b)| < π]

Re a ≥ 0,

a = 0]

4.

[a > 0,

b > 0,

1.

12

2.3



[Re β > 0]

ET I 147(37)



1 − e−x

ν−1

 μ+ν β exp βe−x − μx dx = B(μ, ν)β − 2 e 2 M ν−μ , ν+μ−1 (β) 2



1 − e−x

ν−1

exp (−βex − μx) dx = Γ(ν)β

∞



1 − e−x

ν−1 

1 − λe−x

−

2

μ−1 2

β

e− 2 W

1−μ−2ν 2

3. 3.33411

[Re μ > 0,

∞

∞





Re ν > 0]

ET I 147(38)

(β) , −μ 2 Re ν > 0]

ET I 147(39)

 exp βe−x − μx dx = B(μ, ν) Φ1 (μ, , ν, λ, β)

e−μx dx = Γ(μ) ζ(μ) −x ) − 1 −∞ exp (e  ∞  e−μx dx = 1 − 21−μ Γ(μ) ζ(μ) −x )+1 −∞ exp (e = ln 2 

BI (27)(12)

exp (−βex − μx) dx = β μ Γ(−μ, β)

0

3.333

c > 0]

ET I 147(36)

[Re β > 0, 3.332

NH 145(14)

[Re μ > 0]

0



LI (26)(5)

[Re μ > 0]

[Re μ > 0, ∞



 exp −βe−x − μx dx = β −μ γ(μ, β)

0



b)n

Re ν > 0,

|arg(1 − λ)| < π]

[Re μ < 1] [Re μ > 0,

ET I 147(40)

ET I 121(24)

μ = 1]

[μ = 1] ET I 121(25)

tanh(x) 7 ζ(3) 1 dx = − 2 3 2 x π2 x cosh (x) 0    ∞ β ν−1 β − μx dx = Γ(μ − ν + 1)e 2 β 2 W ν−2μ−1 ,− ν (β) (ex − 1)ν−1 exp − x 2 2 e −1 0 [Re β > 0, Re μ > Re ν − 1] ET I 137(41)

3.338

Exponentials of more complicated arguments

341

Exponentials of hyperbolic functions  ∞  νx e + e−νx cos νπ exp (−β sinh x) dx = −π [Eν (β) + Y ν (β)] 3.335 0

3.336



[Re β > 0]

∞

exp (−νx − β sinh x) dx = π cosec νπ [Jν (β) − J ν (β)]  π π |arg β| < and |arg β| = for Re ν > 0; 2 2

1. 0



∞

exp (nx − β sinh x) dx =

2. 0



 ν is not an integer

exp (−nx − β sinh x) dx =

3.

WA 341(2)

1 [S n (β) − π En (β) − π Y n (β)] 2 [Re β > 0,

∞

EH II 35(34)

0

n = 0, 1, 2, . . .]

WA 342(6)

1 (−1)n+1 [S n (β) + π En (β) + π Y n (β)] 2 [Re β > 0, n = 0, 1, 2, . . .] EH II 84(47)

3.337



∞

1. −∞  ∞

2. −∞  ∞

3. −∞

 π |arg β| < 2

exp (−αx − β cosh x) dx = 2 K α (β) exp (−νx + iβ cosh x) dx = iπe

iνπ 2

exp (−νx − iβ cosh x) dx = −iπe−

WA 201(7)

H (1) ν (β)

[0 < arg β < π]

EH II 21(27)

iνπ 2

[−π < arg β < 0]

EH II 21(30)

H (2) ν (β)

Exponentials of trigonometric functions and logarithms 3.338



π

1. 0



π

2.



{exp i [(ν − 1)x − β sin x] − exp i [(ν + 1)x − β sin x]} dx = 2π Jν (β) + i Eν (β)

exp [±i (νx − β sin x)] dx = π [Jν (β) ± i Eν (β)]

[Re β > 0]

EH II 36

[Re β > 0]

EH II 35(32)

[Re γ > 0]

WA 619(4)

0

 3.

∞

10 0



∞

exp [−γ (x − β sin x)] dx =

π

−π

k=1

 a + b sin x + c cos x 2π 1 + p sin x + q cos x dx =  e−α I 0 (β), 2 2 1 + p sin x + q cos x 1−p −q  a 2 − b 2 − c2 bp + cq − a ; β = α2 − ; with α = 2 2 1−p −q 1 − p2 − q 2 

4.6

 γ J k (kβ) 1 +2 γ γ 2 + k2

exp

2

p + q2 < 1

342

Exponential Functions



π/4

5. 0

 3.3396

π

 3.34211

∞  tan2n x exp − n + 12 n=1



√ dx = ln 2

exp (z cos x) dx = π I 0 (z)

0

3.341



3.339

BI (277)(2)a

π 2

exp (−p tan x) dx = ci(p) sin p − si(p) cos(p)  1 ∞  pk−1 exp (−px ln x) dx = x−px dx = kk 0

[p > 0]

BI (271)(2)a

0 1

0

BI (29)(1)

k=1

3.35 Combinations of exponentials and rational functions 3.351 1.



u

8

xn e−μx dx =

0

n! μn+1

− e−uμ

n  n! uk = μ−n−1 γ(n + 1, μu) k! μn−k+1 k=0 [u > 0, Re μ > 0, n = 0, 1, 2, . . .]

ET I 134(5)



∞

2.11 u



xn e−μx dx = e−uμ

n  n! k=0

uk

k! μn−k+1

= μ−n−1 Γ(n + 1, μu) [u > 0,

Re μ > 0, n = 0, 1, 2, . . .] ET I 33(4)

∞

3.

xn e−μx dx = n!μ−n−1

[Re μ > 0]

ET I 133(3)

0



∞ −px

e

4.

xn+1

u

 1

6. 7.9 8.11 9.7

= (−1)n+1

n−1 pn Ei(−pu) e−pu  (−1)k pk uk + n n! u n(n − 1) . . . (n − k) k=0

[p > 0] ∞ −μx

e

5. 

dx

x

dx

= − Ei(−μ)

NT 21(3)

[Re μ > 0]

BI (104)(10)

u

ex dx = li (eu ) = Ei(u) [u < 0] −∞ x  u 1 1 xe−μx dx = 2 − 2 e−μu (1 + μu) [u > 0] μ μ 0  u  2 1 [u > 0] x2 e−μx dx = 3 − 3 e−μu 2 + 2μu + μ2 u2 μ μ 0  u  6 1 x3 e−μx dx = 4 − 4 e−μu 6 + 6μu + 3μ2 u2 + μ3 u3 μ μ 0 [u > 0]

3.352



1. 0

u −μx

e

dx = eμb [Ei(−μu − μb) − Ei(−μb)] x+b

[ u ≥ 0,

|arg b| < π]

ET II 217(12)

3.354

Exponentials and rational functions



∞ −μx

e

2. u

 3.12

v −μx

e

dx = eaμ {Ei[−(a + v)μ] − Ei[−(a + u)μ]} x+a

e

0

 5.

6.8 

1.

[−a < u or − a > v,

Re μ > 0]

dx = −ebμ Ei(−μb) x+b

[|arg b| < π,

Re μ > 0]

ET II 217(11)

∞ −px

e

ET II 251(37) ∞ −μx

e

0 ∞

dx = e−μa Ei(aμ) a−x

[a < 0,

ipx

e dx = iπeiap −∞ x − a

3.353

Re μ > 0]

dx = e−pa Ei(pa − pu) a − x u

p > 0, a < u; for a > u, one should replace Ei(pa − pu) in this formula with Ei(pa − pu)



7.

|arg(u + b)| < π,

ET I 134 (7) ∞ −μx

4. 7

[u ≥ 0,

ET I 134(6), JA

u



dx = −ebμ Ei(−μu − μb) x+b

343



∞

12 u

Re μ > 0]

BI (91)(4)

[p > 0]

ET II 251(38)

n−1

 (k − 1)!(−μ)n−k−1 e−μx dx (−μ)n−1 bμ −uμ e Ei[−(u + b)μ] = e − (x + b)n (n − 1)!(u + b)k (n − 1)! k=1 [n > 2, |arg(u + b)| < π,

Re μ > 0] ET I 134(10)



∞

2.7 0



n−1  e−μx dx 1 (−μ)n−1 bμ n−k−1 −k e Ei(−bμ) = (k − 1)!(−μ) b − (x + b)n (n − 1)! (n − 1)! k=1 [n > 2, |arg b| < π,

Re μ > 0]

ET I 134(9), BI (92)(2) ∞

3. 0

e−px dx 1 = peap Ei(−ap) + (a + x)2 a

[p > 0,

a > 0] LI (281)(28), LI (281)(29)

 4. 0

 5.7

0

1

xex e dx = − 1. (1 + x)2 2

∞

BI (80)(6) n

 xn e−μx dx = (−1)n−1 bn ebμ Ei(−bμ) + (k − 1)!(−b)n−k μ−k x+b k=1 [|arg b| < π,

Re μ > 0] BI (91)(3)a, LET I 135(11)

3.354



1. 0

∞ −μx

e dx 1 = [ci(bμ) sin bμ − si(bμ) cos bμ] b2 + x2 b

[Re b > 0,

Re μ > 0]

BI (91)(7)

344

2. 3.12 4.

5.8

Exponential Functions



∞



∞

3.355

xe−μx dx = − ci(bμ) cos bμ − si(bμ) sin bμ [Re b > 0, Re μ > 0] BI (91)(8) b2 + x2 0  ∞ −μx

e dx 1 −bμ e PV = Ei(bμ) − ebμ Ei(−bμ) [|arg (±b)| < π, Re μ > 0] BI (91)(14) 2 2 b −x 2b 0  ∞ −μx

xe dx 1 −bμ e = Ei(bμ) + ebμ Ei(−bμ) 2 2 2 0 b − x

|arg (±b)| < π, Re μ > 0; for b > 0 one should replace Ei(bμ) in this formula with Ei(bμ) BI (91)(15)

e−ipx dx π = e−|ap| 2 + x2 a a −∞

3.355  1.

∞

0



xe

−μx

=

+

dx

2 x2 )

e

−px

dx

−

2 x2 )

(a2

0

∞

4.3

xe

−px

(a2

0

∞

0

=

1 {1 − bμ [ci(bμ) sin bμ − si(bμ) cos bμ]} 2b2

−

dx

2 x2 )

=

=

∞

2. 0



p>0



 1

−2 + ap e−ap Ei(ap) − eap Ei(−ap) 4a2  2 Im a > 0,

p>0

3.

 4. 0

[p > 0]

x e 1 dx = (−1)n a2n−1 [ci(ap) sin ap − si(ap) cos ap] + 2n−1 2 2 a +x p

e 1 1 dx = a2n eap Ei(−ap) + e−ap Ei(ap) − 2n 2 −x 2 p

2n+1 −px

LI (92)(9)

x

n 

k−1  (2n − 2k)! −a2 p2

k=1

BI (91)(11)

n 

k−1  (2n − 2k + 1)! a2 p2

k=1

[p > 0] ∞

BI (92)(8)

BI (91)(12)

2n −px

a2

0

BI (92)(7)

x2n+1 e−px dx = (−1)n−1 a2n [ci(ap) cos ap + si(ap) sin ap] a2 + x2 n k−1  1  + 2n (2n − 2k + 1)! −a2 p2 p

[p > 0] ∞

Re μ > 0]

1 (ap − 1)eap Ei(−ap) + (1 + ap)e−ap Ei(ap) 4a3  2 Im a > 0,

k=1



ET I 118(1)a

1 {ci(bμ) sin(bμ) − si(bμ) cos(bμ) − bμ [ci(bμ) cos(bμ) + si(bμ) sin(bμ)]} 2b3

[Re b > 0, ∞

3.3

3.356  1.

+

(b2

0



x2 )2

(b2

p real]

LI (92)(6) ∞

2.12 

e−μx dx

[a = 0,

BI (91)(17)

n k−1

 x2n e−px 1 2n−1 −ap 1  ap a e dx = Ei(ap) − e Ei(−ap) − (2n − 2k)! a2 p2 a2 − x2 2 p2n−1 k=1

[p > 0]

BI (91)(16)

3.358

Exponentials and rational functions

3.357  1.

∞

a3

0

 0





∞

a3

6. 0

∞

0



4. 0

a > 0]

∞

 x e dx 1 −ap e = Ei(ap) − eap Ei(−ap) − 2 ci(ap) sin ap + 2 si(ap) cos ap 4 4 a −x 4a

[p > 0,

0

BI (92)(23)

a > 0]

BI (91)(18)

BI (91)(19)

2 −px

[p > 0, ∞

BI (92)(22)

 e−px 1

dx = 3 e−ap Ei(ap) − eap Ei(−ap) + 2 ci(ap) sin ap − 2 si(ap) cos ap 4 4 a −x 4a  xe−px dx 1

= 2 eap Ei(−ap) + e−ap Ei(ap) − 2 ci(ap) cos ap − 2 si(ap) sin ap 4 4 a −x 4a

3. 

a > 0]

∞

0



a > 0]

x e dx 1 = {ci(aμ) (cos aμ − sin aμ) a3 − a2 x + ax2 − x3 2  + si(aμ) (cos aμ + sin aμ) + e−aμ Ei(aμ)

[p > 0,

2.

BI (92)(21)

2 −μx

[Re μ > 0, 3.358  1.

a > 0]

xe−μx dx 1 {− ci(aμ) (sin aμ + cos aμ) = − a2 x + ax2 − x3 2a  − si(aμ) (sin aμ − cos aμ) + e−aμ Ei(aμ) [Re μ > 0,

∞

BI (92)(20)

1 e dx = 2 {ci(aμ) (sin aμ − cos aμ) 2 2 3 − a x + ax − x 2a  − si(aμ) (sin aμ + cos aμ) + e−aμ Ei(aμ) [Re μ > 0,

0

a > 0]

−μx

a3

5.

BI (92)(19)

x e dx 1 = {− ci(aμ) (sin aμ + cos aμ) 2 2 3 + a x + ax + x 2 − si(aμ) (sin aμ − cos aμ) − eaμ Ei(−aμ)} [Re μ > 0,

∞

0

a > 0]

2 −μx

a3

4.

BI (92)(18)

xe dx 1 {ci(aμ) (sin aμ − cos aμ) = a3 + a2 x + ax2 + x3 2a − si(aμ) (sin aμ + cos aμ) + eaμ Ei(−aμ)} [Re μ > 0,

0

a > 0]

−μx

∞

3.



e−μx dx 1 = 2 {ci(aμ) (sin aμ + cos aμ) + a2 x + ax2 + x3 2a + si(aμ) (sin aμ − cos aμ) − eaμ Ei(−aμ)} [Re μ > 0,

∞

2.12



345

a > 0]

BI (91)(20)

 x3 e−px dx 1 ap e Ei(−ap) + e−ap Ei(ap) + 2 ci(ap) cos ap + 2 si(ap) sin ap = 4 4 a −x 4 [p > 0,

a > 0]

BI (91)(21)

346

Exponential Functions



∞

5. 0



1 x4n e−px dx = a4n−3 e−ap Ei(ap) − eap Ei(−ap) + 2 ci(ap) sin ap − 2 si(ap) cos ap 4 4 a −x 4 n k−1  1  − 4n−3 (4n − 4k)! a4 p4 p k=1



∞

6. 0

∞

7. 0

x

∞

8. 0

3.359

BI (91)(22)

[p > 0,

a > 0]

BI (91)(23)



x4n+2 e−px 1 dx = a4n−1 e−ap Ei(ap) − eap Ei(−ap) − 2 ci(ap) sin ap + 2 si(ap) cos ap 4 4 a −x 4 n k−1  1  − 4n−1 (4n − 4k + 2)! a4 p4 p [p > 0,

a > 0]

BI (91)(24)



x4n+3 e−px 1 dx = a4n eap Ei(−ap) + e−ap Ei(ap) + 2 ci(ap) cos ap + 2 si(ap) sin ap 4 4 a −x 4 n k−1  1  − 4n (4n − 4k + 3)! a4 p4 p k=1



a > 0]



e 1 dx = a4n−2 eap Ei(−ap) + e−ap Ei(ap) − 2 ci(ap) cos ap − 2 si(ap) sin ap a4 − x4 4 n k−1  1  − 4n−2 (4n − 4k + 1)! a4 p4 p

k=1



[p > 0,

4n+1 −px

k=1



3.359

∞

(i − x)n e−ipx dx = (−1)n−1 2πpe−p Ln−1 (2p) n 2 −∞ (i + x) i + x =0

[p > 0,

a > 0]

BI (91)(25)

for p > 0; for p < 0. ET I 118(2)

3.36–3.37 Combinations of exponentials and algebraic functions 3.361 1.8 2.8 3.8 3.362 1. 2.



u −qx



π √ Φ ( qu) q 0   ∞ −qx e π √ dx = x q 0  ∞ −qx  e π q √ dx = e q 1 + x −1 

e √ dx = x

∞ −μx

e dx √ = x−1

[q > 0] [q > 0]

BI(98)(10)

[q > 0]

BI (104)(16)



π −μ e μ 1   ∞ −μx   e dx π bμ  √ e 1−Φ = bμ μ x+b 0

[Re μ > 0] [Re μ > 0,

BI (104)(11)a

|arg b| < π]

ET I 135(18)

3.374

Exponentials and algebraic functions

3.363  1.

∞√

x − u −μx e dx = x

u



∞

2. u

3.364  1.

2

0

 2. 3.

√ π −uμ √ e − π u [1 − Φ ( uμ)] μ

e−μx dx π √ √ = √ [1 − Φ ( uμ)] u x x−u

e−px dx  = πe−p I 0 (p) x(2 − x)

e2x dx √ = π I 0 (2) 2 −1 1 − x  ∞ −px  ap  ap e dx  = e 2 K0 2 x(x + a) 0

3.365  1. 2.



[u > 0,

Re μ > 0]

ET I 136(23)

[u > 0,

Re μ ≥ 0]

ET I 136(26)

[p > 0]

GW (312)(7a)

1

u

xe−μx dx πu √ [L1 (μu) − I 1 (μu)] + u = 2 − x2 2 u 0  ∞ −μx xe dx √ = u K 1 (uμ) 2 x − u2 u

3.366  1.

347

BI (277)(2)a

[a > 0,

p > 0]

[u > 0,

Re μ > 0]

ET I 136(28)

[u > 0,

Re μ > 0]

ET I 136(29)

(u − x)e−μx dx √ = πue−uμ I 1 (uμ) [Re μ > 0] 2ux − x2 0  ∞ (x + b)e−μx dx √ 2. [Re μ > 0, |arg b| < π] = bebμ K 1 (bμ) x2 + 2bx 0  ∞ −μx   xe dx bπ π √ [H1 (bμ) − Y 1 (bμ)] − b 3. = |arg b| < , Re μ > 0 2 2 x2 + b2 0      ∞ t u exp 2μ cos2 2 e−μx dx √ t − sin t = 3.367 K 0 (v)e−v cos t dv 2 sin t 0 (1 + cos t + x) x + 2x 0 [Re μ > 0]  ∞   π π 1 e−μx dx √ |arg b| < , Re μ > 0 3.368 = [H1 (bμ) − Y 1 (bμ)] − 2 2 2bμ b μ 2 x2 + b2 0 ∞ x + −μx e dx 2 √ √ aμ 11  3.369 = √ − 2 πμe (1 − Φ ( aμ)) [|arg a| < π, Re μ > 0] 3 a (x + a) 0  ∞ √ 1 3 1 2n − 1 −n− 1 2 μ 3.37111 xn− 2 e−μx dx = π · · . . . 2 2 2 0 √ = π2−n μ−n−1/2 (2n − 1)!! [n ≥ 0] 2u

[Re μ > 0] (2n − 1)!! x (2 + x) e−px dx = ep K n (p) [p > 0, n = 0, 1, 2, . . .] n p 0  ∞        n n x + x2 + b2 + x − x2 + b2 e−μx dx = 2bn+1 O n (bμ) 

3.372 3.373

GW (312)(8a)

0

∞

n− 12

n− 12

[Re μ > 0]

ET I 136(31) ET I 136(30) ET I 136(27)

ET I 136(33) ET I 136(32) ET I 135(20)

ET I 135(17) GW (312)(8)

WA 05(1)

348

3.374 1.

2.

Exponential Functions

3.381

√ n x + 1 + x2 1 √ e−μx dx = [S n (μ) − π En (μ) − π Y n (μ)] 2 2 1+x 0 [Re μ > 0] √  n  ∞ x − 1 + x2 1 √ e−μx dx = − [S n (μ) + π En (μ) + π Y n (μ)] 2 2 1+x 0

ET I 37(35)

[Re μ > 0]

ET I 137(36)



∞



3.38–3.39 Combinations of exponentials and arbitrary powers 3.381



u

1.

xν−1 e−μx dx = μ−ν γ(ν, μu)

[Re ν > 0]

EH I 266(22), EH II 133(1)

0



u

2.

xp−1 e−x dx =

0

∞ 

(−1)k

k=0

= e−u

∞  k=0



u

0 ∞

5. 0



xν−1 e−μx dx = μ−ν Γ(ν, μu)

xν−1 e−μx dx =

u  ∞

7.

Re μ > 0]

[Re μ > 0,

Re ν > 0]

  − ν  q xν−1 e−(p+iq)x dx = Γ(ν) p2 + q 2 2 exp −iν arctan p [p > 0, Re ν > 0 and p = 0, ν u e dx = u− 2 e− 2 W − ν , (1−ν) (u) ν 2 2 x

FI II 779

0 < Re ν < 1]

EH I 12(32)

[u > 0]

Γ(k) (−iμ)k

xk−1 eiμx dx =

0

WH

[0 < Re(k) < 1,

μ = 0] GH2 62 (313.14)

u

8. 

1 Γ(ν) μν

∞ −x

6.



[u > 0,

EH I 256(21), EH II 133(2) ∞

4. 

up+k p(p + 1) . . . (p + k) AD 6.705

∞

3.8 

up+k k!(p + k)

n

xm e−bx dx =

0 ∞

9. u  ∞

10. u

γ (v, bu ) nbv

n

xm e−βx dx = n

n

xm e−βx dx =

v=

m+1 n

Γ (v, βun ) m+1 v= v nβ n

[u > 0,

Re v > 0,

[Re n > 0,

Re n > 0,

Re β > 0]

Re β > 0]

Γ (v, βun ) nβ v v=

m+1 n

[u > 0,

Re n > 0,

Re β > 0]

(See also 3.326 2)

3.383

Exponentials and arbitrary powers



∞

m −βxn

11. −∞

x e

 dx = 2

∞

n

xm e−βx dx =

0

2 Γ(v) nβ v v=

m+1 n

3.382  u (u − x)ν e−μx dx = (−μ)−ν−1 e−uμ γ(ν + 1, −uμ) 1.6 0



∞

2. u

 

(x − u)ν e−μx dx = μ−ν−1 e−uμ Γ(ν + 1)



Re m > − 21 ,

[Re ν > −1, [u > 0,

Re n > 0,

u > 0]

Re ν > −1,

Re β > 0

ET I 137(6)

Re μ > 0]

ET I 137(5), ET II 202(11) ∞

3. 0

349

μ

ν

(1 + x)−ν e−μx dx = μ 2 −1 e 2 W − ν , (1−ν) (μ)

[Re μ > 0]

(x + b)ν e−μx dx = μ−ν−1 ebμ Γ(ν + 1, bμ)

[|arg b| < π,

2

∞

4.

2

WH

Re μ > 0]

0



ET I 137(4), ET II 233(10) u

5. 0



(a + x)μ−1 e−x dx = ea [γ(μ, a + u) − γ(μ, a)]

∞

6. −∞

(b + ix)−ν e−ipx dx = 0 =



[Re μ > 0] [for p > 0]

2π(−p)ν−1 ebp Γ(ν)

[for p < 0] [Re ν > 0,

∞

7. −∞

(b − ix)−ν e−ipx dx =

EH II 139

Re b > 0]

ET I 118(4)

Re b > 0]

ET I 118(3)

[Re μ > 0, Re ν > 0]     βu βu Γ(μ) I μ− 12 exp 2 2

ET II 187(14)

ν−1 −bp

e 2πp Γ(ν)

[for p > 0] [for p < 0]

=0

[Re ν > 0, 3.383  u xν−1 (u − x)μ−1 eβx dx = B(μ, ν)uμ+ν−1 1 F 1 (ν; μ + ν; βu) 1.11 0



u

2.11

xμ−1 (u − x)μ−1 eβx dx =

0

√

π

 μ− 12 u β

[Re μ > 0] 1       μ− 2 ∞ u βu 1 βu μ−1 μ−1 −βx K μ− 12 x (x − u) e dx = √ Γ(μ) exp − β 2 2 π u

ET II 187(13)

 3.



∞

4.11 u

xν−1 (x − u)μ−1 e−βx dx = β −

μ+ν 2

u

μ+ν−2 2

[Re μ > 0, Re βu > 0]   βu W ν−μ , 1−μ−ν (βu) Γ(μ) exp − 2 2 2 [Re μ > 0,

Re βu > 0]

ET II 202(12)

ET II 202(13)

350

Exponential Functions

 5.

∞

12 0

3.384

e−px xq−1 (1 + ax)−ν dx   p p q−ν  p ν  p q L − Lν−q π2 −q −ν a a − = q p Γ(ν) a sin(πν) sin(π(q − ν))Γ(1 − q) a sin(πq) sin(π(p − ν))Γ(1 − ν) [ν = ±1, ±2, . . .] Γ(q) = q p



∞

6.

[ν = 0]

1

1

0



[|arg b| < π, ∞

7.

1

xν−1 (x + b)−ν+ 2 e−μx dx = 2ν− 2 Γ(ν)μ− 2 e

1

bμ 2

Re ν > 0, 1

xν−1 (x + b)−ν− 2 e−μx dx = 2ν Γ(ν)b− 2 e

bμ 2

0

[Re q > 0,   D 1−2ν 2bμ Re μ ≥ 0,

D −2ν

Re p > 0,

μ = 0]

Re a > 0]

ET I 39(20), EH II 119(2)a

  2bμ [|arg b| < π,

Re ν > 0,

Re μ ≥ 0]

ET I 139(21), EH II 119(1)a



∞

8.

ν−1

x

(x + b)

0



e

1 dx = √ π

 ν− 12   bμ b bμ 2 e Γ(ν) K 12 −ν μ 2 [|arg b| < π,

Re μ > 0,

Re ν > 0]

ET II 233(11), EH II 19(16)a, EH II 82(22)a ∞

9. u



ν−1 −μx

(x − u)ν e−μx dx = uν Γ(ν + 1) Γ(−ν, uμ) x

[u > 0,

Re ν > −1,

Re μ > 0] ET I 138(8)

∞

10. 0

ν−1 −μx

x

e x+b

dx = bν−1 ebμ Γ(ν) Γ(1 − ν, bμ)

[|arg b| < π,

Re μ > 0,

Re ν > 0] EH II 137(3)

3.384



1

1. −1



[Re ν > 0, v

2. u



(1 − x)ν−1 (1 + x)μ−1 e−ipx dx = 2μ+ν−1 B(μ, ν)eip 1 F 1 (μ; ν + μ; −2ip)

u

ET I 119(13)

(x − u)2μ−1 (v − x)2ν−1 e−px dx

∞

3.

Re μ > 0]

  u+v M μ−ν,μ+ν− 12 (vp − up) = B(2μ, 2ν)(v − u)μ+ν−1 p−μ−ν exp −p 2 [v > u > 0, Re μ > 0, Re ν > 0] ET I 139(23)   (u + b)ν+ −1 (b − u)μ Γ(2) W ν− ,ν+ − 12 (uμ + bμ) exp μν+ 2 [u > 0, |arg(b + u)| < π, Re μ > 0, Re  > 0] ET I 139(22)

(x + b)2ν−1 (x − u)2 −1 e−μx dx =

3.386

Exponentials and arbitrary powers



∞

4. u



∞

5. u



  (b + u)μ 1 − (b+u)μ 2 e dx = νπ cosec(νπ)e k2ν μ 2 [ν = 0, u > 0, |arg(u + b)| < π, Re μ > 0, Re ν < 1]

−ν −μx

(x + b) (x − u)

1 1 1 √ (x − u)ν−1 (x + u)−ν+ 2 e−μx dx = √ 2ν− 2 Γ(ν) D 1−2ν (2 uμ) μ [u > 0, Re μ > 0,

u

1 1 1 √ (x − u)ν−1 (x + u)−ν− 2 e−μx dx = √ 2ν− 2 Γ(ν) D −2ν (2 uμ) u [u > 0, Re μ ≥ 0,

∞

−μ

(b − ix)

−∞

2πe−bp pμ+ν−1 1 F 1 (ν; μ + ν; (b − c)p) Γ(μ + ν)

(c − ix)−ν e−ipx dx =

[Re b > 0, ∞

6 −∞

∞

9.6 −∞

Re c > 0,

[for p > 0]

Re(μ + ν) > 1]

(b + ix)−μ (c + ix)−ν e−ipx dx = 0

ET I 119(10)

[for p > 0] cp

= 

Re ν > 0]

[for p < 0]

=0

8.

Re ν > 0]

ET I 139(19)

7.6



ET I 139(17)

ET I 139(18) ∞

6.



ν

351

μ+ν−1

2πe (−p) [for p < 0] 1 F 1 [μ; μ + ν; (b − c)p] Γ(μ + ν) [Re b > 0, Re c > 0, Re(μ + ν) > 1] ET I 19(11)

(b + ix)−2μ (c − ix)−2ν e−ipx dx

  pμ+ν−1 b−c exp p W ν−μ, 12 −ν−μ (bp + cp) [for p > 0] Γ(2ν) 2   (−p)μ+ν−1 b−c exp p W μ−ν, 12 −ν−μ (−bp − cp) [for p < 0] = 2π(b + c)−μ−ν Γ(2μ) 2

Re b > 0, Re c > 0, Re(μ + ν) > 12 ET I 19(12) = 2π(b + c)−μ−ν

 3.385

1

11

xν−1 (1 − x)λ−1 (1 − bx)− e−μx dx = B(ν, λ)Φ1 (ν, , λ + ν, −μ, b)

0

[Re λ > 0,

3.386 

∞

1. −∞

(ix)ν0

n 1 k=1



(βk + ix)νk e−ipx dx β0 − ix

Re ν0 > −1,

= 2πe−β0 p β0ν0 Re βk > 0,

n  k=0

n 1

Re ν > 0,

(β0 + βk )

|arg(1 − b)| < π]

νk



k=1

Re νk < 1,

ET I 39(24)

π arg ix = sign x, 2

p>0

ET I 118(8)

352

Exponential Functions



∞

(ix)ν0

n 1

(βk + ix)

k=1

2.



−∞

νk

e−ipx dx =0

β0 + ix

Re ν0 > −1,

n 

Re βk > 0,

Re νk < 1,

k=0

3.387 1.



1

6



2 ν−1 −μx

1−x

−1

 2.

1

6



−1

e

2 ν−1 iμx

1−x

e

3.387

√ dx = π

√ dx = π

 ν− 12 2 Γ(ν) I ν− 12 (μ) μ

π arg ix = sign x, 2

 Re ν > 0,

p>0

ET I 119(9)

π 2

WA 172(2)a

|arg μ|
0]

 ν− 12 ∞ ν−1 −μx 1 2 2 x −1 e dx = √ Γ(ν) K ν− 12 (μ) π μ 1



WA 25(3), WA 48(4)a

 3.



∞

4.



x2 − 1

ν−1

 π |arg μ| < , 2

Re ν > 0

 WA 190(4)a

eiμx dx

1

√  ν− 12 π 2 (1) [Im μ > 0, Re ν > 0] EH II 83(28)a Γ(ν) H 1 −ν (μ) 2 2 μ ν− 12 √  π 2 (2) [Im μ < 0, Re ν > 0] EH II 83(29)a − = −i Γ(ν) H 1 −ν (−μ) 2 2 μ √  ν− 12  u    2 π 2u 2 ν−1 μx u −x e dx = Γ(ν) I ν− 12 (uμ) + Lν− 12 (uμ) 2 μ 0 =i

5.

[u > 0, 

∞

6. u



x2 − u2

ν−1

1 e−μx dx = √ π



2u μ

Re ν > 0]

ET II 188(20)a

ν− 12 Γ(ν) K ν− 12 (uμ) [u > 0,

Re μ > 0,

Re ν > 0] ET II 203(17)a

√  ν− 12 ∞   π 2u 2 2 ν−1 −μx x +u e dx = Γ(ν) Hν− 12 (uμ) − Y ν− 12 (uμ) 2 μ 0

 7.11

[|arg u| < π, 3.388



2u

1. 0



2ux − x2

ν−1

e−μx dx =

√

 π

2u μ

ν− 12

Re μ > 0]

ET I 138(10)

e−uμ Γ(ν) I ν− 12 (uμ) [u > 0,

Re ν > 0]

ET I 138(14)

3.389

Exponentials and arbitrary powers



∞

2.



2 ν−1 −μx

2βx + x

0

e

1 dx = √ π



2β μ

ν− 12

353

eβμ Γ(ν) K ν− 12 (βμ) [|arg β| < π,

Re ν > 0,

Re μ > 0] ET I 138(13)



∞

3.



ν−1 −μx x2 + ix e dx = −

0



∞

4.



ν−1 −μx x2 − ix e dx =

0

3.389 1.

2.7

2μ

ν− 12

√ iμ i πe− 2 1

2μν− 2

(2)

Γ(ν) H ν− 1 2

(1)

Γ(ν) H ν− 1 2

μ 2 [Re μ > 0,

Re ν > 0]

ET I 138(15)

[Re μ > 0,

Re ν > 0]

ET I 138(16)

μ 2

  μ2 u2 1 1 2ν+2 −2 u −x x e dx = B(ν, )u 1 F 2 ν; , ν + ; 2 2 4 0    1 3 1 μ + B ν + ,  u2ν+2 −1 1 F 2 ν + ; , ν +  + 2 2 2 2 [Re  > 0, Re ν > 0]  2 2   ∞  2 μ u  1 − ν u2ν+2 −2 2ν−1 2 −1 −μx 31 u +x x e dx = √ G 2 π Γ(1 − ) 13  4  1 −  − ν, 0, 12 0 π |arg u| < , Re μ > 0, 2





u

u

3.7 0



2ν−1

2

2 −1 μx

∞

∞

6. 0

 7. 0

∞



ET II 188(21)

Re ν > 0



ET II 234(15)a

ET II 188(19)a

ν−1 −μx  1 √ −1 1 1 x x2 − u2 e dx = 2ν− 2 π μ 2 −ν uν+ 2 Γ(ν) K ν+ 12 (uμ) −ν −ipx

(ix) e dx = πb−ν−1 e−|p|b 2 2 b +x −∞



1 μ2 u 2 ; 2 4

2ν

[Re ν > 0] ∞

u





√  1    2 π μ 2 −ν ν+ 1 u 2 ν−1 μx + x u −x e dx = u 2 Γ(ν) I ν+ 12 (μu) + Lν+ 12 (μu) 2ν 2 2

4.

5.

√ iμ i πe 2

[Re(uμ) > 0]  |ν| < 1, Re b > 0,

ET II 203(16)a

arg ix =

  x e (ν − 1)π 1 ν−1 exp iμb + i dx = Γ(ν)b b2 + x2 2 2    (ν − 1)π Γ (1 − ν, −ibμ) × Γ(1 − ν, ibμ) + exp −ibμ − i 2 [Re b > 0, Re μ > 0, Re ν > −1] ν −μx



xν−1 e−μx dx = π cosec(νπ)Vν (2μ, 0) 1 + x2

[Re μ > 0,

Re ν > 0]

 π sign x 2

ET I 118(5)

ET II 218(22)

ET I 138(9)

354

Exponential Functions

 8.

∞

π (b + ix)−ν e−ipx dx = (b + c)−ν e−pc 2 2 c +x c −∞ [Re ν > −1,

 9.6

p > 0,

Re b > 0,

(b − ix)−ν e−ipx π dx = (b + c)−ν ecp 2 2 c +x c −∞ [p < 0,

3.391

∞

0



Re b > 0,

 ν  1 ν x + 1 + x2 e−μx dx = S 1,ν (μ) + S 0,ν (μ) μ μ

∞

 ν 1 ν 1 + x2 − x e−μx dx = S 1,ν (μ) − S 0,ν (μ) μ μ

0

Re c > 0,

[Re μ > 0] √ ν x + 1 + x2 √ e−μx dx = π cosec νπ [J−ν (μ) − J −ν (μ)] 1 + x2 0 [Re μ > 0] ν  ∞ √ 1 + x2 − x √ e−μx dx = S 0,ν (μ) − ν S −1,ν (μ) [Re μ > 0] 2 1 + x 0

ET I 140(26)

0

 ∞

 3.393

∞

3.394

2.

ET I 140(27), EH II 35(33) ET I 140(28)

√ 2ν x + x2 + 4b2 √ e−μx dx x3 + 4b2 x





μπ 3 = 2ν+3/2 2ν J ν+1/4 (bμ) Y ν−1/4 (bμ) − J ν−1/4 (bμ) Y ν+1/4 (bμ) 2 b [Re b > 0, Re μ > 0] ET I 140(33) √ ν+1/2  ∞     2   √ 1+ 1+x √ e−μx dx = 2 Γ(−ν) D ν 2iμ D ν −2iμ ν+1 2 x 1+x 0 ET I 140(32) [Re μ ≥ 0, Re ν < 0]

0

1.

ET I 118(7)

ET I 140(25)

2.

3.395

Re ν > −1]

[Re μ > 0]



4.

ET I 118(6)

   √ 2ν  √ 2ν −μx ν e x + 2β + x − x + 2β − x dx = 2ν+1 β ν eβμ K ν (βμ) μ [|arg β| < π, Re μ > 0] ET I 140(30)

∞

1.

3.

Re c > 0]

∞



3.392

3.391

√ ν √ −ν x2 − 1 + x + x2 − 1 + x √ e−μx dx = 2 K ν (μ) x2 − 1 1 ET I 140(29) [Re μ > 0]  √ √ 2ν  2ν  ∞ μ μ x + x2 − 1 + x − x2 − 1 2μ  K ν+1/4 K ν−1/4 e−μx dx = π 2 2 x (x2 − 1) 1



∞

[Re μ > 0]

ET I 140(34)

3.411

Rational functions of powers and exponentials



∞



x+

3. 0

355

√ √ ν  −ν x2 + 1 + cos νπ x + x2 + 1 √ e−μx dx = −π [Eν (μ) + Y ν (μ)] x2 + 1 [Re μ > 0]

EH II 35(34)

3.41–3.44 Combinations of rational functions of powers and exponentials 3.411



∞

1. 0



∞

2.12 0



∞

3.12 0



∞

4.12 0



0 ∞

6.8 0

[Re μ > 0,

2n

[n = 1, 2, . . .]

 xa−1 dx = 1 − 21−a Γ(a) ζ(a) x e +1

[Re a > 0]

 x2n−1 2n |B2n | dx = 1 − 21−2n (2π) x e +1 4n

[n = 1, 2, . . .]

 7.

∞

0

FI II 792a

FI II 721a FI II 792a, WH BI(83)(2), EH I 39(25)

x dx π2 = −x 1−e 12

BI (104)(5)

∞  xν−1 e−μx dx = Γ(ν) (μ + n)−ν bn = Γ(ν) Φ(b, ν; μ) 1 − be−x n=0

[Re μ > 0 and either |b| ≤ 1,

11

Re ν > 1]

x2n−1 (2π) |B2n | dx = ex − 1 4n

ln 2

5. 

xν−1 dx 1 = ν Γ(ν) ζ(ν) eμx − 1 μ

 μ xν−1 e−μx 1 dx = Γ(ν) ζ ν, 1 − e−bx bν b

b = 1,

Re ν > 0; or b = 1, [Re b > 0,

Re ν > 1]

Re μ > 0,

EH I 27(3)

Re ν > 1] ET I 144(10)



∞

8. 0



∞

0 ∞

10. 0



∞

11. 0



∞

12.11 0

 13.

11 0

[p > −1,

xe−x dx π2 = −1 x e −1 6

(cf. 4.231 3)

BI (82)(1)

xe−2x dx π2 = 1 − e−x + 1 12

(cf. 4.251 6)

BI (82)(2)

xe−3x π2 3 dx = − −x e +1 12 4

(cf. 4.251 5)

BI (82)(3)

2n−1  (−1)k−1 xe−(2n−1)x π2 + dx = − x 1+e 12 k2

(cf. 4.251 6)

BI (82)(5)

xe−2nx π 2  (−1)k + dx = 1 + ex 12 k2

(cf. 4.251 5)

BI (82)(4)

n = 1, 2, . . .]

BI (83)(9)

k=1

9. 

∞

 (−1)k−1 xn−1 e−px dx = (n − 1)! 1 + ex (p + k)n

k=1

∞

2n

k=1

356

Exponential Functions



∞

14.7 0



k=n

∞

15.7 0



17.



∞

 (−1)n+k x2 e−nx dx = 2 = (−1)n+1 1 + e−x k3

∞

∞

∞

∞

 n dx =− e−px e−x − 1 x

∞

∞

0

xn−1

(−1)k

k=0

n k

LI (82)(13)

ln(p + n − k)

LI (89)(10)

LI (89)(15)

m

 1 1 − e−mx dx = −(n − 1)! x 1−e kn

(cf. 4.272 11)

LI (83)(8)

∞

k=1

[p > 0,

r > 0,

−1 < q < 1] BI (83)(5)

∞

[|arg b| < π,

0 < Re μ < 1] BI (101)(5), ET I 120(16)a

∞

xe

−∞

eνx

μx

−1

dx =

π ν

cosec

μπ 2 ν

[Re ν > Re μ > 0]

(cf. 4.254 2) LI (101)(3)

∞

25.

x

0



k=1

 qk xp−1 1 Γ(p) = Γ(p)r−p Φ(q, p, 1) dx = erx − q qrp kp

24.

27.

n 

xeμx dx = πbμ−1 cosec(μπ) [ln b − π cot(μπ)] x −∞ b + e





k=1

7



n−1

 (−1)k 7 4 π +6 120 k4

n    n dx  k n (p + n − k) ln(p + n − k) e−px e−x − 1 = (−1) x2 k

0

26.



BI (82)(12)

k=0

21.12

23.

ET I 120(17)a

(cf. 4.262 4)

0



(cf. 4.261 11)

(cf. 4.262 5)

 (−1)n+k x3 e−nx dx = 6 = (−1)n+1 −x 1+e k4 k=n

20.9

22.

n−1

[0 < Re μ < 1]

k=1

0



BI (82)(9)



 (−1)k 3 ζ(3) + 2 2 k3 k=1 [n = 1, 2, . . .]

∞

19.9



(cf. 4.261 12)

LI (82)(10)

0





k=n

11



[n = 1, 2, . . .]

k=1

 x2 e−μx 3 3 2 dx = π cos μπ 2 − sin μπ −x −∞ 1 + e  ∞ 3 −nx n−1  1 π4 x e − 6 dx = −x 15 k4 0 1−e

16.

18.

  ∞ n−1   1 x2 e−nx 1 dx = 2 = 2 ζ(3) − 1 − e−x k3 k3

3.411

∞

−x

π 1+e dx = −1 ex − 1 3 2

1 − e−x −x 2π 2 e dx = −3x 1+e 27 0     ∞ Γ μ2 + 1 √ 1 − e−μx dx π  = ln 1 + ex x Γ μ+1 0 2

(cf. 4.231 4)

x

BI (82)(6) LI (82)(7)a

[Re μ > −1]

BI (93)(4)

3.415

Rational functions of powers and exponentials

  Γ ν2 Γ μ+1 − e−μx dx 2 = ln  μ  ν+1 e−x + 1 x Γ 2 Γ 2 0  ∞ px  qx pπ qπ  e − e dx = ln tan cot rx x 2r 2r −∞ 1 + e 

28. 29.

 30.

∞ −νx

e

[Re μ > 0, [|r| > |p|,

357

Re ν > 0] |r| > |q|,

BI (93)(6)

rp > 0,

rq > 0] BI (103)(3)

∞

px

qx

e −e rx −∞ 1 − e

 dx pπ qπ  = ln sin cosec x r r

[|r| > |p|,

|r| > |q|,

rp > 0,

rq > 0] BI (103)(4)



∞ −qx

e

31. 0



+ e(q−p)x x dx = 1 − e−px



qπ π cosec p p

2 [0 < q < p]

BI (82)(8)

∞ −px

pπ − e(p−q)x dx = ln cot [0 < p < q] −qx + 1 e x 2q 0   ∞ a+b p a + be−px dx a + be−qx = ln 3.412 − qx px −px −qx ce + g + he ce + g + he x c+g+h q 0 [p > 0, q > 0] 3.413   ∞ 1 − e−βx (1 − e−γx ) e−μx dx Γ(μ) Γ (β + γ + μ) = ln 1. −x 1−e x Γ(μ + β) Γ(μ + γ) 0 [Re μ > 0, Re μ > − Re β, Re μ > − Re γ, Re μ > − Re(β + γ)] e

32.

BI (93)(7)

BI (96)(7)

(cf. 4.267 25) BI (93)(13)

2 ∞ 1 − e(q−p)x dx qπ = ln cosec qx − e(q−2p)x x 2p e 0  ∞ −px e − e−qx 1 + e−(2n+1)x dx 1 + e−x x 0  

2. 3.

= ln  3.414

∞

0

3.415



1. 0

∞

[0 < q < p]

BI (95)(6)

 q(q + 2)(q + 4) · · · (q + 2n)(p + 1)(p + 3) · · · (p + 2n − 1) p(p + 2)(p + 4) · · · (p + 2n)(q + 1)(q + 3) · · · (q + 2n − 1) [Re p > −2n, Re q > −2n] (cf. 4.267 14) BI (93)(11)



 1 − e−βx (1 − e−γx ) 1 − e−δx e−μx dx Γ(μ) Γ(μ + β + γ) Γ(μ + β + δ) Γ(μ + γ + δ) = ln 1 − e−x x Γ(μ + β) Γ(μ + γ) Γ(μ + δ) Γ(μ + β + γ + δ) [2 Re μ > |Re β| + |Re γ|+| Re δ|] (cf. 4.267 31) BI (93)(14), ET I 145(17)

     x dx 1 bμ π bμ = ln − −ψ 2 2 μx (x + b ) (e − 1) 2 2π bμ 2π [Re b > 0,

Re μ > 0] BI (97)(20), EH I 18(27)

358

Exponential Functions

 2.

∞

x dx

11

(x2

0

+

b2 )2

(e2πx

1 1  1 ψ (b) − 2+ 3 8b 4b 4b − 1) ∞  |B2k+2 | 1 ∼ 4 4b b2k =−

k=0



∞

3.11 0



∞

4.8 0

5.∗



6.∗

  x dx 1 1  1 ψ b + − = 4b2 4b 2 (x2 + b2 )2 (e2πx + 1)

∞

7.∗

BI(97)(22), EH I 22(12)

[Re b > 0,

Re μ > 0]

[Re b > 0,

Re μ > 0]

x3 dx 1 1 1 1 b = ln b − + − ψ(b) − ψ  (b) 2 2 2 2πx (x + b ) (e − 1) 2 8b 4 2 2

∞

[b > 0]   xdx 1 2 3 1  1  = − 4 − 5 + 3 ψ (b) − 2 ψ (b) (x2 + b2 )3 (e2πx − 1) 16 b 2b b b

∞

[b > 0]   x dx 1 2 1 1  1  1  = − 4 − 5 + 3 ψ (b) − 2 ψ (b) − ψ (b) (x2 + b2 )4 (e2πx − 1) 16 3b 4b 2b 2b 6b

0



[asymptotic expansion for Re b > 0]

     x dx 1 bμ 1 bμ = ψ + − ln (x2 + b2 ) (eμx + 1) 2 2π 2 2π

0



3.416

3

0

[b > 0] 3.416



∞

1. 0



∞

2. 0

 3.

∞

8 0

(1 + ix)2n − (1 − ix)2n dx 1 2n − 1 = i e2πx − 1 2 2n + 1

[n = 1, 2, . . .]

BI (88)(4)

(1 + ix)2n − (1 − ix)2n dx 1 = i eπx + 1 2n + 1

[n = 1, 2, . . .]

BI (87)(1)

 1  (1 + ix)2n−1 − (1 − ix)2n−1 dx 2n = 1 − 2 B 2n i eπx + 1 2n [n = 1, 2, . . .]

3.417



∞

1. −∞



∞

2. −∞

3.418



1.6 0

∞

x dx b π ln = 2 −x +b e 2ab a

[ab > 0]

x dx π2 = 2 −x −b e 4ab

(cf. 4.231 10)

a2 e x a2 e x

    1 2 2 x dx 1  = ψ − π = 1.1719536193 . . . ex + e−x − 1 3 3 3

(cf. 4.231 8)

BI (87)(2)

BI (101)(1)

LI (101)(2)

LI (88)(1)

3.421

Rational functions of powers and exponentials

 2.

∞

6 0



0

3.419 1.



    xe−x dx 1 5 2 1  = ψ − π = 0.3118211319 . . . ex + e−x − 1 6 3 6

ln 2

3.

359

LI (88)(2)

x dx π = ln 2 ex + 2e−x − 2 8

∞

BI (104)(7)

2

x dx (ln b) = x ) (1 + e−x ) (b + e 2(b − 1) −∞

[|arg b| < π]

(cf. 4.232 2) BI (101)(16)

 2.

π 2 + (ln b)2 x dx = x −x ) 2(b + 1) −∞ (b + e ) (1 − e

 3.

∞

∞

x2 dx = − e−x ) −∞ (b +

[|arg b| < π]

BI (101)(17)

  π 2 + (ln b)2 ln b

ex ) (1

(cf. 4.232 3)

[|arg b| < π]

3(b + 1)

(cf. 4.261 4) BI (102)(6)

 4.

∞

x3 dx = − e−x ) −∞ (b +

 2 π 2 + (ln b)2

ex ) (1

[|arg b| < π]

4(b + 1)

(cf. 4.262 3) BI (102)(9)

 5.12

6.11

 x4 dx π 2 + (ln b)2  2 2 = 7π ln b + 3 (ln b) x −x ) 15(b + 1) −∞ (b + e ) (1 − e

(cf. 4.263 1)  2  ∞  π 2 + (ln b)2  x5 dx 2 2 = 3π + (ln b) x −x ) 6(b + 1) −∞ (b + e ) (1 − e

BI (102)(10)

(cf. 4.264 3)

BI (102)(11)

 7.

∞

∞

(x − ln b) x dx = (b − ex ) (1 − e−x ) −∞

  2 − 4π 2 + (ln b) ln b

[|arg b| < π]

6(b − 1)

(cf. 4.257 4) BI (102)(7)

3.421



1. 0

∞



e−νx − 1

n 

e−ρx − 1

m

e−μx

dx x2 =

n 

(−1)k

k=0

m n  m (−1)l k l l=0

× {(m − l)ρ + (n − k)ν + μ} ln [(m − l)ρ + (n − k)ν + μ] [Re ν > 0,

Re μ > 0,

Re ρ > 0]

BI (89)(17)

360

Exponential Functions



∞

2.



1 − e−νx

3.422

n n dx 1  (ρ + kν + 1)2 1 − e−ρx e−x 3 = (−1)k x 2 k k=0 n n 1 (kν + 1)2 ln(kν + 1) × ln(ρ + kν + 1) + (−1)k−1 2 k

n 

0

k=1

[n ≥ 2,

Re ρ > 0]

  π bμ−1 ln b − cμ−1 ln c π 2 bμ−1 − cμ−1 cos μπ xe−μx dx = + −x ) (c + e−x ) (b − c) sin μπ (c − b) sin2 μπ −∞ (b + e [|arg b| < π, |arg c| < π, b = c. 0 < Re μ < 2]

 3.

Re ν > 0,

BI (89)(31)

∞



∞

4.



e−px − e−qx

0



∞

5.12



1 − e−px

0

 −rx dx (p + s + 1)(q + r + 1) = ln e − e−sx e−x x (p + r + 1)(q + s + 1) [p + s > −1, p + r > −1, q > p] (cf. 4.267 24)

ET I 120(19)

BI (89)(11)

  dx 1 − e−qx 1 − e−rx e−x 2 x = (p + q + 1) ln(p + q + 1) +(p + r + 1) ln(p + r + 1) + (q + r + 1) ln(q + r + 1) −(p + 1) ln(p + 1) − (q + 1) ln(q + 1) − (r + 1) ln(r + 1) −(p + q + r) ln(p + q + r) [p > 0,



r > 0]

(cf. 4.268 3)



∞

1.

2

∞

2.6

∞

0

[Re ν > 2]

∞

BI (102)(8)a

ET I 313(10)

dx = Γ(ν) [ζ(ν − 1, μ + 2) − (μ + 1) ζ(ν, μ + 2)]

xν−1 e−μx 2

(1 − be−x )

Re ν > 2]

ET I 313(11)

∞

xq e−px dx (1 − ae−px )2

0



dx = Γ(ν) [ζ(ν − 1) − ζ(ν)]

[Re μ > −2,

3.8 4.7

xν−1 e−μx (ex − 1)2

0



xν−1 (ex − 1)

0



BI (89)(14)

∞

x(x − a)eμx dx −π 2 = cosec2 μπ [(eaμ + 1) ln μ − 2π cot μπ (eaμ − 1)] x −x ) ea − 1 −∞ (b − e ) (1 − e [a > 0, |arg b| < π, |Re μ| < 1] (cf. 4.257 5)

3.422

3.423

q > 0,

=

Γ(q + 1)  ak apq+1 kq

[a < 1,

q > −1,

p > 0]

BI (85)(13)

k=1

dx = Γ(ν) [Φ(b; ν − 1; μ) − (μ − 1) Φ(b; ν; μ)] [Re ν > 0,

Re μ > 0,

|arg(1 − b)| < π]

(cf. 9.550)

ET I 313(12)

3.427

Rational functions of powers and exponentials

 5.

∞

xex dx

−∞ (b

+

1 ln b b

=

ex ) 2

361

[|arg b| < π]

(cf. 4.231 5) BI (101)(10)

 6.

t

x5

0

3.424 1.



∞

7

e−x (1 −

(1 − ex )2 ∞

2.

(1 + a)ex + a (1 + ex )

0

 3.

∞

∞

−∞

(a2 ex

1.

7

2

(ex − 1)

∞

−∞



3.427 1. 2.7

∞



a2 e x − e

[a > −1,

n = 1, 2, . . .]

BI (85)(14)

dx =

π2 2ab

[ab > 0]

BI (102)(3)a

dx =

b π ln ab a

[ab > 0]

BI (102)(1)

x2 dx =

∞

−x

2 2 π −2 3

BI (85)(7)

x2 dx

=−

p+1

(ex − ae−x ) x2 dx 2

2

+ ex ) (1 + e−x ) 2

1

B

ap+1

=

(ln a) a−1

=

π 2 + (ln a) a+1

∞

[a > 0,

p > 0]

BI (102)(5)

BI (102)(12) 2

2

+ ex ) (1 − e−x )

 e−μx e−x + −x dx = ψ(μ) x e −1 0   ∞ 1 1 e−x dx = C − −x 1 − e x 0



p p , ln a 2 2

2

(ex − ae−x ) x2 dx

−∞ (a

n > 0] BI(101)(13), LI(101)(13)

(a2 ex + e−x )

−∞ (a

 2.

BI (85)(15)

∞

2.7

1.

n = 1, 2, . . .]

  √   π Γ n − 12 xex dx a 1 2 ln −C−ψ n− = 2 2 2x n 4a2n−1 b Γ(n) 2b 2 −∞ (a + b e ) [ab > 0,





3.426

+

2 2x 2 −x b e )

e − e−x + 2

0

3.425

−

∞  (−1)k (a + k)n

[a > −1,

k=1

2 2x 2 −x b e )

∞ x

5.

e−ax xn dx = n!

a2 ex − b2 e−x

4. 

2

e−ax xn dx = n! ζ(n, a)

a2 ex + b2 e−x

−∞ (a2 ex



e−kt  5 y + 5y 4 + 20y 3 + 60y 2 + 120y + 120 5 k k=1 ∞ 5 −t/2  t e e−kt  4 = 120 ζ(5) − −5 y + 4y 3 + 12y 2 + 24y + 24 5 2 sinh(t/2) k k=1 [y = kt]

dx = 120 ζ(5) −

(1 + a)ex − a

0



2 e−x )

∞ 

BI (102)(13)



[Re μ > 0] (cf. 4.281 1)

(cf. 4.281 4)

WH BI (94)(1)

362

Exponential Functions



∞



3.428



1 π e−2x dx = ln x 2 4 0  −μx    ∞ 1 1 1 e 1 1 − + x dx = ln Γ(μ) − μ − ln μ + μ − ln(2π) 2 x e − 1 x 2 2 0

3. 4.



∞

1 1 − 2 1 + e−x

[Re μ > 0]





WH

dx 1 1 −2x 1 e = − ln π − x 2 e +1 x 2 0  −x  ∞  μx e −1 e dx = − ln Γ(μ) − ln sin(πμ) + ln π −μ −x 1 − e x 0

5. 6.



∞

7. 0



∞



8. 0

10.

e−νx e − 1 − e−x x

e−μx n − x 1 − e−x/n



dx = ln μ − ψ(ν)

BI (94)(6)

[Re μ < 1]

EH I 21(6)

(cf. 4.281 5)

BI (94)(3)

e−x dx = n ψ(nμ + n) − n ln n

[Re μ > 0, n = 1, 2, . . .]   −x −μx 1−e e dx = ln Γ(μ + 1) [Re μ > −1] μ− −x 1 − e x 0   ∞ Γ(μ + ν + 1) e−μx − e−(μ+ν)x dx = ln νe−x − x−1 e x Γ(μ + 1) 0 

9.

 −μx



BI (94)(5)

BI (94)(4)

∞

∞

11.

  −1 (1 − ex ) + x−1 − 1 e−xz dx = ψ(z) − ln z

[Re μ > −1,

Re ν > 0]

[Re z > 0]

WH

BI (94)(8) EH I 18(24)

0

3.428 1.



∞

12 0



  1 1 −x 1 e−x − e−μνx dx −μx = ln Γ(μν) − ν ln μ νe − e − −x μ μ 1−e x μ

∞

2. 0



∞

3. 0

 4. 0

∞

n−1 n−1 e(1−μ)x e−nμx + + + 2 1 − e−x 1 − ex/n 1 − e−x

 n n−1 − nμ − 2 1 − e−x 

−νx

−μνx



[Re μ > 0,

e−x

BI (94)(18) Re ν > 0]   dx n−1 1 = ln 2π − nμ + ln n x 2 2

[Re μ > 0, n = 1, 2, . . .]    n−1  e(1−μ)x e−x k dx = + 1 − ln Γ μ − x n 1 − ex/n

BI (94)(14)

k=0

x

μx

e e e e − − + x μx x 1−e 1−e 1−e 1 − eμx



[Re μ > 0,

n = 1, 2, . . .]

BI (94)(13)

Re ν > 0]

LI (94)(15)

dx = ν ln μ x [Re μ > 0,

3.434

5.

6.12

Rational functions of powers and exponentials

   μe−μx 1 μ + 1 −μx −x dx − e + aμ − + (1 − aμ)e ex − 1 1 − e−μx 2 x 0   1 μ−1 ln(2π) + − aμ ln μ = 2 2 [Re μ > 0] BI (94)(16)     ∞  −νx −μνx −μx μ−1 e (μ − 1)e μ − 1 −μx dx 1 e e = ln(2π) + − μν ln μ − − − 1 − e−x 1 − e−μx 1 − e−μx 2 x 2 2 0 [Re μ > 0, Re ν > 0] (cf. 4.267 37) BI (94)(17)



∞



∞

7. 0



  (1 − e−νx ) (1 − e−μx ) dx −x = ln B(μ, ν) 1−e − 1 − e−x x [Re μ > 0,

 3.429 3.431

363

∞

0



∞

1. 0



∞

2. 0

−x

dx = ψ(μ) e − (1 + x)−μ x

Re ν > 0]

BI (94)(12)

[Re μ > 0]

NH 184(7)

  1 2 2 −1 −μx e − 1 + μx − μ x xν−1 dx = Γ(ν + 3) 2 ν(ν + 1)(ν + 2)μν [Re μ > 0, −2 > Re ν > −3] LI (90)(5)

       1 1 1 ln 1 + x−1 − x−2 (x + 2) 1 − e−x e−px dx = −1 + p + 2 2 p [Re p > 0]

3.432



∞

1. 0



n n   n 1 xν−1 e−mx e−x − 1 dx = Γ(ν) (−1)k k (n + m − k)ν k=0

∞

2.12 0

[n = 0, 1, . . . , Re ν > 0]    ν−1 Γ(μ) Γ(ν) = Γ(ν) − B(μ, ν) xν−1 e−x − e−μx 1 − e−x dx = Γ(ν) − Γ(μ + ν) [Re μ > 0,

 3.433

ET I 144(6)

∞

 xp−1 e−x +

0

n 

(−1)k

k=1

xk−1 (k − 1)!

LI (90)(10)

Re ν > 0]

LI (81)(14)

 dx = Γ(p)

[−n < p < −n + 1,

n = 0, 1, . . .] FI II 805

3.434



1.

∞ −νx

e

xρ+1

0

 2. 0

− e−μx

dx =

μρ − ν ρ Γ(1 − ρ) ρ

[Re μ > 0,

Re ν > 0,

Re ρ < 1] BI (90)(6)

∞ −μx

e

−e x

−νx

dx = ln

ν μ

[Re μ > 0,

Re ν > 0]

FI II 634

364

Exponential Functions

3.435  1.

3.435

∞

x  dx = 1 − ln 2 (x + 1)e−x − e− 2 x 0  ∞

1 − e−μx 1 11 ln (βμ) + C − eβμ Ei(−βμ) 2. [|arg β| < π, Re μ > 0] dx = β 0 x(x + β)   ∞ 1 dx − e−x =C 3. 1+x x 0   ∞ dx a 1 = ln − C [a > 0, Re μ > 0] e−μx − 4. 1 + ax x μ 0   ∞  −npx − e−nqx e−mpx − e−mqx dx e m − [p > 0, q > 0] 3.436 = (q − p) ln 2 n m x n 0   ∞ 1 − e−px dx = p ln p − p [p > 0] pe−x − 3.437 x x 0 3.438     ∞ ln 2 − 1 1 1 1 x dx + e−x − e− 2 = 1. 2 x x x 2 0   ∞ 2 2 −px p p2 11 p −x p dx 1−e e − − 2− = ln p − p3 2.7 3 6 2x x x x 6 36 0 [p > 0]   dx 1 = 1 − ln 2 e−x − e−2x − e−2x x x 0      ∞  dx 1 1 −x x + 2  −px −x 2 e + e = p− (ln p − 1) p− −e 2 2x x 2 0  3. 4.

LI (89)(19) ET II 217 (18) FI II 7 95, 802 BI (92)(10)

BI (89)(28) BI (89)(24)

BI (89)(19)

BI (89)(33)

∞

BI (89)(25)

BI (89)(22) [p > 0]    ∞ dx r 1  −mpx e = p ln p − q ln q − (p − q) 1 + ln (p − q)e−rx + − e−mqx mx x m 0 [p > 0, q > 0, r > 0] LI(89)(26), LI(89)(27)  ∞

 dx (p − r)e−qx + (r − q)e−px + (q − p)e−rx = (r − q)p ln p + (p − r)q ln q + (q − p)r ln r x2 0 [p > 0, q > 0, r > 0] (cf. 4.268 6) BI (89)(18)

 3.439

3.441

3.442  1.

∞

0

 2. 3.

∞

    dx 1 q+1 x+2  1 − e−x e−qx = −1 + q + ln 1− 2x x 2 q 

−x



−1 1 + x 1 + x 0   ∞ 1 e−px − 1 + a2 x2 0 e

[q > 0] dx =C−1 x a dx = −C + ln x p

BI (89)(23) BI (92)(16)

[p > 0]

BI (92)(11)

3.454

3.443

Powers and algebraic functions of exponentials



∞



1. 0



∞

2. 0



∞

3.

p 1 − e−px e−x p2 − + 2 x x2 n

(1 − e−px ) e−qx x3



1 − e−px

2

365



p2 3 dx = ln p − p2 [p > 0] x 2 4 n n 1 (q + kp)2 ln(q + kp) dx = (−1)k−1 2 k

e−qx

0

k=2

[n > 2,

q > 0,

BI (89)(32)

pn + q > 0]

(cf. 4.268 4)

dx = (2p + q) ln(2p + q) − 2(p + q) ln(p + q) + q ln q x2 [q > 0, 2p > −q]

BI (89)(30)

(cf. 4.268 2) BI (89)(13)

3.45 Combinations of powers and algebraic functions of exponentials 3.451 1. 2. 3.452 1. 2. 3. 4. 5. 3.453

 4 − ln 2 xe 1− 3 0    ∞  π 1 + ln 2 xe−x 1 − e−2x dx = 4 2 0





∞

e−x dx

4 = 3



BI (99)(1)

(cf. 4.241 9)

x dx √ = 2π ln 2 ex − 1 0    ∞ 2 x dx π2 2 √ x = 4π (ln 2) + 12 e −1 0  ∞ −x π xe dx √ x = [2 ln 2 − 1] 2 e −1 0  ∞ −x xe dx √ = 1 − ln 2 e2x − 1 0    ∞ −2x xe dx 7 3 √ x = π ln 2 − 4 12 e −1 0



∞

0



[a2 ex

0



11 0

BI (99)(2)

∞

FI II 643a,BI(99)(4) BI (99)(5) BI (99)(6) BI (99)(8) BI (99)(7)

  b 2π xex dx √ ln 1 + = a2 ex − (a2 − b2 ) ex − 1 ab a

∞

2.

1.

√

∞

1.

3.454

−x

xex dx b 2π √ arctan = 2 2 x ab a − (a + b )] e − 1

xe−2nx dx (2n − 1)!! π √ = 2x (2n)!! 2 e −1

 ln 2 +

2n  (−1)k k=1

k

[ab > 0]

(cf. 4.298 17)

BI (99)(16)

[ab > 0]

(cf. 4.298 18)

BI (99)(17)

 LI (99)(10)

366

Exponential Functions



∞

2. 0

3.455 1.

2.



xe−(2n−1)x dx (2n − 2)!! √ =− 2x (2n − 1)!! e −1

ln 2 +

2n−1  k=1

(−1)k k

3.457  1. 

∞

3.458

BI (99)(13)

(cf. 4.244 3)

(cf. 4.241 5) ∞

xex dx

2 [ln(4a) − C − 2 ψ(2n) + ψ(n)] n+1/2 (2n + 1)a +  ∞ μ μ x dx 1 , ln a [a > 0, Re μ > 0] = − B 2 x −x )μ 2aμ 2 2 −∞ (a e + e 

ln 2

7

n+3/2 ex )

x



=

p−1

x

xe (e − 1)

0

2.

BI (99)(12)

BI (99)(14)

 n−1/2 (2n − 1)!! π [C + ψ(n + 1) + 2 ln 2] xe−x 1 − e−2x dx = 4 · (2n)!!

−∞ (a

3.

BI (99)(11)

∞

0

2.12

LI (99)(9)

∞

  x dx π π √ √ √ = ln 3 + 3 3x e −1 3 3 3 3 0    ∞ x dx π π  = √ ln 3 − √ 3 2 3 3 3 3 0 (e3x − 1)

2.



x2 ex dx  = 8π ln 2 0 (ex − 1)3    ∞ x3 ex dx π2  = 24π (ln 2)2 + 12 3 0 (ex − 1)

3.456  1.

1.



3.455

  ∞  (−1)k−1 1 ln 2 + dx = p p+k+1

BI (99)(3) BI (101)(12)

BI (101)(14)

BI (104)(4)

k=0

∞

xex dx ν+1

−∞ (a + ex )

1 [ln a − C − ψ(ν)] νaν   ν−1 1 1 = ν ln a − νa k

=

[a > 0] [a > 0,

ν = 1, 2, . . .]

k=1

BI (101)(11)

3.46–3.48 Combinations of exponentials of more complicated arguments and powers 3.461



∞ −p2 x2

1. u

e

x2n

dx =

√ (−1)n 2n−1 p2n−1 π [1 − Φ(pu)] (2n − 1)!! 2 2 n−1 (−1)k 2k+1 (pu)2k e−p u  + 2n−1 2u (2n − 1)(2n − 3) · · · (2n − 2k − 1) k=0

[p > 0]

NT 21(4)

3.462

Exponentials of complicated arguments and powers

 2.

∞

12

2n −px2

x e

  1 1 − n+ 2 2p

dx =

0



∞

2b.*

1 Γ 2p(m+1)/2

2

xm e−px dx =

0



∞

3.12 0



∞

6.

∞

2

(x + ai)2n e−x dx =

n! 2pn+1

[p > 0]

n

(2n − 1)!! √  (2a)2k n! k π (−1) 2n (2k)!(n − k)!  π |arg μ| < , 2



Re b > 0]

[Re a > 0,    3 4 90b 15b x6 exp −a x2 + b2 dx = 4 K 3 (ab) + 3 K 2 (ab) a a

Re b > 0]

∞

[Re a > 0,

Re b > 0]

 γ √ 2β [Re β > 0,

Re ν > 0]

0

∞

1.

xν−1 e−βx

2

−γx

dx = (2β)−ν/2 Γ(ν) exp

0



u>0

[Re a > 0,    2 3 12b 3b x4 exp −a x2 + b2 dx = 3 K 2 (ab) + 2 K 1 (ab) a a

9.



2

∞

0



BI (100)(12)

∞





γ2 8β



 D −ν

EH II 119(3)a, ET I 313(13) n−1





2 π d qeq /p [p > 0] n−1 p dq −∞  k   n n/2  2 π q 1 p [p > 0] = n!eq /p p p (n − 2k)!(k)! 4q 2 k=0      ∞ q q2 ν −β 2 x2 +iqx − ν2 √ −ν−1 √ (ix) e dx = 2 πβ exp − 2 D ν 8β β 2 −∞

Re β 2 > 0, Re ν > −1, arg ix = π2 sign x



ET I 135(19)a

   2b b2 K 0 (ab) exp −a x2 + b2 dx = 2 K 1 (ab) + a a

∞

8.

FI II 743

BI (81)(7)

Re b > 0]

0



n = 0, 1, . . .]

[Re a > 0,

7.

3.12

[p > 0,

2 dx 1 √ √ = e−μu − μπ [1 − Φ (u μ)] 2 x u u  ∞    exp −a x2 + b2 dx = b K 1 (ab)



2.8

π p

k=0

e−μx

0

3.462

m+1 2





2

−∞





x2n+1 e−px dx = 12 p−(n+1) Γ (n + 1) =

4. 5.11

(2n − 1)!!  Γ n + 12 = 2(2p)n

367

∞

4. −∞

xn e−px

2

+2qx

dx =

1 2n−1 p

√ xn exp −(x − β)2 dx = (2i)−n π H n (iβ)

BI (100)(8) LI (100)(8)

ET I 121(23)

EH II 195(31)

368

Exponential Functions

 5.

∞

11

xe

−μx2 −2νx

0



6. 7.11

8. 9. 10. 11. 12. 13.

14.

15. 16.

17.

18.

∞

ν 1 − dx = 2μ 2μ 



   π νμ2 ν e 1 − erf √ μ μ





3.462

|arg ν|
0

ET I 146(31)a

π q exp [Re p > 0] BI (100)(7) p p −∞      ∞ 2 π 2ν 2 + μ νμ2 ν ν e 1 − erf x2 e−μx −2νx dx = − 2 + √ 2μ μ5 4 μ 0

|arg ν| < π2 , Re μ > 0 ET I 146(32)     ∞ 2 ν2 π ν2 1 1+2 eμ x2 e−μx +2νx dx = [|arg ν| < π, Re μ > 0] BI (100)(8)a 2μ μ μ −∞    ∞ n e±a 1 [Re β > 0, Re n > 0] e−βx ±a dx = Γ 1/n n nβ 0  ∞ (1 − aβ) (x − a)e−β(x−a) dx = eaβ [Re β > 0] β2 0  ∞ (1 − aβ) (x − a)e−β(x+a) dx = e−aβ [Re β > 0] β2 0    ∞   

am e±pb/a pb p > 0, arg ab  < π (ax ± b)m e−px dx = Γ m + 1, ± m+1 p a 0    ∞ am e±pb/a pb (ax ± b)m e−px dx = Γ m + 1, pu ± pm+1 a u  

 p > 0, arg ab ± u  < π       u pb pb am e±pb/a − Γ m + 1, pu ± Γ m + 1, ± (ax ± b)m e−px dx = pm+1 a a 0  

 u > 0, p > 0, arg ab ± u  < π    ∞   

e−px pn−1 e±pb/a pb p > 0, arg ab  < π dx = Γ −n + 1, ± n n (ax ± b) a a 0    ∞ e−px pn−1 e±pb/a pb dx = Γ −n + 1, pu ± n an a u (ax ± b)  

 u > 0, p > 0, arg ab ± u  < π       u e−px pb pb pn−1 e±pb/a − Γ −n + 1, pu ± Γ −n + 1, ± dx = n an a a 0 (ax ± b)  

 u > 0, p > 0, arg ab ± u  < π   a k   j+1 j k    ∞ b Γ , β −b k x−a x−a dx = exp −β b b kβ (j+1)/k 0 

arg − ab > 0, Re b > 0, Re β > 0, Re k > 0 xe−px

2

+2qx

dx =

q p

2

3.469

Exponentials of complicated arguments and powers

 Γ(z, βun ) Re β > 0, dx = xm nβ z u  √  ∞ exp −a x2 + b2 √ dx = K 0 (ab) [Re a > 0, x2 + b2 0  √  ∞ 2 x exp −a x2 + b2 b √ dx = K 1 (ab) [Re a > 0, 2 2 a x +b 0  √  ∞ 4 x exp −a x2 + b2 3b2 √ dx = 2 K 1 (ab) [Re a > 0, a x2 + b2 0  √  ∞ 6 x exp −a x2 + b2 15b3 √ dx = 3 K 3 (ab) [Re a > 0, a x2 + b2 0  √  n  ∞ 2n x exp −a x2 + b2 b √ dx = (2n − 1)!! K n (ab) 2 + b2 a x 0 

19.

12

20.12 21.12 22.12 23.12 24.12



∞ −βxn

e



∞











exp −px a2 p a2 p 1 √ K0 dx = exp 2 2 2 2 2 a +x 0  ∞  2 1 dx = C e−x − e−x 3.463 x 2 0 ∞   dx √ √ √ −μx2 −νx2 e 3.464 −e = π ν− μ 2 x  0 ∞  −μx2 μ+b π 2 1 + 2bx e 3.465 dx = 2 μ3 0 3.466  ∞ −μ2 x2 e π 2 2 1. dx = [1 − Φ(bμ)] eb μ 2 + b2 x 2b 0 √  ∞ 2 −μ2 x2 x e π πb 2 2 − eμ b [1 − Φ(bμ)] 2. dx = 2 + b2 x 2μ 2 0  1 x2 ∞  e −1 1 3. dx = 2 x k!(2k − 1) 0 2

25.

 3.467 3.468

 2 e−x −

0



1 1 + x2



2. 3.469



12 0

Re n > 0,



Re b > 0] Re b > 0] Re b > 0] Re b > 0]

[Re a > 0,

Re b > 0]

[Re a > 0,

Re b > 0]

[Re μ > 0,

Re ν > 0]

FI II 645

[Re μ > 0]

ET I 136(24)a

Re b > 0,

|arg μ|
0,

|arg μ|
0]

 K 14

ν2 2μ

NT 33(17)

[Re μ > 0,

a > 0]

NT 19(11)

[Re μ ≥ 0,

ν > 0]

ET I 146(23)



370

Exponential Functions

 2. 3. 3.471

  dx 4 3 = C e−x − e−x x 4 0  ∞  4 2 1 dx = C e−x − e−x x 4 0 

∞

u

1.12 0



u

2.

    β β dx 1 exp − exp − = x x2 β u ν−1

x

μ−1 − β x

(u − x)

e

dx = β

ν−1 2

u

u

3.

u

4. 0





β

x−μ−1 (u − x)μ−1 e− x dx = β −μ uμ−1 Γ(μ) exp −

β β 1 1 x−2μ (u − x)μ−1 e− x dx = √ β 2 −μ e− 2u πu

∞

5. u



∞

6. u



β

β

x−2μ (x − u)μ−1 e x dx =

(see 3.476 2)

BI (89)(6)

ET II 188(22)

β u

π 1 −μ β 2 Γ(μ) exp u



ET II 187(18)

ET II 187(16)

Re μ > 0]

  β 1 − μ − ν; 1 − ν; 1F 1 u [0 < Re μ < Re(1 − ν),

β 2u

u > 0]



[Re μ > 0, u > 0]   β Γ(μ) K u− 12 2u [u > 0, Re β > 0,

xν−1 (x − u)μ−1 e x dx = B(1 − μ − ν, μ)uμ+ν−1



BI (89)(7)

    β β Γ(μ) W 1−2μ−ν , ν exp − 2 2 2u u [Re μ > 0, Re β > 0,

2μ+ν−1 2

0



(see 3.476 2)

[β > 0]

0



3.471



 I μ− 12

β 2u

ET II 187(17)

u > 0] ET II 203(15)



[Re μ > 0,

∞

7.

β

xν−1 (x + γ)μ−1 e− x

0

ET II 202(14) u > 0]   β ν−1 ν−1 β dx = β 2 γ 2 +μ Γ(1 − μ − ν)e 2γ W ν−1 +μ,− ν 2 2 γ [|arg γ| < π, Re(1 − μ) > Re ν > 0] ET II 234(13)a

 8. 0

u

 μ−1 − β 1 x−2μ u2 − x2 e x dx = √ π

  2 β

μ− 12

3

  β u [Re β > 0, u > 0,

uμ− 2 Γ(μ) K μ− 12

Re μ > 0] ET II 188(23)a



∞

9.

β

xν−1 e− x −γx dx = 2

0

 10.12 0

∞

     β K ν 2 βγ γ ν 2

[Re β > 0,

Re γ > 0] ET II 82(23)a, LET I 146(29)

  iνπ β2 iμ ν−1 x− dx = 2β ν e 2 K −ν (βμ) x exp 2 x 

Im μ > 0, Im β 2 μ > 0; note that K −ν ≡ K ν 

EH II 82(24)

3.474

Exponentials of complicated arguments and powers



∞

11.

ν−1

x

0

12.

13.

14. 15. 16.



iμ exp 2

  β2 iνπ (1) x+ dx = iπβ ν e− 2 H −ν (βμ) x Im μ > 0,

371



Im β 2 μ > 0 EH II 21(33)

   μ ν μ2 ν−1 dx = 2 x exp −x − K −ν (μ) 4x 2 0  π |arg μ| < , Re μ2 > 0; note that K −ν ≡ K ν WA 203(15) 2    ∞ ν−1 − β x e x β ν−1 β γ dx = γ [|arg γ| < π, Re β > 0, Re ν < 1] e Γ(1 − ν) Γ ν, x+γ γ 0 

∞



1





ET II 218(19) ν

−x exp 1 − dx = ψ(ν) [Re ν > 0] x(1 − x) 0   ∞ 1 π −2√βγ e x− 2 e−γx−β/x dx = [Re β ≥ 0, γ 0  ∞ √  √ ∂n  1 xn− 2 e−px−q/x dx = (−1)n π n p−1/2 e−2 pq ∂p 0 [Re p > 0, 1 x

BI (80)(7)

Re γ > 0]

ET 245 (5.6.1)

Re q > 0] PBM 344 (2.3.16(2))

3.472



∞

1. 0



∞

2. 0

 3. 4. 5.

   a  √ 1 π −μx2 [exp (−2 aμ) − 1] exp − 2 − 1 e dx = x 2 μ   a 1 x2 exp − 2 − μx2 dx = x 4



[Re μ > 0,

ET I 146(30)

[Re μ > 0,

Re a > 0]

ET I 146(26)

[Re μ > 0,

a > 0]

π √ √ (1 + 2 aμ) exp (−2 aμ) μ3



 dx  a √ 1 π exp (−2 aμ) exp − 2 − μx2 = 2 x x 2 a 0      ∞ aπ dx 1 1 2 x + 2 (1 + a)e−1/a exp − = 4 2a x x 2 0   ∞ π ∂ n −2√pq x−n−1/2 e−px−q/x dx = (−1)n e p ∂q n 0 ∞

Re a > 0]

ET I 146(28)a

[a > 0]

BI (98)(14)

[Re p > 0,

Re q > 0] PBM 344 (2.3.16(3))



∞

3.47312 3.474

exp (−xn ) x(m+1/2)n−1 dx =

0



1. 0

1



xnp n exp (1 − x−n ) − n 1−x 1−x



(2m − 1)!! √ π 2m n

[n > 0,

m = 0, 1, 2, . . .]

BI (98)(6)

  n 1 k−1 dx = ψ p+ x n n k=1

[p > 0]

BI (80)(8)

372

Exponential Functions



 ! dx n exp (1 − x−n ) exp 1 − x1 = − ln n − 1 − xn 1−x x

1

2. 0

3.475 1.

12

2. 3. 3.476

 0



∞

∞

∞

2.

  exp −x2 −

[exp (−νxp ) − exp (−μxp )] [exp (−xp ) − exp (−xq )]

0

1.12 2.12

BI (80)(9)

 C 1 dx =− 2n+1 x 2 1 + x 0   ∞   n dx 1 = −2−n C exp −x2 − 2 1+x x 0  ∞    dx  n = 1 − 2−n C exp −x2 − e−x x 0 

1.

3.477

3.475



[n ∈ Z]

BI (92)(14) BI (92)(13) BI (89)(8)

1 μ dx = ln x p ν

[Re μ > 0,

p−q dx = C x pq

[p > 0,

Re ν > 0]

BI (89)(3)

q > 0]

∞

e−a|x| dx = eau Ei(−au) − e−au Ei(au) [Re a > 0, x − u −∞  ∞ sign x exp (−a|x|) dx = −eau Ei(−au) − e−au Ei(au) x−u −∞

Im u = 0,

BI (89)(9)

arg u = 0]

[a > 0] 3.478



∞

1.

ν−1

x

0

1 ν exp (−μx ) dx = μ− p Γ p p

  ν p

  ∞ ν 1 ν xν−1 [1 − exp (−μxp )] dx = − μ− p Γ |p| p 0 [Re μ > 0 and − p < Re ν < 0 for p > 0,

[Re μ > 0,

 3.11 0

 4. 0

u

Re ν > 0,

0 < Re ν < −p for p < 0]

p > 0]

ET I 313(18, 19)



ν +n−1 ν ν +1 , ,..., ; n n n μ+ν μ+ν +1 μ+ν +n−1 , ,..., ; βun n n n [Re μ > 0, Re ν > 0, n = 2, 3, . . .]

xν−1 (u − x)μ−1 exp (βxn ) dx = B(μ, ν)uμ+ν−1 n F n

∞

ET II 251(36)

BI(81)(8)a, ET I 313(15, 16)

 2.

ν−1

x

 2 exp −βxp − γx−p dx = p

MC

ET II 187(15)

ν   2p    γ K νp 2 βγ β

[Re β > 0,

Re γ > 0]

ET I 313(17)

3.487

3.479

Exponentials of complicated arguments and powers



∞

1. 0

 √   12 −ν xν−1 exp −β 1 + x β 2 √ dx = √ Γ(ν) K 12 −ν (β) π 2 1+x

[Re β > 0, √  √ 1−ν ∞ ν−1 ν  x exp iμ 1 + x2 π μ 2 (1) √ H 1−ν (μ) dx = i Γ 2 2 2 2 2 1+x 0 [Im μ > 0,

 2.11

3.481



∞

1. −∞  ∞

3.482 1.

xex exp (−μex ) dx = −

1 (C + ln μ) μ

 1 xex exp −μe2x dx = − [C + ln(4μ)] 4 −∞

2.



∞

3

exp (nx − β sinh x) dx =

0

 0

∞

3.

π μ

BI (100)(13)

[Re μ > 0]

BI (100)(14)

exp (−νx − β sinh x) dx =

ET I 168(12)

[Re β > 0]

ET I 168(13)

π [Jν (β) − J ν (β)] sin νπ

∞

0

0

3.487

0

ET I 168(11)

[Re β > 0]

for a > 0, for a < 0

qx  px   a a q dx = (ea − 1) ln [p > 0, 1+ 3.484 − 1+ qx px x p 0 π/2  πe [1 − Φ(1)] 3.485 exp − tan2 x dx = 2 0 1  1 ∞  3.4866 x−x dx = e−x ln x dx = k −k = 1.2912859970627 . . .

1.

EH II 83(30)

[Re μ > 0]

⎧   iνπ ⎪ ⎪ 2 exp − K ν (a) ⎨ ∞ exp (ν arcsinh x − iax)  2 √ dx = iνπ ⎪ 1 + x2 −∞ ⎪ K ν (−a) ⎩2 exp 2



Re ν > 0]

1 exp (−nx − β sinh x) dx = (−1)n+1 [S n (β) + π En (β) + π Y n (β)] 2





ET I 313(14)

1 [S n (β) − π En (β) − π Y n (β)] 2

0

3.483



Re ν > 0]

[Re β > 0] ∞

2. 

373

π/4

 exp −

k=1

 ∞   tan2k+1 x k=0

k+

1 2

dx = ln 2

[|Re ν| < 1] ET I 122(32)

q > 0]

BI (89)(34)

FI II 483

374

Hyperbolic Functions

3.511

3.5 Hyperbolic Functions 3.51 Hyperbolic functions 3.511 1.



∞

12 0



2.12 3.12 4.12 5.12 6.12 7.12 8.11 9. 10.12 3.512

 πa  sinh ax π dx = tan sinh x 2 2 0    ∞  sinh ax π πa  a+1 dx = sec −β cosh x 2 2 2 0  ∞   cosh ax π πa dx = sec cosh x 2 2 0  ∞ sinh ax cosh bx π sin πa dx = sinh x 2 cos πa + cos πb 0  ∞ πb sin πa cosh ax cosh bx 2 cos 2 dx = π cosh x cos πa + cos πb 0  ∞ πb sin πa sinh ax sinh bx 2 sin 2 dx = π cosh x cos πa + cos πb 0  ∞ dx =1 2 0 cosh x  ∞ sinh2 ax dx = 1 − aπ cot aπ 2 −∞ sinh x  ∞ πa sinh ax sinh x πa dx = sec 2 2 2 cosh x 0 

∞

∞

1.12 

∞

2.12 0

1. 2.

cosh 2ax (cosh x)

0

3.513

dx π = cosh x 2

μ

2ν

dx = 4ν−1 B (ν + a, ν − a)

sinh x 1 dx = B coshν x 2



μ+1 ν −1 , 2 2

∞

BI (27)(10)a

[|a| < 1]

GW (351)(3b)

[|a| < 1]

BI (4)(14)a

[|a| + |b| < 1]

BI (27)(11)

[|a| + |b| < 1]

BI (27)(5)a

[|a| + |b| < 1]

BI (27)(6)a BI (98)(25)

2

a 0,

a > 0] LI(27)(17)a, EH I 11(26)



√ dx 1 a + b + a2 + b 2 √ =√ ln a2 + b 2 a + b − a2 + b 2 0 a + b sinh x √  ∞ dx b 2 − a2 2 =√ arctan a√ +b b 2 − a2 0 a + b cosh x a + b + a2 − b 2 1 √ ln =√ 2 2 a −b a + b − a2 − b 2



[|a| < 1]

[Re μ > 0,

[ab = 0]

Re(μ − ν) > 0]

EH I 11(23)

GW (351)(8)

2

b > a2 2

b < a2 GW (351)(7)

3.516

Hyperbolic functions



∞

3. 0



∞

4. 0

√ dx b 2 − a2 2 = √ arctan 2 2 a sinh x + b cosh x a√ +b b −a a + b + a2 − b 2 1 √ ln = √ a2 − b 2 a + b − a2 − b 2 dx = a + b cosh x + c sinh x

=

= =

375



b 2 > a2 a2 > b 2

GW (351)(9)

 √ 2 b 2 − a2 − c2 √ + π arctan a+b+c b 2 − a2 − c2 ⎧ ⎤ ⎡ =0 for (b − a)(a + b + c) > 0 ⎪ ⎪ ⎪ ⎨|| = 1 for (b − a)(a + b + c) < 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢when b2 > a2 + c2 ; and ⎪ ⎦ ⎣  = 1 for a < b + c ⎪ ⎪ ⎩  = −1 for a > b + c √ 1 a + b + c + a 2 − b 2 + c2 √ √ ln a2 − b2 + c2 a + b + c − a2 − b2 + c2

b2 < a2 + c2 , a2 = b2 1 a+c ln c a [a = b = 0, c = 0] 2(a − b) c(a − b − c)

2 b = a2 + c2 , c(a − b − c) < 0 

GW (351)(6)

3.514



∞

1. 0



∞

2.12 0



∞

3.

cosh ax dx (cosh x + cos t)

∞

[0 < t < π,

a > 0]

BI (27)(22)a

a(π−t2 ) cosh ax − cos t1 π − t2 π sin b − dx = cos t1 a cosh bx − cos t2 b sin t2 sin b π b sin t2

2

0



dx t = cosec t cosh ax + cos t a

=

sinh ax sinh bx

π (− cos t sin at + a sin t cos at) sin3 t sin aπ

[0 < |a| < b,

0 < t2 < π]

0 < a2 < 1,

0 −1,





∞

ν = −1, −2, −3, . . .]

EH I 157(12)

1

2μ e−iμπ Γ(ν − 2μ + 1) Γ μ + 2 sinh2μ x dx Q μν−μ (b) μ √ ν+1 = √ π (b2 − 1) 2 Γ(ν + 1) b + b2 − 1 cosh x [Re(ν − 2μ + 1) > 0,



4. 0

Re(ν + 1) > 0] EH I 155(2)

3.517



∞

1.

(b + cosh x)

0

 2. 0

3.518





a

cosh γ +

∞

∞

2.10

∞

0

0

=

x dx ν+ 12

− ν Γ(ν + γ + 1) Γ(ν − γ) P −ν π 2 γ (b)  b −1 2 1 2 Γ ν+2 [Re(ν − γ) > 0, 

=

∞

1

sinh2μ+1 x dx

ν+1

Re(ν + γ + 1) > 0] EH I 156(11)



πΓ 2 −ν P νγ (cosh a) 2 sinhν a

sinh2μ x dx

Re ν < 12 ,

a>0

EH I 156(8)

 2μ e−iμπ Γ(ν − 2μ + 1) Γ μ + 12 Q μν−μ (cosh a) = √ π sinhμ a Γ(ν + 1) [Re(ν + 1) > 0, Re(ν − 2μ + 1) > 0, a > 0] EH I 155(3)a

 μ−ν = 2μ b2 − 1 2 Γ(ν − 2μ) Γ(μ + 1) P μ−ν (b) μ

(b + cosh x) [Re(ν − μ) > Re μ > −1,

3. 

2

ν+1

0

4.7

1



(cosh a + sinh a cosh x)

0



ν+ 12

(cosh a − cosh x)

1.



 cosh γ + 12 x dx

b does not lie on the ray (−∞, +1) of the real axis]

sinh2μ−1 x cosh x dx 1  ν = a−μ B(μ, ν − μ) 2 2 1 + a sinh x ν−1

sinhμ−1 x (cosh x + 1) (b + cosh x)

[Re ν > Re μ > 0,

EH I 155(1)

a > 0]

EH I 11(22)

  1 μ,  + 2 − μ − ν = 2μ+ν−ρ B   2 1 1 1 × 2 F 1 ,  + 2 − μ − ν; 2 − μ − ν; − b 2 2 2 [Re μ > 0, Re( − μ − ν) > −2, |arg(1 + b)| < π]

EH I 115(11)

dx

3.522

Hyperbolic and algebraic functions



sinhμ−1 x (cosh x − 1) (b + cosh x)

∞

sinhμ−1 x coshν−1 x  dx = cosh2 x − b

0

 6.

7 0

 3.51912

  μ 1−b = 2−(2−μ−ν+ ) 2 F 1 , 2 − μ − ν + ; 1 +  − ; 2 2  μ × B 2 − μ − ν + , −1 + ν + 2 [b ∈ (−∞, −1) , Re(2 + ) Re(μ + ν), Re(2ν + μ) > 2] EH I 115(10) ν−1

∞

5.6

π/2

0

377

dx

    ν μ μ+ν μ+ν ;1 + − ;b 2B ,1 + − 2 F 1 , 1 +  − 2 2 2 2 [b ∈ (1, ∞) , Re μ > 0, 2 Re(1 + ) > Re(μ + ν)] EH I 115(9) ∞

 1 sinh [(r − p) tan x] pkπ dx = π sin sinh (r tan x) kπ + r r

2

p < r2

BI (274)(13)

k=1

3.52–3.53 Combinations of hyperbolic functions and algebraic functions 3.521 1.



∞

12 0



∞

2. 0



∞

3. 1



∞

1



∞

0 ∞

0



4. 5. 6. 7.

LI III 225(103a), BI(84)(1)a

∞

 dx = −2 Ei[−(2k + 1)a] x sinh ax

[a > 0]

LI (104)(14)

[a > 0]

LI (104)(13)

∞

 dx =2 (−1)k+1 Ei[−(2k + 1)a] x cosh ax

∞

 (−1)k x dx π = +π 2 2 (b + x ) sinh ax 2ab ab + kπ

[a > 0,

x dx 1 = − β(b + 1) (b2 + x2 ) sinh πx 2b

[b > 0]

b > 0]

k=1

2. 3.

π  x dx = 2G = π ln 2 − 4 L cosh x 4

k=0

1. 

GW (352)(2b)

k=0

4. 3.522

x π2 dx = sinh x 4

∞ 2π  dx (−1)k−1 = 2 2 b 2ab + (2k − 1)π 0 (b + x ) cosh ax k=1    ∞ dx 1 1 = β b+ 2 2 b 2 0 (b + x ) cosh πx  ∞ x dx 1 = ln 2 − 2 2 0 (1 + x ) sinh πx  ∞ dx π =2− 2 ) cosh πx (1 + x 2 0  ∞ x dx π −1 πx = 2 2 0 (1 + x ) sinh 2 ∞

[a > 0, [b > 0]

BI(97)(16), GW(352)(8)

b > 0]

BI (97)(5)

BI (97)(4) BI (97)(7) BI (97)(1) BI (97)(8)

378

Hyperbolic Functions



∞

8. 0

 9. 10. 3.523 1.



∞

0



3.12

dx = ln 2 (1 + x2 ) cosh πx 2

BI (97)(2)

√  x dx 1  √ π + 2 ln = 2 + 1 −2 πx 2 2 0 (1 + x ) sinh 4  ∞ √  dx 1  √ π − 2 ln = 2 + 1 πx 2 2 0 (1 + x ) cosh 4 ∞

12

2.12

3.523

xb−1 dx 2b − 1 = b−1 Γ(b) ζ(b) sinh x 2

∞

x2n−1 22n − 1 2n dx = π |B2n | 2n 0 sinh x    ∞ β−1 x 2 1 dx = Γ(β) Φ −1, β, (2a)β 2 0 cosh ax ∞ k  2 (−1) = β Γ(β) β a (2k + 1) k=0



∞

4.12 0

 0

∞

6. 0



∞

7. 0



∞

8. 0



∞

9. 0



∞

10. 0



∞

11. 0

 12. 0

 π 2n+1 2

|E2n |

BI (97)(3)

[Re b > 1]

WH

[n = 1, 2, . . .]

[Re β > 0,

WH, GW(352)(2a)

a > 0]

EH I 35, ET I 322(1)

[n = 0, 1, 2, . . .] BI(84)(12)a, GW(352)(1a)

∞

5. 

x dx = cosh x 2n

BI (97)(9)

x dx π = cosh x 8

(cf. 4.261 6)

BI (84)(3)

x3 dx π4 = sinh x 8

(cf. 4.262 1 and 2)

BI (84)(5)

2

3

x4 dx 5 5 = π cosh x 32

BI (84)(7)

π6 x5 dx = sinh x 4

BI (84)(8)

x6 61 7 dx = π cosh x 128

BI (84)(9)

x7 17 8 dx = π sinh x 16

BI (84)(10)

∞

x1/2 dx √  1 = π (−1)k cosh x (2k + 1)3/2

BI (98)(7)a

k=0

∞

∞

√  (−1)k dx = 2 π x1/2 cosh x (2k + 1)1/2 k=0

BI (98)(25)a

3.524

Hyperbolic and algebraic functions

3.524  1.

∞

xμ−1

0



2.11 3.

sinh βx Γ(μ) dx = sinh γx (2γ)μ

379

       β 1 β 1 1− − ζ μ, 1+ ζ μ, 2 γ 2 γ [Re γ > |Re β|, Re μ > −1] ET I 323(10)

sinh ax π d2m  aπ  dx = [b > |a|] tan 2m sinh bx 2b da 2b 0   ∞ ∞   sinh ax dx 1 1 = Γ(1 − p) − sinh bx xp [b(2k + 1) − a]1−p [b(2k + 1) + a]1−p 0 ∞

x2m

BI (112)(20)a

k=0

[b > |a|, p < 1] BI (131)(2)a   sinh ax π d aπ dx = [b > |a|] sec x2m+1 BI (112)(18)a 2m+1 cosh bx 2b da 2b 0         ∞ cosh βx Γ(μ) β 1 β 1 dx = 1− + ζ μ, 1+ ζ μ, xμ−1 μ sinh γx (2γ) 2 γ 2 γ 0 [Re γ > |Re β|, Re μ > 1] 

4.11 5.

 6. 7.

∞

2m+1

ET I 323(12)

cosh ax π d2m  aπ  dx = [b > |a|] sec 2m cosh bx 2b da 2b 0    ∞ ∞  cosh ax dx 1 1 · p = Γ(1 − p) (−1)k + cosh bx x [b(2k + 1) − a]1−p [b(2k + 1) + a]1−p 0 ∞

x2m

BI(112)(17)

k=0



[b > |a|, ∞

8.

x2m+1

0



9.8 10.

∞

∞

11.

p < 1]

[b > |a|]

x2

x6

0



13.

aπ 

sinh ax aπ π3 aπ dx = 3 sin sec3 [b > |a|] sinh bx 4b 2b 2b 0  ∞ π sinh ax aπ  aπ 5 aπ  dx = 8 sec 2 + sin2 x4 sin sinh bx 2b 2b 2b 2b 0 

12.



cosh ax π d dx = tan sinh bx 2b da2m+1 2b 2m+1

π

aπ 7

sinh ax dx = 16 sec sinh bx 2b 2b

[b > |a|]  aπ aπ  aπ 45 − 30 cos2 + 2 cos4 sin 2b 2b 2b [b > |a|]

∞

aπ π sinh ax aπ dx = 2 sin sec2 [b > |a|] cosh bx 4b 2b 2b 0  ∞ π sinh ax aπ  aπ 4 aπ  dx = sec 6 − cos2 x3 sin cosh bx 2b 2b 2b 2b 0 

14. 0

∞

x

x5

BI(131)(1)a BI (112)(19)a BI (84)(18)

BI (82)(17)a

BI (82)(21)a

2

π

aπ 6

sinh ax dx = sec cosh bx 2b 2b

[b > |a|]  aπ aπ  aπ 120 − 60 cos2 + cos4 sin 2b 2b 2b [b > |a|]

BI (84)(15)a

BI (82)(14)a

BI (82)(18)a

380

Hyperbolic Functions



∞

15.

x7

0

 16. 17. 18.

∞

∞

19. 0



∞

∞

22. 0



∞

23. 0

3.525

π

[b > |a|]

aπ 2

BI (82)(22)a

x

x7

π

aπ 8 

cosh ax dx = 16 sec sinh bx 2b 2b

[b > |a|] 315 − 420 cos2

BI (84)(16)a BI (82)(15)a

BI (82)(19)a

aπ 

aπ aπ + 126 cos4 − 4 cos6 2b 2b 2b

[b > |a|]   cosh ax aπ π aπ dx = 3 2 sec3 − sec [b > |a|] x2 cosh bx 8b 2b 2b 0  ∞ π aπ 5  cosh ax aπ aπ  dx = sec + cos4 24 − 20 cos2 x4 cosh bx 2b 2b 2b 2b 0 

21.

π sinh ax aπ aπ aπ  aπ 8 aπ  dx = sec 5040 − 4200 cos2 + 546 cos4 − cos6 sin cosh bx 2b 2b 2b 2b 2b 2b

cosh ax dx = sec [b > |a|] sinh bx 2b 2b 0  ∞ π cosh ax aπ  aπ 4  dx = 2 sec [b > |a|] 1 + 2 sin2 x3 sinh bx 2b 2b 2b 0  ∞ π cosh ax aπ aπ  aπ 6  dx = 8 sec + 2 cos4 15 − 15 cos2 x5 sinh bx 2b 2b 2b 2b 0 

20.

3.525



∞

1. 0



∞

2. 0



∞

3. 0



∞

4. 0

 5. 0

[b > |a|]   7 aπ cosh ax aπ aπ aπ  dx = sec + 182 cos4 − cos6 720 − 840 cos2 x6 cosh bx 2b 2b 2b 2b 2b

BI (84)(17)a

BI (82)(16)a

π

 aπ

π

sinh ax dx = ln tan + cosh bx x 4b 4

[b > |a|]

BI (82)(20)a

[b > |a|]

BI (95)(3)a

[π ≥ |a|]

BI (97)(10)a

[π ≥ 2|a|]

BI (97)(11)a

sinh ax dx 1 a = − cos a + sin a ln [2 (1 + cos a)] 2 sinh πx 1 + x 2 2 sinh ax dx 1 1 − sin a π = sin a + cos a ln sinh π2 x 1 + x2 2 2 1 + sin a

cosh ax x dx 1 1 = (a sin a − 1) + cos a ln [2 (1 + cos a)] 2 sinh πx 1 + x 2 2 cosh ax x dx 1 1 + sin a π = cos a − 1 + sin a ln sinh π2 x 1 + x2 2 2 1 − sin a

[π > |a|]

BI (97)(12)a

π

BI (97)(13)a

2 ∞

BI(82)(23)a

3

> |a|

sinh ax x dx a+π a π = −2 sin + sin a − cos a ln tan cosh πx 1 + x2 2 2 4 [π > |a|]

GW (352)(12)

3.527

Hyperbolic and algebraic functions



∞

6. 0



∞

7. 0



∞

0



∞

0

GW (352)(11)

[b ≥ |a|]

BI (97)(18)

∞

 cos π cosh ax x dx π b +π = 2 2 sinh bx c + x 2bc bc + kπ

[b > |a|]

BI (97)(19)

k(b−a)

  sinh ax cosh bx dx 1 (a + b + c)π (b + c − a)π = ln tan cot cosh cx x 2 4c 4c [c > |a| + |b|]

∞

2. 0



∞ k(b−a) sinh ax dx π  sin b π = sinh bx c2 + x2 c bc + kπ

[π > |a|]

k=1

1. 

cosh ax dx a+π a π = 2 cos − cos a − sin a ln tan cosh πx 1 + x2 2 2 4

k=1

8. 3.526

381

∞

3. 0

sinh ax dx 1 a = ln sec π sinh bx x 2 b 2

xμ−1 Γ(μ) dx = sinh βx cosh γx (2γ)μ

[b > |2a|]

BI (93)(10)a BI (95)(5)a

       β 1 β 1 1+ + Φ −1, μ, 1− Φ −1, μ, 2 γ 2 γ [Re γ > |Re β|, Re μ > 0] ET I 323(11)

3.527



∞

1.12 0



2.12 3.12 4. 5. 6. 7. 8. 9.

xb−1 dx = 22−b Γ(b) ζ(b − 1) sinh2 x

[Re b > 2]

BI (86)(7)a

∞

x2n dx = π 2n |B2n | [n = 1, 2, . . .] 2 sinh x 0  ∞ b−1  x dx = 22−b 1 − 22−b Γ(b) ζ(b − 1) [Re b > 0, b = 2] 2 0 cosh x  ∞ x dx ln 2 = 2 [a = 0] 2 a cosh ax 0  2m  ∞ 2 − 2 π 2m x2m |B2m | dx = [a > 0, m = 1, 2, . . .] 2 (2a)2m a 0 cosh ax  ∞ ∞ sinh ax (−1)k 2 Γ(μ)  dx = xμ−1 [Re μ > 1, a > 0] aμ (2k + 1)μ−1 cosh2 ax 0 k=0  ∞ x sinh ax π dx = 2 [a > 0] 2 2a cosh ax 0  ∞ sinh ax 2m + 1  π 2m+1 dx = x2m+1 |E2m | [a > 0, m = 0, 1, . . .] 2 a 2a cosh ax 0  ∞ cosh ax 22m+1 − 1 dx = x2m+1 (2m + 1)! ζ(2m + 1) a2 (2a)2m sinh2 ax 0 [a = 0,

m = 1, 2, . . .]

BI(86)(5)a BI (86)(6)a LO III 396

BI(86)(2)a

BI (86)(15)a

BI (86)(8)a BI (86)(12)a

BI (86)(13)a

382

Hyperbolic Functions

 10.

12

11.12 12.12 13.12 14.11 15.10 16.12 3.528

∞

 cosh x dx = 22n − 1 π 2n |B2n | 2 sinh x 0 √  ∞ x sinh x π Γ(μ)  dx = 2μ+1 4μ Γ μ + 12 0 (cosh x)  ∞ 2 x dx π2 = 2 6 0 sinh x  ∞ cosh x π2 dx = x2 2 2 cosh x 0  ∞ sinh x dx = 4G x2 cosh2 x 0  ∞ tanh x2 dx = ln 2 cosh x 0  ∞ cosh ax 2 Γ (μ) ζ (μ − 1)  dx = 1 − 21−μ xμ−1 2 μ a sinh ax 0 

∞

1. 0



∞

2. 0

3.529



∞

1. 0



3. 3.531



∞

0



∞

2.10 0

 3. 4.

(1 + xi)2n − (1 − xi)2n dx = (−1)n+1 2|E2n | + 2 i sinh πx 2 

1 1 − sinh x x



∞

[Re u > 0]

LI (86)(9)

BI (86)(11)a BI (86)(10)a BI (93)(17)a

BI (87)(8)

[n = 0, 1, . . .]

BI (87)(7)

BI (94)(10)a

[b > |a|]

GW (352)(66) BI (94)(11)a

 π  x dx 4 π =√ ln 2 − L = 1.1719536193 . . . 2 cosh x − 1 3 3 3 [see 8.26 for L(x)] x dx t ln 2 − L(t) = cosh 2x + cos 2t sin 2t

t π 2 − t2 x2 dx = 3 sin t 0 cosh x + cos t    ∞ x4 dx t π 2 − t2 7π 2 − 3t2 = 15 sin t 0 cosh x + cos t

BI (86)(14)a

BI (102)(2)a

dx = − ln 2 x

∞

7

[n = 1, 2, . . .]

(1 + xi)2n−1 − (1 − xi)2n−1 dx = 2 i sinh πx 2

cosh ax − 1 dx aπ = − ln cos sinh bx x 2b 0   ∞ b dx a − = (b − a) ln 2 sinh ax sinh bx x 0

2.

1.

x2n

3.528

LI (88)(1) LO III 402

[0 < t < π]

BI (88)(3)a

[0 < t < π]

BI (88)(4)a

3.533

Hyperbolic and algebraic functions



∞

5.3 0



∞

 sin 2kaπ x2m dx = 2(2m)! cosec 2aπ cosh x − cos 2aπ k 2m+1 k=1  = 2 22m−1 − 1 π 2m |B2m |

0

a=

1 2

a =

1 2



xμ−1 dx cosh x − cos t

 i Γ(μ) −it  −it e Φ e , μ, 1 − eit Φ eit , μ, 1 sin t  = 2 − 23−μ Γ(μ) ζ(μ − 1)

[μ = 2,

t = π]

= 2 ln 2

[μ = 2,

t = π]

[Re μ > 0,

=

∞

7. 0





0 < a < 1,

BI (88)(5)a ∞

6.3





383

0 < t < 2π,

t = π] ET I 323(5)

∞

xμ dx sin kt 2 Γ(μ + 1)  = (−1)k−1 μ+1 cosh x + cos t sin t k

[μ > −1,

0 < t < π]

BII (96)(14)a

k=1

u

8. 0

x dx 1 = cosec 2t [L(θ + t) − L(θ − t) − 2 L(t)] cosh 2x − cos 2t 2 [θ = arctan (tanh u cot t) ,

t = nπ] LO III 402

3.532



∞

1.11 0

 2. 0

3.533



k=0

u

∞

0

0

3. 0

b−a b+a

k

     x cosh x dx π t (π − t) = cosec t ln 2 − L −L cosh 2x − cos 2t 2 2 2 [t = mπ]

∞

2.6 



[a > 0, b > 0, n > −1] GW (352)(5)        x cosh x dx 1 θ+t θ−t ψ+t = cosec t L −L +L π− cosh 2x − cos 2t 2  2 2  2    t π−t ψ−t −2L −2L +L 2 2  2  θ u t ψ u t tan = tanh cot , tan = coth cot ; t = nπ LO III 288a 2 2 2 2 2 2

1. 

∞

xn dx 1 2n!  = a cosh x + b sinh x a+b (2k + 1)n+1

∞

x

sinh ax dx (cosh ax − cos t)2

x3

=

π−t cosec t a2 

sinh x dx 2

(cosh x + cos t)

=

t π2 − t sin t

2

[a > 0,

LO III 403

0 < t < π]

(cf. 3.5141) BI (88)(11)a

[0 < t < π]

(cf. 3.531 3) BI (88)(13)

384

Hyperbolic Functions



∞

4.11

x2m+1

0

3.534

∞  sin 2kaπ 2 k 2m+1 (cosh x − cos 2aπ) k=1  = 2(2m + 1) 22m−1 − 1 π 2m |B2m |

sinh x dx

= 2(2m + 1)! cosec 2aπ

0 < a < 1,

a =

1 2



a = 12 BI (88)(14)

3.534

 1 π 1. I 1 (a) 1 − x2 cosh ax dx = WA 94(9) 2a 0  1 cosh ax π √ 2. dx = I 0 (a) WA 94(9) 2 2 1−x 0  1 x π arcsin (tanh a) dx √ = √ [a > 0] 3.535 BI (80)(11) sinh ax sinh a cosh 2a − cosh 2ax 2 2a2 0 3.536  ∞ x2 π2 11 dx = BI (98)(7) 1. 2 12 0 cosh x √   ∞ 2 ∞ x tanh x2 dx π (−1)k √ = 2. BI (98)(8) 2 2 2k + 1 cosh x 0 k=0      ∞ νπ sin μπ 1−μ−ν 1−μ+ν xμ−1 2 sin 2 Γ(μ) Γ ×Γ 3. sinh (ν arcsinh x) √ dx = 2μ π 2 2 1 + x2 0 

νπ cos μπ xμ−1 2 cos 2 cosh (ν arccosh x) √ dx = 2μ π 1 + x2

∞

4. 0

ET I 324(14) [−1 < Re μ < 1 − |Re ν|]     1−μ+ν 1−μ−ν ×Γ Γ(μ) Γ 2 2

[0 < Re μ < 1 − |Re ν|]

ET I 324(15)

3.54 Combinations of hyperbolic functions and exponentials 3.541



∞

1.

e−μx sinhν βx dx =

0



∞

2. 0



e−μx

e−x

4. 0

0

ν μ − ,ν + 1 2β 2

 [Re β > 0,

Re ν > −1, Re μ > Re βν] EH I 11(25), ET I 163(5)

[Re (μ + b ± β) > 0] ∞

−∞  ∞

5.

2ν+1 β

 B

     sinh βx 1 1 μ+β 1 μ−β dx = ψ + −ψ + e−μx sinh bx 2b 2 2b 2 2b

3.



1

∞

sinh μx π μπ dx = tan sinh βx 2β β

sinh ax 1 π aπ dx = − cot sinh x a 2 2

e−px dx (cosh px)

2q+1

=

1 22q−2 B(q, q) − p 2qp

[Re β > 2|Re μ|] [0 < a < 2] [p > 0,

q > 0]

EH I 16(14)a BI (18)(6) BI (4)(3) LI (27)(19)

3.546

Hyperbolic functions and exponentials

 μ+1 e 2 0  ∞ μ 1 − e−μx tanh x dx = β 2 μ 0  ∞ −μx   e μ −1 dx = β 2 2 cosh x 0  ∞ 1 sinh μx dx = (1 − ln 2) e−μx 2 μ cosh μx 0  ∞ sinh px 1 π pπ dx = − cot e−qx sinh qx p 2q 2q 0 

6. 7. 8. 9. 10. 3.542



∞

∞

1. 0



∞

2. 0

−μx

e

−μx

dx =β cosh x

385



e−μx (cosh x − cosh u)ν−1

ET I 163(7)

[Re μ > 0]

ET I 163(9)

[Re μ > 0]

ET I 163(8)

[Re μ > 0]

LI (27)(15)

[0 < p < 2q]

BI (27)(9)a

 μ − ν, 2ν + 1 β   1 Re β > 0, Re ν > − , Re μ > Re βν ET I 163(6) 2  1 1 2 iπν 2 −ν e Γ(ν) sinhν− 2 u Q μ− dx = −i 1 (cosh u) 2 π [Re ν > 0, Re μ > Re ν − 1]

1 (cosh βx − 1) dx = ν B 2 β ν

[Re μ > −1]



EH I 155(4), ET I 164(23)

3.543 1.



∞

 e−ibx dx iπeitb =− cosh πb − e−2itb sinh πb cosh t −∞ sinh x + sinh t



[t > 0] ∞

2. 0

 3. 4.12

e−μx dx = 2 cosec t cosh x − cos t

∞  k=1

sin kt μ+k

[Re μ > −1,

t = 2nπ]

BI (6)(10)a

∞

 cos kt 1 − e−x cos t −(μ−1)x e dx = 2 μ+k 0 cosh x − cos t k=0    ∞ epx + cos t 1 1 2 dx = p t cosec t + 1 + cos t 0 (cosh px + cos t) ∞

ET I 121(30)

[Re μ > 0,

t = 2nπ]

[p > 0]   ∞ exp − n + 12 x  3.544 dx = Q n (cosh u) , 2 (cosh x − cosh u) u 3.545  ∞ sinh ax π aπ 1 dx = cosec − 1. px +1 2p p 2a 0 e  ∞ sinh ax 1 π aπ dx = − cot 2. px − 1 e 2a 2p p 0

BI (6)(9)a

BI (27)(26)a

[u > 0]

EH II 181(33)

[p > a,

p > 0]

BI (27)(3)

[p > a,

p > 0]

BI (27)(9)

386

3.546

Hyperbolic Functions



∞

1.

e

−βx2

0

 2. 3. 4. 3.547

∞

1 sinh ax dx = 2

 

π a2 exp Φ β 4β



a √ 2 β



π a2 exp β 4β 0     ∞ a2 1 π 2 −βx2 exp −1 e sinh ax dx = 4 β β 0     ∞ a2 1 π 2 −βx2 exp +1 e cosh ax dx = 4 β β 0 

∞

1.

2

e−βx cosh ax dx =

1 2

exp (−β sinh x) sinh γx dx =

0

 0

∞

3. 0

 0

∞

5. 0



ET I166(38)a

[Re β > 0]

FI II 720a

[Re β > 0]

ET I 166(40)

[Re β > 0]

ET I 166(41)

γπ π π cot [J γ (β) − Jγ (β)] − [Eγ (β) + Y γ (β)] = γ S −1,γ (β) 2 2 2

exp (−β cosh x) sinh γx sinh x dx = exp (−β sinh x) cosh γx dx =

WA 341(5), ET I 168(14)a

γ K γ (β) β

πγ π π tan [Jγ (β) − J γ (β)] − [Eγ (β) + Y γ (β)] = S 0,γ (β) 2 2 2 [Re β > 0, γ not an integer] ET I 168(16)a, WA 341(4), EH II 84(50)

∞

4. 

[Re β > 0]

[Re β > 0] ∞

2. 

3.547

∞

6.

exp (−β cosh x) cosh γx dx = K γ (β) exp (−β sinh x) sinh γx cosh x dx =

γ S 0,γ (β) β

[Re β > 0]

ET I 168(16)a, WA 201(5)

[Re β > 0]

ET I 168(7), EH II 85(51)

exp (−β sinh x) sinh[(2n + 1)x] cosh x dx = O 2n+1 (β)

0



[Re β > 0] ∞

7. 0



∞

8.

exp (−β sinh x) cosh γx cosh x dx =

exp (−β sinh x) cosh 2nx cosh x dx = O 2n (β)

0



∞

9. 0



10.11

11.

∞

1 S 1,γ (β) β

1 exp (−β cosh x) sinh2ν x dx = √ π

ET I 167(5)

[Re β > 0] [Re β > 0]

 ν   2 1 K ν (β) Γ ν+ β 2 Re β > 0,

ET I 168(6)

Re ν > − 21

1 ν β Γ (μ − ν) W −μ,ν− 12 (4β) 2 0 [Re β > 0, Re μ > Re ν]    ∞     β2 exp − sinh x sinhν−1 x coshν x dx = −π D ν βeiπ/4 D ν βe−iπ/4 2 0

Re ν > 0, |arg β| ≤ π4

EH II 82(20)

exp [−2 (β coth x + μx)] sinh2ν x dx =

EH II 120(10)

3.548

Hyperbolic functions and exponentials

 12.

13.

387

 exp (2νx − 2β sinh x) 1 3  √ π β J ν+ 14 (β) J ν− 14 (β) + Y ν+ 14 (β) Y ν− 14 (β) dx = 2 sinh x 0 EH I 169(20) [Re β > 0]  ∞  exp (−2νx − 2β sinh x) 1 3  √ π β J ν+ 14 (β) Y ν− 14 (β) − J ν− 14 (β) Y ν+ 14 (β) dx = 2 sinh x 0 

∞

∞

14. 0

[Re β > 0]



exp (−2β sinh x) sinh 2νx 1 π 3 β  νπi (1) (2) √ e H 1 +ν (β) H 1 −ν (β) dx = 2 4i 2 sinh x 2 (1) (2) −νπi −e H 1 −ν (β) H 1 +ν (β) 2



∞

15. 0



∞

16. 0



∞

17.8 0

2

[Re β > 0]

ET I 170(24)

exp (−2β sinh x) cosh 2νx 1 π 3 β  νπi (1) (2) √ e H 1 +ν (β) H 1 −ν (β) dx = 2 2 4 2 sinh x  (1) (2) −νπi +e H 1 −ν (β) H 1 +ν (β) 2



ET I 169(21)

exp (−2β cosh x) cosh 2νx √ dx = cosh x



2

[Re β > 0]

ET I 170(25)

β K 1 (β) K ν− 14 (β) π ν+ 4

exp [−2β (cosh x − 1)] cosh 2νx √ dx = cosh x



[Re β > 0]

ET I 170(26)

β 2β e K ν+ 14 (β) K ν− 14 (β) π

ET I 170(27) [Re β > 0]



 ∞ cos ν + 14 π exp (−2νx − 2β sinh x) + sin ν + 14 π exp (2νx − 2β sinh x) √ dx sinh x 0    1 π 3 β J 14 +ν (β) J 14 −ν (β) + Y 14 +ν (β) Y 14 −ν (β) = 2 [Re β > 0] ET I 169(22)    ∞ 1 1 sin ν + 4 π exp (−2νx − 2β sinh x) − cos ν + 4 π exp (2νx − 2β sinh x) √ dx sinh x 0   1 3 π β J 14 +ν (β) Y 14 −ν (β) − J 14 −ν (β) Y 14 +ν (β) = 2 [Re β > 0] ET I 169(23)

 18.

19.

3.548 1. 2.

 2  2 a a a exp I 14 [Re μ > 0, e 2μ 8μ 8μ 0  2  2   ∞ 4 π a a a exp I − 14 e−μx cosh ax2 dx = 4 2μ 8μ 8μ 0



∞

−μx4

π sinh ax dx = 4



2

[Re μ > 0,

a ≥ 0]

ET I 166(42)

a > 0]

ET I 166(43)

388

3.549

Hyperbolic Functions



∞

1.

e−βx sinh [(2n + 1) arcsinh x] dx = O 2n+1 (β)

3.549

[Re β > 0]

(cf. 3.547 6)

0



ET I 167(5) ∞

2.

e−βx cosh (2n arcsinh x) dx = O 2n (β)

[Re β > 0]

(cf. 3.547 8)

0



ET I 168(6) ∞

3. 0



∞

4.

e−βx sinh (ν arcsinh x) dx =

ν S 0,ν (β) β

[Re β > 0]

(cf. 3.5475)

e−βx cosh (ν arcsinh x) dx =

1 S 1,ν (β) β

[Re β > 0]

(cf. 3.547 7)

0

ET I 168(7)

A number of other integrals containing hyperbolic functions and exponentials, depending on arcsinh x or arccosh x can be found by first making the substitution x = sinh t or x = cosh t.

3.55–3.56 Combinations of hyperbolic functions, exponentials, and powers 3.551



∞

1.12

xμ−1 e−βx sinh γx dx =

0



∞

xμ−1 e−βx cosh γx dx =

0

∞

3. 0

 4.



1 Γ(μ) (β − γ)−μ + (β + γ)−μ 2 [Re μ > 0,

5.11 0

 6.

xn e−(p+mq)x sinhm qx dx = 2−m n!



1

m   m k=0

1 −βx

e

x

k

(−1)k (p + 2kq)n+1 [p > 0,

Re β > 0]

q > 0,

  1 β+γ sinh γx dx = ln + Ei(γ − β) − Ei(−γ − β) 2 β−γ [β > γ]

∞ −βx

e

x

0

7.

ET I 164(19)

   β μ−1 −βx 1−μ −μ −β x e coth x dx = Γ(μ) 2 ζ μ, 2 [Re μ > 1,

∞

Re β > |Re γ|]



0



Re β > |Re γ|] ET I 164(18)

2.





1 Γ(μ) (β − γ)−μ − (β + γ)−μ 2 [Re μ > −1,

∞ −βx

e

x

sinh γx dx =

1 β+γ ln 2 β−γ

cosh γx dx =

1 [− Ei(γ − β) − Ei(−γ − β)] 2

ET I 164(21)

m < p + qm] LI (81)(4)

BI (80)(4)

[Re β > |Re γ|]

ET I 163(12)

[Re β > |Re γ|]

ET I 164(15)

3.554

Hyperbolic functions, exponentials, and powers

 8.

∞

6

xe−x coth x dx =

0



∞

9. 0



∞

10.6

π2 −1 4

0

3.552 1. 2. 3.

BI (82)(6)

β dx = ln + 2 ln e−βx tanh x x 4 xe−x coth(x/2) dx =

  Γ β4   1 β + Γ 4 2



∞

ET I 164(16)

[Re μ > 1, [a > 0,

Re β > −1]

m = 1, 2, . . .]

[Re μ > 0,

ET I 164(20) EH I 38(24)a

μ = 1]

[if μ = 1] EH I 32(5)

∞

4. 0



[Re β > 0]

π2 −1 3

  xμ−1 e−βx 1 dx = 21−μ Γ(μ) ζ μ, (β + 1) sinh x 2 0  ∞ 2m−1 −ax   1 x e π 2m dx = |B2m | sinh ax 2m a 0  ∞ μ−1 −x  x e dx = 21−μ 1 − 21−μ Γ(μ) ζ(μ) cosh x 0 = ln 2 

389

∞

5. 0

2m−1 −ax

x

e cosh ax

1−2 2m

1−2m

dx =

|B2m |

 π 2m a

[a > 0,

m = 1, 2, . . .]

EH I 39(25)a

∞

 x2 e−2nx 1 dx = 4 sinh x (2k + 1)3

[n = 0, 1, 2, . . .]

(cf. 4.261 13)

k=n

BI(84)(4)



∞

6.11 0

n

 x3 e−2nx 1 π4 dx = − 12 sinh x 8 (2k − 1)4

[n = 0, 1, . . .]

(cf. 4.262 6)

k=1

BI (84)(6)

3.553



∞

1. 0

 2.

∞

11 0

3.554

2.

[a < 1]

BI (95)(7)

sinh2 x2 e−x dx 1 4 = ln cosh x x 2 π

(cf. 4.267 2)

BI (95)(4)

  Γ β+3 4 β dx  − ln = 2 ln  e−βx (1 − sech x) β+1 x 4 0 Γ 4      ∞ β+1 β 1 −βx − cosech x dx = ψ − ln e x 2 2 0 

1.11

sinh2 ax e−x dx 1 = ln (aπ cosec aπ) sinh x x 2

∞

[Re β > 0]

ET I 164(17)

[Re β > 0]

ET I 163(10)

390

Hyperbolic Functions





∞

3. 0

3.555

  sinh 12 − β x dx −x = 2 ln Γ(β) − ln π + ln (sin πβ) − (1 − 2β)e sinh x2 x

[0 < Re β < 1]    β β 1 1 − coth x dx = ψ − ln + [Re β > 0] e−βx x 2 2 β 0     ∞ 1 sinh qx dx −x = 2 ln Γ q + + ln cos πq − ln π − + 2qe sinh x2 x 2 0 2

q < 12    ∞ β xμ−1 e−βx (coth x − 1) dx = 21−μ Γ(μ) ζ μ, + 1 2 0 

4. 5.

6.

[Re β > 0, 3.555 1.







∞

1 sinh2 (ax) dx = ln x e −1 x 4

∞

sinh ax dx 1 = − ln (aπ cot aπ) x e +1 x 4

12 0

EH I 21(7)



∞

2πa sin 2πa

ET I 163(11)

WH

Re μ > 1]



0 < |a|< 12

ET I 164(22)

(cf. 3.545 2) BI (93)(15)

 2. 0

2



0 < |a|< 12

(cf. 3.545 1) BI (93)(9)

3.556



∞

1. −∞



∞

2. 0

3.557



1. 0

∞

x

pπ π2 1 − epx dx = − tan2 sinh x 2 2

1 − e−px 1 − e−(p+1)x dx = 2p ln 2 sinh x x

[p < 1]

(cf. 4.255 3)

[p > −1]

BI (101)(4)

BI (95)(8)

e−px − e−qx dx cosh x − cos m nπ x

     Γ n+q+k Γ p+k   m  n−1  2n 2n km    π π ln  (−1)k−1 sin = 2 cosec q+k n n Γ Γ n+p+k k=1  2n   2n  n−1  Γ n+q−k Γ p+k  2 m   n n km    π π ln  = 2 cosec (−1)k−1 sin q+k n n Γ Γ n+p−k k=1 n n [p > −1,

q > −1]

[m + n odd]

[m + n even] BI (96)(1)

3.558

Hyperbolic functions, exponentials, and powers

 2.

∞

391

2

(1 − e−x ) dx m cosh x + cos π x 0 n  n+k+1 2  k+2  k    m  n−1  Γ Γ 2n Γ 2n km k−1 π π × ln  2n 2  (−1) sin = 2 cosec  k+1 n+k n n Γ 2n Γ 2n Γ n+k+2 k=1 2n n−1  n−k+1 2  k+2  k   2 m   Γ Γ n Γ n km k−1 n = 2 cosec π π × ln  (−1) sin  n−k  n−k+2 k+1 2 n n Γ n Γ n Γ k=1 n

   Γ p+n+k  n−1 m   2n km   π ln(2n) + 2 π ln (−1)k−1 sin = tan p+k 2n n Γ k=1 2n   n−1 p+n−k   2 m  Γ  n km   π ln n + 2 π ln = tan (−1)k−1 sin p+k 2n n Γ k=1



∞

4. 0

1 + e−x dx a = 2 sec Γ(p) cosh x + cos a x1−p 2

[m + n odd]

[m + n even]

n



(−1)k−1

k=1

cos k − kp

1 2

BI (96)(3)

a

[p > 0] ∞ 1 ∞ q x e cosh x2 Γ(q + 1)  k−1 cos k − 2 λ dx = (−1) k q+1 cos λ2 0 cosh x + cos λ

 5.

∞ 

[m + n even] BI (96)(2)

 ∞ e−px sin m m dx nπ π− e−x tan m 2n cosh x + cos π x 0 n

 3.

[m + n odd]



−x 2

LI (96)(5)

k=1



[q > −1] ∞

6.

x

0



∞

7.

a2 π2 e−x − cos a dx = |a|π − − cosh x − cos a 2 3



∞

0 ∞

2. 0

 3.

4.

 cos kaπ e−x − cos aπ dx = 2 · (2m + 1)! cosh x − cos aπ k 2m+2

BI (88)(6)

k=1

1. 

BI (88)(8) ∞

x2m+1

0

3.558

LI (96)(5)a

x

n−1  n−k 1 − e−nx 2nπ 2 dx = − 4 2 x 3 k2 sinh 2 k=1

BI (85)(3)

n−1  n−k 1 − (−1)n e−nx nπ 2 +4 x dx = (−1)k 2 2 x 3 k cosh 2 k=1

n−1  n−k 1 − e−nx dx = 8n ζ(3) − 8 x k3 0 sinh2 k=1 2  ∞ ∞ n−1   n−k 1 − e−2nx 1 dx = 8n x2 ex − 8 2 3 (2k − 1) (2k − 1)3 sinh x 0 ∞

x2

k=1

k=1

LI (85)(1)

BI (85)(5)

LI (85)(6)

392

Trigonometric Functions



∞

5.

x2

n−1  n−k 1 + (−1)n e−nx dx = 6n ζ(3) − 8 2 x k3 cosh 2 k=1

LI (85)(4)

x3

n−1  n−k 1 − e−nx 4 4 nπ dx = − 24 15 k4 sinh2 x2 k=1

BI (85)(9)

x3

n−1  1 + (−1)n e−nx n−k 7 4 nπ dx = + 24 (−1)k 4 2 x 30 k cosh 2 k=1

BI (85)(8)

0



∞

6. 0



∞

7. 0

 3.559

∞

 e

−x

0

 3.561 3.562 1.

2.

3.559

1 (1 − e−x ) (1 − ax) − xe−x (2−a)x a− + e 2 4 sinh2 x2



1 1 dx = a − + ln Γ(a) − ln(2π) x 2 2

[a > 0] BI (96)(6)

tanh x2 π dx = 2 ln √ x cosh x 2 2

∞ −2x

e

0

BI (93)(18)



    γ γ D −2μ − √ − D −2μ √ 2β 2β 0

1 ET I 166(44) Re μ > − 2 , Re β > 0      2  ∞ 2 γ γ 1 γ D −2μ − √ + D −2μ √ x2μ−1 e−βx cosh γx dx = Γ(2μ)(2β)−μ exp 2 8β 2β 2β 0



∞



∞

3.

2

x2μ−1 e−βx sinh γx dx =

2

xe−βx sinh γx dx =

0

 4.

∞

xe

0

−βx2

γ 4β

γ cosh γx dx = 4β

1 Γ(2μ)(2β)−μ exp 2

 

π exp β π exp β

 

γ 4β 2

γ2 4β



γ2 8β

[Re μ > 0,



[Re β > 0] 

 Φ

γ √ 2 β

 +

Re β > 0]

ET I 166(45)

BI(81)(12)a,ET I 165(34)

1 2β

[Re β > 0]  √     ∞ π 2β + γ 2 γ γ2 γ 2 −βx2 √ √ Φ exp − 2 x e sinh γx dx = 2 4β 4β 8β β 2 β 0

ET I 166(35)

 5.12

√   2 ∞ π 2β + γ 2 γ 2 −βx2 √ exp x e cosh γx dx = 2 4β 8β β 0

 6.

[Re β > 0]

ET I 166(36)

[Re β > 0]

ET I 166(37)

3.6–4.1 Trigonometric Functions 3.61 Rational functions of sines and cosines and trigonometric functions of multiple angles 3.611



2π

1. 0

(1 − cos x)n sin nx dx = 0

BI (68)(10)

3.613

Rational functions of sines and cosines



2π

2. 0



π

3.

n

(1 − cos x) cos nx dx = (−1)n 

n

(cos t + i sin t cos x) dx =

0

3.612



π

BI (68)(11)

2n−1 −n−1

(cos t + i sin t cos x)

dx = π P n (cos t)

EH I 158(23)a

0 π

1.12 0



π

393

sin nx cos mx dx = 0 sin x =π

for n > m > 0,

if m + n is odd and positive

=0

for n > m > 0,

if m + n is even

for 0 < n ≤ m;

LI (64)(3) π

2.12 0

sin nx dx = 0 sin x =π

[n > 0 and even] [n > 0 and odd] BI (64)(1, 2)

 3.12

π/2

sin(2n − 1)x π dx = [n > 0] sin x 2    π/2 sin 2nx 1 1 (−1)n−1 dx = 2 1 − + − · · · + [n > 0] sin x 3 5 2n − 1 0    π/2  π sin 2nx sin 2nx (−1)n−1 1 1 n−1 dx = 2 dx = (−1) 4 1 − + − ··· + cos x cos x 3 5 2n − 1 0 0

FI II 145

0

4.12 5.12



π

6.12 0



π/2

7.12 0

3.613



π

1.6 0

 2.

π

6 0

 3. 0

cos(2n + 1)x dx = 2 cos x

 0

π 2

[n > 0]

GW (332)(22a)

cos(2n + 1)x dx = (−1)n π [n ≥ 0] cos x

GW (332)(22b)

[n > 0]

LI (45)(17)

sin 2nx cos x π dx = sin x 2

cos nx dx π =√ 1 + a cos x 1 − a2

GW (332)(21b)

n √ 1 − a2 − 1 a

cos nx dx πan = 1 − 2a cos x + a2 1 − a2 π = 2 (a − 1) an

2 a < 1, 2 a < 1, 2 a > 1,

n≥0 n≥0 n≥0

BI (64)(12)

BI (65)(3)

π

sin nx sin x dx π = an−1 1 − 2a cos x + a2 2 π = n+1 2a

2 a < 1, 2 a > 1,

n≥1 n≥1

BI(65)(4), GW(332)(34a)

394

Trigonometric Functions

 4.

π

10 0

 5. 6. 7.

π

π 1 + a2 n−1 cos nx cos x dx = a 1 − 2a cos x + a2 2 1 − a2 2 π a +1 = n+1 2 2a a −1 πa = 1 − a2 π = a (a2 − 1) 

π

8. 0

 0

0

π

11.6 0



n−1

sin(2n − 1)x sin x dx π a = · 1 − 2a cos 2x + a2 2 1+a 1 π = · 2 (1 + a)an n−1

cos(2n − 1)x cos x dx π a = · 1 − 2a cos 2x + a2 2 1−a 1 π = · 2 (a − 1)an

π

sin nx − a sin(n − 1)x sin mx dx = 0 1 − 2a cos x + a2 π = am−n 2  cos nx − a cos(n − 1)x π  |m|−n a cos mx dx = −1 2 1 − 2a cos x + a 2

sin nx − a sin[(n + 1)x] dx = 0 1 − 2a cos x + a2 0  π cos nx − a cos[(n + 1)x] 13. dx = 2πan 1 − 2a cos x + a2 0  π sin x sin px · dx 7 3.614 · 2 2 1 − 2ap cos px + a2p 0 a − 2ab cos x + b πbp−1 = p+1 2a (1 − bp ) p−1 πa = 2b (bp − a2p ) 12.

2

a > 1, n ≥ 1

n = 0, a2 < 1

n = 0, a2 > 1 2

a = 1

BI (65)(9, 10)

2

a = 1

BI (65)(12)

2

a = 1

BI (65)(6, 7)

2

a 1 2

a 1 BI (65)(11)

π

10.



n≥1

BI (65)(8) π

9.



2 a < 1,

BI(65)(5), GW(332)(34b) π

cos(2n − 1)x dx cos 2nx cos x dx = =0 2 2 0 1 − 2a cos 2x + a 0 1 − 2a cos 2x + a  π cos(2n − 1)x cos 2x dx =0 1 − 2a cos 2x + a2 0  π  π sin 2nx sin x dx sin(2n − 1)x sin 2x dx = =0 2 1 − 2a cos 2x + a2 0 1 − 2a cos 2x + a 0 

3.614

for m < n for m ≥ n 2

a 0]

3.623

Powers of trigonometric functions

397

3.62 Powers of trigonometric functions 3.621



π/2

1.

sinμ−1 x dx =

0



π/2

2.12



π/2

3.





π/2



sin2m+1 x dx =

0



π/2

0

0 π/2

cosμ−1 x dx = 2μ−2 B

0

sin2m x dx =

0

4.

π/2

sin3/2 x dx =

0





π/2

FI II 789

 1 cos3/2 x dx = √ Γ 14 6 2π cos2m x dx =

π/2

(2m − 1)!! π (2m)!! 2

cos2m+1 x dx =

0

5.

μ μ , 2 2

sinμ−1 x cosν−1 x dx =

0

FI II 151

(2m)!! (2m + 1)!!

1 μ ν  B , 2 2 2

FI II 151

[Re μ > 0,

LO V 113(50), LO V 122, FI II 788



6. 7. 3.622 1. 2. 3.

  2  π/2 √ 2 3 Γ sin x dx = π 4 0   1 2  π/2 Γ 4 dx √ √ = 2 2π sin x 0 

π/2

μπ π sec 2 2 0    π/4 μ+1 1 tanμ x dx = β 2 2 0  π/4 n−1 π  (−1)k tan2n x dx = (−1)n + 4 2n − 2k − 1 0 

tan±μ x dx =

π/4

tan2n+1 x dx = (−1)n

0



π/2

π/4



tanμ x sin2 x dx =

1+μ β 4

tanμ x cos2 x dx =

1−μ β 4

0

0

BI (34)(1)

BI (34)(2)

n−1

π/2

BI (34)(3)

μ 1 μ B ,ν − 2 2 2 [0 < Re μ < 2 Re ν] BI(42)(6), BI(45)(22)

cotμ−1 x sin2ν−2 x dx =

0

2.6 

[Re μ > −1]

ln 2  (−1)k + 2 2n − 2k

tanμ−1 x cos2ν−2 x dx =

0

3.6

BI (42)(1)

k=0

1.



[|Re μ| < 1]

k=0

4.11 3.623

Re ν > 0]

π/4

 

μ+1 2 μ+1 2

 −

1 4

[Re μ > −1]

BI (34)(4)

+

1 4

[Re μ > −1]

BI (34)(5)



398

3.624 1. 2.12

3.12 4.8 5.

6.6 3.625

Trigonometric Functions



π/4



π/4

sinp x 1 dx = [p > −1] GW (331)(34b) p+2 cos x p+1 0  π/2  π/2 1 1  π sinμ− 2 x cosμ− 2 x 21/2−μ √ dx = (1 + 2μ) Γ(1 − μ)Γ(μ − 12 ) cos dx = − cos2μ−1 x 4 π sin2μ−1 x 0 0 1

− 2 < Re μ < 1 LI (55)(12)  π/4 1 cosn− 2 (2x) (2n)! dx = π BI (38)(3) 2n+1 2n+1 cos (x) 2 (n!)2 0  π/4 cosμ 2x dx = 22μ B(μ + 1, μ + 1) [Re μ > −1] BI (35)(1) 2(μ+1) x cos 0   π/4 2μ−2 Γ μ − 12 Γ(1 − μ) sin x 1−2μ √ dx = 2 B(2μ − 1, 1 − μ) = cosμ 2x 2 π 0

1 BI (35)(4) 2 < Re μ < 1 2  π/2  aπ 1 sin ax − sin πa [2a β(a) − 1] [a > 0] dx = sin x 2 2 0

1. 0



π/4

2. 0



π/4

3. 0



π/4

4.8 0

3.626

3.624



π/4

1. 0



π/4

2. 0

 3.627

3.62811 0

sin2n x cosp 2x dx = cos2p+2n+2 x

π 2

1 2

 B n + 12 , p + 1

[p > −1]

(cf. 3.251 1)

BI (35)(2) BI (35)(3)

1

sin2n−1 x cosm− 2 2x (2n − 2)!!(2m − 1)!! dx = cos2n+2m x (2n + 2m − 1)!!

BI (38)(6)

1

sin2n x cosm− 2 2x (2n − 1)!!(2m − 1)!! π dx = · cos2n+2m+1 x (2n + 2m)!! 2

BI (38)(7)

sin2n−1 x √ (2n − 2)!! cos 2x dx = 2n+2 cos x (2n + 1)!!

(cf. 3.251 1)

BI (38)(4)

sin2n x √ (2n − 1)!! π cos 2x dx = 2n+3 cos x (2n + 2)!! 2

(cf. 3.251 1)

BI (38)(5)

 Γ(μ) Γ 12 − μ cotμ x μπ √ dx = sin μ μ sin x 2 π 0 2

−1 < Re μ < 12 

1 sec2p x sin2p−1 x dx = √ Γ(p) Γ 12 − p 0 < p < 12 2 π

π/2

0



sin2n−1 x cosp 2x (n − 1)! Γ(p + 1) dx = cos2p+2n+1 x 2 Γ(p + n + 1) 1 (n − 1)! = B(n, p + 1) = 2(p + n)(p + n − 1) · · · (p + 1) 2 [p > −1] (cf. 3.251 1)

tanμ x dx = cosμ x



π/2

BI (55)(12)a WA 691

3.631

Powers of trigonometric functions

399

3.63 Powers of trigonometric functions and trigonometric functions of linear functions 3.631  1.

π

sinν−1 x sin ax dx =

0



π/2

2.12 0



π

3.6 0



π

4.

2ν−1 ν B

sinν−2 x sin νx dx =

νπ 1 cos 1−ν 2

sinν x sin νx dx = 2−ν π sin



π

νπ 2

[Re ν > 0]

LO V 121(67a), WA 337a

[Re ν > 1]

GW(332)(16d), FI I 152

[Re ν > −1]

LO V 121(69)

sinn x sin 2mx dx = 0

GW (332)(11a)

0

5.

π sin aπ 2  ν +a+1 ν −a+1 , 2 2



 sin2n x sin(2m + 1)x dx =

0

π/2

sin2n x sin(2m + 1)x dx

0

(−1)m 2n+1 n!(2n − 1)!! (2n − 2m − 1)!!(2m + 2n + 1)!! (−1)n 2n+1 n!(2m − 2n − 1)!!(2n − 1)!! = (2m + 2n + 1)!! =

[m ≤ n] ∗ [m ≥ n] ∗ GW (332)(11b)



π

6.



π/2

sin2n+1 x sin(2m + 1)x dx = 2

sin2n+1 x sin(2m + 1)x dx   (−1)m π 2n + 1 = 2n+1 2 n−m

0

0

[n < m]

=0 

BI(40)(12), GW(332)(11c) π

7. 0

 8.

π

sinn x cos(2m + 1)x dx = 0 sinν−1 x cos ax dx =

0



2ν−1 ν B

π/2

cosν−1 x cos ax dx =

0

10. 0 ∗ In

GW (332)(12a)



π cos aπ 2  ν+a+1 ν −a+1 , 2 2 [Re ν > 0]

9.



[n ≥ m]

π/2

sinν−2 x cos νx dx =

2ν ν B



LO V 121(68)a, WA 337a

π  ν +a+1 ν −a+1 , 2 2

νπ 1 sin ν−1 2

3.631.5, for m = n we should set (2n − 2m − 1)!! = 1

[Re ν > 0]

GW (332)(9c)

[Re ν > 1]

GW(332)(16b), FI II 15 2

400

Trigonometric Functions

 11. 12.12

π

3.632

π νπ [Re ν > −1] cos ν 2 2 0    π  π/2 2n (−1)m sin2n x cos 2mx dx = 2 sin2n x cos 2mx dx = 22n n−m 0 0 sinν x cos νx dx =

[n ≥ m] [n < m]

=0 

LO V 121(70)a

BI(40)(16), GW(332)(12b) π

13.12 0

sin2n+1 x cos 2mx dx  π/2 sin2n+1 x cos 2mx dx = =2

(−1)m 2n+1 n!(2n + 1)!! (2n − 2m + 1)!!(2m + 2n + 1)!! 0 (−1)n+1 2n+1 n!(2m − 2n − 3)!!(2n + 1)!! = (2m + 2n + 1)!!

[n ≥ m − 1] [n < m − 1] GW (332)(12c)



π/2

14.

1 ν−1

cosν−2 x sin νx dx =

0



π

15.

cos x sin nx dx = 1 − (−1)m+n m

0

⎢ ⎣r = π/2

16.

cosn x sin nx dx =

0



π

17.11

cosn x sin mx dx =

0



π

18.6 0





π/2

GW(332)(16c), FI II 152

cosm x sin nx dx

! m! (m + n − 2k − 2)!! m!(n − m − 2)!! +s (m − k)! (m + n)!! (m + n)!! k=0 ⎧ ⎤ ⎪ ⎨2 if n − m = 4l + 2 > 0 m if m ≤ n ⎥ s = 1 if n − m = 2l + 1 > 0 ⎦ GW (332)(13a) ⎪ n if m ≥ n ⎩ 0 if n − m = 4l or n − m < 0

= 1 − (−1)m+n ⎡



[Re ν > 1]

0 r−1 

n  2k

1 2n+1

k=1

1 + (−1)m+n 0

FI II 153

k π

n

2n+1

k

if m ≤ n and n − m = 2k otherwise

  a+m a+m (−1) sin aπ m ;1− ; −1 cos x cos ax dx = m 2 F 1 −m, − 2 (m + a) 2 2

GW (332)(15a)

m

[a = 0, ±1, ±2, . . .] π/2

19.

cosν−2 x cos νx dx = 0

WA 313

[Re ν > 1]

GW(332)(16a), FI II 152

[Re n > −1]

LO V 122(78), FI II 153

0



π/2

20.10 0

3.632



1. 0

π

cosn x cos nx dx =

π 2n+1

 p+a    π Γ 2 Γ p−a 2 − x dx = 2p−1 Γ(p) sinp−1 x cos a 2 Γ(p − a) Γ(p + a) 2

p < a2

BI (62)(11)

3.634

Powers of trigonometric functions

 2.

3.10

π 2

−π 2

  π  dx = cosp−1 x sin a x + 2



2p−1 p B

401

π sin aπ 2  p+a+1 p−a+1 , 2 2

[Re p > 0]  π/2 n−1  (−1)k 2k  n − 1  cosp x sin[(p + 2n)x] dx = (−1)n−1 p+k+1 k 0

WA 337a

k=0



π

4. −π

cosn−1 x cos[m(x − a)] dx = 1 − (−1)n+m = =



π/2

5.

cosp+q−2 x cos[(p − q)x] dx =

0



[n > 0] π 2

−π 2

LI (41)(12)

cosn−1 x cos[m(x − a)] dx

[1 − (−1)n+m ] π cos ma   n+m+1 n−m+1 , 2n−1 n B 2 2 [n ≥ m] LO V 123(80), LO V 139(94a)

π 2p+q−1 (p + q − 1) B(p, q) [p + q > 1]

3.633



π/2

1. 0



π/2

2.

cosp−1 x sin ax sin x dx =

2p+1 p(p + 1) B

cosn x sin nx sin 2mx dx =

0





π/2

π/2

cosn−1 x cos[(n + 1)x] cos 2mx dx =

0

π/2

4.

cosp+q x cos px cos qx dx =

0



π/2

5.6 0

cosp+q x sin px sin qx dx =

aπ  p−a p+a + 1, +1 2 2

cosn x cos nx cos 2mx dx =

0

3. 





π 2n+1

WH

n−1 m−1



LO V 150(110)

π

n

2n+2

m

[n > m − 1]



BI (42)(21)

1 π 1+ 2p+q+2 (p + q + 1) B(p + 1, q + 1) π 2p+q+2

∞   k=1

BI (42)(19, 20)



GW (332)(10c) [p + q > −1]   Γ(p + q + 1) p  q  π −1 = p+q+2 2 Γ(p + 1) Γ(q + 1) k k

[p + q > −1] 3.634



1. 0

π/2

sinμ−1 x cosν−1 x sin(μ + ν)x dx = sin

μπ B(μ, ν) 2

[Re μ > 0,

BI (42)(16)

Re ν > 0] BI(42)(23), FI II 814a

402

Trigonometric Functions



π/2

2.

sinμ−1 x cosν−1 x cos(μ + ν)x dx = cos

0

3.635

μπ B(μ, ν) 2

[Re μ > 0,

Re ν > 0] BI(42)(24), FI II 814a



π/2

3.

cosp+n−1 x sin px cos[(n + 1)x] sin x dx =

0

π 2p+n+1

Γ(p + n) n! Γ(p) [p > −n]

3.635 1. 2.12

3. 3.636

    μ  μ+1 1 ψ −ψ [Re μ > 0] cos 2x tan x dx = 4 2 2 0  π/2 ∞    n Γ(p + n − k) π cosp+2n x sin px tan x dx = p+2n+1 2 Γ(p) (n − k)! k 0 k=0 Γ(p + 2n) pπ = p+2n+1 2 Γ(n + 1) Γ(p + n + 1) [p > −2n]  π/2 π cosn−1 x sin[(n + 1)x] cot x dx = 2 0





π/4

π/2

1.

μ−1

tan±μ x sin 2x dx =

0



π/2

2.

tan±μ x cos 2x dx = ∓

0

 3.

π/2

11 0

3.637



π/2

1.

μπ μπ cosec 2 2

tan2μ x dx = cos x

 0

π/2

μπ μπ sec 2 2

tanp x sinq−2 x sin qx dx = − cos

tanp x sinq−2 x cos qx dx = sin

0

3. 0

cotp x cosq−2 x sin qx dx = cos

BI (42)(22) BI (45)(18)

BI (45)(20)a

BI (45)(21)

(cf. 3.251 1) BI (45)(13, 14)

GW (332)(15d)

(p + q)π B(p + q − 1, 1 − p) 2 [p + q > 1 > p]

π/2

BI (34)(7)

(p + q)π B(p + q − 1, 1 − p) 2 [p + q > 1 > p]

π/2

2. 

[|Re μ| < 1]

 Γ μ + 12 Γ(−μ) cot2μ x √ dx = sin x 2 π 1

− 2 < Re μ < 1

0



[0 < Re μ < 2]

BI (42)(15)

GW (332)(15b)

pπ B(p + q − 1, 1 − p) 2 [p + q > 1 > p]

GW (332)(15c)

3.642

Trigonometric functions: powers and rational functions



π/2

4.

cotp x cosq−2 x cos qx dx = sin

0

403

pπ B(p + q − 1, 1 − p) 2 [p + q > 1 > p]

3.638



π/4

1.

1 cosμ+ 2

0



π/4

2. 0



sin2μ x dx

π/2

3. 0

2x cos x

=



|Re μ| < 12

π sec μπ 2

(cf. 3.192 2) BI (38)(8)

1



μ− 12

GW (332)(15a)

sin 2x dx 2 Γ μ + 2 Γ(1 − μ) √ = sin cosμ 2x cos x 2μ − 1 π



cosp−1 x sin px π dx = sin x 2



2μ − 1 π 4 1

− 2 < Re μ < 1 [p > 0]

BI (38)(17) GW(332)(17), BI(45)(5)

3.64–3.65 Powers and rational functions of trigonometric functions 3.641



π/2

1. 0



π/2

2.

sinp−1 x cos−p x dx = a cos x + b sin x 1−p

sin

(sin x + cos x)

0

3

π/2

0



p

x cos x



dx = 0

π cosec pπ sin−p x cosp−1 x dx = a sin x + b cos x a1−p bp [ab > 0,

π/2

p

sin x cos

1−p

x 3

(sin x + cos x)

dx =

0 < p < 1]

(1 − p)p π cosec pπ 2 [−1 < p < 2]

3.642 1.

2. 3.



GW (331)(62)

BI(48)(5)

π/2

sin2μ−1 x cos2ν−1 x dx 1 [Re μ > 0, Re ν > 0] BI (48)(28) μ+ν = 2μ 2ν B(μ, ν) 2 2a b 0 a2 sin x + b2 cos2 x   π/2 B n2 , n2 sinn−1 x cosn−1 x dx  [ab > 0] GW (331)(59a) n = 2(ab)n a2 cos2 x + b2 sin2 x 0   π/2 sin2n x dx sin2n x dx 1 π =   n+1 2 0 a2 cos2 x + b2 sin2 x n+1 0 a2 cos2 x + b2 sin2 x   π/2 cos2n x dx cos2n x dx 1 π (2n − 1)!!π = =   n+1 = n+1 n+1 2 2 2 2 2 2 2 2 2 2 n!ab2n+1 0 0 a sin x + b cos x a sin x + b cos x 

 4. 0

π/2



n  

cosp+2n x cos px dx n+1 = π a2 cos2 x + b2 sin2 x k=0

2n − k n



[ab > 0] GW (331)(58)  p+k−1 bp−1 (2a)2n−k+1 (a + b)p+k k [a > 0, b > 0, p > −2n − 1] GW (332)(30)

404

3.643

Trigonometric Functions



π/2

1. 0



π/2

2.12 0

3.644

3.643

2 a < 1,

π (1 + a)p−1 cosp x cos px dx = 1 − 2a cos 2x + a2 2p+1 1 − a

p > −1

 m−1 m−k−1   sin2n x cosμ x cos βx β 2n (−1)n π(1 − a)2n−2m+1   dx = m 22m−β−1 (1 + a)2m+β+1 k l (1 − 2a cos 2x + a2 ) k=0 l=0   2m − k − l − 2 × (−2)l (a − 1)k m − 1 2

a < 1, β = 2m − 2n − μ − 2, μ > −1

GW (332)(33c)

GW (332)(33)

  2 ν−1  k k  2  sinm x p − q2 p − q2 m + 1 − 2ν m + 1 − 2ν m−2 p dx = 2 , + B A 1. q 2 ν=1 −4q 2 2 2 −q 2 0 p + q cos x  ⎧   ⎪ πp q2 ⎪ ⎪ if m = 2k + 2 ⎪ ⎨ q 2 1 − 1 − p2

k ≥ 1, q = 0, p2 − q 2 ≥ 0 where A = ⎪ ⎪ 1 p+q ⎪ ⎪ if m = 2k + 1 ⎩ ln q p−q    π sinm x m−1 m+1 dx = 2m−1 B , [m ≥ 2] 2. 2 2 0 1 + cos x    π sinm x m−1 m+1 m−1 3. dx = 2 , [m ≥ 2] B 2 2 0 1 − cos x     π sin2 x pπ q2 dx = 2 1 − 1 − 2 4. q p 0 p + q cos x    π sin3 x p p2 p+q 1 dx = 2 2 + 1 − 2 ln 5. q q q p−q 0 p + q cos x  k  π n  cosn x dx a+b π k (2n − 2k − 1)!!(2k − 1)!! √ (−1) 3.645 n+1 = n (n − k)!k! a−b 2 (a + b)n a2 − b2 k=0 0 (a + b cos x) 2

2 a >b LI (64)(16) 3.646  n   π/2 2

π 1 cosn x sin nx sin 2x 1+a a 0,

b > 0]

BI (47)(20)

3.652

3.648

Trigonometric functions: powers and rational functions



π/4

1. 0

405

tanl x dx 1 + cos m n π sin 2x

     n−1 m  n+l+k l+k 1 km cosec π π ψ −ψ (−1)k−1 sin 2n n n 2n 2n k=0 n−1      2 1 m  n+l−k l+k km = cosec π π ψ −ψ (−1)k−1 sin n n n n n

[m + n is odd]

=

[m + n is even]

k=0



π/2

2. 0

3.649



π/2

1. 0



π/2

2. 0

3.651 1. 2. 3.652

tan±μ x dx = π cosec t sin μt cosec(μπ) 1 + cos t sin 2x   μ  tan±μ x sin 2x dx μπ 1−a π cosec 1 − = 1 ∓ 2a cos 2x + a2 4a 2   1 + a μ  μπ a−1 π cosec 1+ = 4a 2 a+1 tan±μ x (1 ∓ a cos 2x) μπ π dx = sec 2 1 ∓ 2a cos 2x + a 4 2 μπ π = sec 4 2



π/4

π/2

1. 0



π/2

2. 0

 3. 0

π/2

tanμ x dx = (sin x + cos x) sin x tanμ x dx = (sin x − cos x) sin x μ+ 12

cot x dx = (sin x + cos x) cos x



π/2

0

BI (36)(5)

|Re μ| < 1,

BI (47)(4)

π/2

0



2

a >1

0

BI (50)(3)

BI (50)(4)

[Re μ > −1]

BI (36)(3)

[Re μ > −1]

BI (36)(4)a

cotμ x dx = π cosec μπ (sin x + cos x) cos x BI (49)(1)

cotμ x dx = −π cot μπ (cos x − sin x) cos x [0 < Re μ < 1]

π/2

2

a −1] (cf. 3.651 1 and 2) LI (36)(10) tanμ−1 x cos2 x dx = 1 − sin2 x cos2 x



π/2

0

cotμ−1 x sin2 x dx π μπ cosec = √ cosec 6 1 − sin2 x cos2 x 4 3 [0 < Re μ < 4]



2+μ π 6



LI (47)(26)

3.66 Forms containing powers of linear functions of trigonometric functions 3.661



2π

1.

(a sin x + b cos x)2n+1 dx = 0

BI (68)(9)

0



2π

2.

2n

(a sin x + b cos x)

0



π

3.

n

(a + b cos x) dx =

0

n (2n − 1)!!  2 2π a + b2 (2n)!!

dx =

1 2



2π

BI (68)(8)

n  n (a + b cos x) dx = π a2 − b2 2 P n



0 n/2

a √ 2 a − b2



k π  (−1)k (2n − 2k)! n−2k  2 a a − b2 2n k!(n − k)!(n − 2k)! k=0 2

a > b2 GW (332)(37a)    2π dx a 1 π √ = = n+1 P n 2 − b2 2 0 (a + b cos x)n+1 2 2 2 a (a − b )  k n  (2n − 2k − 1)!!(2k − 1)!! a + b π √ = (n − k)!k! a−b 2n (a + b)n a2 − b2 =

 4. 0

π

dx (a + b cos x)

n+1

k=0

3.662



π/2

1.



μ

0



π/2

3.

(cosec x − 1)μ sin 2x dx = (1 − μ)μπ cosec μπ μ

0



π/2

(sec x − 1) tan x dx =

0

4.

μ

(cosec x − 1) cos x dx = μπ cosec μπ [|Re μ| < 1]

π/2

0



GW(332)(38), LI(64)(14)

0

2. 

π/2

(sec x − 1) sin x dx =

[a > |b|]

[−1 < Re μ < 2]

BI (55)(13) BI (48)(7)

μ

(cosec x − 1) cot x dx = −π cosec μπ

0

π/4

(cot x − 1)μ

dx π = − cosec μπ sin 2x 2

[−1 < Re μ < 0]

BI (46)(4,6)

[−1 < Re μ < 0]

BI (38)(22)a

408

Trigonometric Functions



π/4

5.

μ

(cot x − 1)

0

3.663



u

1.

dx = μπ cosec μπ cos2 x

1

(cos x − cos u)ν− 2 cos ax dx =

0



3.663

[|Re μ| < 1]

BI (38)(11)a

  π 1 sinν u Γ ν + P −ν 1 (cos u) 2 2 a− 2 Re ν > − 12 , a > 0,

0 −1, 0 < u < π]

EH I 178(23)

3.664



π



1.

z+

0



π

2.



z+

0

q  z 2 − 1 cos x dx = π P q (z)  Re z > 0, dx √ q = π P q−1 (z) z 2 − 1 cos x 

   π arg z + z 2 − 1 cos x = arg z for x = 2

Re z > 0,



π



π



3.

z+

q  z 2 − 1 cos x cos nx dx =

0

 4.

SM 482

   π arg z + z 2 − 1 cos x = arg z for x = 2

WH

π P nq (z) (q + 1)(q + 2) · · · (q + n)     π Re z > 0, arg z + z 2 − 1 cos x = arg z for x = , 2  z lies outside the interval (−1, 1) of the real axis WH, SM 483(15)

μ  z + z 2 − 1 cos x sin2ν−1 x dx

0

22ν−1 Γ(μ + 1) [Γ(ν)]2 ν C μ (z) Γ(2ν + μ)  √ 1 1−ν π Γ(ν) Γ(2ν) Γ(μ + 1) ν π 2 ν 2 −ν  z − 1 4 2 Γ(ν) P μ+ν− C μ (z) = 2 = 1 (z) 1 2 2 Γ(2ν + μ) Γ ν + 2 EH I 155(6)a, EH I 178(22) [Re ν > 0]  2π       ν ν−1 β + β 2 − 1 cos(a − x) γ + γ 2 − 1 cos x dx 0     = 2π P ν βγ − β 2 − 1 γ 2 − 1 cos a =

5.

[Re β > 0,

Re γ > 0]

EH I 157(18)

3.667

3.665

Trigonometric functions: powers of linear functions



π

1. 0



π

2. 0

μ μ sinμ−1 x dx 2μ−1 , B μ =  μ 2 2 (a + b cos x) (a2 − b2 )

[Re μ > 0,



π

1.

(β + cos x)

μ−ν− 12

0



π

2.6

3.

π

μ+ν

sin−2ν x dx =

(cosh β + sinh β cos x)

√

μ

2π

4.

1

[cos t + i sin t cos(a − x)]ν cos mx dx =

0

 5.10

2π

ν

[cos t + i sin t cos(a − x)] sin mx dx =

0

3.667



π/4

1.

(cos x + sin x)2μ

0



sinμ−1 2x dx

π/4

2.

=

FI II 790a

EH I 81(9)

sinμ x dx

Re ν > − 21

EH I 155(5)a

 sinhν (β) Γ 12 − ν P νμ (cosh β)

Re ν < 12

1 √ 1 2 −ν π sin 2 −ν t Γ(ν) P μ+ν− 1 (cos t) 2

Re ν > 0, t2 < π 2

EH I 156(7)

EH I 158(23)

i3m 2π Γ(ν + 1) cos ma P m ν (cos t) Γ(ν + m + 1)

0 < t < π2

EH I 159(25)

i 2π Γ(ν + 1) sin ma P m ν (cos t) Γ(ν + m + 1)

0 < t < π2

EH I 159(26)

3m

√ π Γ(μ)  2μ+1 Γ μ + 12

(cos x − sin x)μ+1 cos x

0

π

2ν

(cos t + i sin t cos x) sin2ν−1 x dx = 2ν− 2

0



|a| < 1]

 μ  1 2ν+ 2 e−iμπ β 2 − 1 2 Γ ν + 12 Q μν− 1 (β) 2  sin2ν x dx = Γ ν + μ + 12 Re ν + μ + 12 > 0,

0



0 < b < a]

 1  sin2μ−1 x dx 2 1 1 F ν, ν − μ + = B μ, ; μ + ; a ν 2 2 2 (1 + 2a cos x + a2 ) [Re μ > 0,

3.666

409

= −π cosec μπ

[Re μ > 0] [−1 < Re μ < 0]

BI (37)(1)

(cf. 3.192 2) BI (37)(16)



π/4

3. 0



π/4

4. 0



π/4

5. 0

 6.

0

π/4

μ

(cos x − sin x) π dx = − cosec μπ μ sin x sin 2x 2

[−1 < Re μ < 0]

π sinμ x dx = cosec μπ μ (cos x − sin x) sin 2x 2

[0 < Re μ < 1]

sinμ x dx = μπ cosec μπ μ (cos x − sin x) cos2 x

[|Re μ| < 1]

BI (37)(17)

[|Re μ| < 1]

BI(35)(24), BI(37)(18)

sinμ x dx (cos x − sin x)

μ−1

cos3

x

=

1−μ μπ cosec μπ 2

BI (35)(27)

LI (37)(20)a

410

Trigonometric Functions



π/2

7.

(sin x + cos x)

0

3.668 1.

2.



sinμ−1 x cosν−1 x



π 4

cos x + sin x cos x − sin x

μ+ν

3.668

dx = B(μ, ν)

[Re μ > 0,

Re ν > 0]

BI (48)(8)

cos 2t

π 2 sin (π cos2 t)  μ−1  v 1 − 2a cos u + a2 (cos u − cos x)μ−1 sin x dx π = μ 2 2 )μ sin μπ (cos x − cos v) 1 − 2a cos x + a (1 − 2a cos v + a u

0 < Re μ < 1, a2 < 1 −π 4

 3.669

π/2

0

dx =

sinp−1 x cosq−p−1 x dx = q (a cos x + b sin x)

 0

π/2

FI II 788

BI (73)(2)

sinq−p−1 x cosp−1 x B(p, q − p) q dx = aq−p bp (a sin x + b cos x) [q > p > 0, ab > 0]

BI (331)(9)

3.67 Square roots of expressions containing trigonometric functions 3.671



     α+1 β+1 α+1 1 α+β+2 2 1 , F ,− ; ;k sinα x cosβ x 1 − k 2 sin2 x dx = B 2 2 2 2 2 2 [α > −1, β > −1, |k| < 1]

π/2

1. 0

GW (331)(93)



π/2

2. 0

sinα x cosβ x 1  dx = B 2 2 2 1 − k sin x



α+1 β+1 , 2 2



 F



α+1 1 α+β+2 2 , ; ;k 2 2 2 [α > −1, β > −1,

|k| < 1] GW (331)(92)



π

3. 0

 4.12

∞ sin2n x dx π  (2j − 1)!! (2n + 2j − 1)!! 2j  k = n 22j j!(n + j)! 1 − k 2 sin2 x 2 j=0  j ∞ (2n − 1)!!π  [(2j − 1)!!]2 k2 √ = 2n 1 − k 2 j=0 22j j!(n + j)! k 2 − 1

π√

 a ± b cos x dx =

0



π

5.12 0

6.∗

 0

π

√

dx = a ± b cos x



π/2

−π/2

√ √ a ± b sin x dx = 2 a + bE

π/2

dx 2 √ =√ K a+b −π/2 a ± b sin x

 √  2 p dx 2  K = 1+p 1+p 1 + 2p cos x + p2







2b a+b

k2 < 1 k2
b > 0]  2b a+b [a > b > 0]

1 2

LI (67)(2)

3.677

3.672

Trigonometric functions and square roots



π/4

1. 0



π/4

2. 0

 3.673 3.674

π 2

u

 0



π

2. 0



π

3.12 0

3.675 1. 2. 3.676 1.

BI (39)(5)

(2n − 1)!! sinn x dx  = π cosn+1 x sin x (cos x − sin x) (2n)!!

BI (39)(6)

  √ dx π − 2u = 2K sin 4 sin x − sin u

dx 2  K = 1+p 1 ± 2p cos x + p2



2

p ≤1 2

p ≥1 BI (67)(6)

2

π

π/2



sin x dx

=

WH

FI II 684, WH

1 arctan p p

BI (60)(5)

1 + p2 sin x  π/2  tan2 x 1 − p2 sin2 x dx = ∞ π/2

0

2

dx

1  = K 2 2 2 2 p p cos x + q sin x

BI (53)(8)



p2 − q 2 p



√  √  √ sin2 x dx 2 2 1  −√ K = 2E 2 2 2 2 0 1 + sin x  √   √   π/2 √ cos2 x dx 2 2  −E = 2 K 2 2 2 0 1 + sin x



BI (67)(5)

 √   √  2 p 2 p 2 cos x dx 1 1+p  − (1 + p)E = [K(p) − E(p)] = K 2 p p 1+p 1+p 1+p 1 − 2p cos x + p

2 BI (67)(7) p 0]

2π

413

414

Trigonometric Functions

3.679(2)

3.679(2)*

 Define Δ = (sinh α sin γ − sinh β sin x)2 + (cosh α cos γ − cosh β cos x)2 ,     p = sinh2 α + sin2 γ = cosh2 α − cos2 γ, q = sinh2 β + sin2 γ = cosh2 β − cos2 γ, I1 = = = I2 = = =  1. 2. 3. 4. 5.

6.

7.

 √   √  2 pq 2 pq 1 1 E + K p−q p+q p+q p+q     2 2q p p + K E p2 − q 2 q q q   2p q E p2 − q 2 p  √   √  2 pq 2 pq 1 1 − E K p−q p+q p+q p+q   2q p E p2 − q 2 q     2p 2 q p − K E p2 − q 2 q p p

[|α| = |β|] [|α| < |β|] [|α| > |β|] [|α| = |β|] [|α| < |β|] [|α| > |β|]

dx = 2(I1 − I2 ) Δ     0 2π dx 2 sin2 γ cos2 γ I1 2 I2 2 2 I1 2 2 I2 sinh + = α cosh α + sinh β cosh β − Δ3 p2 − q 2 p4 q4 p2 q 2 p2 q2 0       2π sin x 2 cos2 γ I1 2 I2 2 I1 2 I2 cosh α 4 + cosh β 4 − dx = sinh α sinh β sin γ 2 − 2 Δ3 p − q2 p q p2 q 2 p2 q 0       2π 2 cos x 2 sin γ I1 2 I2 2 I1 2 I2 sinh + dx = cosh α cosh β cos γ α + sinh β − Δ3 p2 − q 2 p4 q4 p2 q 2 p2 q2 0    2π 2 sin x 2 sin2 γ I1 I2 cosh2 α sinh2 β 4 + sinh2 α cosh2 β 4 dx = 3 2 2 Δ p −q p q 0   2 sinh2 α sinh2 β cos2 γ I1 I2 + − 2 p2 q 2 p2 q    2π cos2 x 2 cos2 γ 2 2 I1 2 2 I2 sinh dx = α cosh β + cosh α sinh β Δ3 p2 − q 2 p4 q4 0   2 2 2 2 cosh α cosh β sin γ I1 I2 + − 2 p2 q 2 p2 q  2π cos x sin x dx = sinh α cosh α sinh β cosh β sin γ cos γ Δ3      0 2 I2 I2 2 I1 I1 × 2 − + − p − q 2 p4 q4 p2 q 2 p2 q2 2π

3.679(3)*  π/2  1. 0

dx a cos2

x + b sin2 x

= RF (0, a, b)

[Re a > 0,

Re b > 0] DLMF

(19.23.1)

3.683

Trigonometric functions: various powers



π/2 

a cos2 x + b sin2 x dx = 2RG (0, a, b)

2.

[Re a > 0,

415

Re b > 0]

DLMF

(19.23.2)

DLMF

(19.23.3)

DLMF

(19.23.7)

0

 3. 4.

π/2

1 sin2 x dx = RD (0, a, b) [Re a > 0, Re b > 0] 2 2 3/2 3 (a cos x + b sin x) 0    ∞ x b c a √ + + dx = 4RG (a, b, c) √ √ x+a x+b x+c x+a x+b x+c 0 [a, b, c ∈ C\(−∞, 0]]

3.68 Various forms of powers of trigonometric functions 3.681



π/2

1. 0



π/2

2. 0

 3.12

4.8

 sin2μ−1 x cos2ν−1 x dx 1  = B(μ, ν) F , μ; μ + ν; k 2 2 2 2 1 − k sin x sin2μ−1 x cos2ν−1 x dx B(μ, ν)  μ+ν = 2 2 2 (1 − k 2 )μ 1 − k sin x

[Re μ > 0,

Re ν > 0]

EH I 115(7)

[Re μ > 0,

Re ν > 0]

EH I 10(20)

π/2

sinμ x dx  μ −1 0 cosμ−3 x 1 − k 2 sin2 x 2     Γ μ+1 Γ 2 − μ2 1 − (μ − 3)k + k 2 1 + (μ − 3)k + k 2 2 √ = 3 − (1 + k)μ−3 (1 − k)μ−3 k (μ − 1)(μ − 3)(μ − 5) π [−1 < Re μ < 4] BI (54)(10)    π/2   μ+1 sin x dx 1−μ μ (1 − k)−μ − (1 + k)−μ √ Γ Γ 1+ μ+1 =  2 2 2kμ π 0 cosμ x 1 − k 2 sin2 x 2 [−2 < Re μ < 1] 

3.682

π/2

0

1 sinμ x cosν x B dx = 2 2a (a − b cos x)



μ+1 ν+1 , 2 2



 F

BI (61)(5)



ν +1 μ+ν b , ; + 1; 2 2 a [Re μ > −1, Re ν > −1,

a > |b| ≥ 0] GW (331)(64)

3.683



1. 0

2.

π/4

n

(sin 2x − 1) tan

π 4

  + x dx =

π/4 0

(cosn 2x − 1) cot x dx = −

n

11 2 k k=1

1 = − [C + ψ(n + 1)] 2 [n ≥ 0] BI(34)(8), BI(35)(11)  π/4  π/4  π + x dx = (sinμ 2x − 1) cosecμ 2x tan (cosμ 2x − 1) secμ 2x cot x dx 4 0 0 1 = [C + ψ(1 − μ)] 2 [Re μ < 1] BI (35)(20)

416

Trigonometric Functions

 3.

π 4

3.684

 π/4  π  2μ + x dx = sin2μ 2x − 1 cosecμ 2x tan cos 2x − 1 secμ 2x cot x dx 4 0 π 1 + cot μπ =− 2μ 2



0

BI (35)(21)



π/4

4.

(1 − secμ 2x) cot x dx =



0

 3.684

π/4

0

π/4

0

(cotμ x − 1) dx = (cos x − sin x) sin x



π/2

0

 1 + x dx = [C + ψ(1 − μ)] (1 − cosecμ 2x) tan 4 2 [Re μ < 1] BI (35)(13) π

(tanμ x − 1) dx = −C − ψ(1 − μ) (sin x − cos x) cos x

[Re μ < 1] BI (37)(9)

3.685



π/4



μ−1

sin

1.

ν−1

2x − sin

0



π/2

2. 0



π/2

3. 0

 4. 0

π/4

 π/4 π   μ−1 cos 2x tan 2x − cosν−1 2x cot x dx + x dx = 4 0 1 = [ψ(ν) − ψ(μ)] 2 [Re μ > 0, Re ν > 0] BI(34)(9), BI(35)(12)



dx = sinμ−1 x − sinν−1 x cos x

(sinμ x − cosecμ x)

dx = cos x

(sinμ 2x − cosecμ 2x) cot



π 4



π/2

0

 μ  dx 1  ν  = ψ −ψ cosμ−1 x − cosν−1 x sin x 2 2 2



[Re μ > 0, π/2

(cosμ x − secμ x)

0

 + x dx =

0

6.

7.

π/4

π μπ dx = − tan sin x 2 2 [|Re μ| < 1]

BI (46)(2)

BI (46)(1, 3)

(cosμ 2x − secμ 2x) tan x dx

π 1 − cosec μπ 2μ 2 [|Re μ| < 1] BI (35)(19, 22)  π/4  π/4  π + x dx = (sinμ 2x − cosecμ 2x) tan (cosμ 2x − secμ 2x) cot x dx 4 0 0 π 1 + cot μπ =− 2μ 2 [|Re μ| < 1] BI (35)(14)  π/4    μ−1 π + x dx sin 2x + cosecμ 2x cot 4 0  π/4  μ−1 π = cos 2x + secμ 2x tan x dx = cosec μπ 4 0 [0 < Re μ < 1] BI (35)(18, 8)  π/4  π/4  π  μ−1  μ−1 π + x dx = sin cos 2x − cosecμ 2x tan 2x − secμ 2x cot x dx = cot μπ 4 2 0 0 [0 < Re μ < 1] BI(35)(7), LI(34)(10) =

5.



Re ν > 0]

3.688

Trigonometric functions: various powers

 3.686 3.687

π/2

0



π/2

1. 0

tan x dx = cosμ x + secμ x



π/2

0

417

cot x dx π = sinμ x + cosecμ x 4μ

sinμ−1 x + sinν−1 x dx = cosμ+ν−1 x



π/2

0

BI(47)(28), BI(49)(14)

 μ ν  cos ν−μ π 4 B  , ν +μ 2 2 π 2 cos 4 [Re μ > 0, Re ν > 0, Re(μ + ν) < 2]

cosμ−1 x + cosν−1 x dx = sinμ+ν−1 x

BI (46)(7)



π/2

2. 0

μ−1

sin

ν−1

x − sin cosμ+ν−1 x

x



π/2

dx =

cos

μ−1

0

ν−1

x − cos sinμ+ν−1 x

  sin ν−μ π x μ ν 4  ν+μ B , dx = 2 2 2 sin 4 π [Re μ > 0, Re ν > 0, Re(μ + ν) < 4]

BI(46)(8)



π/2

3. 0

sinμ x + sinν x cot x dx = sinμ+ν x + 1



π 2

0



cosμ x + cosν x π μ−ν π tan x dx = sec cosμ+ν x + 1 μ+ν μ+ν 2 [Re μ > 0, Re ν > 0]



BI (49)(15)a, BI (47)(29)



π/2

4. 0

sinμ x − sinν x cot x dx = sinμ+ν x − 1

 0

π 2



cosμ x − cosν x π μ−ν π tan x dx = tan cosμ+ν x − 1 μ+ν μ+ν 2 [Re μ > 0, Re ν > 0]



BI(149)(16)a, BI(47)(30)

 5. 6. 3.688

μ π

π/2

cosμ x + secμ x π tan x dx = sec ν x + secν x cos 2ν ν2 0  π/2  μ μ cos x − sec x π μ π tan x dx = tan cosν x − secν x 2ν ν2 0 

π/4

1. 0



π/4

2. 0



π/4

3. 0



π/4

4. 0



π/4

5. 0



π/4

6. 0

 7.

0

π/4

[|Re ν| > |Re μ|]

BI (49)(12)

[|Re ν| > |Re μ|]

BI (49)(13)

tanν x − tanμ x dx = ψ(μ) − ψ(ν) cos x − sin x sin x

[Re μ > 0,

BI (37)(10)

tanμ x − tan1−μ x dx = π cot μπ cos x − sin x sin x

[0 < Re μ < 1]

(tanμ x + cotμ x) dx =

μπ π sec 2 2

(tanμ x − cotμ x) tan x dx =

π μπ 1 − cosec μ 2 2

Re ν > 0]

[|Re μ| < 1]

BI (37)(11)

BI (35)(9)

[0 < Re μ < 2]

BI (35)(15)

π μπ tanμ−1 x − cotμ−1 x dx = cot cos 2x 2 2

[|Re μ| < 2]

BI (35)(10)

tanμ x − cotμ x 1 π μπ tan x dx = − + cot cos 2x μ 2 2

[−2 < Re μ < 0]

BI (35)(23)

tanμ x + cotμ x dx = π cosec t cosec μπ sin μt 1 + cos t sin 2x

[t = nπ,

|Re μ| < 1]

BI (36)(6)

418

Trigonometric Functions



π/4

8. 0



π/4

9. 0



π/4

10. 0



π/4

11. 0



π/4

12. 0

 13. 14.

tanμ−1 x + cotμ x dx = π cosec μπ (sin x + cos x) cos x

[0 < Re μ < 1]

BI (37)(3)

tanμ x − cotμ x 1 dx = −π cosec μπ + (sin x + cos x) cos x μ

[0 < Re μ < 1]

BI (37)(4)

tanν x − cotμ x dx = ψ(1 − μ) − ψ(1 + ν) (cos x − sin x) cos x

[Re μ < 1,

BI (37)(5)

tanμ−1 x − cotμ x dx = π cot μπ (cos x − sin x) cos x

[0 < Re μ < 1]

BI (37)(7)

tanμ x − cotμ x 1 dx = π cot μπ − (cos x − sin x) cos x μ

[0 < Re μ < 1]

BI (37)(8)

Re ν > −1]

π/4

1 dx π = [Re μ = 0] μ μ tan x + cot x sin 2x 8μ 0 √  π/2  π/2 1 dx 1 dx π Γ(ν)  = = ν ν μ μ μ μ 2ν+1 μ Γ ν + 1 tan x sin 2x 2 (tan x + cot x) (tan x + cot x) 0 0 2 

[ν > 0] π/4

15.

(tanμ x − cotμ x) (tanν x − cotν x) dx =

0



π/4

16. 

π/4

17. 0



π/4

18. 0



π/4

19. 0



π/2

20. 3.689  1.

π/2

0



BI(49)(25), BI(49)(26)

νπ 2π sin μπ 2 sin 2 cos μπ + cos νπ [|Re μ| < 1,

BI (35)(17)

2π cos cos cos μπ + cos νπ [|Re μ| < 1,

|Re ν| < 1]

BI (35)(16)

(tan x − cot x) (tan x + cot x) sin μπ dx = −π cos 2x cos μπ + cos νπ [|Re μ| < 1,

|Re ν| < 1]

BI (35)(25)

(tanμ x + cotμ x) (tanν x + cotν x) dx =

μ

μ

ν

μπ 2

νπ 2

ν

tanν x − cotν x dx π νπ = tan tanμ x − cotμ x sin 2x 4μ 2μ

[0 < Re ν < 1]

BI (37)(14)

tanν x + cotν x dx π νπ = sec tanμ x + cotμ x sin 2x 4μ 2μ

[0 < Re ν < 1]

BI (37)(13)

[μ > 0,

BI (49)(29)

ν

(1 + tan x) − 1 (1 + tan x)

0

BI (37)(12)

|Re ν| < 1]

0

2.

3.689

μ+ν

dx = ψ(μ + ν) − ψ(μ) sin x cos x

(sinμ x + cosecμ x) cot x dx π μπ μt = cosec t cosec sin ν ν sin x − 2 cos t + cosec x ν ν ν [μ < ν]

ν > 0]

LI (50)(14)

π/2

sinμ x − 2 cos t1 + cosecμ x π μt2 t2 μπ cot x dx = cosec t2 cosec sin − cosec t2 cos t1 ν νx sin x + 2 cos t + cosec ν ν ν ν 2 0 [(ν > μ > 0) or (ν < μ < 0) or (μ > 0, ν < 0, and μ + ν < 0) or (μ < 0, ν > 0, and μ + ν > 0)] BI (50)(15)

3.691

Trigonometric functions: complicated arguments

419

3.69–3.71 Trigonometric functions of more complicated arguments 3.691 1.





∞

12

2

sin(x ) dx = 0



0

1

2.



sin ax

2



 dx =

0



3. 4.

∞



1 cos(x ) dx = 2 2

1

 cos ax2 dx =

∞

5. 0



∞

6.

FI II 743a, ET I 64(7)a

π √ S a 2a

 1 sin ax2 cos 2bx dx = 2  cos ax2 sin 2bx dx =

0



∞

7. 0



[a > 0]

cos ax2 cos 2bx dx =

1 2





π b b cos − sin 2a a a 2

2



ET I 8(5)a

[a > 0, b > 0]   2  1 π b π = cos + 2 a a 4 [a > 0, b > 0]

ET I 82(1)a

ET I 82(18), BI(70)(13) GW(334)(5a)

     π b2 b b b2 sin C √ − cos S √ 2a a a a a





 2

π b2 b cos + sin 2a a a

[a > 0,

b > 0]

[a > 0,

b > 0]

ET I 83(3)a

GW(334)(5a), BI(70)(14), ET I 24(7) ∞

8. 0

10.

π 2

√ π C a [a > 0] 2a 0        ∞  2 π b2 b b b2 cos C √ + sin S √ sin ax sin 2bx dx = 2a a a a a 0 

9.





∞



∞

 (cos ax + sin ax) sin b2 x2 dx    2   2   a   a  1 π  a a 1+2C cos sin − 1−2S = 2b 2 2b 4b2 2b 4b2 [a > 0, b > 0] ET I 85(22)

 (cos ax + sin ax) cos b2 x2 dx 0    2   2   a   a  a a 1 π  1+2C sin cos + 1−2S = 2b 2 2b 4b2 2b 4b2 [a > 0, b > 0] ET I 25(21)   2 √  ∞  π 2bc b + c2 π sin 2 cos sin a2 x2 sin 2bx sin 2cx dx = − 2 2a a a 4 0

11. 0

 sin a2 x2 cos 2bx cos 2cx dx =

√

π 2bc cos 2 cos 2a a



[a > 0, b 2 + c2 π + a2 4 [a > 0,

b > 0, 

c > 0]

ET I 84(15)

b > 0,

c > 0]

ET I 84(21)

420

Trigonometric Functions



∞

12.



cos a x

2 2

0



∞

13. 0

15. 16. 17.12

√

2bc π sin 2 sin sin 2bx sin 2cx dx = 2a a 

  1 π sin ax2 cos bx2 dx = 4 2 1 π = 4 2



π b 2 + c2 − 2 a 4



[a > 0, b > 0,  1 1 √ +√ [a > b > 0] a − b  a+b 1 1 √ −√ [b > a > 0] b+a b−a 

b > 0]

BI (178)(3)

b > 0]

BI (178)(5)

[a > 0, b > 0]      π π π π 1 + + − a b 32 2a 2b

BI (178)(2)

[a > 0,      √  4 2 π π 1 8+ 2 − cos ax − cos4 bx2 dx = 64 a b

b > 0]

BI (178)(4)

[a > 0,

b > 0]

BI (178)(6)

∞

20.

1 cos4 ax2 − sin4 bx2 dx = 8

 sin2n ax2 dx =

0

∞

cos2n ax2 dx = ∞

BI (177)(5, 6)

0 ∞

21.

2n+1

sin



ax

2



dx =

0



BI (178)(1)

∞

0



b > 0]



19. 

∞

22.

 cos2n+1 ax2 dx =

0

1 22n+1

1 22n+1

n 

n+k



(−1)

k=0 n   k=0

2n + 1 k

2n + 1 k





π 2(2n − 2k + 1)a [a > 0]

∞

1.12

  sin a − x2 + cos a − x2 dx =

0

 2.



∞

cos 0

π x2 − 2 8



 cos ax dx =

π cos 2

 

BI (70)(9)

π 2(2n − 2k + 1)a [a > 0]



ET I 25(19)

∞

0





c > 0]

BI (177)(21)



∞

18.

3.692



   2 2 π π 1 − [a > 0, sin ax − sin2 bx2 dx = 8 b a 0     ∞  2 2 π π 1 2 2 + [a > 0, cos ax − sin bx dx = 8 b a 0     ∞  2 2 π π 1 2 2 − [a > 0, cos ax − cos bx dx = 8 a b 0     ∞ √   4 2 1  π π 8− 2 − sin ax − sin4 bx2 dx = 64 b a 0 

14.



3.692

π sin a 2

π a2 − 2 8

BI(177)(7)a, BI(70)(10)

GW(333)(30c), BI(178)(7)a

 [a > 0]

ET I 24(8)

3.695

Trigonometric functions: complicated arguments



∞

3. 0



∞

4. 0



∞

5. 0



−∞



 1 cos a 1 − x2 cos bx dx = 2

∞

1.



  π b2 π cos a + + a 4a 4

[a > 0]   2 π b π sin a + + a 4a 4

ET I 23(2)

[a > 0]       ∞ 2 2 1 π b b 2 2 cos 2bx dx = cos 2bx dx = sin ax + cos ax + a a 2 2a 0 ∞      cos x2 − 1 − cos x2 + 1 dx = n=0



sin ax + 2bx dx =



2

0





[a > 0] ∞

6.8 3.693

 1 sin a 1 − x2 cos bx dx = − 2

∞

2.

 cos ax2 + 2bx dx =

0



π 2 24n+1 [(2n)!]



1.

2.

3.695



ET I 24(10)

BI (70)(19, 20)

n+

1 2



  2    2  2 2 π  cos ba 12 − S 2 ba − sin ba 12 − C 2 ba 2a [a > 0]

BI (70)(3)

  2    2  2 2 π  cos ba 12 − C 2 ba + sin ba 12 − S 2 ba 2a [a > 0]

3.694

421

BI (70)(4)



  2    2   2 π 1 b 1 b cos ba sin c + cos c − C − S 2 2 2 a 2 a 2a 0         2 2 2 π 1 sin ba sin c − 12 − C 2 ba cos c − S 2 ba + 2 2a GW (334)(4a) [a > 0]   ∞         2 π 1 b2 1 b2 cos ba cos c − sin c cos ax2 + 2bx + c dx = − C − S 2 2 2 a 2 a 2a 0         2 2 2 π 1 sin ba cos c + 12 − C 2 ba sin c − S 2 ba + 2 2a GW (334)(4b) [a > 0]



∞

∞

1. 0

 2. 0

∞



2



3 3



sin ax + 2bx + c dx =

sin a x



π sin(bx) dx = 6a



 π cos a3 x3 cos(bx) dx = 6a

       √    b b b 3 b 2b 2b 2b J 13 + J − 13 − K 13 3a 3a 3a 3a 3a π 3a 3a





b J 13 3a



2b 3a



b 3a



[a > 0,  + J − 13 [a > 0,

b > 0] ET I 83(5)      √ b 3 b 2b 2b + K 13 3a 3a π 3a 3a b > 0]

ET I 24(11)

422

3.696

Trigonometric Functions



∞

1.



sin ax

4



sin bx

0



∞

2. 0



∞

3. 0



∞

4. 0



2



π dx = − 4

  π sin ax4 cos bx2 dx = − 4   π cos ax4 sin bx2 dx = 4   π cos ax4 cos bx2 dx = 4







b sin 2a b sin 2a

b cos 2a



b cos 2a









3.696

  2 3 b2 b − π J 14 8a 8 8a b2 π − 8a 8



[a > 0, b > 0]  2 b J − 14 8a

[a > 0,  2 b2 3 b 1 − π J4 8a 8 8a

b > 0]

ET I 84(19)

[a > 0, b > 0]  2 b J − 14 8a

ET I 83(4), ET I 25(24)



b2 π − 8a 8



[a > 0,  3.697 3.698



∞

sin 0





∞

sin

1. 0





∞

2.12

sin 0



a2 x2

cos 0





∞

4.

cos 0





 2

a x2



∞

3.

a2 x

 2

a x2

 2

a x2

 √  aπ sin(bx) dx = √ J 1 2a b 2 b  1 sin b2 x2 dx = 4b



 1 cos b2 x2 dx = 4b  1 sin b2 x2 dx = 4b  1 cos b2 x2 dx = 4b

[a > 0,

1. 2. 3.

b > 0]

b > 0]

ET I 83(6)







[a > 0,

b > 0]

ET I 83(9)

b > 0]

ET I 24(13)

b > 0]

ET I 84(12)

b > 0]

ET I 24(14)

π sin 2ab + cos 2ab − e−2ab 2 [a > 0,

π sin 2ab + cos 2ab + e−2ab 2 [a > 0,

π cos 2ab − sin 2ab + e−2ab 2

√   2π b2 2 2 (cos 2ab + sin 2ab) [a > 0, b > 0] sin a x + 2 dx = x 4a 0 √    ∞ 2π b2 2 2 (cos 2ab − sin 2ab) [a > 0, b > 0] cos a x + 2 dx = x 4a 0 √      ∞  ∞ 2π b2 b2 2 2 2 2 sin a x − 2ab + 2 dx = cos a x − 2ab + 2 dx = x x 4a 0 0 [a > 0, b > 0]



ET I 25(25)

π sin 2ab − cos 2ab + e−2ab 2

[a > 0, 3.699

ET I 83(2)

∞

BI (70)(27)

BI (70)(28)

BI(179)(11, 12)a, ET I 83(6)

3.715

Trigonometric functions: complicated arguments

√   2π −2ab b2 2 2 e 4. sin a x − 2 dx = [a > 0, x 4a 0 √    ∞ 2π −2ab b2 2 2 e 5. cos a x − 2 dx = [a > 0, x 4a 0  u       πau 3.711 sin a u2 − x2 cos bx dx = √ J 1 u a2 + b 2 2 a2 + b 2 0 

∞

423

b > 0]

GW (334)(9b)a

b > 0]

GW (334)(9b)a

[a > 0,

b > 0,

u > 0] ET I 27(37)

3.712 

  ∞

1.

sin (axp ) dx =

0



∞

2.

cos (axp ) dx =

Γ



∞

1. 0

π sin 2p 1

pa p   π Γ p1 cos 2p

0

3.713

1 p

1

pa p ∞

1  (−b)k − kq+1 a p Γ sin (ax + bx ) dx = p k! p

q



k=0

kq + 1 p

[a > 0,

p > 1]

EH I 13(40)

[a > 0,

p > 1]

EH I 13(39)





 k(q − p) + 1 sin π 2p [a > 0, b > 0, p > 0,

q > 0] BI (70)(7)



∞

2.

cos (axp + bxq ) dx =

0

∞

1  (−b)k −(kq+1)/p a Γ p k!



k=0





k(q − p) + 1 kq + 1 cos π p 2p [a > 0, b > 0, p > 0,

 q > 0] BI (70)(8)

3.714  1. 0



∞

∞

2. 0



∞

3. 0

 4. 5.

3.715

∞

cos (z sinh x) dx = K 0 (z) sin (z cosh x) dx =

π J 0 (z) 2

cos (z cosh x) dx = −

π Y 0 (z) 2

[Re z > 0]

WA 202(14)

[Re z > 0]

MO 36

[Re z > 0]

MO 37

μπ K μ (z) [Re z > 0, 2 0   μ   π √ 2 1 I μ (z) cos (z cosh x) sin2μ x dx = π Γ μ+ z 2 0 Re z > 0, 

1. 0

π

cos (z sinh x) cosh μx dx = cos

sin (z sin x) sin ax dx = sin aπ s 0,a (z) = sin aπ

∞  k=1

(12

−

|Re μ| < 1]

Re μ > − 12

WA 202(13)

WH

(−1)k−1 z 2k−1 − a2 ) . . . [(2k − 1)2 − a2 ]

a2 ) (32

[a > 0]

WA 338(13)

424

Trigonometric Functions



π

2. 0



1 sin (z sin x) sin nx dx = 2

π/2



π

4.

π

sin (z sin x) sin nx dx  π/2 π = [1 − (−1)n ] sin (z sin x) sin nx dx = [1 − (−1)n ] J n (z) 2 0 [n = 0, ±1, ±2, . . .] WA 30(6), GW(334)(153a)

3. 0



sin (z sin x) sin 2x dx =

−π

2 (sin z − z cos z) z2

= (1 + cos aπ)

∞  k=1

π

5. 0



π

6.

LI (43)(14)

sin (z sin x) cos ax dx = (1 + cos aπ) s 0,a (z)

0



(−1)k−1 z 2k−1 (12 − a2 ) (32 − a2 ) . . . [(2k − 1)2 − a2 ] [a > 0]

sin (z sin x) cos[(2n + 1)x] dx = 0

GW (334)(53b)

∞

 (−1)k−1 z 2k 1 = −a (1 − cos aπ) − 2 + 2 2 2 a a (2 − a ) (42 − a2 ) . . . [(2k)2 − a2 ] k=1

π

7. 0



π

8.

WA 338(12)

cos (z sin x) cos ax dx = −a sin aπ s −1,a (z) ∞

 (−1)k−1 z 2k 1 − 2+ 2 2 2 a a (2 − a ) (42 − a2 ) . . . [(2k)2 − a2 ] k=1

π

cos (z sin x) cos nx dx =

0

1 2

!

GW (334)(54a)

= −a sin aπ

9.

[a > 0]

cos (z sin x) sin 2nx dx = 0

0



WA 338(14)

cos (z sin x) sin ax dx = −a (1 − cos aπ) s −1,a (z)

0



3.715



[a > 0]

!

WA 338(11)

π

cos (z sin x) cos nx dx  π/2 π n = [1 + (−1) ] cos (z sin x) cos nx dx = [1 + (−1)n ] J n (z) 2 0 −π

GW (334)(54b)



π/2

10.8

cos (z sin x) cos2n x dx =

0

 11.

0

π/2

sin (z cos x) sin 2x dx =

π (2n − 1)!! J n (z) 2 zn

2 (sin z − z cos z) z2

[n = 0, 1, 2, . . .]

FI II 486, WA 35a

LI (43)(15)

3.716

Trigonometric functions: complicated arguments



π/2

12.8 0

aπ aπ π s 0,a (z) = cosec [Ja (z) − J−a (z)] sin (z cos x) cos ax dx = cos 2 4 2 aπ π [Ea (z) + E−a (z)] = − sec 4 4 ∞ (−1)k−1 z 2k−1 aπ  = cos 2 (12 − a2 ) (32 − a2 ) . . . [(2k − 1)2 − a2 ] k=1



π

13.

sin (z cos x) cos nx dx =

0



π/2

14.

1 2



π

−π



π/2

sin (z cos x) cos[(2n + 1)x] dx = (−1)n sin (a cos x) tan x dx = si(a) +

0



π/2

16. 0



π/2

17.7 0

sin (z cos x) sin

2ν

[a > 0]

sin (z cos x) cos nx dx = π sin

0

15.11

18.

π



π 2

20.

21.

3.716

WA 30(8)

[a > 0]

BI (43)(17)

1 2



π

−π

[a > 0]

π/2

1.

cos (z cos x) cos nx dx = π cos

nπ J n (z) 2

cos (z cos x) cos 2nx dx = (−1)n

sin (a tan x) dx =

1 −a e Ei(a) − ea Ei(−a) 2

[a > 0]

cos (a tan x) dx =

π −a e 2

[a ≥ 0]

0

 2.

0

π/2

WA 358(1)

WA 339

π J 2n (z) 2 0  √  ν   π/2 π 2 1 2ν J ν (z) cos (z cos x) sin x dx = Γ ν+ 2 z 2 0

Re ν > − 12   μ   π √ 2 1 2μ J μ (z) cos (z cos x) sin x dx = π Γ μ+ z 2 0

Re μ > − 12 

GW (334)(55b)

aπ s −1,a (z) cos (z cos x) cos ax dx = −a sin 2 aπ aπ π π [Ja (z) + J−a (z)] = cosec [Ea (z) − E−a (z)] = sec 4 2  4 2  ∞  (−1)k−1 z 2k 1 aπ − 2+ = −a sin 2 a a2 (22 − a2 ) (42 − a2 ) . . . [(2k)2 − a2 ]

cos (z cos x) cos nx dx =

π/2

nπ J n (z) 2

 √  ν  π 2 1 Hν (z) x dx = Γ ν+ 2 z 2

Re ν > − 12

0

19.

WA 339

π J 2n+1 (z) 2

k=1



425

(cf. 3.723 1)

GW (334)(56b)

WA 30(9)

WA 35, WH

WH

BI (43)(1)

BI (43)(2)

426

Trigonometric Functions



π/2

3.

sin (a tan x) sin 2x dx =

aπ −a e 2

[a ≥ 0]

BI (43)(7)

cos (a tan x) sin2 x dx =

1 − a −a πe 4

[a ≥ 0]

BI (43)(8)

cos (a tan x) cos2 x dx =

1 + a −a πe 4

[a ≥ 0]

BI (43)(9)

[a > 0]

BI (43)(5)

0



π/2

4. 0



π/2

5. 0



π/2

6.

sin (a tan x) tan x dx =

0



π/2

7.

π −a e 2

cos (a tan x) tan x dx = −

0



π/2

sin (a tan x) sin2 x tan x dx =

0 π/2

9. π/2

10. 

π/2



π/2

(cf. 3.742 1)

BI (43)(3)

cos2 (a tan x) dx =

π 1 + e−2a 4

[a ≥ 0]

(cf. 3.742 3)

BI (43)(4)

sin2 (a tan x) cot2 x dx =

1 − sec2 x cos (tan x)

0



π/2

13.

BI (43)(11)

[a ≥ 0]

0

12.

[a > 0]

BI (43)(6)

π 1 − e−2a 4

0

11.

2 − a −a πe 4

(cf. 3.723 5)

sin2 (a tan x) dx =

0



1 −a e Ei(a) + ea Ei(−a) 2 [a > 0]

8. 

3.717

[a ≥ 0]

BI (43)(19)

dx =C tan x

sin (a cot x) sin 2x dx =

0

π  −2a e + 2a − 1 4

BI (51)(14)

aπ −a e 2

[a ≥ 0]

(cf. 3.716 3.)

and in general, formulas 3.716 remain valid if we replace tan x in the argument of the sine or cosine with cot x if we also replace sin x with cos x, cos x with sin x, hence tan x with cot x, cot x with tan x, sec x with cosec x, and cosec x with sec x in the factors. Analogously,  π/2  π/2 π dx dx 3.717 = = sin a [a ≥ 0] sin (a cosec x) sin (a cot x) sin (a sec x) sin (a tan x) cos x sin x 2 0 0 BI (52)(11, 12)

3.718



π/2

sin

1. 0



π/2

2.

π 2

  p − a tan x tanp−1 x dx = 0

sin (a tan x − νx) sinν−2 x dx = 0

π/2

cos

π

 π p − a tan x tanp x dx = e−a 2 2 2

p < 1, p = 0, a ≥ 0 [Re ν > 0,

a > 0]

BI (44)(5, 6) NH 157(15)

0

 3.

0

π/2

sin (n tan x + νx)

π cosν−1 x dx = sin x 2

[Re ν > 0]

BI (51)(15)

3.722

Trigonometric and rational functions



π/2

4.

cos (a tan x − νx) cosν−2 x dx =

0

πe−a aν−1 Γ(ν)

427

[Re ν > 1,

a > 0] LO V 153(112), NT 157(14)



π/2

5.

cos (a tan x + νx) cosν x dx = 2−ν−1 πe−a

[Re ν > −1,

a ≥ 0]

BI (44)(4)

0



π/2

6.

ν

cos (a tan x − γx) cosν x dx =

0



π/2

7. 0

(2a) πa 2 W γ2 ,− ν+1 2  ν γ+ν +1 22 Γ 1+ 2  a > 0, Re ν > −1,

sin nx − sin (nx − a tan x) cosn−1 x dx = sin x

 ν +γ = −1, −2, . . . 2

π/2 [n = 0, −a π (1 − e ) [n = 1,

EH I 274(13)a

a > 0] , a ≥ 0] LO V 153(114)

3.719 1.



π

6 0



π

2. 0

 3.

π

sin (νx − z sin x) dx = π Eν (z)

WA 336(2)

cos (nx − z sin x) dx = π J n (z)

WH

cos (νx − z sin x) dx = π Jν (z)

WA 336(1)

0

3.72–3.74 Combinations of trigonometric and rational functions 3.721



∞

1. 0



∞

2. 1

 3.

∞

8 1

3.722



∞

1. 0



2.11 3.

sin(ax) π dx = sign a x 2

FI II 645

sin(ax) dx = − si(a) x

BI 203(1)

cos(ax) dx = − ci(a) x

BI 203(5)

sin(ax) dx = ci(ab) sin(ab) − cos(ab) si(ab) x+b

[|arg b| < π,

a > 0] BI(16)(1), FI II 646a

∞

sin(ax) dx = πeiab −∞ x + b  ∞ cos(ax) dx = − sin(ab) si(ab) − cos(ab) ci(ab) x+b 0

[a > 0,

Im b > 0]

[|arg b| < π,

a > 0] ET I 8(7), BI(160)(2)

428

Trigonometric Functions

 4.

8

5.10

cos(ax) dx = −iπeiab −∞ x + b  ∞ sin(ax) dx = sin(ba) ci(ba) − cos(ba) [si(ba) + π] b−x 0 

6.8 7.12

∞

0

 0

0

4. 5.12 6. 7. 8.

b not real and positive]

[a > 0,

Im b > 0]

[a > 0,

b not real and positive]

sin(ax) 1 −ab e Ei(ab) − eab Ei(−ab) dx = b2 + x2 2b

[a > 0,

Im b > 0]

[a > 0,

Re b > 0]

cos(ax) π dx = e−ab 2 2 b +x 2b

[a ≥ 0,

x sin(ax) π dx = e−ab b2 + x2 2

[a > 0,



∞

10. 0

 11.12

0

Re b > 0]

BI (202)(10)

Re b > 0]

BI (160)(6)

b > 0,

c > 0]

LI (202)(9)

b > 0,

c > 0]

LI (202)(11)a

[|arg b| < π, ∞

9. 0

Re b > 0] FI II 741, 750, ET I 65(15), WH

∞

x sin(ax) dx = πe−ab [a > 0, 2 2 −∞ b + x  ∞

x cos(ax) 1 dx = − e−ab Ei(ab) + eab Ei(−ab) [a > 0, 2 2 b +x 2 0  ∞ sin[a(b − x)] π dx = e−ac sin(ab) [a > 0, 2 + x2 c c −∞  ∞ cos[a(b − x)] π dx = e−ac cos(ab) [a > 0, 2 2 c +x c −∞  ∞  sin(ax) π  1 sin(ab) ci(ab) − cos(ab) si(ab) + dx = 2 2 b 2 0 b −x 

Re b > 0] FI II 741, 750, ET I 8(11), WH

∞

3. 

[a > 0,

ET I 65(14), BI(160)(3) ∞

2. 

Im b > 0]

ET I 8(8), BI(161)(2)a ∞

cos(ax) dx = iπe−iab b − x −∞

3.723  1.12

[a > 0,

FI II 646, BI(161)(1) ∞

sin(ax) dx = −πeiab −∞ b − x  ∞ cos(ax) dx = cos(ab) ci(ab) + sin(ab) [si(ab) + π] b−x 0 

8.11

∞

3.723

∞

cos(ax) π sin(ab) dx = 2 2 b −x 2b

[a > 0,

x sin(ax) π dx = − cos(ab) b2 − x2 2

[a > 0]

a > 0]

b > 0]

BI (161)(3)

BI(161)(5), ET I 9(15) FI II 647, ET II 252(45)

 x cos(ax) π dx = cos(ab) ci(ab) + sin(ab) si(ab) + 2 2 b −x 2 [|arg b| < π,

a > 0]

BI (161)(6)

3.726

Trigonometric and rational functions

 12.

12

3.724 1.

2.

3.

PV

3.

sin(ax) cos(ab) − 1 dx = π x(x − b) b −∞

[a > 0,

b > 0]

 √ 2 cq − b  sin(aq) + c cos(aq) πe−a p−q p − q2

a > 0, p > q 2    ∞ √ 2 b + cx b − cq cos(ax) dx =  cos(aq) + c sin(aq) πe−a p−q 2 p − q2 −∞ p + 2qx + x

a > 0, p > q 2  ∞ cos[(b − 1)t] − x cos(bt) cos(ax) dx = πe−a sin t sin (bt + a cos t) 2 1 − 2x cos t + x −∞

a > 0, t2 < π 2



∞

b + cx sin(ax) dx = p + 2qx + x2 −∞

3.725  1. 2.12

∞

429

∞

ET II 252(44)



sin(ax) dx π  = 1 − e−aβ 2 + x2 ) 2 x (β 2β 0  ∞ sin(ax) dx π = 2 (1 − cos(ab)) PV 2 2 2b 0 x (b − x )  ∞ sin(ax) cos(bx) π −βb dx = e sinh(aβ) 2 2 x (x + β ) 2β 2 0 π π = − 2 e−aβ cosh(bβ) + 2 2β 2β

[Re β > 0, [a > 0]

a > 0]

BI (202)(12)

BI (202)(13)

BI (202)(14)

BI (172)(1) BI (172)(4)

[0 < a < b] [a > b > 0] ET I 19(4)

3.726 In each of the following integrals, take both lower (−) signs or both upper (+) signs.  ∞ x sin(ax) dx 1.11 3 ± b2 x + bx2 ± x3 b 0  1  −ab π  e Ei(ab) − eab Ei(−ab) − 2 ci(ab) sin(ab) + 2 cos(ab) si(ab) + =± 4b 2 πe−ab − π cos(ab) + 4b [a > 0, b > 0; if the lower sign is taken, then the integral is a principal value integral]  2.7 0

ET I 65(21)a, BI(176)(10, 13) ∞

x sin(ax) dx b3 ± b2 x + bx2 ± x3 2

= [a > 0,

b > 0;

 1  ab π  e Ei(−ab) − e−ab Ei(ab) + 2 ci(ab) sin(ab) − 2 cos(ab) si(ab) + 4 2  −ab ±π e + cos(ab)

if the lower sign is taken, then the integral is a principal value integral] ET I 66(22), BI(176)(11, 14)

430

3.727

Trigonometric Functions



∞

1. 0

√    cos(ax) ab ab π 2 ab cos √ + sin √ dx = exp − √ b4 + x4 4b3 2 2 2

 2.12

∞

PV 0

 3.12

PV

6.12

cos(ax) π dx = 3 e−ab + sin(ab) b4 − x4 4b [a > 0,



∞

7. 0

∞

PV



11.12 12.12

BI (161)(12)

BI (161)(16)

b > 0]

BI (160)(23)a

b > 0]

BI (161)(13)

π 2

(cf. 3.723 5 and 3.723 11)

[a > 0,

0

10.

b > 0]

BI (161)(17)

√    x2 cos(ax) ab ab ab π 2 √ √ √ exp − cos − sin dx = b4 + x4 4b 2 2 2



9.12

b > 0]

BI(160)(25)a, ET I 9(19)

(cf. 3.723 2 and 3.723 9)

∞

[a > 0,

8.12

b > 0]

  x sin(ax) π ab ab √ dx = exp − sin √ [a > 0, 4 + x4 2 b 2b 2 2 0  ∞

x sin(ax) π PV dx = 2 e−ab − cos(ab) [a > 0, 4 4 b −x 4b 0   ∞  x cos(ax) 1 2 cos(ab) ci(ab) + 2 sin(ab) si(ab) + PV dx = b4 − x4 4b2 0  − e−ab Ei(ab) − eab Ei(−ab)



5.12

[a > 0, b > 0]   sin(ax) π 1 dx = 3 2 sin(ab) ci(ab) − 2 cos(ab) si(ab) + 4 4 b −x 4b 2  −ab ab +e Ei(ab) − e Ei(−ab) [a > 0,

∞

0

4.

3.727

∞

b > 0]

x2 sin(ax) dx 1 2 sin(ab) ci(ab) = 4 4 b −x 4b   π − e−ab Ei(ab) + eab Ei(−ab) −2 cos(ab) si(ab) + 2 [a > 0, b > 0]

x cos(ax) π  sin(ab) − e−ab [a > 0, b > 0] dx = 4 − x4 b 4b 0    ∞ 3 x sin(ax) ab π ab √ exp − dx = cos √ [a > 0, b > 0] 4 + x4 b 2 2 2 0  ∞ 3

x sin(ax) −π −ab e PV dx = − cos(ab) [a > 0, b > 0] 4 4 b −x 4 0  ∞ 3  x cos(ax) dx 1  π PV = 2 cos(ab) ci(ab) + 2 sin(ab) si(ab) + 4 4 b −x 4 2 0  −ab ab +e Ei(ab) + e Ei(−ab)

BI (160)(26)a

BI (161)(14)

2

PV

[a > 0,

b > 0]

BI (161)(18) BI (160)(24) BI (161)(15)

BI(161)(19)

3.729

Trigonometric and rational functions



∞

13.

3

(x2 + b2 )

0



∞

14.

4

∞

0



∞

2. 0



x3 sin ax (x2 + b2 )

0

3.728  1.

x3 sin ax

∞

3. 0

dx =

πe−ab  3a − ba2 16b

dx =

πe−ab a  3 + 3ab − a2 b2 96b3

 π βe−aγ − γe−aβ cos(ax) dx = (β 2 + x2 ) (γ 2 + x2 ) 2βγ (β 2 − γ 2 )  π e−aβ − e−aγ x sin(ax) dx = (β 2 + x2 ) (γ 2 + x2 ) 2 (γ 2 − β 2 )  π βe−aβ − γe−aγ x cos(ax) dx = (β 2 + x2 ) (γ 2 + x2 ) 2 (β 2 − γ 2 ) 2

 ∞ π β 2 e−aβ − γ 2 e−aγ x3 sin(ax) dx = 2 2 2 2 2 (β 2 − γ 2 ) 0 (β + x ) (γ + x )

[a > 0,

431

Re β > 0,

Re γ > 0] BI (175)(1)

[a > 0,

Re β > 0,

Re γ > 0] BI (174)(1)

[a > 0,

Re β > 0,

Re γ > 0] BI (175)(2)

 4.

∞

PV

[a > 0,



∞

PV 0

 7.12

(b2 ∞

PV 

∞

PV 0

 9.12



∞

2.

+

x sin ax π e−ac − cos ba dx = 2 2 2 − x ) (c + x ) 2 a 2 + c2

2 x2 )

+

x2 )2

∞

3.

cos(px) 0

 4.

0

∞

=

x sin(ax) dx (b2

0



 π b2 cos(ab) − c2 cos(ac) x sin(ax) dx = (b2 − x2 ) (c2 − x2 ) 2 (b2 − c2 )

cos(ax) dx (b2

0

[a > 0]

BI (175)(3) BI (174)(3)

[a > 0,

b > 0,

c > 0]

BI (175)(4)

[a > 0,

b > 0,

c > 0]

BI (174)(4)

[a > 0,

c > 0,

b real]

[a > 0,

b > 0]

BI (170)(7)

[a > 0,

b > 0]

BI (170)(3)

3

(b2

0 ∞

c > 0]

x2 cos(ax) dx π (c sin(ac) − b sin(ab)) = − x2 ) (c2 − x2 ) 2 (b2 − c2 )

∞

PV

3.729  1.

x sin(ax) dx π (cos(ab) − cos(ac)) = 2 2 2 − x ) (c − x ) 2 (b2 − c2 )

b > 0,

(b2

0

8.12

Re γ > 0]

cos(ax) dx π (b sin(ac) − c sin(ab)) = 2 2 2 − x ) (c − x ) 2bc (b2 − c2 )

(b2

0

6.12

Re β > 0,

BI (174)(2)

 5.12

[a > 0,

=

+

π −ab ae 4b

1 − x2 (1 + x2 )

x3 sin(ax) dx (b2

π (1 + ab)e−ab 4b3

2 x2 )

=

2

dx =

πp −p e 2

π (2 − ab)e−ab 4

BI (43)(10)a

[a > 0,

b > 0]

BI (170)(4)

432

3.731 1.

Trigonometric Functions

√ √ Notation: 2A2 = b4 + c2 + b2 , 2B 2 = b4 + c2 − b2 ,  ∞ cos(ax) dx π e−aA (B cos(aB) + A sin(aB)) √ = 2 2 2 2 2c b 4 + c2 0 (x + b ) + c 

∞

2. 

x sin(ax) dx (x2

0 ∞

3.





∞



x x +b

∞



0

∞

2. 0

∞

2



0

4. 0



=

sin(ax) dx

b > 0,

c > 0]

BI (176)(3)

[a > 0,

b > 0,

c > 0]

BI (176)(1)

[a > 0,

b > 0,

c > 0]

BI (176)(4)

[a > 0,

b > 0,

c > 0]

BI (176)(2)

π e−aA (A cos(aB) − B sin(aB)) √ 2 b 4 + c2

=

π −aA e cos(aB) 2

ET I 65(16)

π 1 1 cos(ax) dx = e−aβ cos(aγ) + 2 β 2 + (γ − x)2 β + (γ + x)2 β 

[a > 0,



∞

1. 0



 3. 0

γ + iβ is not real] LI (175)(17)



|Im a| < Re β]

π sin (t + ab sin t) cos(ax) dx = 3 exp (−ab cos t) x4 + 2b2 x2 cos 2t + b4 2b sin 2t a > 0, b > 0, x4

0

ET I 8(13)

γ+x γ−x cos(ax) dx = πe−aβ sin(aγ) + 2 β 2 + (γ + x)2 β + (γ − x)2

∞

2.

|Im γ| < Re β]

γ+x γ−x sin(ax) dx = πe−aβ cos(aγ) − 2 2 + (γ + x) β + (γ − x)2 [a > 0, Re β > 0,

[a > 0, 3.733

γ + iβ is not real]



β2

∞

π −aA e sin(aB) 2c

[a > 0,

 1 π 1 sin(ax) dx = e−aβ sin(aγ) − 2 2 2 2 β + (γ − x) β + (γ + x) β [a > 0, Re β > 0,



3.



2

2

1.



=

(x2 + b2 ) + c2

0



+

c2

(x2 + b2 )2 + c2

4.



+

2 b2 )

x2 + b2 cos(ax) dx

0

3.732

3.731

x sin(ax) dx π sin (ab sin t) = 2 exp (−ab cos t) 2 2 4 + 2b x cos 2t + b 2b sin 2t a > 0,

b > 0,

LI (176)(21)

|t|
0,

|t|
0, b > 0, |t| <    ∞ sin(ax) dx π sin (2t + ab sin t) = 4 1 − exp (−ab cos t) 4 2 2 4 2b sin 2t 0 x (x + 2b x cos 2t + b ) a > 0, b > 0, |t|
0, 4 4 2b4 2 2 0 x (b + x )  ∞

sin(ax) dx π 12 = 4 2 − e−ab − cos(ab) 2. PV [a > 0, 4 − x4 ) x (b 4b 0  ∞

sin x π −a 3.73512 (2 + a) [a > 0] 2 dx = 4a4 2 − e 2 2 0 x (a + x ) 3.736  ∞

cos(ax) dx π = 5 sin(ab) + (2 + ab)e−ab 1.12 PV 2 2 4 4 8b 0 (b + x ) (b − x ) [a > 0,  ∞

x sin(ax) dx π = 4 (1 + ab)e−ab − cos(ab) 2.12 PV 2 2 4 4 8b 0 (b + x ) (b − x ) [a > 0,  ∞ 2

x cos(ax) dx π = 3 sin(ab) − abe−ab [a > 0, 3.12 PV 2 + x2 ) (b4 − x4 ) (b 8b 0  ∞

π x3 sin(ax) dx = 2 (1 − ab)e−ab − cos(ab) 4.12 PV 2 2 4 4 8b 0 (b + x ) (b − x ) [a > 0,  ∞

x4 cos(ax) dx π = sin(ab) + (ab − 2)e−ab 5.12 PV 2 2 4 4 8b 0 (b + x ) (b − x ) [a > 0,  ∞ 5

x sin(ax) dx π = (ab − 3)e−ab − cos(ab) 6.12 PV 2 + x2 ) (b4 − x4 ) (b 8 0 [a > 0, 1.12

3.737

433



∞

PV



1.12 0

∞

π 2

π 2



BI (176)(6)

BI (176)(22)

b > 0]

BI (172)(7)

b > 0]

BI (172)(10) WH, BI (172)(22)

b > 0]

BI (176)(5)

b > 0]

BI (174)(5)

b > 0]

BI (175)(6)

b > 0]

BI (174)(6)

b > 0]

BI (175)(7)

b > 0]

BI (174)(7)

n−1

 (2n − k − 2)!(2ab)k cos(ax) dx πe−ab n = 2 2 2n−1 (b + x ) (2b) (n − 1)! k!(n − k − 1)!  k=0  −ab√p  n−1 n−1 π d (−1) e = 2n−1 √ b (n − 1)! dpn−1 p p=1    n−1 n−1 −abp π d e (−1) = 2n−1 2b (n − 1)! dpn−1 (1 + p)n p=1 [a > 0, b > 0]

GW(333)(67b), WA 209, WA 192

434

Trigonometric Functions

 2.

∞

12

x sin(ax) dx (x2 +

0

n+1 b2 )

n−1 πae−ab  (2n − k − 2)!(2ab)k 22n n!b2n−1 k!(n − k − 1)! k=0 π = e−ab 2

=

[a > 0,

Re b > 0]

[n = 0,

a > 0,

b ≥ 0] GW (333)(66c)



 ∞ sin(ax) dx e−ab π = 2n+2 1 − n Fn (ab) 2 2 n+1 2b 2 n! 0 x (b + x )

 (z) a > 0, Re b > 0, F0 (z) = 1, F1 (z) = z + 2, . . . , Fn (z) = (z + 2n)Fn−1 (z) − zFn−1

 3.

3.738



GW (333)(66e) ∞

4.

(b2

0



∞

5.

+

∞

6.

3 x2 )

x sin(ax) dx (b2 +

0



x sin(ax) dx

4 x2 )

x3 sin ax (x2 + b2 )n+1

0

=

πa (1 + ab)e−ab 16b3

[a > 0,

b > 0]

BI(170)(5), ET I 67(35)a

πa  3 + 3ab + a2 b2 e−ab [a > 0, b > 0] BI(170)(6), ET I 67(35)a 5 96b  πe−ab dx = 2n 2n−2 2n−1 (2n − 3)!!(2 − ba) 2 n!b  n−1  (2n − k − 2)!2k (ba)k−1

k(k + 1) − 2(k + 1)ba + b2 a2 − k!(n − k − 1)!

=

k=1

3.738



∞

1.12 0

  n xm−1 sin(ax) πbm−2n  (2k − 1)π dx = − exp −ab sin x2n + b2n 2n 2n k=1   (2k − 1)mπ (2k − 1)π × cos + ab cos 2n 2n  [m is even] ,

=0

a > 0,

|arg b|
0,

|arg b|
0,

∞

2.

(x2

0

+

12 ) (x2

n ≥ 0]

LI(174)(8)

cos(ax) dx + 32 ) . . . [x2 + (2n + 1)2 ]   n π  (−1)n k 2n + 1 e(2k−2n−1)a (−1) = (2n + 1)! 22n+1 k =



435

[a ≥ 0,

n ≥ 0]

[a = 0,

n ≥ 0]

k=0

π2−2n−1 2

(2n + 1) (n!)

BI(175)(8) ∞

3.

(x2

0

+

12 ) (x2

x sin(ax) dx + 32 ) . . . [x2 + (2n + 1)2 ]   n  π(−1)n k 2n + 1 (2n − 2k + 1)e(2k−2n−1)a (−1) = (2n + 1)!22n+1 k k=0

[a > 0, n ≥ 0]   n ∞ cos ax dx π21−2n  2n k = (−1) k e−2ak 2 2 2 2 2 2 (2n)! n−k 0 (x + 2 ) (x + 4 ) . . . (x + 4n )

LI (174)(9)

 4.

k=1

3.741



∞

1. 0



∞

2. 0

 3. 0

sin(ax) sin(bx) 1 dx = ln x 4 sin(ax) cos(bx) π dx = x 2 π = 4 =0



a+b a−b

[n ≥ 1,

a ≥ 0]

[a > 0,

b > 0,

2 a = b]

FI II 647

[a > b ≥ 0] [a = b > 0] [b > a ≥ 0] FI II 645

∞

sin(ax) sin(bx) aπ dx = x2 2 bπ = 2

[0 < a ≤ b] [0 < b ≤ a] BI (157)(1)

436

3.742

Trigonometric Functions



∞

1. 0



[c > 0,

b ≥ a ≥ 0]

 cos(ax) cos(bx) π  −|a−b|c −(a+b)c e dx = + e c2 + x2 4c π = e−ac cosh bc 2c π = e−bc cosh ac 2c

Re c > 0]

BI (162)(3)

[a > 0,

b > 0,

Re c > 0]

[c > 0,

a ≥ b ≥ 0]

[c > 0,

b ≥ a ≥ 0] BI(163)(1)a, GW(333)(71c)

∞

4. 0

 x cos(ax) cos(bx) 1 dx = − eac ebc Ei [−c(a + b)] + e−bc Ei [c(b − a)] c2 + x2 4

 1 − e−ac ebc Ei [c(a − b)] + e−bc Ei [c(a + b)] 4 =∞

[a = b] [a = b] BI (163)(2)

∞

5. 0

x sin(ax) cos(bx) π dx = e−ac cosh(bc) x2 + c2 2 π = e−2ac 4 π = − e−bc sinh(ac) 2

 PV

 PV 0

[0 < b < a] [0 < b = a] [0 < a < b] BI (162)(4)

∞

0

7.12

a ≥ b ≥ 0]

∞

0

6.12

[c > 0,

 sin(ax) cos(bx) 1 dx = e−ac ebc Ei [c(a − b)] + e−bc Ei [c(a + b)] 2 2 c +x 4c

 1 − eac ebc Ei [−c(a + b)] + e−bc Ei [c(b − a)] 4c

3.



b > 0,

BI(162)(1)a, GW(333)(71a)

0



[a > 0,

∞

2.



 sin(ax) sin(bx) π  −|a−b|c e dx = − e−(a+b)c 2 2 c +x 4c π = e−ac sinh bc 2c π = e−bc sinh ac 2c

3.742

sin(ax) sin(bx) π dx = − cos(ap) sin(bp) p2 − x2 2p π = − sin(2ap) 4p π = − sin(ap) cos(bp) 2p

[a > b > 0] [a = b > 0] [b > a > 0] BI (166)(1)

∞

sin(ax) cos(bx) π x dx = − cos(ap) cos(bp) p2 − x2 2 π = − cos(2ap) 4 π = sin(ap) sin(bp) 2

[a > b > 0] [a = b > 0] [b > a > 0] BI (166)(2)

3.747

Trigonometric and rational functions

 8.

12

∞

PV 0

cos(ax) cos(bx) π sin(ap) cos(bp) dx = p2 − x2 2p π sin(2ap) = 4p π cos(ap) sin(bp) = 2p

437

[a > b > 0] [a = b > 0] [b > a > 0] BI (166)(3)

3.743 1.

12



∞

PV 0

 2.12

∞

PV 0

 3.12

∞

PV 0

 4.

12

∞

PV 0

 5.6

∞

PV 0



3.74512 

ET I 80(21)

sin(ax) x dx π sinh(ac) =− 2 2 cos(bx) x + c 2 cosh(bc)

[0 < a < b,

Re c > 0]

ET I 81(30)

π cosh(ac) cos(ax) x dx = sin(bx) x2 + c2 2 sinh(bc)

[0 < a < b,

Re c > 0]

ET I 23(37)

cos(ax) dx π cosh(ac) = cos(bx) x2 + c2 2c cosh(bc)

[0 < a < b,

Re c > 0]

ET I 23(36)

sin(ax) dx =0 sin x b2 − x2 π = sin(a − 1)b b

if 0 ≤ a ≤ 1 if 1 ≤ a ≤ 2 [b real, b/π ∈ Z]

∞

xn+1

0



∞

2. 0



π/2



π/2

∞

12 0

 4.

0

ET I 82(32)

[0 < a < b,

c > 0]

ET I 82(31)

k=1

⎡ ⎣a >

n 

 ak ,

ak > 0

k=1 n  k=1

|ak | +

m 

FI II 646

⎤ |bj |⎦

WH

j=1

  ∞  π m 1  xm 22k−1 − 1 7 dx = + ζ(2k) = 2πG − ζ(3) sin x 2 m 42k−1 (m + 2k) 2 k=1

2. 0

Re β > 0]

a0 >

k=1

k=1

0



k=0

[0 < a < b,



n π 1 sin (ak x) = ak 2

n m n 1 1 π 1 sin(ax) dx sin (a x) cos (b x) = ak k j xn+1 2 j=1

1.7

3.

Re c > 0]

n dx 1

∞

1.

3.747

[0 < a < b,

sin(ax) dx π sinh(aβ) = 2 + β2) 2 cosh(bβ) cos(bx) x (x 2β 0 ∞ sin(ax) dx =0 PV 2 − x2 ) cos(bx) x (c 0

3.74412 PV

3.746

sin(ax) dx π sinh(ac) = sin(bx) x2 + c2 2c sinh(bc)

π

x dx = sin x



 π/2 π 2

0

− x dx = 2G cos x

π x dx = (x2 + a2 ) sin x 2 sinh a

x tan x dx = −π ln 2

[m = 2]

LI (206)(2)

BI(204)(18), BI(206)(1), GW(333)(32)

[a > 0]

GW (333)(79c) BI (218)(4)

438

Trigonometric Functions



π/2

5.

3.748

x tan x dx = ∞

BI (205)(2)

0



π/4

6.

x tan x dx = −

0



π/2

7.

x cot x dx =

0

 8. 9. 10. 11. 3.748 1.

2.

3.

π/4

1 π ln 2 + G 8 2

BI (204)(1)

π ln 2 2

FI II 623

1 π ln 2 + G 8 2 0   π/2    1 π π π π − x tan x dx = − x tan x dx = ln 2 2 2 0 2 2 0  ∞ π dx = [a > 0] tan ax x 2 0  π/2 x cot x π dx = ln 2 cos 2x 4 0 x cot x dx =

BI (204)(2)

GW(333)(33b), BI(218)(12) LO V 279(5)

BI (206)(12)

∞  1  π m  4k − 1 ζ(2k) 2 4 42k−1 (m + 2k) 0 k=1    π/2 ∞  π p 1  1 p −2 ζ(2k) x cot x dx = 2 p 4k (p + 2k) 0 k=1    π/4 ∞  ζ(2k) 1  π m 2 m − x cot x dx = 2 4 m 42k−1 (m + 2k) 0



π/4

xm tan x dx =

LI (204)(5)

LI (205)(7)

LI (204)(6)

k=1

3.749



∞

1. 0

 2. 3.

π x tan(ax) dx = 2ab x2 + b2 e +1

[a > 0,

b > 0]

GW (333)(79a)

∞

x cot(ax) dx π [a > 0, b > 0] = 2ab 2 + b2 x e −1 0  ∞  ∞  ∞ x tan(ax) dx x cot(ax) dx x cosec(ax) dx = = =∞ 2 2 2 2 b −x b −x b2 − x2 0 0 0

GW (333)(79b) BI (161)(7, 8, 9)

3.75 Combinations of trigonometric and algebraic functions 3.751



∞

1. 0

 2.9 0

∞

sin(ax) dx √ = x+b cos(ax) dx √ = x+b





√  √   π  cos(ab) − sin(ab) + 2 C ab sin(ab) − 2 S ab cos(ab) 2a [a > 0,

ET I 65(12)a |arg b| < π] √  √   π  cos(ab) + sin(ab) − 2 C ab cos(ab) − 2 S ab sin(ab) 2a

[a > 0,

|arg b| < π]

ET I 8(9)a

3.755

Trigonometric and algebraic functions



∞

3. u  ∞

4. u

3.752 1.



1

8

sin(ax) √ dx = x−u cos(ax) √ dx = x−u

 



1

1

1.8 0

 2. 3. 4.12 5. 3.754 1. 2.

[a > 0,

u > 0]

ET I 8(10)

(−1)k a2k+1 π = H1 (a) (2k − 1)!!(2k + 3)!! 2a



KU 65(6)a

∞

BI (149)(9)

1

∞

π sin(ax) dx √ = [I 0 (ab) − L0 (ab)] 2 + x2 2 b 0  ∞ cos(ax) dx √ = K 0 (ab) b2 + x2 0

WA 30(7)a

[a > 0]

WA 200(14)

[a > 0]

WA 200(15)

WA 30(6)

[a > 0,

Re b > 0]

[a > 0,

Re b > 0]

ET I 66(26)

WA 191(1), GW(333)(78a) ∞

x sin(ax)  dx = a K 0 (ab) (b2 + x2 )3

√  x2 + b2 − b sin(ax) dx π −ab √ e = 2 2 2a x +b 0   ∞ √ 2 x + b2 + b cos(ax) dx π −ab √ e = 2 2 2a x +b 0



BI (149)(6)

sin(ax) dx  (−1)k a2k+1 π √ = 2 = 2 H0 (a) 1 − x2 [(2k + 1)!!] k=0

0

2.

π [cos(au) − sin(au)] 2a

 π J 1 (a) cos(ax) 1 − x2 dx = 2a

3.

1.

ET I 65(13)

cos(ax) dx π √ = J 0 (a) 2 2 1−x 0  ∞ sin(ax) dx π √ = J 0 (a) 2−1 2 x 1  ∞ cos(ax) π √ dx = − Y 0 (a) 2 2 x −1 1  1 x sin(ax) π √ dx = J 1 (a) 2 2 1−x 0



3.755

u > 0]

[a > 0]

0



[a > 0,

k=0

2. 3.753

π [sin(au) + cos(au)] 2a

∞   sin(ax) 1 − x2 dx =

0

439

∞

[a > 0,

Re b > 0]

[a > 0] [a > 0,

ET I 66(27)

ET I 66(31)

Re b > 0]

ET I 10(25)

440

3.756

Trigonometric Functions



∞

1. 0



∞

2.

x



∞

0

2.

ak > 0,

a>



cos (ak x) dx = 0

∞

12 0

sin x √ dx = x cos x √ dx = x

 

n 

 ak

k=2

ak > 0,

a>

k=1

1.12 

n 1

cos(ax)

0

3.757



n sin(ax) 1 sin (ak x) dx = 0 n x 2 −1 k=2 n 2 −1

3.756

n 

ET I 80(22)

 ak

ET I 22(26)

k=1

π 2

BI (177)(1)

π 2

BI (177)(2)

3.76–3.77 Combinations of trigonometric functions and powers 3.761



1

1.

xμ−1 sin(ax) dx =

0



u

∞

3. 1



1F 1

(μ; μ + 1; −ia)] [a > 0,

Re μ > −1,

μ = 0] ET I 68(2)a

∞

2.8 

−i [ 1 F 1 (μ; μ + 1; ia) − 2μ

π i − π2 iμ e xμ−1 sin x dx = Γ(μ, iu) − e 2 iμ Γ(μ, −iu) 2 sin(ax) a2n−1 dx = x2n (2n − 1)!

2n−1  (2n − k − 1)! k=1

a2n−k

[Re μ < 1]



 π + (−1)n ci(a) sin a + (k − 1) 2 [a > 0]

∞

4.

xμ−1 sin(ax) dx =

0

EH II 149(2)

μπ 2

π sec Γ(μ) μπ = μ sin aμ 2 2a Γ(1 − μ)

[a > 0,

LI (203)(15)

0 < |Re μ| < 1] FI II 809a, BI(150)(1)



π

5.10 0



1

6.8

xμ−1 cos(ax) dx =

0



m/2 m! (−1)n+1  (nπ)m−2k (−1)k nm+1 (m − 2k)! k=04 5 4 5 m! m−2 m m/2 2 −1 −(−1) nm+1

xm sin(nx) dx =

1 [ 1 F 1 (μ; μ + 1; ia) + 2μ

1F 1

(μ; μ + 1; −ia)] [a > 0,

∞

7. u

GW(333)(6)

Re μ > 0]

ET I 11(2)

π 1 − π iμ e 2 Γ(μ, iu)s + e 2 iμ Γ(μ, −iu) xμ−1 cos x dx = 2 [Re μ < 1]

EH II 149(1)

3.763

Trigonometric functions and powers



∞

8. 1



441

 2n   cos(ax) a2n  (2n − k)! π + (−1)n+1 ci(a) dx = cos a + (k − 1) x2n+1 (2n)! a2n−k+1 2 k=1

[a > 0] ∞

9.8

xμ−1 cos(ax) dx =

0

μπ 2

π cosec Γ(μ) μπ = μ cos μ a 2 2a Γ(1 − μ)

[a > 0,

LI (203)(16)

0 < Re μ < 1] FI II 809a, BI(150)(2)

 10.

π

0



(−1)n nm+1

(m−1)/2



m! (nπ)m−2k−1 (m − 2k − 1)! k=0 (m+1)/2 2 (m + 1)/2 − m +(−1) m! nm+1

xm cos(nx) dx =

π/2

11.

m/2 m

x cos x dx =

0



(−1)k

k=0

(−1)k

GW (333)(7)

 π m−2k   6m7 m! − m m! + (−1)m/2 2 (m − 2k)! 2 2 GW (333)(9c)



2nπ

12.

xm cos kx dx = −

0

3.762



m−1  j=0

∞

1.

j! k j+1

xμ−1 sin(ax) sin(bx) dx =

0



 m j+1 π (2nπ)m−j cos 2 j

BI (226)(2)

  μπ 1 −μ cos Γ(μ) |b − a| − (b + a)−μ 2 2 [a > 0, b > 0,

a = b,

−2 < Re μ < 1]

(for μ = 0, see 3.741 1, for μ = −1, see 3.741 3) 

BI(149)(7), ET I 321(40) ∞

2. 0



[a > 0, ∞

3.

  μπ 1 −μ sin Γ(μ) (a + b)−μ + |a − b| sign(a − b) 2 2 b > 0, |Re μ| < 1] (for μ = 0 see 3.741 2) BI(159)(8)a, ET I 321(41)

xμ−1 sin(ax) cos(bx) dx =

xμ−1 cos(ax) cos(bx) dx =

0

  μπ 1 cos Γ(μ) (a + b)−μ + |a − b|−μ 2 2 [a > 0, b > 0,

0 < Re μ < 1] ET I 20(17)

3.763



1. 0

∞

 sin(ax) sin(bx) sin(cx) νπ 1 ν−1 cos Γ(1 − ν) (c + a − b) dx = − (c + a + b)ν−1 xν 4 2  − |c − a + b|ν−1 sign(a − b − c) + |c − a − b|ν−1 sign(a + b − c) [c > 0,

0 < Re ν < 4,

ν = 1, 2, 3, . . . ,

a ≥ b > 0]

GW(333)(26a)a, ET I 79(13)

442

Trigonometric Functions



∞

2. 0



∞

3. 0



∞

4. 0

3.764  1.

sin(ax) sin(bx) sin(cx) dx = 0 x π = 8 π = 4

2.

3.765  1.12

∞

[c = a − b and c = a + b] [a − b < c < a + b] [a ≥ b > 0,

c > 0]

∞

2. 0

FI II 645

πbc sin(ax) sin(bx) sin(cx) [0 < c < a − b and c > a + b] dx = x3 2 2 πbc π(a − b − c) − [a − b < c < a + b] = 2 8 [a ≥ b > 0, c > 0] BI(157)(20), ET I 79(12)

xp sin(ax + b) dx =

1

ap+1

[a > 0,

−1 < p < 0]

GW (333)(30a)

[a > 0,

−1 < p < 0]

GW (333)(30b)

sin ax i Γ(1 − ν) iab dx = e Γ(ν, iab) − e−iab Γ(ν, −iab) ν + b) 2b [Im a = 0, −1 < Re b < 2,

xν (x

0



[c < a − b and c > a + b]

sin(ax) sin(bx) sin(cx) 1 dx = (c + a + b) ln(c + a + b) 2 x 4 1 1 − (c + a − b) ln(c + a − b) − |c − a − b| ln |c − a − b| 4 4 1 × sign(a + b − c) + |c − a + b| ln |c − a + b| sign(a − b − c) 4 [a ≥ b > 0, c > 0] BI(157)(8)a, ET I 79(11)

 pπ  Γ(1 + p) cos b + 2 0  ∞  1 πp  xp cos(ax + b) dx = − p+1 Γ(1 + p) sin b + a 2 0 ∞

3.764

cos(ax) Γ(1 − ν) iab dx = e Γ(ν, iab) + e−iab Γ(ν, −iab) ν ν x (x + b) 2b [a > 0,

|Re ν| < 1,

arg b = π]

MC

|arg b| < π] ET II 221(52)

3.766 1.



∞

 μπ    xμ−1 sin ax 1 π sinh a −a−iπ(1−μ) a  sin Γ(mu) e + dx = γ(1 − μ, −a) − e γ(1 − μ, a) μπ 1 + x2 2 2 2 cos 2 MC [Im a = 0, −1 < Re μ < 3]

∞

xμ−1 cos(ax) μπ π cosh a dx = cosec 1 + x2 2 2   μπ 1 Γ(μ) e−a−iπ(1−μ) γ(1 − μ, −a) − ea γ(1 − μ, a) + cos 2 2 [a > 0, 0 < Re μ < 3] ET I 319(24)

12 0

 2.

12 0

3.768

Trigonometric functions and powers

 3.

∞

π x2μ+1 sin(ax) dx = − b2μ sec(μπ) sinh(ab) 2 2 x +b 2 sin(μπ) + Γ(2μ) [ 1 F 1 (1; 1 − 2μ; ab) + 1 F 1 (1; 1 − 2μ; −ab)] 2a2μ

a > 0, − 23 < Re μ < 12 ET II 220(39)

∞

x2μ+1 cos(ax) dx x2 + b2

9 0

 4.

12 0

cosh(ab) cos(μπ) π + = − b2μ Γ(2u) [1 F1 (1; 1 − 2μ; ab) + 1 F1 (1; 1 − 2μ; −ab)] 2 sin(μπ) 2a2μ

a > 0, −1 < Re μ < 12 ET II 221(56)

3.767  1.

∞ xβ−1

 sin ax − γ2

0

2.

+

∞ xβ−1

 π dx = − γ β−2 e−aγ 2

βπ 2



sin ax −

  ∞ xβ cos ax − 4.

dx =

βπ 2

βπ 2



x2 − b2

0

Re γ > 0,

0 < Re β < 2] BI (160)(20)



x2 − b2

0

[a > 0,



γ 2 + x2

0

3.

βπ 2

x2

  ∞ xβ cos ax −



443

π β−1 −aγ γ e 2

[a > 0,

Re γ > 0,

|Re β| < 1] BI (160)(21)

  π β−2 πβ dx = b cos ab − 2 2

[a > 0,

b > 0,

0 < Re β < 2] BI (161)(11)

  π πβ dx = − bβ−1 sin ab − 2 2

[a > 0,

b > 0,

|β| < 1] GW (333)(82)

3.768 1. 2. 3.11



 Γ(μ) μπ  [a > 0, 0 < Re μ < 1] sin au + aμ 2 u  ∞  Γ(μ) μπ  [a > 0, 0 < Re μ < 1] (x − u)μ−1 cos(ax) dx = μ cos au + a 2 u  1 1 Γ(ν + 1) (1 − x)ν sin(ax) dx = − C ν (a) = a−ν−1/2 s ν+1/2,1/2 (a) a aν+1 0 [a > 0, Re ν > −1] ∞

(x − u)μ−1 sin(ax) dx =

ET II 203(19) ET II 204(24)

Here C ν (a) is the Young’s function given by: C ν (a) =

1 ν 2a

Γ(ν + 1)

[ 1 F 1 (1; ν + 1; ia) +

1F 1

(1; ν + 1; −ia)] =

∞  (−1)n aν+2n Γ(ν + 2n + 1) n=0

ET I 11(3)a

444

Trigonometric Functions



1

4.3

   i i −ν−1 a exp (νπ − 2a) γ (ν + 1, −ia) 2 2    i − exp − (νπ − 2a) γ (ν + 1, ia) 2  2 n ∞  −a = Γ(ν + 1) Γ(ν + 2 + 2n) n=0

(1 − x)ν cos(ax) dx =

0



[a > 0,

u

5.

xν−1 (u − x)μ−1 sin(ax) dx =

0



u

6.

xν−1 (u − x)μ−1 cos(ax) dx =

0

 0

8.

9.

Re ν > −1]

ET I 11(3)a

uμ+ν−1 B(μ, ν) [ 1 F 1 (ν; μ + ν; iau) − 1 F 1 (ν; μ + ν; −iau)] 2i [a > 0, Re μ > 0, Re ν > −1, ν = 0] ET II 189(26) uμ+ν−1 B(μ, ν) [ 1 F 1 (ν; μ + ν; iau) + 1 F 1 (ν; μ + ν; −iau)] 2 [a > 0, Re μ > 0, Re ν > 0] ET II 189(32)

u

7. 

3.768

 au  √  u μ−1/2 au Γ(μ) J μ−1/2 xμ−1 (u − x)μ−1 sin(ax) dx = π sin a 2 2 [Re μ > 0]

ET II 189(25)

∞

xμ−1 (x − u)μ−1 sin(ax) dx u √  μ−1/2  au   au   π u au au J 1/2−μ − sin Y 1/2−μ = Γ(μ) cos 2 a 2 2 2 2 a > 0, 0 < Re μ < 12 ET II 203(20)  u  au  √  u μ− 12 au Γ(μ) J μ− 12 xμ−1 (u − x)μ−1 cos(ax) dx = π cos a 2 2 0 ET II 189(31) [Re μ > 0] √ 1 ∞  au   au   π  u μ− 2 au au J 12 −μ − cos Y 12 −μ xμ−1 (x − u)μ−1 cos(ax) dx = − Γ(μ) sin 2 a 2 2 2 2 u

1 ET II 204(25) a > 0, 0 < Re μ < 2  1 i xν−1 (1 − x)μ−1 sin(ax) dx = − B(μ, ν) [ 1 F 1 (ν; ν + μ; ia) − 1 F 1 (ν; ν + μ; −ia)] 2 0 [a > 0, Re μ > 0, Re ν > −1, ν = 0] ET I 68 (5)a, ET I 317(5)

 10.

11.12



1

12.12

xν−1 (1 − x)μ−1 cos(ax) dx =

0

 13. 0

1

xμ (1 − x)μ sin(2ax) dx =

1 B(μ, ν) [ 1 F 1 (ν; ν + μ; ia) + 1 F 1 (ν; ν + μ; −ia)] 2 [a > 0, Re μ > 0, Re ν > 0]

√

ET I 11(5)

π

1 (2a)μ+ 2

Γ(μ + 1) J μ+ 12 (a) sin a [a > 0,

Re μ > −1]

ET I 68(4)

3.771

Trigonometric functions and powers

 14.

1

xμ (1 − x)μ cos(2ax) dx =

0

√ π 1

(2a)μ+ 2

Γ(μ + 1) J μ+ 12 (a) cos a [a > 0,

3.769



∞

1. 0





πiaν−1 e−ab (b + ix)−ν − (b − ix)−ν sin(ax) dx = − Γ(ν) [a > 0,

πa (b + ix)−ν + (b − ix)−ν cos(ax) dx =

0

∞

0

0

∞

0



i (−1) (2n)!πaν−2n−1 e−ab Lν−2n−1 x2n (b − ix)−ν − (b + ix)−ν sin(ax) dx = (ab) 2n Γ(ν) [a > 0, Re b > 0, 0 ≤ 2n ≤ Re ν] n+1



(−1)n (2n)!πaν−2n−1 e−ab Lν−2n−1 x2n (b + ix)−ν + (b − ix)−ν cos(ax) dx = (ab) 2n Γ(ν) [a > 0, Re b > 0, 0 ≤ 2n < Re ν] 

x2n+1 (b + ix)−ν + (b − ix)

−ν



n+1

(−1) (2n + 1)!πaν−2n−2 e−ab Lν−2n−2 (ab) 2n+1 Γ(ν) [a > 0, Re b > 0, −1 ≤ 2n + 1 < Re ν] ET I 70(18)

sin(ax) dx =



1 2 ν− 2

b +x 2

0

0

Re ν > 0]

∞

1.

2.

Re ν > 0]

  (−1)n+1 i −ν (2n + 1)!πaν−2n−2 e−ab Lν−2n−2 cos(ax) dx = x2n+1 (b + ix)−ν − (b − ix) (ab) 2n+1 Γ(ν) [a > 0, Re b > 0, 0 ≤ 2n < Re ν − 1] ET I 13(21)

0



Re b > 0,

∞

7.



[a > 0,

πaν−2 (1 − ab) −ab e x (b + ix)−ν + (b − ix)−ν sin(ax) dx = − Γ(ν) [a > 0, Re b > 0,

0

3.771

e Γ(ν)

ET I 13(20) ∞

6.



ν−1 −ab

ET I 70(17)

5.



Re ν > 0]

ET I 70(16) ∞

4.12



Re b > 0,

ET I 11(4)

ET I 13(19)

3.12



Re μ > −1]

ET I 70(15) ∞

2.



445

∞



1 2 ν− 2

b +x 2

√  ν  π 2b sin(ax) dx = Γ ν + 12 [I −ν (ab) − Lν (ab)] 2 a

a > 0, Re b > 0, Re ν < 12 , ν = − 12 , − 32 , − 25 , . . . 1 cos(ax) dx = √ π



2b a

ν

 cos(πν) Γ ν + 12 K −ν (ab) a > 0, Re b > 0,

EH II 38a, ET I 68(6)

Re ν
0, Re ν > − 12    u  2 a2 u 2 1 1 2μ+2ν−2 2ν−1 2 μ−1 u −x x cos(ax) dx = u B(μ, ν) 1 F 2 ν; , μ + ν; − 2 2 4 0  a 2μ+2ν−1 = u B μ, ν + 2

0

4.

3.771



∞

5.7

ν− 12  x x2 + b2

0



u

6.



u2 − x2

ν− 12



∞

7.



x2 − u2

u





u2 − x2

ν− 12

0

∞

9. u



u > 0,

Re ν > − 21

ET I 69(7), WA 358(1)a

u > 0,

|Re ν|
0,

ν− 12

u > 0,

√  ν  π 2u cos(ax) dx = − Γ ν + 12 Y −ν (au) 2 a a > 0, u > 0,

Re ν > − 21

|Re ν|
0,

EH II 81(12)a, ET I 69(8), WA 187(3)a

u

8.



ET II 189(29)

ν

√  ν  π 2u sin(ax) dx = Γ ν + 12 J −ν (au) 2 a a > 0,

ν− 12



ET II 190(35) [Re μ > 0, Re ν > 0]   2b 1 1 K ν+1 (ab) sin(ax) dx = √ b cos νπ Γ ν + π  a 2 ν √ 1 2b 1 K ν+1 (ab) = πb a Γ 2 −ν [a > 0, Re b > 0, Re ν < 0] ET I 69(11)



sin(ax) dx =

0

1 a2 u 2 ;− 2 4

∞

11. u

ν− 12  x x2 − u2

√  ν  π 2u u sin(ax) dx = Γ ν + 12 J ν+1 (au) 2 a a > 0, u > 0, √  ν  π 2u u sin(ax) dx = Γ ν + 12 Y −ν−1 (au) 2 a a > 0, u > 0,

Re ν > − 21

ET I 69(9)

− 12 < Re ν < 0

ET I 69(10)

3.772

Trigonometric functions and powers

 12.

u

12 0



ν− 12  uν+1 x u2 − x2 cos(ax) dx = − ν s ν,ν+1 (au) a −1 √  ν  π 1 2u 1 2ν+1 ν+ u = u − Γ ν + 12 Hν+1 (au) 2 2 a 2

a > 0, u > 0, Re ν > − 12 ET I 12(10)

∞

13. u

3.772





x x −u

∞



∞



1.

2

2 ν−1/2

√  ν  πu 2u cos(ax) dx Γ ν + 12 J −ν−1 (au) 2 a a > 0, u > 0,

ν−1/2 x2 + 2bx sin(ax) dx =

0

 2.

447

0 < Re ν
0, |arg b| < π, 12 > Re ν > − 32 ET I 69(12)

ν−1/2 x2 + 2bx cos(ax) dx

√  ν  π 2b Γ ν + 12 [Y −ν (ab) cos(ab) − J −ν (ab) sin(ab)] 2 a

a > 0, |Re ν| < 12 ET I 12(13)  ν  2u  ν−1/2  √ 2u 2ux − x2 sin(ax) dx = π Γ ν + 12 sin(au) J ν (au) a 0

a > 0, u > 0, Re ν > − 21 0

=−

3.



ET I 69(13)a

∞

4. 2u



2u

5. 0



∞

6. 2u

√  ν  2 ν−1/2  π 2u x − 2ux sin(ax) dx = Γ ν + 12 [J −ν (au) cos(au) − Y −ν (au) sin(au)] 2 a

a > 0, u > 0, |Re ν| < 12 ET I 70(14) 

2ux − x2

ν−1/2

cos(ax) dx =

√ π



2u a

ν

 Γ ν + 12 J ν (au) cos(au) a > 0, u > 0,

Re ν > − 21

ET I 12(4)

√  ν  2 ν−1/2  π 2u x − 2ux cos(ax) dx = − Γ ν + 12 [J −ν (au) sin(au) + Y −ν (au) cos(au)] 2 a

a > 0, u > 0, |Re ν| < 12 ET I 12(12)

448

3.773 1.

Trigonometric Functions



∞

12

x2ν (x2 + b2 )μ+1

0

3.773

sin(ax) dx   3 b 2 a2 1 2ν−2μ b a B (1 + ν, μ − ν) 1 F 2 ν + 1; ν + 1 − μ, ; 2√ 2 4   πa2μ−2ν+1 b 2 a2 3 Γ(ν − μ)  + + 1 F 2 μ + 1; μ − ν + , μ − ν + 1; 4μ−ν+1 2 4 Γ μ − μ + 32   √ 2 2  −ν + 1 π a b  2 b2ν−2μ−1 G 21 =  13 2 Γ(μ + 1) 4  μ − ν + 12 , 12 , 0 [a > 0, Re b > 0, −1 < Re ν < Re μ + 1] ET I 71(28)a, ET II 234(17)

=

 2.

∞

8

(z +

0



∞

3.12 0

 0

∞

5.

dx =

(−1)n+m π dn  m −a√z  z e n! 2 dz n

[a > 0,

√ x sin(ax) dx π d2m+1 n (−1)  = [a K n (ab)] 1 n+ 2n bn Γ n + 12 da2m+1 (b2 + x2 ) 2 [a > 0,

0 ≤ m ≤ n,

|arg z| < π] ET I 68(39)

m+1

2m+1

Re b > 0,

−1 ≤ m < n]

    x cos(ax) dx 1 2ν−2μ−1 1 1 1 b 2 a2 1 1 = b B ν + ,μ− ν + 1F 2 ν + ; ν − μ + , ; 2 2 2 2 2 2 4 (x2 + b2 )μ+1   √ 2μ−2ν+1  Γ ν − μ − 12 3 b 2 a2 πa + 1 F 2 μ + 1; μ − ν + 1, μ − ν + ; 4μ−ν+1 Γ(μ − ν +1)  2 4  √ 2 2  −ν + 1 π a b  2 b2ν−2μ−1 G 21 =  13 2 Γ(μ + 1) 4  μ − ν + 12 , 0, 12

a > 0, Re b > 0, − 12 < Re ν < Re μ + 1 ET I 14(29)a, ET II 235(19) 2ν

x2m cos(ax) dx n+1

(z + x2 )

0



n+1 x2 )

ET I 67(37)

∞

4.



x2m+1 sin(ax)

∞

6.7 0

= (−1)m+n

π dn  m− 1 −a√z  z 2e 2n! dz n

[a > 0,

√ x2m cos(ax) dx d2m (−1)m π 2m [an K n (ab)]  = 1 1 n n n+ 2 b Γ n + 2 da (b2 + x2 ) 2 a > 0,

n + 1 > m ≥ 0,

|arg z| < π] ET I 10(28)

Re b > 0,

0≤m 0, b > 0, Re ν > −1] ET I 70(19)

3.775

Trigonometric functions and powers



∞

2. 0



449

  cos(ax) dx 1 π 1 νπ Jν (iab) + Jν (−iab) − cos I ν (ab) √ √  ν = ν b sin(νπ) 2 2 2 x2 + b2 x + x2 + b2 [a > 0, b > 0, Re ν > −1] √ ν      x + x2 + b2 aπ ν ab ab  b I 14 − ν2 K 14 + ν2 sin(ax) dx = 2 2 2 2 2 x (x + b ) a > 0, Re b > 0,

ET I 12(15)

 ∞

3. 0

√ ν      ∞ x2 + b2 − x aπ ν ab ab  b I − 14 + ν2 K − 14 − ν2 cos(ax) dx = 2 + b2 ) 2 2 2 x (x 0 a > 0, Re b > 0,

Re ν
0, Re b > 0,

Re ν > − 23

ET I 12(17)

 ∞ 5. 0



Re ν
0, Re b > 0, Re ν < 12

 ∞

6. 0

ET I 12(18)

3.775 1.



12 0

2.12

∞

√ ν √ ν x2 + b2 + x − x2 + b2 − x νπ √ K ν (ab) sin(ax) dx = 2bν sin 2 2 2 x +b [a > 0, Re b > 0,

ν √ ν  ∞ √ 2 x + b2 + x + x2 + b2 − x νπ √ K ν (ab) cos(ax) dx = 2bν cos 2 2 2 x +b 0 [a > 0, Re b > 0,

4.12

ET I 70(20)

Re ν < 1] ET I 13(22)

√ √ ν  ν  x + x2 − u2 + x − x2 − u2 νπ νπ  √ sin(ax) dx = πuν J ν (au) cos − Y ν (au) sin 2 2 x2 − u2 u [a > 0, u > 0, Re ν < 1] ET I 70(22) √ √ ν  ν  ∞  x + x2 − u2 + x − x2 − u2 νπ νπ  √ + J ν (au) sin cos(ax) dx = −πuν Y ν (au) cos 2 2 x2 − u2 u  ∞

3.12

Re ν < 1]

 5. 0

[a > 0, u > 0, Re ν < 1] ET I 13(25) √ √ ν  ν x + i u2 − x2 + x − i u2 − x2 π νπ √ [Jν (au) − J−ν (au)] sin(ax) dx = uν cosec 2 2 2 2 u −x ET I 70(21) [a > 0, u > 0]

 u

450

Trigonometric Functions



u

√ √ ν  ν x + i u2 − x2 + x − i u2 − x2 π νπ √ [Jν (au) + J−ν (au)] cos(ax) dx = uν sec 2 2 u2 − x2 [a > 0, u > 0, |Re ν| < 1]



6. 0



ET I 13(24)

√ √ ν  ν x + x2 − u2 + x − x2 − u2  sin(ax) dx 2 − u2 ) x (x    au   au   au   au  π 3 ν Y 1/4−ν/2 + J 1/4−ν/2 Y 1/4+ν/2 au J 1/4+ν/2 =− 2 2 2 2

2 ET I 71(25) a > 0, u > 0, Re ν < 32

 ∞

7.6 u

 8.



√ √ ν  ν x2 − u2 + x − x2 − u2  cos(ax) dx 2 2   x (x  − u )  au   au   au   au  π 3 ν Y −1/4−ν/2 + J −1/4−ν/2 Y −1/4+ν/2 au J −1/4+ν/2 =− 2 2 2 2 2

ET I 13(26) a > 0, u > 0, Re ν < 32

∞



√ √ ν  ν x2 + 2bx + x + b − x2 + 2bx √ sin(ax) dx x2 + 2bx    νπ  νπ  + J ν (ba) cos ba − = πbν Y ν (ba) sin ba − 2 2 [a > 0, |arg b| < π, Re ν < 1] ET I 71(26)

∞



√ √ ν  ν x2 + 2bx + x + b − x2 + 2bx √ cos(ax) dx x2 + 2bx    νπ  νπ  − Y ν (ba) cos ba − = πbν J ν (ba) sin ba − 2 2 [a > 0, |arg b| < π, Re ν < 1] ET I 13(23)

∞

x+

6 u



x+b+

9. 0

 10. 0



2u

11. 0

3.776



∞

1. 0

 2.

0

3.776

∞

x+b+

√ 4ν √ 4ν √ √ 2u + x + i 2u − x + 2u + x − i 2u − x √ cos(ax) dx 4u2 x − x3  a 2ν 3/2 J ν−1/4 (au) J −ν−1/4 (au) = (4u) π 2 ET I 14(27) [a > 0, u > 0]

a2 (b + x)2 + p(p + 1) a sin(ax) dx = p (b + x)p+2 b

[a > 0,

b > 0,

p > 0]

BI (170)(1)

a2 (b + x)2 + p(p + 1) p cos(ax) dx = p+1 p+2 (b + x) b

[a > 0,

b > 0,

p > 0]

BI (170)(2)

3.784

Rational and trigonometric functions

451

3.78–3.81 Rational functions of x and of trigonometric functions 3.781 1. 2. 3.782

 1 sin x − x 1+x 0   ∞ 1 cos x − 1+x 0





1. 0



∞

u

1 − cos x dx − x

∞

2. 0

 3.

12

3.783 1. 2. 3.784

∞

u

(cf. 3.784 4 and 3.781 2)

cos x dx = C + ln u x

1 − cos ax aπ dx = x2 2

BI (173)(8)

[u > 0]

GW (333)(31)

[a ≥ 0]

BI (158)(1)

Im b = 0,

Re b2 ≤ 0

 1 3 1 dx cos x − 1 = C− + 2 x 2(1 + x) x 2 4 0   ∞ 1 dx = −C cos x − 2 1 + x x 0





∞

∞



∞

2. 0



∞

3. 0



∞

4. 0



 0



BI (173)(19) EH I 17, BI(273)(21)

[a > 0,

b > 0]

FI II 635, GW(333)(20)

a sin bx − b sin ax a dx = ab ln 2 x b

[a > 0,

b > 0]

FI II 647

(b − a)π cos ax − cos bx dx = x2 2

[a ≥ 0,

b ≥ 0]

BI(158)(12), FI II 645

sin x − x cos x dx = 1 x2

BI (158)(3)



 b ci(aβ) cos aβ + si(aβ) sin aβ − ci(bβ) cos bβ − si(bβ) sin bβ + ln a [a > 0, b > 0, |arg β| < π] ET II 221(49)

∞

cos ax + x sin ax dx = πe−a 1 + x2

6.

ET II 253(48)

cos ax − cos bx b dx = ln x a

cos ax − cos bx 1 dx = x(x + β) β

0



∞

5.

BI (173)(7)

dx = −C x

1 − cos x πi  ib dx = − e −1 x(x − b) b −∞

0

8.



dx =1−C x

∞

1.

7.



[a > 0]

GW (333)(73)

∞

sin ax − ax cos ax π dx = a2 sign a 3 x 4 0

 ∞ π (b − a)β + e−bβ − e−aβ cos ax − cos bx dx = x2 (x2 + β 2 ) 2β 3 0

LI (158)(5)

[a > 0,

b > 0,

|arg β| < π] BI(173)(20)a, ET II 222(59)

452

Trigonometric Functions



∞

9.10 0

n  cos mx π √ = e−m sin u sinh φ (cos β sin u cosh φ + sin β cos u sinh φ) 1 + a2 T n (x) 2n 1 + a2 k=1

[u = (2k − 1) π/ (2n) ,  3.785

∞

0

3.786



∞

1. 0



∞

2.11 0

 3.

∞

11 0

3.785

n

n

k=1

k=1

 1 ak cos bk x dx = − ak ln bk x

φ = arcsinh(1/a), 

β = m cos u cosh φ,  n  bk > 0, ak = 0

0 < |a| < 1] FI II 649

k=1

(1 − cos ax) sin bx b b 2 − a2 a a+b dx = ln + ln 2 x 2 b2 2 a−b (1 − cos ax) cos bx dx = ln x



|a2 − b2 | b

(1 − cos ax) cos bx π dx = (a − b) x2 2 =0

[a > 0,

b > 0]

[a > 0,

b > 0,

ET I 81(29)

a = b]

FI II 647

[a < b ≤ 0] [0 < a ≤ b] ET I 20(16)

3.787



∞

1. 0

(cos a − cos nax) sin mx π dx = (cos a − 1) x 2 π = cos a 2

[m > na > 0] [na > m] BI(155)(7)



∞

2. 0



sin2 ax − sin2 bx 1 a dx = ln x 2 b

[a > 0,

b > 0]

GW (333)(20b)

∞

x3 − sin3 x 13 π dx = 5 x 32 0  ∞ 3 − 4 sin2 ax sin2 ax 1 dx = ln 2 4. x 2 0   π/2  π 1 − cot x dx = ln 3.788 x 2 0 π/2 2 4x cos x + (π − x)x dx = π 2 ln 2 3.789 sin x 0 3.791  π/2 x dx = ln 2 1. 1 + sin x 0  π x cos x dx = π ln 2 − 4G 2. 0 1 + sin x  π/2 x cos x dx = π ln 2 − 2G 3. 1 + sin x 0 3.

BI (158)(6)

[a real, a = 0]

HBI (155)(6)

GW (333)(61)a LI (206)(10)

GW (333)(55a) GW (333)(55c)

GW (333)(55b)

3.792

Rational and trigonometric functions



π

π

 π/2  π − x cos x 2 − x cos x dx = 2 dx = π ln 2 + 4G 1 − sin x 1 − sin x 0

2

4. 0



π/2

5. 0

 6. 7.

453

BI(207)(3), GW(333)(56c)

x2 dx π2 =− + π ln 2 + 4G 1 − cos x 4

BI (207)(3)

π

x2 dx = 4π ln 2 0 1 − cos x    π/2 p+1 ∞  π p+1  π p x dx 1 2  =− − ζ(2k) + (p + 1) 1 − cos x 2 2 p 42k−1 (p + 2k) 0

BI (219)(1)

k=1



[p > 0] π/2

8. 0



π/2

9. 0



π

10. 0



π

11. 0



2.

3.12

GW (333)(55a)

x sin x dx π = ln 2 + 2G 1 − cos x 2

GW (333)(56a)

x − sin x π dx = + 1 − cos x 2

π/2

0

3.792  1.

x dx π = − ln 2 1 + cos x 2

x sin x dx = 2π ln 2 1 − cos x

12.

LI (207)(4)

GW (333)(56b)

 0

π/2

x − sin x dx = 2 1 − cos x

GW (333)(57a)

x sin x π dx = − ln 2 + 2G 1 + cos x 2

π

GW (333)(55b)

2 2π dx a = 2 2 1−a −π 1 − 2a cos x + a  π/2 ∞  x cos x dx a2k π ln(1 + a) − = (−1)k 2 1 + 2a sin x + a 2a (2k + 1)2 0 k=0 2 a  π 2 π x sin x dx a = ln(1 + a) 2 1 − 2a cos x + a a   0 1 π a2 = ln 1 + a a

1 n−1 

a = 0



 −n x sin nx dx a−k − ak 2π a − an ln(1 − a) + = 2 2 1 − 2a cos x + a 1−a n−k k=1 2

a < 1, a = 0

BI (223)(4)

BI (223)(5)

454

Trigonometric Functions



∞

6. 0

   sin x dx π  1 + a  = −1 1 − 2a cos x + a2 x 4a  1 − a 

3.792

[a real,

a = 0,

a = 1] GW (333)(62b)



∞

7.8 0

 0

∞

9. 0



∞

10.3 0



∞

11. 0



∞

12. 0



∞

13. 0

 0

∞

15. 0

 16. 0

[b = 1, 2, . . .] ;

[0 < a < 1]

sin x cos bx dx π = ab 1 − 2a cos x + a2 x 2(1 − a) π π ab + ab−1 = 2(1 − a) 4 [0 < a < 1,

[b = 0, 1, 2, . . .] [b = 1, 2, 3, . . .] ; b > 0] ; (for b = 0, see 3.792 6)

(1 − a cos x) sin bx dx π 1 − a[b]+1 = 1 − 2a cos x + a2 x 2 1−a πab π 1 − ab + = 2 1−a 4

ET I 19(5)

[b = 1, 2, 3, . . .] [b = 1, 2, 3, . . .] [0 < a < 1,

b > 0]

ET I 82(33)

−bβ

1 dx 1 + ae π = 2 2 2 2 1 − 2a cos bx + a β + x 2β (1 − a ) 1 − ae−bβ

2 a < 1,

b≥0

aπ 1 dx sin bβ = 1 − 2a cos bx + a2 β 2 − x2 β (1 − a2 ) 1 − 2a cos bβ + a2 2 a < 1,

b>0

sin bcx x dx e−βbc − ac π = 1 − 2a cos bx + a2 β 2 + x2 2 (1 − ae−bβ ) (1 − aebβ ) 2 a < 1,

b > 0,

2 a < 1, 2 a > 1,

sin bx x dx 1 π = 1 − 2a cos bx + a2 β 2 + x2 2 ebβ − a 1 π = 2a aebβ − 1

b>0 b>0

BI (192)(1)

BI (193)(1)

c>0

BI (192)(8)

BI (192)(2)

∞

14. 

[b = 0, 1, 2, . . .]

ET I 81(26) ∞

8.



sin bx dx π 1 + a − 2ab+1 = 1 − 2a cos x + a2 x 2 (1 − a2 ) (1 − a) π 1 + a − ab − ab+1 = 2 (1 − a2 ) (1 − a)

∞

c

sin bcx x dx a − cos βbc π = 2 2 2 1 − 2a cos bx + a β − x 2 1 − 2a cos βb + a2 

2

2 a < 1,

b > 0,

c>0

1 − a sin βbc + 2ac+1 sin βb π cos bcx dx = 1 − 2a cos bx + a2 β 2 − x2 2β (1 − a2 ) 1 − 2a cos βb + a2 2

a < 1, b > 0, c > 0 b

1 − a cos bx dx π e = 1 − 2a cos bx + a2 1 + x2 2 eb − a

2 a < 1,

b>0

BI (193)(5)

BI (193)(9) FI II 719

3.794

Rational and trigonometric functions



∞

17. 0



|a| < 1,

Re β > 0] ET I 21(21)

π sinh bβ sin bx sin x dx [0 ≤ b < 1] = 1 − 2a cos x + a2 x2 + β 2 2β eβ − a   π am eβ(m+1−b) − e(1−b)β = 4β (aeβ − 1)   π m −(m+1−b)β −(1−b)β a [m ≤ b ≤ m + 1] e − e − 4β (ae−β − 1) [0 < a < 1, Re β > 0] ET I 81(27)

∞

(cos x − a) cos bx dx π cosh βb = 1 − 2a cos x + a2 x2 + β 2 2β (eβ − a)

0

19. 0



[0 ≤ b < 1,

∞

18.



 π eβ−βb + aeβb cos bx dx = 1 − 2a cos x + a2 x2 + β 2 2β (1 − a2 ) (eβ − a)

455

[0 ≤ b < 1,

|a| < 1,

Re β > 0] ET I 21(23)

∞

20.

sin x (1 − 2a cos 2x +

0

n+1 a2 )

dx x



∞

= 0

 =

0

∞

tan x (1 − 2a cos 2x + tan x

a2 )n+1 n+1

(1 − 2a cos 4x + a2 )

dx x n  2  π dx n = a2k 2n+1 2 x k 2 (1 − a ) k=0 BI (187)(14)

3.793



2π

1.3 0



k=1

2π

2.12 0

3.794 1.3

2.12

3.3



  n  sin nx − a sin[(n + 1)x] 1 x dx = −2πan ln(1 − a) + 1 − 2a cos x + a2 kak cos nx − a cos[(n + 1)x] x dx = 2π 2 an 1 − 2a cos x + a2 ∞

[|a| < 1]

BI (223)(9)

2

a 0]

GW (333)(53b)

2

a > b2

2 a < b2 BI (181)(1)

 3.795

3.796



∞

−∞



π/2

1. 0



π/4

2. 0

3.797 1. 2. 3. 3.798



 b2 + c2 + x2 x sin ax − b2 − c2 − x2 c sinh ac dx = π [x2 + (b − c)2 ] [x2 + (b + c)2 ] (cos ax + cosh ac) 2π = ab e +1 [a > 0]



∞

1.8 0

2.8 0

BI (202)(18)

BI (207)(8, 9)

cos x − sin x π 1 x dx = ln 2 − G cos x + sin x 4 2

BI (204)(23)

π

tan x dx π = √ 2 a + b cos 2x x 2 a − b2

BI (204)(8)

BI (204)(19)

BI (204)(20)

[0 < b < a] [0 < a < b]

=0 

[b > c > 0]

cos x ± sin x π x dx = ∓ ln 2 − G cos x ∓ sin x 4

 1 π2 π π − x tan x tan x dx = ln 2 + − + ln 2 4 2 32 4 8 0  π/4  π π 1 4 − x tan x dx = − ln 2 + G cos 2x 8 2 0  π/4 π π 1 4 − x tan x dx = ln 2 + G cos 2x 8 2 0 π/4

[c > b > 0]

BI (181)(2) ∞

tan x dx π = √ 2 a + b cos 4x x 2 a − b2

[0 < b < a] [0 < a < b]

=0

BI (181)(3)

3.799



1. 0

π/2

x dx (sin x + a cos x)

2

=

ln a a π − 2 1+a 2 1 + a2

[a > 0]

BI (208)(5)

3.812

Rational and trigonometric functions



π/4

2. 

a cos x + b (a + b cos x)

0

3.811 π

1. 0



π/2

0 π/4

3. 0



π/4

4. 0



π/4

5. 0

3.812



π

1. 0

2. 3. 4.11

=

1−a 1 1+a π ln √ + 1 + a2 4 (1 + a) (1 + a2 ) 2

2 2x

dx =

2(a − b) 2π √ ln b a + a2 − b 2

5.7 0

π

[a > |b| > 0]

GW (333)(58a)

x dx π = ln 2 + G (cos x ± sin x) sin x 4

BI (222)(5) BI (208))(16, 17)

x dx π = − ln 2 + G (cos x + sin x) sin x 8

BI (204)(29)

x dx π = ln 2 (cos x + sin x) cos x 8

BI (204)(28)

sin x x dx π π 1 = − ln 2 + − ln 2 2 sin x + cos x cos x 8 4 2

BI (204)(30)

 x sin x dx b π = √ arctan 2 a + b cos x a√ ab √ a + −b π √ = √ ln √ 2 −ab a − −b

√  π/2 π 1+ 1+a x sin 2x dx = ln 1 + a cos2 x a 2 0  √  π/2 x sin 2x dx π 2 1+a− 1+a = ln a 2 1 + a sin2 x 0  π 2 x dx π = √ 2 − cos2 x a 2a a2 − 1 0



BI (204)(24)

t1 sin x x dx t1 + t2 t1 − t2 1 + tan 2 = π cosec cosec ln t2 1 − cos t1 cos x 1 − cos t2 cos x 2 2 1 + tan 2 (cf. 3.794 4)

2. 

2

[a > 0] π

3.



x dx (cos x + a sin x)

0

457

[a > 0,

b > 0]

[a > −b > 0] GW (333)(60a)

[a > −1,

a = 0]

BI (207)(10)

[a > −1,

a = 0]

BI (207)(2)

2 a >1

=0

principal value for 0 < a2 < 1

= divergent

[a = 0]

x sin x dx π = ln a2 − cos2 x 2a



1+a 1−a

BI (219)(10)

 [0 < a < 1]

divergent if a = 0 BI (219)(13)

458

Trigonometric Functions

 6.

π

11 0

 x sin 2x dx = π ln 4 1 − a2 a2 − cos2 x     = 2π ln 2 1 − a2 + a a2 − 1 = divergent

3.813



principal value for 0 ≤ a2 < 1 2

a >1 [|a| = 1] BI (219)(19)



π/2

7. 0

 8. 9. 10. 11. 12.

13.

BI (207)(1)

π

x sin x dx = π(π − 2t) cosec 2t 2 2 0 1 − cos t sin x  π ∞  x cos x dx sin(2k + 1)t = 4 cosec t 2 2 (2k + 1)2 0 cos t − cos x k=0  π π x sin x dx = (π − 2t) cot t 2 2 2 0 tan t + cos x  ∞ x (a cos x + b) sin x dx t = 2aπ ln cos + πbt tan t 2 2x 2 cot t + cos 0     π x sin x cos x a−1 dx = −π ln 2 + ln 1 + a a − sin2 x 0  π/2  √ ln a − sin2 x dx = −π ln 2 + iπ ln arccos a 0

14.

∞  x sin x dx sin(2k + 1)t = −2 cosec t 2 2 (2k + 1)2 cos t − sin x k=0



BI (219)(12)

BI (219)(17)

BI (219)(14) BI (219)(18)

[a > 1] [0 < a < 1]

π/2

  ln a − sin2 x dx = −π ln 2

[0 < a < 1]

π/2

  ln a − cos2 x dx = −π ln 2

[0 < a < 1]

PV 0

 15.

PV 0

3.813



1. 0



π

x dx 1 = 2 2 2 2 4 a cos x + b sin x

∞

2. 0



0

x dx π2 = 2ab a2 cos2 x + b2 sin2 x [a > 0,

GW(333)(81), ET II 222(63)

sin x dx π = 2 2 2 2ab sin x + b cos x

[ab > 0]

sin2 x dx π = 2 2 2 2 2b(a + b) x a cos x + b sin x

[a > 0,

x

0



0

2π

GW (333)(36) b > 0]   1 γ 2 dx β π sinh(2aδ)  − − = β sinh(2aδ) β 2 sin2 ax + γ 2 cos2 ax x2 + δ 2 4δ β 2 sinh2 (aδ) −γ 2 cosh2 (aδ) γ    arg β  < π, Re δ > 0, a > 0  γ

∞

3. 4.



∞



a2



BI (181)(8)

b > 0]

BI (181)(11)

3.814

Rational and trigonometric functions



459

π/2

x sin 2x dx π a+b [a > 0, b > 0, a = b] GW (333)(52a) = 2 ln a − b2 2b a2 cos2 x + b2 sin2 x  π x sin 2x dx 2π a+b [a > 0, b > 0, a = b] GW (333)(52b) = 2 ln 2 2 cos2 x + b2 sin2 x a − b 2a a 0  ∞ sin 2x π dx = [a > 0, b > 0] BI (182)(3) 2 2 2 2 x a(a + b) 0 a cos x + b sin x    ∞ sin 2ax x dx β−γ π −2aδ  − e = 2 2 2 2 2 2 β+γ 2 β 2 sinh2 (aδ) − γ2 cosh2 (aδ) 0 β sin ax + γ cos ax x + δ      β   a > 0, arg  < π, Re δ > 0 γ

5.

0

6. 7. 8.



ET II 222(64), GW(333)(80) ∞

9. 0



∞

12 0

3.814



b > 0]

BI (182)(7)a

sin x cos2 x π dx = 2 2 2 2a(a + b) cos x + b sin x x

[a > 0,

b > 0]

BI (182)(4)

[a > 0,

b > 0]

BI (182)(1)

a2

0

11.

[a > 0,

∞

10. 

(1 − cos x) sin x dx π = 2b(a + b) a2 cos2 x + b2 sin2 x x

sin3 x π 1 dx = 2 2 2 2 x 2b a+b a cos x + b sin x

π/2

1. 0



π/4

2. 0

 

5.

tan x π dx = 2 2 2ab x + b sin x x

b > 0]

BI (181)(9)

[a > 0,

b > 0]

LI (208)(20)

[a > 0,

b > 0]

GW (333)(59)

[a > 0,

b > 0]

BI (182)(6)

tan x π dx = 2ab a2 cos2 2x + b2 sin2 2x x

[a > 0,

b > 0]

BI (181)(10)a

sin2 2x tan x π 1 dx = 2 2 2 2 2b a + b a cos 2x + b sin 2x x

[a > 0,

b > 0]

BI (182)(2)a

π 1 cos2 2x tan x dx = 2a a + b a2 cos2 2x + b2 sin2 2x x

[a > 0,

b > 0]

BI (182)(5)a

cos2

π/2

x cot x dx π a+b = 2 ln 2 2 2 2a b cos x + b sin x 0  π π  π/2  π 1 2 − x tan x dx 2 − x tan x dx = a2 cos2 x + b2 sin2 x 2 0 a2 cos2 x + b2 sin2 x 0 π a+b = 2 ln 2b a a2



∞

6. 

∞

7. 0



∞

8. 0

 9.

0

∞

sin x tan x π dx = 2 2 2 2b(a + b) cos x + b sin x x 2

a2

0

BI (204)(30)

[a > 0,

a2

0

BI (206)(9)

x tan x dx π π 1 = − ln 2 + − ln 2 (sin x + cos x) cos x 8 4 2

∞

3. 4.

(1 − x cot x) dx π = 2 4 sin x

460

Trigonometric Functions



∞

10. 0

 11. 12. 13. 14. 15. 3.815

sin2 x cos x π a dx =− · 2 2 2 2 2 8b a + b2 a cos 2x + b sin 2x x cos 4x

∞

sin x π b 2 − a2 dx = 2 2 2 2 2ab b2 + a2 0 a cos x + b sin x x cos 2x  ∞ sin x cos x π b dx 2 x cos 2x = 2a a2 + b2 2 2 2 0 a cos x + b sin x  ∞ sin x cos2 x π b2 dx 2 x cos 2x = 2ab a2 + b2 2 2 2 0 a cos x + b sin x  ∞ sin3 x π a dx 2 x cos 2x = − 2b a2 + b2 2 2 2 0 a cos x + b sin x  ∞ 1 − cos x π dx = · 2 2 2 2 2ab 0 a cos x + b sin x x sin x 

π/2

1. 0



π/2

2. 0

x sin 2x dx π   = ln 2 2 a − b 1 + a sin x 1 + b sin x



3.815

[a > 0,

b > 0]

BI (186)(12)a

[a > 0,

b > 0]

BI (186)(4)a

[a > 0,

b > 0]

BI (186)(7)a

[a > 0,

b > 0]

BI (186)(8)a

[a > 0,

b > 0]

BI (186)(10)

[a > 0,

b > 0]

BI (186)(3)a

√ √  1+a 1+ 1+b √ ·√ 1+ 1+a 1+b [a > 0, b > 0]

√ √ 1+ 1+b 1+a x sin 2x dx π  √ ln = a + ab + b 1+ 1+a 1 + a sin2 x (1 + b cos2 x) [a > 0, b > 0]

(cf. 3.812 3) BI (208)(22)



√ π/2 x sin 2x dx π 1+ 1+a √ = ln (1 + a cos2 x) (1 + b cos2 x) a−b 1+ 1+b 0

(cf. 3.812 2 and 3) BI (208)(24)

 3.

[a > 0,

b > 0]

(cf. 3.812 2) BI (208)(23)



π/2

4. 0

t1 cos x sin 2x dx 2π 2   = ln cos2 t1 − cos2 t2 cos t2 1 − sin2 t1 cos2 x 1 − sin2 t2 cos2 x 2 [−π < t1 < π,

−π < t2 < π] BI (208)(21)

3.816 1.



π

x2 sin 2x 2

(a2 − cos2 x)  π 2 a − 1 − sin2 x cos x 0

2.

12

(a2

0

 3.

dx = π 2

11 0

π

−

cos2

2

x)

√ a2 − 1 − a a (a2 − 1)

  π  1 − a  x dx = ln  a 1 + a 2

[a > 1] 2

a >1

LI (220)(9)

(cf. 3.812 5)

 √ √ a cos 2x − sin2 x 2  2 x dx = −2π ln 2 −a + a a + 1 2 a + sin x   √ √ a < −1 and a > 0. When a > 0, can write a a + 1 as a(a + 1).

BI (220)(12)

LI (220)(10)

3.818

Rational and trigonometric functions

 4.11 0

π

 √ √ a cos 2x + sin2 x 2  2 x dx = 2π ln 2 a − a a + 1 2 a− sin x   √ √ a < 0 and a > 1. When a > 1, can write a a + 1 as a(a + 1).

461

(cf. 3.812 6) LI (220)(11)

3.817



∞

0



[ab > 0]

BI (181)(14)

[ab > 0]

BI (182)(11)

[ab > 0]

BI (182)(10)

[ab > 0]

BI (181)(16)

[ab > 0]

BI (182)(13)

dx π a2 + 3b2 = 3 x 16 a5 b3 a2 cos2 x + b2 sin2 x

[ab > 0]

BI (182)(14)

π 3a2 + b2 dx = 3 x 16 a3 b5 a2 cos2 x + b2 sin2 x

[ab > 0]

LI (181)(19)

[ab > 0]

BI (182)(17)

tan x

dx π a2 + b 2 = 2 x 4 a3 b 3 a2 cos2 2x + b2 sin2 2x sin2 x tan x

dx π = 2 x 4ab3 a2 cos2 x + b2 sin x 2

∞

tan x cos2 2x



a2

∞

0

dx π a + 3b = 3 x 16 a5 b3 a2 cos2 x + b2 sin x

∞

sin3 x



4. 0

2

sin x cos2 x



0

2

2

∞

3.

dx π = 3 2 x 4a b sin 2x 2

sin x cos x



0

2x +

b2

dx π 3a4 + 2a2 b2 + 3b4 = 3 x 16 a5 b 5 a2 cos2 x + b2 sin2 x

∞

2.

cos2

sin x



1.

dx π = 3 2 x 4a b sin x 2

π a2 + b 2 dx = 2 x 4 a3 b 3 a2 cos2 x + b2 sin2 x



0

0

BI (181)(13)

x+

b2

tan x

∞

8.

5.

[ab > 0]

cos2



0



BI (182)(9)

∞

7.



[ab > 0]

sin x cos2 x



0



BI (181)(15)

∞

6.



[ab > 0]

2

2

a2

0



dx π = 2 x 4ab3 a2 cos2 x + b2 sin x

x+

b2

sin3 x



5.

3.818

BI (182)(8)

∞

0



[ab > 0]

cos2



4.



dx π = 3 2 x 4a b sin x

sin x cos x

∞

0



BI (181)(12)

a2

3.



[ab > 0]



0



dx π a2 + b 2 = 2 x 4 a3 b 3 a2 cos2 x + b2 sin2 x

∞

2. 

sin x



1.

∞

dx π 3a2 + b2 = 3 x 64 a3 b5 a2 cos2 2x + b2 sin2 2x



sin3 x cos x

462

Trigonometric Functions



∞

0



∞

0

  

dx π a2 + b 2 = 4 x 32 a5 b5 a2 cos2 x + b2 sin2 x sin x cos4 x

dx π a2 + 5b2 = 4 x 32 a7 b3 a2 cos2 x + b2 sin2 x

∞

sin5 x

dx π 5a2 + b2 = 4 x 32 a3 b7 a2 cos2 x + b2 sin2 x



0

0

sin3 x cos2 x



9. 10.

dx π a2 + 5b2 = 4 x 32 a7 b3 a2 cos2 x + b2 sin2 x

∞

0



sin x cos3 x



8. 

dx π a2 + b 2 = 4 x 32 a5 b5 a2 cos2 x + b2 sin2 x

∞

0

2 2

sin3 x cos x



7.

4

2

∞

0

2 2

π 5a + a b + b dx = 4 x 32 a5 b 7 a2 cos2 x + b2 sin x sin x



6.

4

3

∞

0

[ab > 0]

BI (182)(15)

[ab > 0]

BI (181)(20)

[ab > 0]

BI (182)(18)

[ab > 0]

BI (182)(19)

[ab > 0]

BI (181)(23)

[ab > 0]

BI (182)(26)

[ab > 0]

BI (182)(23)

[ab > 0]

BI (182)(27)

[ab > 0]

BI (182)(24)

[ab > 0]

BI (181)(24)

4

dx π a4 + 2a2 b2 + 5b4 = 4 x 32 a7 b 5 a2 cos2 x + b2 sin2 x



5.

BI (181)(18)

sin x cos2 x

∞

0



dx π a + 2a b + 5b = 4 x 32 a7 b 5 a2 cos2 x + b2 sin x



4.

[ab > 0]

2

∞

0



sin x cos x



3.

BI (182)(16)

dx π 5a6 + 3a4 b2 + 3a2 b4 + 5b6 = 4 x 32 a7 b 7 a2 cos2 x + b2 sin2 x

∞

0

[ab > 0]

sin x



2. 

dx π a2 + 3b2 = 3 x 16 a5 b3 a2 cos2 2x + b2 sin 2x 2

∞

0



tan x cos2 2x



0

BI (181)(17)

dx π 3a4 + 2a2 b2 + 3b4 = 3 x 16 a5 b 5 a2 cos2 2x + b2 sin2 2x

∞

9.

[ab > 0]

tan x



0

3.819  1.

dx π 3a2 + b2 = 3 x 16 a3 b5 a2 cos2 x + b2 sin2 x

∞

8. 

sin2 x tan x



7. 

π 3a4 + 2a2 b2 + 3b4 dx = 3 x 16 a5 b 5 a2 cos2 x + b2 sin2 x tan x



6.

3.819

∞

4

dx π 5a4 + 2a2 b2 + b4 = 4 x 128 a5 b 7 a2 cos2 2x + b2 sin2 2x



sin3 x cos x

[ab > 0]

BI (182)(22)

3.821

Powers and trigonometric functions



∞

0



a2

∞

0



dx π a2 + b 2 = 4 x 8 a5 b 5 a2 cos2 2x + b2 sin2 2x

∞

0

cos4 2x tan x

dx π a2 + 5b2 = 4 x 32 a7 b3 a2 cos2 2x + b2 sin2 2x



16.

dx π 5a + b = 4 x 32 a3 b 7 sin x 2

sin3 4x tan x



0

x+

b2

2

[ab > 0]

BI (182)(21)

[ab > 0]

BI (182)(29)

[ab > 0]

BI (182)(29)

[ab > 0]

BI (182)(28)

[ab > 0]

BI (182)(25)

dx π a4 + 2a2 b2 + 5b4 = 4 x 32 a7 b 5 a2 cos2 2x + b2 sin2 2x

∞

15.

cos2

2

cos2 2x tan x



14. 

sin x tan x 4



0

BI (182)(30)

π 5a4 + 2a2 b2 + b4 dx = 4 x 32 a5 b 7 a2 cos2 x + b2 sin2 x

∞

13.

[ab > 0]

sin2 x tan x



0



dx π 5a2 + b2 = 4 x 512 a3 b7 a2 cos2 2x + b2 sin2 2x

∞

12. 

sin5 x cos3 x



11.

463

3.82–3.83 Powers of trigonometric functions combined with other powers 3.821  π π2 Γ(p + 1) 1. x sinp x dx = p+1   2 p 2 0 +1 Γ 2  rπ 2 (2m − 1)!! 2 π r 2. x sinn x dx = 2 (2m)!! 0 (2m)!! r = (−1)r+1 π (2m + 1)!!

[p > −1]

BI(218)(7), LO V 121(71)

[n = 2m] [n = 2m + 1] [r is a natural number]



π/2

3.11

x cosn x dx =

0

=  4. 5. 6.

1 π 2 (n − 1)!! − n−2 8 (n)!! 2 1 π (n − 1)!! − n−1 2 (n)!! 2

n

m−1 

k=0,m−k odd m−1   k=0

n k

k

1 (n − 2k)2

1 (n − 2k)2

GW (333)(8c)

[n = 2m] [n = 2m − 1] GW (333)(9b)

π

π (2m − 1)!! 2 (2m)!! 0  sπ (2m − 1)!! π2  2 s − r2 x cos2m x dx = 2 (2m)!! rπ p √  ∞ p p p sin x π Γ 2  p+1 = 2p−2 B dx = , x 2 Γ 2 2 2 0 [p is a fraction with odd numerator and denominator] x cos2m x dx =

2

BI (218)(10) BI (226)(3)

LO V 278, FI II 808

464

Trigonometric Functions



∞

7. 0



∞

8. 0



∞

9. 0



∞

10. 0



∞

11. 0



∞

3.822

sin2n+1 x (2n − 1)!! π dx = x (2n)!! 2

BI (151)(4)

sin2n x dx = ∞ x

BI (151)(3)

sin2 ax aπ dx = x2 2

[a > 0]

LO V 307, 312, FI II 632

sin2m ax (2m − 3)!! aπ dx = 2 x (2m − 2)!! 2

[a > 0]

GW (333)(14b)

sin2m+1 ax a2 π (2m − 3)!! (2m + 1) dx = 3 x (2m)!! 4

[a > 0]

GW (333)(14d)

sinp x dx xm

12.

 ∞ p−1 sin x p cos x dx m−1 0 xm−1  ∞ p−2  ∞ p sin x sin x p2 p(p − 1) dx − dx = (m − 1)(m − 2) 0 xm−2 (m − 1)(m − 2) 0 xm−2

0

=



∞

sin2n px √ dx = ∞ x

0 ∞

0



π/2

1.

xp cosm x dx = −

0



∞

2.

BI (177)(5)

1 dx sin2n+1 px √ = 2n x 2

14. 3.822

[p > m − 1 > 1] GW (333)(17)

13. 

[p > m − 1 > 0]



p(p − 1) m2

0

 0

∞

xb−1 sin2 x dx = −



k=0

x−1/2 cos2n+1 (px) dx =

3.82312

n

π  (−1)k 2p 

1 22n

π/2

2n + 1 n+k+1



xp−2 cosm x dx +

0



π 2p

n   k=0

2n + 1 n+k+1

1 √ 2k + 1

m−1 m





π/2

BI (177)(7)

xp cosm−2 x dx

0

[m > 1,

p > 1]

1 √ 2k + 1

Γ(b) πb cos 2b+1 2

GW (333)(9a) BI (177)(8)

[−2 < Re b < 0] ET I 319(15), GW(333)(19c)a

3.824 1.

12

2.12 3.7



∞

π  sin2 x 1 − e−2a [Re a > 0] dx = BI (160)(10) 2 2 4a 0 x +a  ∞ cos2 x π  1 + e−2a [Re a > 0] dx = BI (160)(11) 2 2 4a 0 x +a      ∞ m  dx (−1)m π 2m 2m 2m 2m k 2 sinh a − 2 sinh[2(m − k)a] sin x 2 = 2m+1 · (−1) a + x2 2 2 k 0 k=0

[a > 0]

BI (160)(12)

3.824

Powers and trigonometric functions



465

   2m+1 m−1  dx (−1) 2m+1 (2m+1)a k 2m + 1 e e−2ka Ei[(2k − 2m − 1)a] sin x 2 = 2m+2 (−1) a + x2 2 a k k=0    2m+1  2m + 1 e2ka Ei[(2m + 1 − 2k)a] (−1)k−1 + e−(2m+1)a k

∞

4.12 0

k=1

BI (160)(14) [a > 0]   m ∞  2m + 1 2ka x dx π −(2m+1)a e sin2m+1 x 2 = e (−1)m+k 2 2m+1 a + x 2 k 0 k=0   π |arg a| < , m = 0, 1, 2, . . . 2     ∞ m   π dx π 2m 2m + 2m e−2ka cos2m x 2 = 2m+1 2 a + x 2 a m 2 m + k 0

 5.7

6.7

k=1



∞

cos2m+1 x

7. 0

8.

dx π = 2m+1 a2 + x2 2 a

m   k=1

[a > 0]  2m + 1 e−(2k+1)a m+k+1

BI (160)(16)

[a > 0]   ∞ 2m + 1 2ka x dx e−(2m+1)a e Ei[(2m − 2k + 1)a] cos2m+1 x 2 = − 2 2m+2 a +x 2 k 0 k=0  2m+1  e(2m+1)a  2m + 1 −2ka e Ei[(2k − 2m − 1)a] − 2m+2 2 k

BI (160)(17)

2m+1  

k=0



9.12 10.

BI (160)(18) ∞

cos ax π sin 2ab [a > 0, b > 0] dx = 2 − x2 b 4b 0    ∞ 2 sin ax cos2 bx 1 −2(a+b)β π 1 2(b−a)β −2bβ −2aβ 1 − e e dx = + e − − e β 2 + x2 8β 2 2 0

π −4aβ 1−e = 16β  1 π 1 1 − e−2(a+b)β + e−2bβ − e2(a−b)β − e−2aβ = 8β 2 2 [a > 0, b > 0] , (cf. 3.824 1 2

PV

 11. 0

∞

 x sin 2ax cos2 bx π  −2aβ −2(a+b)β 2(b−a)β 2e dx = + e + e β 2 + x2 8

π −4aβ e + 2e−2aβ = 8  π  −2aβ 2e + e−2(a+b)β − e2(a−b)β = 8

BI (161)(10)

[a > b] [a = b] [a < b] and 3)

BI (162)(6)

[a > 0] [a = b] [a < b] LI (162)(5)

466

3.825

Trigonometric Functions



∞

1. 0

 π b − c + ce−2ab − be−2ac sin2 ax dx = (b2 + x2 ) (c2 + x2 ) 4bc (b2 − c2 )

 ∞ π b − c + be−2ac − ce−2ab cos2 ax dx = 2 2 2 2 4bc (b2 − c2 ) 0 (b + x ) (c + x )

 2.



∞

3.3 0



3.825

sin ax dx π (c sin 2ab − b sin 2ac) = (b2 − x2 ) (c2 − x2 ) 4bc (b2 − c2 ) 2

[a > 0,

b > 0,

c > 0]

BI (174)(15)

[a > 0,

b > 0,

c > 0]

BI (175)(14)

[a > 0,

b > 0,

c > 0,

b = c] LI (174)(16)

∞

4.3 0

cos2 ax dx π (b sin 2ac − c sin 2ab) = (b2 − x2 ) (c2 − x2 ) 4bc (b2 − c2 )

[a > 0,

b > 0,

c > 0,

b = c] LI (175)(15)

3.826



∞

1.12 0



∞

2. 0

3.827 1.



∞

12 0

sin2 (ax) π dx = 2a − (1 − e−2a ) 2 2 x (1 + x ) 4

[a > 0]

BI (172)(13)

sin2 ax dx π = (2a − sin 2a) x2 (1 − x2 ) 4

[a > 0]

BII (172)(14)

sin3 x πν 3 − 3ν−1 Γ(1 − ν) cos dx = xν 4 2

[1 < Re ν < 4,

ν = 2, 3] GW (333)(19f)



∞

2.8 0



∞

3. 0

 4.

∞

12 0



∞

5. 0



∞

6. 0



∞

7. 0



∞

8. 0

 9.

0

∞

sin3 ax π dx = x 4

[a > 0]

sin3 ax 3 dx = a ln 3 x2 4

LO V 277

BI (156)(2)

sin3 ax 3 dx = a2 π 3 x 8

[a > 0]

BI(156)(7)a,LO V 312

sin4 ax aπ dx = x2 4

[a > 0]

BI (156)(3)

sin4 ax dx = a2 ln 2 x3 sin4 ax a3 π dx = x4 3

BI (156)(8)

[a > 0]

sin5 ax 5 a (3 ln 3 − ln 5) dx = x2 16 sin5 ax 5 2 a π dx = 3 x 32

BI(156)(11), LO V 312

BI (156)(4)

[a > 0]

BI (156)(9)

3.828

Powers and trigonometric functions



∞

10. 0



∞

11. 0



∞

12. 0



∞

13. 0



∞

14. 0



∞

15. 0

467

sin5 ax 5 3 a (25 ln 5 − 27 ln 3) dx = 4 x 96

BI (156)(12)

sin5 ax 115 4 a π dx = x5 384

[a > 0]

BI(156)(13), LO V 312

sin6 ax 3 aπ dx = x2 16

[a > 0]

BI (156)(5)

sin6 ax 3 2 a (8 ln 2 − 3 ln 3) dx = 3 x 16

BI (156)(10)

sin6 ax 1 4 a (27 ln 3 − 32 ln 2) dx = x5 16

BI (156)(14)

sin6 ax 11 5 a π dx = 6 x 40

[a > 0]

LO V 312

3.828 In 3.828 1–21 the restrictions a > 0, b > 0, c > 0 apply.    ∞ sin ax sin bx 1  a + b  8 dx = ln  [a =  b] 1. x 2 a − b 0  ∞ dx 1 2.8 sin ax sin bx 2 = π min(a, b) x 2 0  ∞ 2 sin ax sin bx π dx = [b < 2a] 3.8 x 4 0 π [b = 2a] = 8 =0 [b > 2a]

FI II 647 BI (157)(1)

BI (151)(10)



∞

4.8 0



∞

sin2 ax cos 2bx 1 dx = π max(0, a − b) x2 2

∞

sin 2ax cos2 bx π dx = x 2 3 = π 8 π = 4

5.8 0



sin2 ax cos bx 1 4a2 − b2 dx = ln x 4 b2

6. 0

[2a = b]

BI (151)(12)

[a > b] [a = b] [a < b] BI (151)(9)

 7.8 0

∞

sin2 ax sin bx sin cx π (|b − 2a − c| − |2a − b − c| + 2c) dx = x2 16 [a > 0, 0 < c ≤ b] BI(157)(9)a, ET I 79(15)

468

Trigonometric Functions

 8.

∞

8 0



∞

9. 0

3.828

  sin2 ax sin bx sin cx 1  (b + c)2 (2a − b + c)(2a + b − c)  dx = ln  x 8 (b − c)2 (2a + b + c)(2a − b − c)  [b = c, 2a + c = b, 2a + b = c, sin2 ax sin2 bx π dx = a 2 x 4 π = b 4

2a = b + c]

LI (152)(2)

[0 ≤ a ≤ b] [0 ≤ b ≤ a] BI (157)(3)



∞

10.8 0



∞

11.8 0



∞

12. 0



∞

13. 0

 14.12 0

 15. 0

∞

 sin2 ax sin2 bx 1 dx = π min a2 , b2 [3 max(a, b) − min(a, b)] x4 6

BI (157)(27)

sin2 ax cos2 bx 1 dx = π [a + max(0, a − b)] x2 4 sin3 ax sin 3bx a3 π dx = 4 x 2

π 3 8a − 9(a − b)3 = 16 9bπ  2 a − b2 = 8 sin3 ax cos bx dx = 0 x π =− 16 π =− 8 π = 16 π = 4

BI (157)(6)

[b > a] [a ≤ 3b ≤ 3a]

BI (157)(28)

[3b ≤ a]

LI (157)(28)

[b > 3a] [b = 3a] [3a > b > a] [b = a] [a > b] [a > 0,

b > 0]

BI (151)(15)

sin ax cos 3bx dx x2 1 3a ln 3 + [(a + b) ln |a + b| + (a − b) ln |a − b| − (a − 3b) ln |a − 3b| − (a + 3b) ln |a + 3b|] = 4 8 [Im a = 0, Im b = 0] MC

∞

3

sin3 ax cos bx π 2 3a − b2 dx = 3 x 8 πb2 = 4 π = (3a − b)2 16 =0

[b < a] [a = b] [a < b < 3a] [3a < b] [a > 0,

b > 0]

BI(157)(19), ET I 19(10)

3.829

Powers and trigonometric functions



∞

16. 0



sin3 ax sin bx bπ  2 9a − b2 dx = 4 x 24

π 24a3 − (3a − b)3 = 48 πa3 = 2

0

sin ax sin bx dx = x = 3

2

∞

18.8 0

0

[2b > 3a]

5π 32

[2b = 3a]

=

3π 16

[3a > 2b > a]

=

3π 32

[2b = a] [a > 2b]

[a > 0, b > 0]    (2a + b)3 (b − 2a)3 (2a + 3b)(3b − 2a)  sin2 ax cos3 bx 1  dx = ln   x 16 9b8 sin ax sin bx sin cx dx x 2

2

2a = 3b]

BI (151)(14)

BI (151)(13)

2

=

π [sign(c − a + b) + sign(c + a − b) − 2 sign(c − a) − 2 sign(c − b)] 32 [Im a = 0, Im b = 0, Im c = 0] MC

∞

sin2 ax sin2 bx sin 2cx dx x2 a−b−c a+b+c a+b−c ln 4(a − b − c)2 − ln 4(a + b + c)2 + ln 4(a + b − c)2 = 16 16 16 a−b+c a+c a−c ln 4(a − b + c)2 + ln 4(a + c)2 − ln 4(a − c)2 − 16 8 8 b+c b−c 1 ln 4(b + c)2 − ln 4(b − c)2 − c ln 2c + 8 8 2 [a > 0, b > 0, c > 0] BI (157)(10)

∞

sin2 ax sin3 bx dx = x3 =

20. 0



[0 < 3a ≤ b]

[2a = b,

∞

19.12



[0 < a ≤ b ≤ 3a]

π 8

=0



[0 < b ≤ a]

ET I 79(16) ∞

17.



469

21.8 0

3b2 π 16

[2a > 3b]

a2 π 12

[2a = 3b]

=

6b2 −(3b−2a)2 π 32

[3b > 2a > b]

=

a2 π 4

[b ≥ 2a] BI (157)(18)

3.829



1. 0

∞

xn − sinn x π dx = n n+2 x 2 (n + 1)!

(n−1)/2



k=0

(−1)k

n (n − 2k)n+1 k

GW (333)(63)

470

Trigonometric Functions



∞

2.



1 − cos

2m−1

0

3.831 1. 2. 3. 4.



dx x 2 = x



∞



1 − cos

2m

0

3.831

dx mπ x 2 = 2m x 2



2m m



∞

sin2n ax − sin2n bx (2n − 1)!! b dx = ln [ab > 0, x (2n)!! a 0    ∞ cos2n ax − cos2n bx (2n − 1)!! b dx = 1 − ln [ab > 0, x (2n)!! a 0  ∞ cos2m+1 ax − cos2m+1 bx b dx = ln [ab > 0, x a 0    ∞ cosm ax cos max − cosm bx cos mbx 1 b dx = 1 − m ln x 2 a 0 [ab > 0,

3.832



π/2

1. 0



x cos

x sin ax dx =

π 2p+1

∞

0

sin2m−1 x sin[(2m − 1)x]

0

FI II

m = 0, 1, . . .]

LI (155)(8)

−(p + 1) < a < p + 1]

sin2m−1 x sin[(2m + 1)x]

m = 0, 1. . . .]

BI (162)(17)

2m−1 π dx (−1) 1 − e−2a = 2 2m +x 2 a

a2

sin2m+1 x sin[3(2m + 1)x]

m = 1, 2, . . .]

BI (162)(11)

2m−1 dx (−1)m−1 π −2a  e 1 − e−2a = 2 2m +x 2 a m = 1, 2, . . .]

BI (162)(12)

m

a2

dx (−1) π −3(2m+1)a e = sinh2m+1 a 2 +x 2a

[a > 0]   2m   x dx (−1) π sin2m x sin[(2m − 1)x] 2 = 2m+1 ea 1 − e−2a − 1 + e−2a 2 a +x 2

BI (162)(18)

[a ≥ 0,   x dx (−1) π  −2a 2m 1 − e sin2m x sin(2mx) 2 = − 1 a + x2 22m+1

m = 0, 1, . . .]

BI (162)(13)

∞

[a > 0,

m = 0, 1, . . .]

BI (162)(14)

m = 0, 1, . . .]

BI (162)(15)

0

7. 0

0

m = 0, 1, . . .]

∞

6.3

8.

a2

[a > 0, ∞

5.



FI II 651

m+1

[a > 0,

4.



n = 0, 1, . . .]

 dx (−1)m π  −2a 2m 1 − e sin2m+1 x sin 2mx 2 = − 1 sinh a a + x2 22m+1 a

0



FI II 651



[a > 0, ∞

3.





n = 1, 2, . . .]

BI (205)(6) ∞

0



Γ(p)

 p+a+1

− ψ p−a+1 2 2  p−a+1  p+a+1 Γ Γ 2 2

ψ

[p > 0,

2.3 

p−1

BI (158)(7, 8)

∞

m

m

 2m x dx (−1) π sin2m x sin[(2m + 2)x] 2 = 2m+1 e−2a 1 − e−2a 2 a +x 2 m

[a > 0,

3.832

Powers and trigonometric functions



∞

9.

x dx (−1)m π −4ma e = sinh2m a a2 + x2 2

sin2m x sin 4mx

0



[a > 0, ∞

10. 0



∞

11.

sin2m x cos x

dx (2m − 3)!! π = x2 (2m)!! 2

sin2m x cos[(2m − 1)x]

0



∞

12.

sin2m x cos(2mx)

0



∞

13.

a2

∞

14.

sin2m x cos 4mx

0



∞

15. 

∞

∞

17.

 dx (−1)m π  −2a 2m−1 1 − e = − 1 sinh a a2 + x2 22m a

∞

18.3 0



m = 0, 1, . . .]

BI (162)(26)

[a > 0,

m = 0, 1, . . .]

BI (162)(27)

[a > 0,

m = 0, 1, . . .]

BI (162)(28)

dx (−1)m π −4ma e = sinh2m a 2 +x 2a

GW (333)(15)

sin2m+1 x cos x

dx (2m − 3)!! π = 3 x (2m)!! 2

[m = 1, 2, . . .]

GW (333)(15b)

sin2m−1 x cos[(2m − 1)x]

 x dx (−1)m π  −2a 2m−1 1 − e = − 1 a2 + x2 22m

[m = 1, 2, . . . , a > 0]     x dx (−1)m−1 π a  −2a 2m+1 −a e 1 − e sin2m+1 x cos 2mx 2 = − 1 − e a + x2 22m+2

sin2m−1 x cos[(2m + 1)x]

∞

cosm x sin mx

a ≥ 0]

BI (162)(29)

a > 0]

BI (162)(24)

2m−1 x dx (−1) π −2a  1 − e−2a = e 2 2m +x 2

a2

x dx (−1)m−1 π −2(2m+1)a e = sinh2m+1 a a2 + x2 2 [m = 0, 1, . . . ,

∞

BI (162)(23)

m

sin2m+1 x cos[2(2m + 1)x]

0

0

[a > 0, m

[m = 1, 2, . . . ,

20.

21.12

BI (162)(25)

[m = 0, 1, . . .]

0



m = 1, 2, . . .]

[m = 0, 1, . . . , ∞

19. 

[a > 0,

 2m dx (−1) π = 2m+1 e−2a 1 − e−2a a2 + x2 2 a

0



GW (333)(15a)

(2m − 1)!! π dx = x (2m + 2)!! 2

0



[m = 1, 2, . . .]

BI (162)(16)

sin2m+1 x cos x

0

16.3

a2

m = 1, 2, . . .]

2m dx (−1)m π  = 2m+1 1 − e−2a 2 +x 2 a

sin2m x cos[(2m + 2)x]

0



471

a > 0]

BI (162)(30)

m

dx 1   m  −2ka e = Ei(2ka) − e2ka Ei(−2ka) a2 + x2 2m+1 a k k=1

[a > 0]

BI (162)(8)

472

Trigonometric Functions



∞

22.

cosn sx sin nsx

0



 x dx π  −2as n 1 + e = − 1 a2 + x2 2n+1 [s > 0,

∞

23.

cosn sx sin nsx

0



cosm−1 x sin[(m + 1)x]

cosm x sin[(m + 1)x]

0 ∞

26.3

cosm x sin[(m − 1)x]

0

27.12 28. 29. 30.

BI (163)(9)

x dx π  −n 2 − cosn as cos nas = 2 −x 2

∞

BI (166)(10)

 m−1 x dx π = m e−2a 1 + e−2a 2 +x 2

a2

[a > 0, ∞

25.



n ≥ 0]

a2

0



Re a > 0,

[n = 0, 1, . . .] ∞

24. 

3.832

x dx π = m+1 e a2 + x2 2

 −a

1+e

x dx π = m cosh a a2 + x2 2

m = 1, 2, . . .]

BI (163)(6)

−2a m

[m = 0, 1, . . . ,  m−1 1 + e−2a −1

a > 0]

BI (163)(10)

[m = 0, 1, . . . ,

a ≥ 0]

BI (163)(7)



x dx π = e−3a coshm a [a > 0, m = 1, 2, . . .] 2 +x 2 0  ∞ n dx π  cosn sx cos nsx 2 = n+1 1 + e−2as [n = 0, 1, . . .] 2 a +x 2 a 0  ∞ dx π cosn as sin nas cosn sx cos nsx 2 = [n = 0, 1, . . .] 2 a −x 2a 0  ∞  m−1 dx π cosm−1 x cos[(m + 1)x] 2 = m e−2a 1 + e−2a 2 a +x 2 a 0 

∞

31.

cosm x sin(3mx)

cosm x cos[(m − 1)x]

0

 32.

cosm x cos[(m + 1)x]

0



∞

33. 0

 34. 0

dx π = m+1 e a2 + x2 2 a

 a

[m = 1, 2, . . . , 1 + e−2a

m

 − 1 − e−2a

[m = 0, 1, . . . , ∞

∞

sinp x cos x

BI (163)(11)

a2

a2

a > 0] 

BI (163)(14)

a > 0]

BI (163)(15)

a > 0]

BI (163)(17)

 m dx π = m+1 e−a 1 + e−2a 2 +x 2 a





[m = 0, 1, . . . ,

∞ sinp−1 x dx p p + 1 ∞ sinp+1 x = dx − dx q q−1 x q−1 0 x  q−1 0 xq−1 ∞ p(p − 1) dx = sinp−2 x cos x q−2 (q − 1)(q − 2) 0 x ∞ (p + 1)2 dx − sinp x cos x q−2 (q − 1)(q − 2) 0 x

cos2m x cos 2nx sin x

BI (163)(16)

dx x= x

 0

∞

cos2m−1 x cos 2nx sin

[p > q − 1 > 0]

[p > q − 1 > 1]

π dx x = 2m+1 x 2



2m m+n



GW (333)(18)

BI (152)(5, 6)

3.836

Powers and trigonometric functions



∞

35. 0



∞

36.

cosp ax sin bx cos x

cosp ax sin pax cos x

0



∞

37. 0

π dx = x 2

dx x2



n 1



pk

cos

k=1

473

[b > ap,

π dx = p+1 (2p − 1) x 2

p > −1]

[p > −1] 

π ak x sin bx sin x = 2

b>

n 

BI (153)(12) BI (153)(2)

 ak p k ,

ak > 0,

pk > 0

k=1

BI (157)(15)

3.833



∞

1.10

sin2m+1 x cos2n x

0

dx = x



∞

sin2m+1 x cos2n−1 x

0

(2m − 1)!!(2n − 1)!! dx = π x 2m+n+1 (m + n)! BI (151)(24, 25)

  1 1 1 = B m + ,n+ 2 2 2

GW (333)(24)



∞

2.

sin2m+1 2x cos2n−1 2x cos2 x

0

3.834



∞

1. 0

π (2m − 1)!!(2n − 1)!! dx = x 2 (2m + 2n)!!

sin2m+1 x dx (−1)m π(1 + a)4m = · 1 − 2a cos x + a2 x 22m+2 a2m+1  2m  k m− − (−1) k k=0



∞

2. 0

sin2m+1 x cosn x dx p (1 − 2a cos x + a2 ) x =

3.835



∞

1. 0



∞

2. 0

3.836



1. 0

∞

LI (152)(4)

   1 − a 2m−1   1 + a k  1  4a 2 (1 + a)2 [|a| = 1]

GW (333)(62a)

n

 (−1)k (2m + 2n − 2k + 1)!!(2m + 2k − 1)!! n!π n+1 2p 2 (2m + n + 1)!(1 + a) k!(n − k)! k=0   4a 3 × F m + n − k + , p; 2m + n + 2; 2 (1 + a)2 [a = ±1] GW (333)(62)

cos2m x cos 2mx sin x dx π b2m−1 = 2 a(a + b)2m a2 cos2 x + b2 sin2 x x

[ab > 0]

BI (182)(31)a

cos2m−1 x cos 2mx sin x dx π b2m−1 = 2 2 2 2 2a (a + b)2m a cos x + b sin x x

[ab > 0]

LI (182)(32)a

[m ≥ n]

LI (159)(12)



sin x x

n

sin mx π dx = x 2

474

Trigonometric Functions



∞



2.11 0

sin x x

n

nπ cos mx dx = n 2

12 (m+n)   k=0

3.837

(−1)k (n + m − 2k)n−1 k!(n − k)!

[0 ≤ m < n] [m ≥ n ≥ 2]

=0 π = 4

[m = n = 1] GI(159)(14), ET I 20(11)



∞

3. 0



∞



4.8 0

sin x x sin x x

n−1 sin nx cos x n

dx π = x 2 ⎡

sin(anx) π⎢ 1 dx = ⎣1 − n−1 x 2 2 n!

[n ≥ 1]

12 n(1+a)  

(−1)k

n

k=0

k

BI (159)(20)

⎤ ⎥ (n + an − 2k)n ⎦

[all real a, n ≥ 1] 5.10

I n (b) =

2 π



∞

0



sin x x

n

n  −1  (n − b − 2k)n−1 cos bx dx = n 2n−1 n! (−1)k k r

k=0

[0 ≤ b < n, 

∞



6.11 0

3.837



π/2

1. 0



π/4

2. 0

 3. 4.

sin x x

ET I 20(11)

n ≥ 1,

r = (n − b)/2]

LO V 340(14)

n cos anx dx = 0

[a ≤ −1 or a ≥ 1,

n ≥ 2;

for n = 1 see 3.741 2]

x2 dx = π ln 2 sin2 x

BI (206)(9)

x2 dx π π2 + ln 2 + G =− 2 16 4 sin x

BI (204)(10)

π/4

x2 dx π2 π = + ln 2 − G 2 cos x 16 4 0    π/4 p+1 ∞   π p 1 1  x 1 π p+1 − ζ(2k) dx = − + (p + 1) 4 4 p 2 42k−1 (p + 2k) sin2 x 0

GW (333)(35a)

k=1



[p > 0] π/2

5. 0



π/2

6. 0



∞

7. 0

 8.

0

∞

LI (204)(14)

x2 cos x π2 + 4G dx = − 2 4 sin x

BI (206)(7)

x3 cos x 3 π3 + π ln 2 dx = − 3 16 2 sin x

BI (206)(8)

cos 2nx dx sin2n x m = 0 cos x x cos 2nx dx sin2n+1 x m = 0 cos x x

 



m−1 , 2

m>0

m−2 , n> 2

m>0

n>

BI (180)(16)

 BI (180)(17)

3.839

Powers and trigonometric functions



1

9. 0

 10.3 11.3

x dx 1 = cosec a · ln sec a cos ax cos[a(1 − x)] a

π

x sin(2n + 1)x 1 dx = π 2 sin x 2 0  π n  x sin 2nx dx = −4 (2k − 1)−2 sin x 0

 π a< 2

475

BI (149)(20)

[n = 0, 1, 2, . . .] [n = 1, 2, 3, . . .]

k=1

3.838  1.

π/2

0



π/4

2. 0

 3.

4.

x cosp−1 x πp π sec dx = p+1 2p 2 sin x x sinp−1 x π 1 dx = − β cosp+1 x 4p 2p

[p < 1] 

p+1 2

BI (206)(13)a

 [p > −1]

m−1 x sin2m−1 x π 1  (−1)k−1 dx = (1 − cos mπ) + cos2m+1 x 8m 2m 2m − 2k − 1 0 k=0    π/4 m−1  (−1)k−1 x sin2m x 1 π dx = + (−1)m−1 ln 2 + cos2m+2 x 2(2m + 1) 2 m−k 0

LI (204)(15)

π/4

BI (204)(17)

BI (204)(16)

k=0

3.839  1.11

π/4

x tan2 x dx =

π2 1 π − − ln 2 4 32 2

BI (204)(3)

x tan3 x dx =

1 π 1 π − + ln 2 − G 4 2 8 2

BI (204)(7)

0



π/4

2. 0

 3. 4. 5.

6.

π/4

x2 tan x 1 π π2 dx = ln 2 − + cos2 x 2 4 16 0    π/4 2 2 x tan x 1 π π π2 dx = 1 − ln 2 − + +G cos2 x 3 4 2 16 0  π/2 Γ(p + 1) π x cosp x tan x dx = p+1   2 p 2 p 0 +1 Γ 2    π/2 p−1 2 p+1 p+1 π p − B , x sin x cot x dx = 2p p 2 2 0 

∞

7. 0

 8. 9. 10.

∞

sin2n x tan x

(cf. 3.839 1)

BI (204)(13)

(cf. 3.839 2)

BI (204)(12)

[p > −1]

BI (205)(3)

[p > −1]

BI (206)(11)

π (2n − 1)!! dx = x 2 (2n)!!

π dx = [s > −1] x 2 0  2n  ∞ cos[(2n − 1)x] sin x 22n − 1 2n−1 2 dx = (−1)n−1 π|B2n | cos x x (2n)! 0  ∞ 2

rπ dx π sec tanhr pq tanr px 2 = r 0]    2   b b b C √ − cos S √ a a a

[a ≥ 0]   π b2 b2 sin − cos 2a a a

BI (150)(5)a

[a > 0, b > 0]   2    2   π b b b b cos C √ + sin S √ 2a a a a a

BI (150)(7)

BI (150)(6)a [a ≥ 0]        2  √ π b b dx bπ b √ √ S −C + aπ sin + sin ax2 cos(bx) 2 = x 2 2 a 2 a 4a 4 [a > 0, b > 0] , (cf. 3.691 7) ET I 23(3)a

3.852



∞

1. 0



∞

2. 0

4.10 5.12 6.12



sin ax

2



∞

[a ≥ 0]

  √  2 dx 1 π √ cos bx = a + b + a − b [a > b > 0] x2 2 2 1√ πa [b = a ≥ 0] = 2  √ 1 π √ a+b− b−a = 2 2 [b > a > 0] ,

 √ sin2 a2 x2 2 π 3 a dx = x4 3 0   ∞ 3  2 2 √  sin a x a π 3 − dx = 3 x2 4 2 0     ∞ sin x2 − x2 cos x2 1 π dx = x4 3 2 0   ∞  2 1 dx 1 =− C cos x − 2 1 + x x 2 0

 3.

  sin ax2 aπ dx = 2 x 2

[a ≥ 0]

Im a2 = 0

BI (177)(10)a

(cf. 3.852 1)

BI (177)(23)

GW (333)(19e)

MC

BI (178)(8) BI (173)(22)

480

7.

∗

Trigonometric Functions



∞

xm cos(axn ) dx =

0

8.∗



∞

xm sin(axn ) dx =

0

3.853



∞

12

1.

0

3.

4.

3.854 1.

12

2.12 3.12 4.12

Γ(γ) sin naγ

 πγ  2

γ=

γ=

m+1 n [a > 0,

γ > 0]

[a > 0,

γ > 0]

m+1 n

   √  2  sin x2 π √ π π 2 2 C (b) − S (b) − sin b dx = 2 sin b + 2 cos b + b2 + x2 2b 4 4

ET II 219(33)a [Re b > 0]       √  cos x π √ π π C (b) − 2 sin b2 + S (b) − cos b2 dx = − 2 cos b2 + 2 2 b +x 2b 4 4 0 ET II 221(51)a [Re b > 0]  2  ∞ 2 x sin ax dx b2 + x2 0   π  √ √ π  √  bπ   2 √ sin ab − 2 sin ab2 + C S ab + 2 cos ab2 + ab = 2  4 4 1 π − 2 2a [a > 0, Re b > 0] ET II 219(32)a  2   ∞ 2     √ √ x cos ax bπ 1 π π − cos ab2 − 2 cos ab2 + C dx = ab 2 2 b +x 2 2a 2 4 0    √ √ π S ab − 2 sin ab2 + 4 [a > 0, Re b > 0] ET II 221(50)a



2.12

 πγ  Γ(γ) cos naγ 2

3.853

∞



2



  cos ax2 − sin ax2 πe−a √ dx = x4 + 1 2 2 0    ∞ cos ax2 + sin ax2 2 πe−a x dx = √ 4 x +1 2 2 0  2  2    ∞ cos ax + sin ax πe−a 1 2 √ x dx = a + 2 2 4 2 (x4 + 1) 0      ∞ cos ax2 − sin ax2 4 πe−a 1 √ − a x dx = 2 4 2 2 (x4 + 1) 0 

∞

[a > 0]

LI (178)(11)a, BI (168)(25)

[a > 0]

LI (178)(12)

[a > 0]

LI (178)(14)

[a > 0]

BI (178)(15)

3.856

3.855 1. 2. 3.12 4.

Trigonometric functions with powers

      sin ax2 1 aπ ab ab √ I 14 K 14 [a > 0, dx = 2 + x4 2 2 2 2 b 0        ∞ cos ax2 ab ab 1 aπ √ I − 14 K 14 [a > 0, dx = 2 4 2 2 2 2 b +x 0     2  u sin a2 x2 a2 u 2 a π3 √ J 14 dx = [a > 0] 4 2 2 u4 − x4 0    2 2  2 2  ∞ sin a2 x2 a u a u a π3 √ J 14 Y 14 dx = − 4 4 4 2 2 2 x −u u



 5. 6. 3.856 1.

2.

3.

4.

5.

6.

∞

u











2 2

481

Re b > 0]

ET I 66(28)

Re b > 0]

ET I 9(22)

[a > 0]

cos a x a2 u a π3 √ J − 14 dx = 4 4 4 2 2 u −x 0   2 2    2 2  ∞ cos a x a2 u 2 a u a π3 √ J − 14 Y − 14 dx = − 4 2 2 2 x4 − u4 u 2 2

 ν   2 2  2 2 β 4 + x4 + x2  2 2 a β a β a π 2ν  β I 14 − ν2 K 14 + ν2 sin a x dx = 4 4 2 2 2 2 β +x 0

3 Re ν < 2 , |arg β| < π4  ν   2 2  2 2  ∞ β 4 + x4 + x2  2 2 a π 2ν a β a β  K − 14 + ν2 cos a x dx = β I − 14 − ν2 4 4 2 2 2 2 β +x 0

3 π Re ν < 2 , |arg β| < 4  ν   2 2  2 2  ∞ β 4 + x4 − x2  2 2 a π 2ν a β a β  K − 14 − ν2 cos a x dx = β I − 14 + ν2 4 4 2 2 2 2 β +x 0

3 Re ν > − 2 , |arg β| < π4  2 2  2 2  ∞ sinh a 2β sin a2 x2 dx a β  = √ K0   2 2 2β 0 β 4 + x4 x2 + β 4 + x4

|arg β| < π4  2 2  2 2  ∞ sinh a 2β cos a2 x2 dx a β  √ = K 1  4 3  2 2 2β 0  4 x2 + β 4 + x4 β + x4

|arg β| < π4  2 2  ∞ √ 4  2 2 b + x4 + x2 a b π − a2 b2 2 √ sin a x dx = √ e I0 4 4 2 b +x 2 2 0

|arg b| < π4



ET I 66(29)

ET I 66(30) ET I 9(23)

ET I 10(24)

∞

ET I 71(23)

ET I 12(16)

ET I 12(17)

ET I 66(32)

ET I 10(27)

ET I 67(33)

482

Trigonometric Functions

3.857  1.

∞

0



∞

2. 0

3.858

x2 R1 R2

x2 R1 R2





3.857

 R2 − R1 1 sin ax2 dx = √ K 0 (ac) sin ab R2 + 2 b  R1   2 2 R1 = c2 + (b − x2 ) , R2 = c2 + (b + x2 ) ,  R2 + R1 1 cos ax2 dx = √ K 0 (ac) cos ab R2 − R 2 b  1   2 2 2 2 R1 = c + (b − x ) , R2 = c2 + (b + x2 ) ,

 a > 0,

c>0

ET I 67(34)

 a > 0,

c>0

ET I 10(26)

√ √ ν  ν  x4 − u4 + x2 − x4 − u4 √ 1. sin a2 x2 dx 4 4 u x − u   2 2  2 2  2 2  2 2  a u a u a u a u a π 3 2ν u Y 14 − ν2 + J 14 − ν2 Y 14 + ν2 J 14 + ν2 =− 4 a 2 2 2 2

Re ν < 32 ET I 71(25) √ ν  ν  ∞ 2 √ 4  x + x − u4 + x2 − x4 − u4 √ 2. cos a2 x2 dx x4 − u4 u    2 2  2 2  2 2  2 2  a u a u a u a u a π 3 2ν u Y − 14 − ν2 + J − 14 − ν2 Y − 14 + ν2 J − 14 + ν2 =− 4 a 2 2 2 2

Re ν < 32 ET I 13(26)   ∞   1 1 dx 2n = − nC cos x − 3.859 BI (173)(24) n+1 2 x 2 1+x 0 3.861   √ m− 1  ∞ n+1   2 dx πa 2 1 2n+1 k−1 2n + 1 (2k − 1)m− 2 ax sin = ± 2n−m+ 1 (−1) 1. 2m x n + k 2 2 (2m − 1)!! k=1 0   the + sign is taken when m ≡ 0 (mod 4) or m ≡ 1 (mod 4), BI (177)(19)a the − sign is taken when m ≡ 2 (mod 4) or m ≡ 3 (mod 4)   √ m− 1  ∞ n   dx πa 2 1 2n k m− 2 2. sin2n ax2 2m = ± 2n−2m+1 (−1)k x 2 (2m − 1)!! n+k 0 k=1   the + sign is taken when m ≡ 0 (mod 4) or m ≡ 3 (mod 4), BI (177)(18)a, LI (177)(18) the − sign is taken when m ≡ 2 (mod 4) or m ≡ 1 (mod 4)    ∞  2√  2 √ sin2 x n cos ax n + sin ax n 3.862 dx x2 0 √ n n   √ n− 12 π √ n − 2k + a n = (−1)k k (2n − 1)!! 2 k=0

√ a> n>0 BI (178)(9) 3.863      2   2  2   ∞   4 π b π b2 b b π b3 2 2 1 3 − J + cos − J sin 1. x cos ax sin 2bx dx = − −4 3 4 8 a 2a 8 2a 2a 8 2a 0 

∞



x2 +

[a > 0,

b > 0]

ET I 25(22)

3.866

Trigonometric functions with powers



∞

2. 0

3.864 1. 2.



  π x cos ax4 cos 2bx2 dx = − 8 2



483

  2    2  2  2  b3 π b π b b b 3 1 + J + cos + J sin −4 −4 a3 2a 8 2a 2a 8 2a

 √   √  dx π b sin ax = Y 0 2 ab + K 0 2 ab x x 2 0  ∞  √   √  dx π b = − Y 0 2 ab + K 0 2 ab cos cos ax x x 2 0 ∞

sin

[a > 0,

b > 0]

ET I 25(23)

[a > 0,

b > 0]

WA 204(3)a

[a > 0,

b > 0] WA 204(4)a, ET I 24 (12)

3.865



u



u2 − x2 x2μ

1. 0



∞

2. u

μ−1

√  μ− 12 a 3 π 2 a sin dx = uμ− 2 Γ(μ) J 12 −μ x 2 a u [a > 0, u > 0,

μ−1

(x − u) x2μ

sin

a dx = x



0 < Re μ < 1] ET II 189(30)

π 1 −μ a a 2 Γ(μ) sin J 1 u 2u μ− 2

a 2u [a > 0,

u > 0,

Re μ > 0] ET II 203(21)

 u

μ−1 √  μ− 12 a u2 − x2 a π 2 μ− 32 1 cos Γ(μ)u Y dx = − 2 −μ x2μ x 2 a u [a > 0, u > 0,

3. 0



∞

4. u

μ−1

(x − u) x2μ

cos

a dx = x



0 < Re μ < 1] ET II 190(36)

a π 1 −μ a 2 Γ(μ) cos J 1 u 2u μ− 2

a 2u [a > 0,

u > 0,

Re μ > 0] ET II 204(26)

3.866



∞

1. 0

 0

3. 0

x

 b2 π sin sin a2 x dx = x 4

 μ b μπ [J μ (2ab) − J −μ (2ab) + I −μ (2ab) − I μ (2ab)] cosec a 2 [a > 0, b > 0, |Re μ| < 1] ET I 322(42)

∞

2.



μ−1

 μ  b2 π b μπ [J μ (2ab) + J −μ (2ab) + I μ (2ab) − I −μ (2ab)] xμ−1 sin cos a2 x dx = sec x 4 a 2 [a > 0, b > 0, |Re μ| < 1]

ET I 322(43)

∞

 μ  2 b π b μπ μ−1 x cos cos a x dx = cosec [J −μ (2ab) − J μ (2ab) + I −μ (2ab) − I μ (2ab)] x 4 a 2 [a > 0, b > 0, |Re μ| < 1] 2

ET I 322(44)

484

3.867

Trigonometric Functions



1

1. 0

 2. 0

3.868  1.

1

cos ax − cos xa 1 dx = 1 − x2 2 cos ax + cos 1 + x2

∞

0

3. 4.

dx =

3.869  1.

∞

∞

0



∞

2. 0

1 2

∞

cos ax − cos ax π dx = sin a 1 − x2 2

∞

a x

0

 0

cos ax + cos 1 + x2

dx =

[a > 0]

GW (334)(7a)

[a > 0]

GW (334)(7b)

π −a e 2

  b2 dx = π J 0 (2ab) sin a2 x + x x

  b2 2 cos a x + x 0    ∞ b2 sin a2 x − x 0    ∞ b2 2 cos a x − x 0 

2.

a x



3.867

[a > 0,

b > 0] GW (334)(11a), WA 200(16)

dx = −π Y 0 (2ab) x

[a > 0,

b > 0]

GW (334)(11a)

dx =0 x

[a > 0,

b > 0]

GW (334)(11b)

dx = 2 K 0 (2ab) x

[a > 0,

b > 0]

GW (334)(11b)

[a > 0,

b > 0,

    x dx b b π sin ax − = exp −aβ − x β 2 + x2 2 β     b π dx b cos ax − = exp −aβ − x β 2 + x2 2β β

Re β > 0] ET II 220(42)

[a > 0,

b > 0,

Re β > 0] ET II 222(58)

3.871  1.

∞

0



∞

2. 0

    b2 μπ μπ  dx = πbμ J μ (2ab) cos − Y μ (2ab) sin xμ−1 sin a x + x 2 2 [a > 0, b > 0, Re μ < 1]     b2 μπ μπ  μ−1 dx = −πbμ J μ (2ab) sin + Y μ (2ab) cos x cos a x + x 2 2 [a > 0, b > 0, |Re μ| < 1]

   ∞ b2 μπ dx = 2bμ K μ (2ab) sin xμ−1 sin a x − x 2 0

ET I 321(35)

 3.

 4. 0

∞

   b2 μπ dx = 2bμ K μ (2ab) cos xμ−1 cos a x − x 2

ET I 319(17)

[a > 0,

b > 0,

|Re μ| < 1] ET I 319(16)

[a > 0,

b > 0,

|Re μ| < 1] ET I 321(36)

3.875

3.872 1.

2.

3.873 1.

2.

3.874

Trigonometric functions with powers

485

      1 dx 1 sin a x − sin a x + x x 1 − x2   0      1 dx 1 ∞ 1 π sin a x − = sin a x + = − sin 2a 2 2 0 x x 1−x 4 [a ≥ 0] BI (149)(15), GW (334)(8a)        1 1 dx 1 cos a x − cos a x + x x 1+ x2   0     1 dx 1 ∞ 1 π cos a x − = cos a x + = e−2a 2 0 x x 1 + x2 4 [a ≥ 0] GW (334)(8b)



1

√

π a2 2 2 dx sin(2ab) + cos(2ab) + e−2ab sin 2 cos b x 2 = √ x x 4 2a 0 [a > 0, √  ∞

dx π a2 cos(2ab) − sin(2ab) + e−2ab cos 2 cos b2 x2 2 = √ x x 4 2a 0 [a > 0,





∞

∞

1. 0

  √  π π b2 dx 2 2 sin 2ab + sin a x + 2 = x x2 2b 4

  √  ∞ π π b2 dx 2 2 cos 2ab + cos a x + 2 = 2 x x 2b 4 0

[a > 0,

b > 0]

ET I 24(15)

b > 0]

ET I 24(16)

b > 0] BI (179)(6)a, GW(334)(10a)

 2.

  √ ∞ π b2 dx √ e−2ab sin a2 x2 − 2 = − 2 x x 2 2b 0   √  ∞ 2 π dx b cos a2 x2 − 2 = √ e−2ab 2 x x 2 2b 0 √ 2   ∞ dx 2π b sin ax − = 2 x x 4b 0 √ 2   ∞ dx 2π b cos ax − = 2 x x 4b 0

[a > 0,

b > 0] GI (179)(8)a, GW(334)(10a)

 3. 4. 5. 6. 3.875



∞

1. u

[a ≥ 0,

b > 0]

GW (335)(10b)

[a ≥ 0,

b > 0]

GW (334)(10b)

[a > 0,

b > 0]

BI (179)(13)a

[a > 0,

b > 0]

BI (179)(14)a

 √    x sin p x2 − u2 π 2 + u2 cosh ab exp −p cos bx dx = a x2 + a2 2 [0 < b < p]

ET I 27(39)

486

Trigonometric Functions



∞

2. u

 3.12

∞

4.∗

0

5.∗



∞

 √ sin p a2 + x2 (a2

x2 )2

(x2 +

2 a2 )



 √ cos p x2 + a2 (x2 +

0

6.∗

 √    x sin p x2 − u2 π −ap 2 − a2 e cos bx dx = cos b u a2 + x2 − u2 2

+  √  ∞ sin p x2 + a2 0

∞

a2 )3/2

cos bx dx =

3/2 a2 )

πp −ab e 2a

cos(bx) dx =

cos(bx) dx =

 √ x cos p x2 + a2 (x2 +

0

3.876

[0 < b < p,

a > 0]

ET I 27(38)

[0 < p < b,

b > a > 0]

ET I 26(29)

pb p3 K1 (ab) − K0 (ab) a 6 [a > 0,

b > p > 0]

[a > 0,

b > p > 0]

b p2 K1 (ab) − K0 (ab) a 2

sin(bx) dx = bK0 (ab) −

ap2 K1 (ab) 2 [a > 0,

3.876



∞

1. 0

 √ ∞    cos p x2 + a2 π √ cos bx dx = − Y 0 a p2 − b2 2  x2 + a2 0  = K 0 a b 2 − p2

 2.

 √    sin p x2 + a2 π √ cos bx dx = J 0 a p2 − b2 2 x2 + a2 =0

b > p > 0]

[0 < b < p] [b > p > 0] [a > 0]

ET I 26(30)

[0 < b < p] [b > p > 0]

[a > 0] √  ∞    cos p x2 + a2 π cos bx dx = e−bc cos p a2 − c2 2 2 x +c 2c 0

ET I 26(34)

 3.

[c > 0, b > p]  √  √ −bc sin p x2 + a2 sin p a2 − c2 π e √ √ cos bx dx = [c = a] 2 2 2 2 2c a 2 − c2 0 (x + c ) x + a p π [c = a] = e−ba 2 a [b > p, c > 0]  √  ∞ cos p x2 + a2 π −ab e cos bx dx = [b > p > 0, a > 0] 2 + a2 x 2a 0  √  ∞ x cos p x2 + a2 π sin bx dx = e−ab [a > 0, b > p > 0] 2 2 x +a 2 0  √  u    cos p u2 − x2 π √ cos bx dx = J 0 u b2 + p2 2 u2 − x2 0 

4.

5.6 6.6 7.

ET I 26(33)

∞

ET I 26(31)a ET I 27(35)a

ET I 85(29)a

ET I 28(42)

3.881

Trigonometric functions with powers



∞

8. u

 √    cos p x2 − u2 √ cos bx dx = K 0 u p2 − b2 x2 − u2    π = − Y 0 u b 2 − p2 2

487

[0 < b < |p|] [b > |p|] ET I 28(43)

3.877



u

1. 0

  √  u   u    sin p u2 − x2 π3 p  J 14 cos bx dx = b2 + p2 − b J 14 b 2 + p2 + b 8 2 2 4 3 (u2 − x2 )

ET I 27(40) [b > 0, p > 0]  √  ∞ u   u     sin p x2 − u2 π3 p  J 14 b − b2 − p2 Y 14 b + b 2 − p2 cos bx dx = − 8 2 2 4 3 u (x2 − u2 )

 2.

[b > p > 0] ET I 27(41)  √  u  u   u    cos p u2 − x2 π3 p  J − 14 cos bx dx = p2 + b2 − b J − 14 p2 + b 2 + b 8 2 2 4 0 (u2 − x2 )3

 3.

4.

ET I 28(44) [u > 0, p > 0]  √   ∞ u   u     cos p x2 − u2 π3 p  J − 14 b − b2 − p2 Y 14 b + b 2 − p2 cos bx dx = − 8 2 2 3 4 u (x2 − u2 )

[b > p > 0] 3.878 1.

2.

3.

  √  2  2     sin p x4 + a4 1  π 3 a a 2 √ p − p2 − b2 J 14 p + p2 − b 2 cos bx dx = b J − 14 2 2 2 2 x4 + a4 0 ET I 26(32) [p > b > 0]  √  ∞ cos p x4 + a4 √ cos bx2 dx 4 + a4 x 0   2  2     a a 1  π 3 p − p2 − b2 Y 14 p + p2 − b 2 =− b J − 14 2 2 2 2 [a > 0, p > b > 0] ET I 27(36)    √     u   cos p u4 − x4 u 2  2 u 2  2 π 3 1 √ cos bx2 dx = b J − 14 p + b2 − p J − 14 p + b2 + p 2 2 2 2 u4 − x4 0



∞

 3.879 3.881

ET I 28(45)

∞

0



1. 0

π/2

sin axp

dx π = x 2p

x sin (a tan x) dx =

[p > 0,

b > 0]

ET I 28(46)

[a > 0,

p > 0]

GW (334)(6)

π −a e C + ln 2a − e2a Ei(−2a) 4 [a > 0]

BI (205)(9)

488

Trigonometric Functions

 2. 3. 4. 5. 6. 7. 8. 9. 10.

∞

π dx = 1 − e−a [a > 0] x 2 0  ∞ π dx = 1 − e−a [a > 0] sin (a tan x) cos x x 2 0  ∞ π dx = e−a cos (a tan x) sin x [a > 0] x 2 0  ∞ 1 + a −a dx = πe sin (a tan x) sin 2x [a > 0] x 2 0  ∞ 1 − a −a dx = πe cos (a tan x) sin3 x [a > 0] x 4 0  ∞ x dx 1 + a −a sin (a tan x) tan cos2 x [a > 0] = πe 2 x 4 0  π/2 x dx π cos (a tan x) = − Ei(−a) [a > 0] sin 2x 4 0  π/2 1 − e−a x dx π [a > 0] = sin (a cot x) 2 2a sin x 0  π/2

π x cos (a tan x) tan x dx = − e−a C + ln 2a + e2a Ei(−2a) 4 0 

∞

11. 0

 12. 13. 14. 15. 16. 17.

3.882

∞

sin (a tan x)

cos (a tan x) tan x

dx π = e−a x 2

1 − a −a dx = πe x 16 0  ∞ π dx = e−a sin (a tan x) tan2 x x 2 0  ∞ π dx = e−a cos (a tan 2x) tan x x 2 0  ∞ 1 + a −a dx = πe sin (a tan 2x) cos2 2x tan x x 4 0  ∞ π dx = e−a sin (a tan 2x) tan x tan 2x x 2 0  ∞ π dx = 1 − e−a sin (a tan 2x) tan x cot 2x x 2 0

3.882  1.

∞

0

 2. 0

cos (a tan x) sin2 x tan x

BI (151)(19) BI (151)(20) BI (152)(11) BI (151)(23) BI (152)(13)

BI (206)(15)

LI (206)(14)

[a > 0]

BI (205)(10)

[a > 0]

BI (151)(21)

[a > 0]

BI (152)(15)

[a > 0]

BI (152)(9)

[a > 0]

BI (151)(22)

[a > 0]

BI (152)(13)

[a > 0]

BI (152)(10)

[a > 0]

BI (180)(6)

 x dx π exp (−a tanh b) − e−a sin a tan2 x 2 = 2 b +x 2 [a > 0,

∞

BI (151)(6)

 cos a tan2 x cos x

b > 0]

dx π cosh b exp (−a tanh b) − e−a sinh b = b2 + x2 2b [a > 0, b > 0]

BI (160)(22)

BI (163)(3)

3.892

Trigonometric functions and exponentials



∞

3. 0

 x dx π exp (−a tanh b) cos a tan2 x cosec 2x 2 = b + x2 2 sinh 2b [a > 0,



−a 

x dx π e cosh b − exp (−a tanh b) sinh b cos a tan2 x tan x 2 = 2 b +x 2 cosh b

∞

x dx π coth b exp (−a tanh b) − e−a cos a tan2 x cot x 2 = 2 b +x 2

0

[a > 0,

5.11 0 ∞

6. 0



1

cos (a ln x)

1. 0



1

2. 1

3. 0

xμ−1 sin (β ln x) dx = − xμ−1 cos (β ln x) dx =

3.884

b > 0]

BI (163)(5)

b > 0]

BI (191)(11)

BI (404)(4)

β β 2 + μ2

μ β 2 + μ2

[Re μ > |Im β|]

ET I 319(19)

[Re μ > |Im β|]

ET I 321(38)

        sin a |x| sign x dx = π exp −a |−b| + exp −a |b| x−b −∞ [a > 0, Im b = 0]



11

BI (163)(4)

dx aπ = (1 + x)2 2 sinh aπ

0



b > 0]

 x dx π coth 2b exp (−a tanh b) − e−a cos a tan2 x cot 2x 2 = b + x2 2 [a > 0,

3.883

BI (191)(10)



[a > 0,



b > 0]

∞

4. 

489

∞

ET II 253(46)

3.89–3.91 Trigonometric functions and exponentials 3.891



2π

1. 0



2π 0



1.11 0

[m = n or m = n = 0]

= πi

2.

3.892

eimx sin nx dx = 0

π

[m = n = 0]

eimx cos nx dx = 0

[m = n]

=π

[m = n = 0]

= 2π

[m = n = 0]

π

eiβx sinν−1 x dx =

2ν−1 ν B



πeiβ 2  ν +β+1 ν −β+1 , 2 2 [Re ν > −1]

NH 158, EH I 12(29)

490

Trigonometric Functions

 2.

π 2

−π 2



eiβx cosν−1 x dx =

π/2

3.6

2ν−1 ν B

ei2βx sin2μ x cos2ν x dx =

0





3.893

π  ν +β+1 ν −β+1 , 2 2 

1 22μ+2ν+1

[Re ν > −1]

GW (335)(19)

 exp iπ β − ν − 12 B (β − μ − ν, 2ν + 1)

 × F (−2μ, β − μ − ν; 1 + β − μ + ν; −1) + exp iπ μ + 12

π

4.

 × B(β − μ − ν, 2μ + 1) F (−2ν, β − μ − ν; 1 + β + μ − ν; −1)

Re μ > − 12 , Re ν > − 12 EH I 80(6) ei2βx sin2μ x cos2ν x dx =

0

π exp [iπ(β − ν)] F (−2ν, β − μ − ν; 1 + β + μ − ν; −1) 4μ+ν (2μ + 1) B(1 − β + μ + ν, 1 + β + μ − ν) EH I 80(8)



π/2

5. 0

3.893  1.8

∞

0



∞

2.8 0



∞

3.

π

ei(μ+ν)x sinμ−1 x cosν−1 x dx = eiμ 2 B(μ, ν)   1 1 1 iμ π 2 F (1 − ν, 1; μ + 1; −1) + F (1 − μ, 1; ν + 1; −1) = μ+ν−1 e 2 μ ν [Re μ > 0, Re ν > 0] EH I 80(7)

e−px sin(qx + λ) dx = e−px cos(qx + λ) dx =



∞ −βx

e

0

sin ax dx = Re sin bx

∞ −2px

e

5.8 0



1 (p cos λ − q sin λ) + q2

[Re p > 0]

BI (261)(4)

p2



BI (261)(7)

     1 a+b β b−a β ψ −i −ψ −i 2bi 2b 2b 2b 2b

e

0



1 sin[(2n + 1)x] dx = + sin x 2p

n  k=1

p p2 + k 2

b = 0]

GW (335)(15)

[Re p > 0]

BI (267)(15)

[Re p > 0]

GW (335)(15c)

n−1

 sin 2nx 1 dx = 2p sin x p2 + (2k + 1)2

∞ −px

8

k=0

∞

7.

e−px cos[(2n + 1)x] tan x dx =

0

n−1

 (−1)k (2k + 1) 2n + 1 n + (−1) 2 p2 + (2n + 1)2 p2 + (2k + 1)2 k=0

 3.894

BI (261)(3)

[Re β > 0,



6.

[Re p > 0]

e−x cos t cos (t − x sin t) dx = 1

0

4.8

1 (q cos λ + p sin λ) p2 + q 2

[p > 0]

 ν  2π Γ(ν + 1) P m ν (β) β + β 2 − 1 cos x einx dx = Γ(ν + m + 1) −π [Re β > 0]

LI (267)(16)

π

ET I 157(15)

3.895

3.895

Trigonometric functions and exponentials



∞

1.

(2m)! β(β 2 + 22 ) (β 2 + 42 ) · · · [β 2 + (2m)2 ]

e−βx sin2m x dx =

0



[Re β > 0] π

2.10

e−px sin2m x dx =

0



π/2

3.12

p (p2

+

=

p (p2

+

4.

e−βx sin2m+1 x dx =

0

5.10

e−px sin2m+1 x dx =

π/2

6.8

=

7.

8.12

(2m + 1)! (β 2 + 12 ) (β 2 + 32 ) · · · [β 2 + (2m + 1)2 ]

(p2

+

(2m + 1)! (1 + e−pπ ) + 32 ) · · · [p2 + (2m + 1)2 ]

12 ) (p2

FI II 615, WA 620a GW (335)(4b)

(2m + 1)! + +32 ) · · · [p2 + (2m + 1)2 ]  2 

! p + 12 p2 + 32 · · · p2 + (2m − 1)2 −pπ p2 + 1 2 + ··· + × 1 − pe 2 1 + 3! (2m + 1)! (p2

12 ) (p2

BI (270)(5) ∞

e−px cos2m x dx =

0



BI (270)(4)

e−px sin2m+1 x dx

0



pπ 2

[Re β > 0] π

0



(2m)! 2 2 (2m)2 ]  + 4 ) · · · [p +  2 

! 2 2 p p + 22 p2 p2 + 22 · · · p2 + (2m − 2)2 p + + ···+ 1+ 2! 4! (2m)! [p = 0]

∞

GW (335)(4a)

22 ) (p2

× 1 − e−



(2m)! (1 − e−pπ ) + 42 ) · · · [p2 + (2m)2 ]

22 ) (p2

FI II 615, WA 620a

e−px sin2m x dx

0



491

(2m)! p (p2 + 22 ) · · · [p2 + (2m)2 ]  

! p 2 p2 + 2 2 p2 p2 + 22 · · · p2 + (2m − 2)2 p2 + + ··· + × 1+ 2! 4! (2m)! [p > 0]

π/2

BI (262)(3)

e−px cos2m x dx

0

=

(2m)! p (p2 + 22 ) · · · [p2 + (2m)2 ]  

! 2 p 2 p2 + 2 2 p2 p2 + 22 · · · p2 + (2m − 2)2 p −p π + + ···+ × −e 2 + 1 + 2! 4! (2m)! [p = 0]

BI (270)(6)

492

Trigonometric Functions

 9.

∞

7

e−px cos2m+1 x dx

0

=



π/2

10.11

(2m + 1)!p (p2 + 12 ) (p2 + 32 ) · · · [p2 + (2m + 1)2 ]  2  2  

! p + 1 2 p2 + 3 2 p + 12 p2 + 32 · · · p2 + (2m − 1)2 p2 + 1 2 + + ···+ × 1+ 3! 5! (2m + 1)!

=

∞

11.8

e−βx sinn ax

0



15.12 16.

3.896  1. 2. 3.

sin bx cos bx

BI (270)(7)



−n−2

dx =

1 2 e 4 (1∓1+2n)πi a(n + 1) ⎧  −1 ⎫  ⎨ b+na+iβ −1 ⎬ b+na−iβ 2a 2a × ± (−1)n ⎩ ⎭ n+1 n+1

a + 2m a (a2 + 4m2 ) 2

∞

e−q

2

x2

sin[p(x + λ)] dx =

k=1

Re β > 0] DW61 (861.06)

−ax

√ p2 π − 4q e 2 sin pλ q −∞ √  ∞ p2 2 2 π − 4q e 2 cos pλ e−q x cos[p(x + λ)] dx = q −∞  2    ∞ 2 b 1 3 b2 b e−ax sin bx dx = exp − ; ; 1F 1 2a 0  2 2 4a  4a 2 b 3 b = 1 F 1 1; ; − 2a 2 4a  2 k−1 ∞ b b  1 − = 2a (2k − 1)!! 2a ∞

b > 0,

2

 a a2 + m 2 + n 2 e cos mx cos nx dx = 2 (a + (m − n)2 ) (a2 + (m + n)2 ) 0   ∞ m a 2 + m2 − n 2 −ax e sin mx cos nx dx = 2 (a + (m − n)2 ) (a2 + (m + n)2 ) 0 6  ∞ 2m2 e−ax sin2 mx dx = [a > 0] a (a2 + 4m2 ) 0  ∞ 2amn e−ax sin mx sin nx dx = 2 [a + (m − n)2 ] [a2 + (m + n)2 ] 0 

14.



e−ax cos2 mx dx =

0

BI (262)(4)

(2m + 1)! 2 2 2] (p2 + 12 ) (p2 +  3 ) · · · [p + (2m + 1)  2

!  2 2 p + 1 p2 + 32 · · · p2 + (2m − 1)2 p +1 −p π + ···+ × e 2 +p 1+ 3! (2m + 1)!

[a > 0, ∞

12. 13.

[p > 0]

e−px cos2m+1 x dx

0



3.896

DW61 (861.15)

DW61 (861.14)

DW61 (861.10) DW61 (861.13)

BI (269)(2) BI (269)(3)

ET I 73(18)

[a > 0]

FI II 720

3.911

Trigonometric functions and exponentials



∞

4.

e

−βx2

0

3.897 1.

12

2.

3.898 1.

1 cos bx dx = 2



 2 b π exp − β 4β



∞

∞

−βx2 −γx

e

−βx2

0



∞

2.

i sin bx dx = − 4



∞

3.8

2

e−βx cos ax cos bx dx =

2

e−px sin2 ax dx =

0



12

∞ −p2 x2

e

0

 2.12



e

0

1 4

ET I 15(16)

  2 2 π − (a−b) − (a+b) 4β 4β e −e β





π e β

2 − (a−b) 4β

+e

2 − (a+b) 4β

 a2 π 1 − e− p p

[Re β > 0, 

a > 0,

b > 0]

BI (263)(4)

[Re β > 0]

BI (263)(5)

[Re p > 0]

BI (263)(6)

 √  n sin[(2n + 1)x] π 1  −( kp )2 dx = + e sin x p 2

2

[p > 0]  √  2n k 2 cos[(4n + 1)x] π 1  − dx = + (−1)k e ( p ) cos x p 2

BI (267)(17)

k=1

∞

0



2

∞ −p x

3.

1.12

1 4





ET I 74(27)

k=1

0

3.911

BI (263)(2)



1 sin ax sin bx dx = 4

0

1.

[Re β > 0]

    π (γ − ib)2 γ − ib √ exp 1−Φ e β 4β 2 β 0    γ + ib (γ + ib)2 √ 1−Φ − exp 4β 2 β [Re β > 0, b > 0]      ∞ (γ − ib)2 γ − ib 1 π −βx2 −γx √ exp 1−Φ e cos bx dx = 4 β 4β 2 β 0    γ + ib (γ + ib)2 √ 1−Φ + exp 4β 2 β [Re β > 0, b > 0] 

12

3.899

493

∞



2 π  e−px dx p 1 1 − 2a cos x + a2 = 1 − a2 2  π  p 1 = 2 a −1 2

sin ax 1 π dx = − x e +1 2a 2 sinh πa

[p > 0]  2  ∞ 2  k k a < 1, + a exp − 4p k=1  2  ∞ 2  k −k a > 1, + a exp − 4p

BI (267)(18)

p>0 p>0



EI (266)(1)

LI (266)(1)

k=1

[a > 0]

BI (264)(1)

494

Trigonometric Functions

 2.

∞

12 0



∞

3.11 0



∞

4. 0



∞

0

0 ∞

0



[a > 0]

BI (264)(2), WH

sin ax x/2 1 e dx = π tanh(aπ) x e −1 2

[a > 0]

ET I 73(13)

n−1

 sin ax −nx a 1 π π + 2πa − e dx = − −x 2 1−e 2 2a e −1 a + k2 sin ax eβx − eγx

[a > 0]      β + ia β − ia 1 ψ −ψ dx = 2i(β − γ) β−γ β−γ [Re β > 0,

∞

6. 3.912  1.

sin ax π 1 dx = coth πa − ex − 1 2 2a

k=1

5. 

∞

2.

sin ax dx i = [ψ(β + ia) − ψ(β − ia)] eβx (e−x − 1) 2

1.

Re γ > 0]

GW (335)(8)

[Re β > −1]

ET 73(15)

a > 0]

ET I 73(17)

     β − ia β + ia 1 B ν, + B ν, cos ax dx = 2γ γ γ [Re β > 0, Re γ > 0, Re ν > 0,

a > 0]

ET I 15(10)

e

−βx



1−e

−γx ν−1

  β μ β μ β2 ν ν −μ, π − − ; 1 + + − ; F 2 1 2 μ  2 2 2 2 2 2 ν   eiβx cosν x β 2 eix + ν 2 e−ix dx = π ν μ β ν μ β −2 ν 2 (ν + 1) B 1 + + − , 1 − + + 2 2 2 2 2 2



 2.11

BI (264)(8)

      ν−1 β − ia β + ia i B ν, − B ν, e−βx 1 − e−γx sin ax dx = − 2γ γ γ [Re β > 0, Re γ > 0, Re ν > 0,

0

3.913

3.912

π 2

[Re ν > −1, π 2

−π 2

3.914  1. 0

∞

|ν| > |β|]

EH I 81(11)a

ν  e−iux cosμ x a2 eix + b2 e−ix dx

  μ−ν−u a2 ; 1 + ; πb2ν 2 F 1 −ν, − u+μ+ν 2 2 2 b   = u + μ +ν u + ν − μ μ ,1 + 2 (μ + 1) B 1 − 2 2   u−μ−ν μ−ν+u b2 2ν πa 2 F 1 −ν, 2 ; 1 + 2 ; a2   = μ+ν−u u+μ−ν ,1 + 2μ (μ + 1) B 1 + 2 2 [Re μ > −1]

e−β

√

γ 2 +x2



for a2 < b2

for b2 < a2 ET I 122(31)a

   βγ cos bx dx =  K 1 γ β 2 + b2 β 2 + b2 [Re β > 0,

Re γ > 0]

ET I 16(26)

3.915

Trigonometric functions and exponentials



∞

γ2

2.

+

x2 e−β

√

γ 2 +x2

0

 3.

4.

∞

∞

5.  0



10.



1

+

1 γ 2 + x2

ET I 16(27)

 e−β

√

γ 2 +x2

cos bx dx =

   1  2 β + b2 K 1 γ β 2 + b2 βγ (6.726(4))

∞

6.

9.

 γ 2β 2 γ K 1 (γA) − A3 A    A = β 2 + b2



β (γ 2 + x2 )3/2

0

8.



√ 2 2 γ 2 + x2 e−β γ +x cos bx dx 0     3βγ 2 4β 3 γ 2 6βγ 8β 3 γ β3γ3 = − 2 + K K 1 (γA) (γA) + − + + 0 5 A A4 A3 A3  A  A = β 2 + b2  ∞ −β √γ 2 +x2    e  [Re β > 0, Re γ > 0, b > 0] cos bx dx = K 0 γ β 2 + b2 γ 2 + x2 0 

7.

β2γ2 cos bx dx = K 0 (γA) + A2

xe−β

√

γ 2 +x2

sin bx dx =

bβγ β 2 + b2 2

   K 2 γ β 2 + b2

ET I 175(35)

√ 2 2  x γ 2 + x2 e−β γ +x sin bx dx 0     bγ 2 4bβ 2 γ 2 2bγ 8bβ 2 γ bβ 2 γ 3 = − 2 + K K 1 (γA) (γA) + − + + 0 5 A A4 A3 A A3    A = β 2 + b2    ∞ √ 2 2  2 12bβγ 2 24bβ 3 γ 2 bβ 3 γ 4 K 0 (γA) γ + x2 e−β γ +x x sin bx dx = − + + 4 A6 A4 0   A 3bβγ 3 8bβ 3 γ 3 24bβγ 48bβ 3 γ K 1 (γA) + − + + − 7 A5 A3  A5 A  A = β 2 + b2 √  ∞ −β γ 2 +x2    xe γb  sin bx dx =  K 1 γ β 2 + b2 ET I 75(36) γ 2 + x2 β 2 + b2 0   ∞    √ 1 1 b −β γ 2 +x2 2 + b2 K e γ + x sin bx dx = β (6.726(3)) 0 3/2 γ 2 + x2 β 0 β (γ 2 + x2 ) ∞

3.915  π 2 ea cos x sin x dx = sinh a 1. a 0  π 2. eiβ cos x cos nx dx = in π J n (β) 0

 3.3

495

π 2

−π 2

eiβ cos x cos2ν x dx =

√

π

  ν  1 2 J ν (β) Γ ν+ β 2

GW (337)(15c) EH II 81(2)

Re ν > − 12

EH II 81(6)

496

Trigonometric Functions

  ν  2 1 I ν (β) e sin Γ ν+ β 2 0   ν   π √ 2 1 J ν (β) eiβ cos x sin2ν x dx = π Γ ν+ β 2 0 

4. 5. 3.916



π

±β cos x

π/2

1.

e

2ν

x√ −p2 tan x sin 2 cos x

sin 2x

0



π/2

2. 0

√ x dx = π

3.916

Re ν > − 12

GW (337)(15b)

Re ν > − 12

WA 34(2), WA 60(6)

 2  2 1 1 dx = C (p) − + S (p) − 2 2

exp (−p tan x) dx 1 = − eap Ei(−ap) sin 2x + a cos 2x + a 2

[p > 0] ,

NT 33(18)a

(cf. 3552 4 and 6) BI (273)(11)



π/2

3. 0

exp (−p cot x) dx 1 = − e−ap Ei(ap) sin 2x + a cos 2x − a 2

[p > 0] ,

(cf. 3.552 4 and 6) BI (273)(12)



π/2

4.

(1 −

0



exp (−p tan x) sin 2x dx 1 = − e−ap Ei(ap) + eap Ei(−ap) 2 2 2 − 2a cos 2x − (1 + a ) cos 2x 4

a2 )

[p > 0] π/2

5.

(1 −

0

BI (273)(13)

exp (−p cot x) sin 2x dx 1 = − e−ap Ei(ap) + eap Ei(−ap) 2 2 2 + 2a cos 2x − (1 + a ) cos 2x 4

a2 )

[p > 0] 3.917 1.

2.12

3.918

BI (273)(14)

     √ π 1 1 x dx = J ν (β) e cos x sin x sin β − ν − Γ ν+ 2 2(2β)ν 2 0

Re ν > − 12 WA 186(7)       √  π/2 π 1 1 x dx = Y ν (β) e−2β cot x cosν−1/2 x sin−(ν+1) x cos β − ν − Γ ν+ ν 2 2(2β) 2 0

Re ν > − 12 WA 186(8)



π/2



π/2

1. 0



π/2

2. 0

 3. 0

π/2

−2β cot x

ν−1/2

−(ν+1)



 cosμ x i γ(β−μx)−2β cot x π iγ (ε) (2β)−μ Γ(μ + 1) H μ+ 1 (β) e dx = 2μ+2 2 2 2β sin x

ε = 1, 2, γ = (−1)ε+1 , Re β > 0, Re μ > −1 1 cosμ x sin(β − μx) −2β cot x e dx = 2 sin2μ+2 x



cosμ x cos(β − μx) −2β cot x 1 e dx = − 2μ+2 2 sin x

GW (337)(16)

π (2β)−μ Γ(μ + 1) J μ+ 12 (β) 2β



[Re β > 0,

Re μ > −1]

WH

π (2β)−μ Γ(μ + 1) Y μ+ 12 (β) 2β [Re β > 0,

Re μ > −1]

GW (337)(17b)

3.922

3.919

Trigonometric functions with exponentials



π/2

1. 0



π/2

2. 0

497

sin 2nx dx n−1 2n − 1 2n+2 exp (2π cot x) − 1 = (−1) 4(2n + 1) sin x

BI (275)(6), LI (275)(6)

sin 2nx n dx n−1 2n+2 exp (π cot x) − 1 = (−1) 2n + 1 sin x

BI (275)(7), LI (275)(7)

3.92 Trigonometric functions of more complicated arguments combined with exponentials 3.9216 1.



∞

12

 e

−γx

cos ax (cos γx − sin γx) dx = 2

0

[a > 0,  1 π exp − tan2n x = − 1 n 2 0 n=1    ∞   π/2  π/2 ∞  1 1 π 2n 2n sin x = cos x = exp − exp − n n 4 0 0 n=1 n=1 

2.10

3.10 3.922 1.

2.

 2 π γ exp − 8a 2a

∞ π/4 1



Re γ ≥ |Im γ|]

  2 π β + a2 − β e−βx e−βx sin ax2 dx = 8 β 2 + a2 0 −∞   √ π a 1 arctan =  sin 4 2 2 2 β 2 β +a [Re β > 0, a > 0]      ∞ 2 π β 2 + a2 + β 1 ∞ −βx2 e−βx cos ax2 dx = e cos ax2 dx = 2 −∞ 8 β 2 + a2 0   √ π a 1 arctan =  cos 4 2 2 2 β 2 β +a [Re β > 0, a > 0]



∞

2

1 sin ax2 dx = 2



∞

ET I 26(28)

2

[In formulas 3.922 3 and 4, a > 0, b > 0, Re β > 0, and    1  2 b2 , B= A= β + a2 + β , 2 2 4 (a + β ) 2

FI II 750, BI (263)(8)

FI II 750, BI (263)(9)

   1  2 C= β + a2 − β 2

If a is complex, then Re β > |Im a|.]   ∞ 2 π 1 e−βx sin ax2 cos bx dx = − e−Aβ (B sin Aa − C cos Aa) 3. 2 + a2 2 β 0     √ π 1 a ab2 βb2 sin arctan − exp − =  4 (β 2 + a2 ) 2 β 4 (β 2 + a2 ) 2 4 β 2 + a2 LI (263)(10), GW (337)(5)

498

Trigonometric Functions



∞

4.

e

−βx2

0

3.923



π e−Aβ (B cos Aa + C sin Aa) β 2 + a2     √ βb2 π 1 a ab2 cos exp − =  arctan − 4 (β 2 + a2 ) 2 β 4 (β 2 + a2 ) 2 4 β 2 + a2

1 cos ax cos bx dx = 2 2

LI (263)(11), GW (337)(5)

3.923 1.

2.



∞

  exp − ax2 + 2bx + c sin px2 + 2qx + r dx −∞   √ a b2 − ac − aq 2 − 2bpq + cp2 π =  exp 4 a2 + p 2 a2 + p2  2   p p q − pr − b2 p − 2abq + a2 r 1 arctan − × sin 2 a a2 + p 2 GW (337)(3), BI (296)(6) [a > 0]  ∞   exp − ax2 + 2bx + c cos px2 + 2qx + r dx −∞   √ a b2 − ac − aq 2 − 2bpq + cp2 π =  exp 4 2 a2 + p 2 a2 +  p  2   p p q − pr − b2 p − 2abq + a2 r 1 arctan − × cos 2 a a2 + p 2 [a > 0]

3.924



∞

1.

e

−βx4

0



∞

2.

π sin bx dx = 4

4

e−βx cos bx2 dx =

0

3.925



∞

1.

e

2

p −x 2

0

 2. 0

∞

e

p2 −x 2



2

π 4

 2  2 b b b 1 exp − I4 2β 8β 8β



1 sin 2a x dx = 2

1 2

[Re β > 0,

b > 0]

ET 73(22)

[Re β > 0,

b > 0]

ET I 15(12)

 2  2 b b b I − 14 exp − 2β 8β 8β



∞

2 2

cos 2a2 x2 dx =

GW (337)(3), BI (269)(7)

−∞



e

∞

−∞

2

p −x 2

e

p2 −x 2

√ π −2ap e sin 2a x dx = (cos 2ap + sin 2ap) 4a 2 2

[a > 0, b > 0] √ π −2ap e cos 2a2 x2 dx = (cos 2ap − sin 2ap) 4a [a > 0,

b > 0]

BI (268)(12)

BI (268)(13)

3.932

3.926

1.

Trigonometric and exponential functions

499

Notation:

  a2 + β 2 + β a2 + β 2 − β , v= u= 2 2   ∞ √ γ 2 π 1 √ √ −2u γ e−(βx + x2 ) sin ax2 dx = e [v cos (2v γ) + u sin (2v γ)] 2 + β2 2 a 0 

∞

2.

e−(βx

2

+ xγ2 )

cos ax2 dx =

0

1 2

[Re β > 0,





∞

0

p

e− x sin2

BI (268)(14)

√ π √ √ e−2u γ [u cos (2v γ) − v sin (2v γ)] 2 +β

a2

[Re β > 0, 3.927

Re γ > 0]

a 2a p p2 dx = a arctan + ln 2 x p 4 p + 4a2

[a > 0,

Re γ > 0]

BI (268)(15)

p > 0]

LI (268)(4)

3.928   a2 b2 1 1 Notation: a2 + p2 > 0, r = 4 a4 + p4 , s = 4 b4 + q 4 , A = arctan 2 , and B = arctan 2 . 2 p 2 q      √  ∞ q2 b2 π −2rs cos(A+B) sin a2 x2 + 2 dx = 1. exp − p2 x2 + 2 sin [A + 2rs sin(A + B)] e x x 2r 0 

BI (268)(22)

∞

2. 0

     √ π −2rs cos(A+B) q2 b2 2 2 2 2 e cos a x + 2 dx = exp − p x + 2 cos [A + 2rs sin(A + B)] x x 2r

BI (268)(23)

 3.929

∞

   √ 2 e−x cos p x + pe−x sin px dx = 1

LI (268)(3)

0

3.93 Trigonometric and exponential functions of trigonometric functions 3.931 1.



π

12

e−p cos x sin (p sin x) dx = Ei(−p) − ci(p)

0



π

2.

e−p cos x sin (p sin x) dx = −

0



π/2

3. 

π

0

3.932



1. 0

π

0

−π

e−p cos x sin (p sin x) dx = −2 shi(p)

GW (337)(11b)

e−p cos x cos (p sin x) dx = − si(p)

0

4.



NT 13(27)

e−p cos x cos (p sin x) dx =

1 2



2π

NT 13(26)

e−p cos x cos (p sin x) dx = π

GW (337)(11a)

0

ep cos x sin (p sin x) sin mx dx =

1 2

 0

2π

ep cos x sin (p sin x) sin mx dx =

π pm 2 m! BI (277)(7), GW (337)(13a)

500

Trigonometric Functions



π

2.

ep cos x cos (p sin x) cos mx dx =

0

1 2



2π

3.933

ep cos x cos (p sin x) cos mx dx =

0

π pm 2 m!

BI (277)(8), GW (337)(13b)



π

3.933 3.934

ep cos x sin (p sin x) cosec x dx = π sinh p



π

1. 0



π

2.

ep cos x sin (p sin x) tan

x dx = π (1 − ep ) 2

BI (271)(8)

ep cos x sin (p sin x) cot

x dx = π (ep − 1) 2

BI (272)(5)

0



π

3.935 0

3.936

BI (278)(1)

0



ep cos x cos (p sin x)

n−1

 p2k+1 sin 2nx dx = π sin x (2k + 1)!

[p > 0]

LI (278)(3)

k=0

2π

1.

ep cos x cos (p sin x − mx) dx = 2

0



π

ep cos x cos (p sin x − mx) dx =

0

2πpm m! BI (277)(9), GW (337)(14a)



2π

2.

ep sin x sin (p cos x + mx) dx =

mπ 2πpm sin m! 2

[p > 0]

GW (337)(14b)

ep sin x cos (p cos x + mx) dx =

mπ 2πpm cos m! 2

[p > 0]

GW (337)(14b)

0



2π

3. 0



2π

4. 0

 5.

π

ecos x sin (mx − sin x) dx = 0

WH

eβ cos x cos (ax + β sin x) dx = β −a sin(aπ) γ(a, β)

EH II 137(2)

0

3.937 Notation: In formulas 3.937 1 and 2, (b−p)2 +(a+q)2 > 0, m = 0, 1, 2, . . . , A = p2 −q 2 +a2 −b2 , B = 2(pq + ab), C = p2 + q 2 − a2 − b2 , and D = −2(ap + bq).  1.11 0

2π

exp (p cos x + q sin x) sin (a cos x + b sin x − mx) dx √ √  

− m  C − iD − (A − iB)m/2 I m C + iD = iπ (b − p)2 + (a + q)2 2 (A + iB)m/2 I m GW (337)(9b)



2π

2. 0

exp (p cos x + q sin x) cos (a cos x + b sin x − mx) dx √ √  

− m  m m C − iD + (A − iB) 2 I m C + iD = π (b − p)2 + (a + q)2 2 (A + iB) 2 I m GW (337)(9a)

 3.12 0

2π

exp (p cos x + q sin x) sin (q cos x − p sin x + mx) dx =

m 2π  2 p + q2 2 m!

  q sin m arctan p

GW (337)(12)

3.944

Trigonometric and exponentials functions and powers



2π

4.12

exp (p cos x + q sin x) cos (q cos x − p sin x + mx) dx =

0

501

  m 2π  2 q p + q 2 2 cos m arctan m! p GW (337)(12)

3.938



π

1.

er(cos px+cos qx) sin (r sin px) sin (r sin qx) dx =

0



∞ π 1 r(p+q)k 2 Γ(pk + 1) Γ(qk + 1) k=1

π

2.

er(cos px+cos qx) cos (r sin px) cos (r sin qx) dx =

0

π 2

 2+

∞  k=1

r(p+q)k Γ(pk + 1) Γ(qk + 1)



BI (277)(14)

BI (277)(15)

3.939



π

1.

eq cos x

0



k=1

π

2.3



eq cos x

0



∞ sin rx π  (pq)kr sin (q sin x) dx = 1 − 2pr cos rx + p2r 2pr Γ(kr + 1)

1 − pr cos rx π 2+ cos (q sin x) dx = 1 − 2pr cos rx + p2r 2 

π/2 p cos 2x

e

3. 0

cos (p sin 2x) dx q−1 π exp p = 2q q+1 cos2 x + q 2 sin2 x



[r > 0, ∞  k=1

0 < p < 1]  (pq)kr Γ(kr + 1)

[r > 0, 0 < p < 1]

BI (278)(15)

BI (278)(16) BI (273)(8)

3.94–3.97 Combinations involving trigonometric functions, exponentials, and powers 3.941  1.

∞

0



∞

2.

e−px sin qx

q dx = arctan x p

e−px cos qx

dx =∞ x

e−px cos px

 √  x dx π = exp −bp 2 b4 + x4 4b2

0

3.942  1.

∞

0



∞

2.

e−px cos px

0



3.943 0

∞

x dx π = 2 e−bp sin bp 4 −x 4b

b4

e−βx (1 − cos ax)

1 a2 + β 2 dx = ln x 2 β2

[p > 0]

BI (365)(1) BI (365)(2)

[p > 0,

b > 0]

BI (386)(6)a

[p > 0,

b > 0]

BI (386)(7)a

[Re β > 0]

BI (367)(6)

3.944  u i i −μ −μ xμ−1 e−βx sin δx dx = (β + iδ) γ [μ, (β + iδ) u] − (β − iδ) γ [μ, (β − iδ) u] 1. 2 2 0 [Re μ > −1]

ET I 318(8)

502

Trigonometric Functions



∞

2. u



xμ−1 e−βx sin δx dx =

xμ−1 e−βx cos δx dx =

0

u ∞

5.11

xμ−1 e−βx cos δx dx =

xμ−1 e−βx sin δx dx =

0



∞

6.12 0

 0

0

 10.

0

11.

1 1 −μ −μ (β + iδ) Γ [μ, (β + iδ) u] + (β − iδ) Γ [μ, (β − iδ) u] 2 2 Γ(μ) (β 2

+ δ2)

μ/2

Γ(μ) (δ 2

μ/2

+ β2)

[Re β > |Im δ|] ET I 320(29)   δ sin μ arctan β [Re μ > −1, Re β > |Im δ|] FI II 812, BI (361)(9)

  δ cos μ arctan β [Re μ > 0,

Re β > |Im δ|]

xμ−1 exp (−ax cos t) sin (ax sin t) dx = Γ(μ)a−μ sin(μt)  Re μ > −1,

a > 0,

|t|
−1,

a > 0,

|t|
0] ∞

4. 

i i (β + iδ)−μ Γ [μ, (β + iδ) u] − (β − iδ)−μ Γ [μ, (β − iδ) u] 2 2 [Re β > |Im δ|]

u

3. 

3.944

∞

xp−1 e−qx sin (qx tan t) dx =

1 Γ(p) cosp t sin pt qp

xp−1 e−qx cos (qx tan t) dx =

1 Γ(p) cosp (t) cos pt qp

 π |t| < , 2

q>0

 LO V 288(16)

  π |t| < , q > 0 LO V 288(15) 2 n+1     2k+1   ∞ n+1 b β k xn e−βx sin bx dx = n! (−1) 2 2 β +b β 2k + 1 0 0≤2k≤n   n ∂ b = (−1)n n ∂β b2 + β 2 [Re β > 0, b > 0] GW (336)(3), ET I 72(3)

3.947

Trigonometric and exponentials functions and powers



∞

12.

n+1     2k b β k n+1 (−1) β 2 + b2 β 2k 0≤2k≤n+1   n ∂ β n = (−1) ∂β n b2 + β 2 [Re β > 0, b > 0] GW (336)(4), ET I 14(5)

xn e−βx cos bx dx = n!

0



∞

13.



 xn−1/2 e−βx sin bx dx = (−1)n

0

14.

⎛  ⎞ 2 + b2 − β β π d ⎝ ⎠  2 dβ n β 2 + b2 n

[Re β > 0, ⎛  ⎞   ∞ 2 + b2 + β n β π d ⎝ ⎠  xn−1/2 e−βx cos bx dx = (−1)n 2 dβ n β 2 + b2 0

b > 0]

ET I 72(6)

[Re β > 0,

b > 0]

ET I 15(6)

3.945  1.12

∞



∞



∞



dx e−βx sin ax − e−γx sin bx r x       r−1 r−1  2 b a 2 2 − a2 + β 2 2 sin (r − 1) arctan = Γ(1 − r) b + γ sin (r − 1) arctan γ β [Re β > 0, Re γ > 0, r < 2, r = 1] BI(371)(6)

0

 2.12

dx e−βx cos ax − e−γx cos bx r x       r−1 r−1  2 a b 2 2 − b2 + γ 2 2 cos (r − 1) arctan = Γ(1 − r) a + β cos (r − 1) arctan β γ [Re β > 0, Re γ > 0, r < 2, r = 1] BI (371)(7)

0

 3. 0

  dx γ β β 1 a2 + γ 2 γ ln 2 arccot − arccot ae−βx sin bx − be−γx sin ax 2 = ab + x 2 b + β2 a a b b [Re β > 0,

3.946

503



∞

1.

e−px sin2m+1 ax

0

Re γ > 0]

  m (2m − 2k + 1)a dx (−1)m  2m + 1 arctan (−1)k = 2m x 2 p k k=0

[m = 0, 1, . . . , 

∞

2. 0

e−px sin2m ax

m+1 m−1 

(−1) dx = x 22m

(−1)k

k=0



2m k





1. 0

∞

e−βx sin γx sin ax

1 β 2 + (a + γ)2 dx = ln 2 x 4 β + (a − γ)2

p > 0]



1 ln p2 + (2m − 2k)2 a2 − 2m 2 [m = 1, 2, . . . ,

3.947

BI (368)(22)

[Re β > |Im γ|,

p > 0]

a > 0]

GW (336)(9a)



2m m

 ln p

GW (336)(9b)

BI (365)(5)

504

Trigonometric Functions

 2.

∞

11 0

 3.

∞

11 0

3.948



∞

1.11

e

−px

    |a + b| |a − b| |a − b| dx |a + b| arctan − arctan sin ax sin bx 2 = x 2  p  2 p 2 2 p + (a − b) p + ln 4 p2 + (a + b)2 [p > 0, for p = 0 see 3.741 3] BI (368)(1), FI II 744

e−px sin ax cos bx

a+b a−b dx = arctan + arctan x p p

e−px (sin ax − sin bx)

0



∞

e−px (cos ax − cos bx)

0 ∞

e−px (cos ax − cos bx)

0

[Re p > 0] ,

0

0

1 b 2 + p2 dx = ln 2 x 2 a + p2

[Re p > 0] ,

2.8

(cf. 3.951 3)

a dx p a2 + p2 b = ln 2 + b arctan − a arctan 2 x 2 b + p2 p p BI (368)(20)

 dx 2b p p + 4a 2a − b arctan − ln 2 e−px sin2 ax − sin2 bx 2 = a arctan x p p 4 p + 4b2 2

2



∞



∞

BI (368)(25)

 dx 2b p p + 4a 2a + b arctan + ln 2 e−px cos2 ax − cos2 bx 2 = −a arctan x p p 4 p + 4b2 2

[p > 0]

1.

(cf. 3.951 2)

[p > 0] ∞

5.

3.949

GW (336)(10b)

[Re p > 0] ∞

4. 

p > 0]

BI (367)(8), FI II 748a

3. 

a b dx = arctan − arctan x p p

[a ≥ 0,

BI (367)(7)

2. 

3.948

2

BI (368)(26)

dx 1 a+b+c 1 a+b−c 1 a−b+c = − arctan + arctan + arctan x 4 p 4 p 4 p 0 −a + b + c 1 + arctan 4 p [p > 0] BI (365)(11)    ∞ 1 b 1 1 2pb dx π = arctan − arctan 2 e−px sin2 ax sin bx +s 2 − b2 x 2 p 2 2 p + 4a 2 0   2 1 for p + 4a2 − b2 < 0 s= 0 for p2 + 4a2 − b2 ≥ 0

3. 0

e−px sin ax sin bx sin cx

e−px sin2 ax cos bx

1 dx = ln x 8

2



p + (2a + b)2 p2 + (2a − b)2

BI (365)(8)

2

(p2 + b2 )

[p > 0]

BI (365)(9)

3.951

Trigonometric and exponentials functions and powers

 4.

∞

8

e

−px

0



  1 a 1 1 2pa dx π = arctan + arctan 2 sin ax cos bx +s x 2 p 2 2 p  + 4b2 − a2 2  2 1 for p + 4b2 − a2 < 0 s= 0 for p2 + 4b2 − a2 ≥ 0 2

BI (365)(10) ∞

5.

e−px sin2 ax sin bx sin cx

0

3.951



∞

1.

√ dx = ln 2 1 − e−x cos x x

∞ −γx

e

2. 0



∞ −γx

e

3. 0

 4.11

∞ −γx

e

0

 5. 6.

[Re β > 0,

Re γ ≥ 0]

BI (367)(3)

− e−βx 1 b2 + β 2 cos bx dx = ln 2 x 2 b + γ2

[Re β > 0,

Re γ ≥ 0]

BI (367)(4)

Re γ > 0]

BI (368)(21)a

− e−βx b b b2 + β 2 b sin bx dx = ln 2 + β arctan − γ arctan 2 2 x 2 b +γ β γ [Re β > 0,

x bπ 1 π cos bx dx = 2 − 2 cosech2 [Re β > 0] −1 2b 2β β 0   ∞ 1 1 1 − cos bx dx = ln b − [ψ(ib) + ψ(−ib)] x−1 e x 2 0

7. 0



9.

(β − γ)b − e−βx sin bx dx = arctan 2 x b + βγ

∞

∞

∞

2

1 − cos ax dx a 1 1−e · = + ln 2πx e −1 x 4 2 a

−a

ET I 15(18)

[b > 0]

ET I 15(9)

[a > 0]

BI (387)(10)



dx 1 a2 + γ 2 = ln e−βx − e−γx cos ax [Re β > 0, x 2 β2 0      ∞ cos px − e−px dx π 1 √ 1 √ = 4 exp − bp 2 sin bp 2 b4 + x4 x 2b 2 2 0 

∞

10. 0

 11.

0

∞

BI (365)(15)

FI II 745

eβx



8.

dx 1 p2 + (b + c)2 = ln 2 x 8 p + (b − c)2  2

 2 2 2 p p + (2a − b + c) + (2a + b − c) 1 + ln 2 16 [p + (2a + b + c)2 ] [p2 + (2a − b − c)2 ] [p > 0]



0



505



cos x 1 − ex − 1 x

Re γ > 0]

[p > 0]



BI (367)(10)

BI (390)(6)

dx = C

NT 65(8)

  dx a a2 + q 2 e−qx a sin ax = ln ae−px − + q arctan − a x x 2 p2 q [p > 0,

q > 0]

BI (368)(24)

506

Trigonometric Functions

  2m 1 x2m sin bx π m ∂ dx = (−1) coth bπ − [b > 0] ex − 1 ∂b2m 2 2b 0    ∞ 2m+1 2m+1 x cos bx 1 π m ∂ dx = (−1) coth bπ − ex − 1 ∂b2m+1 2 2b 0 

12. 13.

3.952



∞

∞

14. 0



∞

15. 0



∞

16.12 0



∞

17. 0

2m x2m sin bx dx m ∂ = (−1) ∂b2m e(2n+1)cx − e(2n−1)cx



[b > 0]

2m x2m sin bx dx m ∂ = (−1) ∂b2m e2ncx − e(2n−2)cx

x2m+1 cos bx dx ∂ 2m+1 = (−1)m 2m+1 2ncx (2n−2)cx ∂b e −e

n

b bπ  π tanh − 2 4c 2c b + (2k − 1)2 c2 

[b > 0] n

bπ  b π tanh − 2 4c 2c b + (2k − 1)2 c2

∞

18.

[b > 0] n−1

 b bπ 1 π coth − − 4c 2c 2b b2 + (2k)2 c2

∞

19. 0



c > 0]

n−1

 b bπ 1 π coth − − 2 4c 2c 2b b + (2k)2 c2

∞

21. 0



∞

1.

bπ 2p

c > 0]



0

GW (336)(14d)

m 

cosh cos ax − cos bx dx 1 b2 + (2k − 1)2 p2 1 = ln ln 2 − aπ (2m−1)px x 2 cosh 2 a + (2k − 1)2 p2 −e k=1 2p GW (336)(16a)

bπ m−1 cos ax − cos bx dx 1 a sinh 2p 1  b2 + 4k 2 p2 = ln − ln 2 2 b sinh aπ 2 a + 4k 2 p2 e2mpx − e(2m−2)px x 2p

sin x sin bx dx 1 (b + 1) sinh[(b − 1)π] = ln · x 1−e x 4 (b − 1) sinh[(b + 1)π]

∞

2

b = 1

sin2 ax dx 1 2aπ = ln · 1 − ex x 4 sinh 2aπ

2

x2

sin ax dx =

2

x2

cos ax dx =

xe−p

0

2.

GW (336)(14c)

k=1

[p > 0] ∞

0

3.952



k=1

20. 

GW (336)(14b)



k=1

[b > 0,

[p > 0]



GW (336)(14a)

e(2m+1)px

0





k=1

[b > 0, 

GW (336)(15b)



k=1

x2m+1 cos bx dx ∂ 2m+1 = (−1)m 2m+1 (2n+1)cx (2n−1)cx ∂b e −e 

GW (336)(15a)

xe−p

GW (336)(16b) LO V 305

LO V 306, BI (387)(5)

  √ a π a2 exp − 4p3 4p2

BI (362)(1)

 2k+1 ∞ 1 a  (−1)k k! a − 2p2 4p3 (2k + 1)! p k=0

[a > 0]

BI (362)(2)

3.953

Trigonometric and exponentials functions and powers



∞

3.12

2

x2 e−p

x2

sin ax dx =

0

4. 5. 6.3

 2k+2 ∞ a 2p2 − a2  (−1)k k! a + 4p4 8p5 (2k + 1)! p k=0

[a > 0]   2 √ 2p − a 2 2 a x2 e−p x cos ax dx = π exp − 2 5 8p 4p 0    ∞ 2 3 √ 6ap − a a2 3 −p2 x2 x e sin ax dx = π exp − 2 16p7 4p 0  2k   √   ∞ ∞ (−1)k a π a a dx π −p2 x2 = e sin ax = Φ x 2p k!(2k + 1) 2p 2 2p 0 

507

∞

2

BI (362)(4)

2

BI (362)(5) BI (362)(6)

BI (365)(21)

k=0



∞

7.

x

e

sin γx dx =

0



∞

8.10

2

xμ−1 e−βx cos ax dx =

0

10.

3.953 1.

2.

γe− 4β 2β

μ+1 2

Γ

1+μ 2



  μ 3 γ2 ; ; 1 − F 1 1 2 2 4β

1 −μ/2  μ  −a2 /4β β e Γ 2 2

[Re β > 0, Re μ > −1] ET I 318(10)   μ 1 1 a2 + ; ; 1F 1 − 2 2 2 4β [Re β > 0, Re μ > 0, a > 0]

    √ π a a2 2n −β 2 x2 n √ x e cos ax dx = (−1) n+1 2n+1 exp − 2 D 2n 2 √β 8β β 2 0   π a a2 exp − 2 H 2n = (−1)n 2n+1 (2β) 4β 2β   π |arg β| < , a > 0 4     √  ∞ 2 2 2 π a a 2n+1 −β x n √ x e sin ax dx = (−1) n+ 3 exp − 2 D 2n+1 8β β 2 2 √2 β 2n+2  0   π a a2 exp − 2 H 2n+1 = (−1)n 2n+2 (2β) 4β  2β  π |arg β| < , a > 0 4 

9.



γ2

μ−1 −βx2

ET I 320(30)

∞



∞



∞

WH, ET I 15(13)

WH, ET I 74(23)

2

xμ−1 e−γx−βx sin ax dx 0            2 iaγ i iaγ γ − ia γ + ia γ − a2 Γ(μ) exp − D −μ √ D −μ √ =− − exp μ exp 8β 4β 4β 2β 2β 2(2β) 2 [Re μ > −1, Re β > 0, a > 0] ET I 318(11) 2

xμ−1 e−γx−βx cos ax dx 0            2 iaγ 1 iaγ γ − ia γ + ia γ − a2 Γ(μ) exp − D −μ √ D −μ √ = + exp μ exp 8β 4β 4β 2β 2β 2(2β) 2 [Re μ > 0, Re β > 0, a > 0] ET I 16(18)

508

Trigonometric Functions

    √  i π γ − ia (γ − ia)2 √ 1−Φ (γ − ia) exp xe sin ax dx =  4β 2 β 8 β3 0      (γ + ia)2 γ + ia √ − (γ + ia) exp 1−Φ 4β 2 β [Re β > 0, a > 0]     √   ∞ 2 π γ − ia (γ − ia)2 √ 1−Φ (γ − ia) exp xe−γx−βx cos ax dx = −  4β 2 β 8 β3 0     (γ + ia)2 γ + ia 1 √ + (γ + ia) exp 1−Φ + 4β 2β 2 β [Re β > 0, a > 0] 

3.

12

4.

3.954

∞

3.954  1.11

∞

−γx−βx2

2

e−βx sin ax

0



∞

2.11 0

ET I 16(17)

       x dx π βγ 2 a a −γa γa √ √ e − e 2 sinh aγ + e = − Φ γ β − Φ γ β + γ 2 + x2 4 2 β 2 β [Re β > 0, Re γ > 0, a > 0]

2

e−βx cos ax

ET I 74(28)

γ2

dx π βγ 2 e = 2 +x 4γ



ET I 74(26)a

      a a −γa γa −e Φ γ β+ √ 2 cosh aγ − e Φ γ β− √ 2 β 2 β [Re β > 0, Re γ > 0, a > 0] ET I 15(15)





 π − β2 π dx = e 4 D ν (β) cos βx − ν [Re ν > −1] EH II 120(4) 2 2 0  ∞ 2 dx √ e − 1 3.956 e−x (2x cos x − sin x) sin x 2 = π BI (369)(19) x 2e 0 3.957    ∞ −β 2 sin ax dx xμ−1 exp 1. 4x 0        πi √   √  μ i i i = μ β μ a− 2 exp − μπ K μ βe 4 a − exp μπ K μ βe−πi/4 a 2 4 4 [Re β > 0, Re μ < 1, a > 0] ET I 318(12) 3.955



∞

∞

2.

xν e−

xμ−1 exp

0

3.958



x2 2

∞

1. −∞



−β 2 4x



cos ax dx         √  √  μ 1 i i = μ β μ a− 2 exp − μπ K μ βeπi/4 a + exp μπ K μ βe−πi/4 a 2 4 4 [Re β > 0, Re μ < 1, a > 0] ET I 320(32)a

n −(ax2 +bx+c)

x e

n   2  n/2  π n! b − p2 −1 exp −c ak sin(px + q) dx = − 2a a 4a (n − 2k)!k! k=0   n−2k   n − 2k  π pb −q+ j bn−2k−j pj sin × 2a 2 j j=0 

[a > 0]

GW (37)(1b)

3.963

Trigonometric and exponentials functions and powers

 2.

∞

n −(ax2 +bx+c)

12 −∞

x e

n 

 2  n/2  π n! b − p2 exp −c ak a 4a (n − 2k)!k! k=0   n−2k   n − 2k  π pb −q+ j bn−2k−j pj cos × 2a 2 j j=0 

cos(px + q) dx =

509

−1 2a

[a > 0]

 3.959

∞

2

xe−p

x2

tan ax dx =

0

3.961



∞

1. 0



GW (337)(1a)

 2 2 √ ∞ a π a k k (−1) k exp − 2 p3 p k=1

[p > 0]

BI (362)(15)

      x dx aγ exp −β γ 2 + x2 sin ax  =  K 1 γ a2 + β 2 γ 2 + x2 a2 + β 2 [Re β > 0, Re γ > 0,

a > 0] ET I 75(36)

∞

2. 0

      dx exp −β γ 2 + x2 cos ax  = K 0 γ a2 + β 2 γ 2 + x2 [Re β > 0,

Re γ > 0,

a > 0] ET I 17(27)

3.962 

∞

1. 0

        2 + β2 γ 2 + x2 − γ exp −β γ 2 + x2 a a exp −γ π   sin ax dx =  2 2 2 2 γ +x β + a2 β + a2 + β 2 [Re β > 0,

Re γ > 0,

a > 0] ET I 75(38)

    ∞ x exp −β γ 2 + x2 2 2    π β+ a +β 2 + β2   exp −γ a cos ax dx =   2 a2 + β 2 0 γ 2 + x2 γ 2 + x2 − γ [Re β > 0, Re γ > 0, a > 0] 

2.





ET I 17(29)

3.963



1.

∞

2

e− tan

x

0



π/2

2.

√ sin x dx π = cos2 x x 2

e−p tan x

0



π/2

3.8

x dx 1 = [ci(p) sin p − cos p si(p)] 2 cos x p

xe− tan

2

x

xe− tan

2

x

0

 4.

8 0

π/2

sin 4x

BI (391)(1)

[p > 0]

(cf. 3.339)

BI (396)(3)

3√ dx =− π 8 cos x 2

BI (396)(5)

√ dx =2 π 8 cos x

BI (396)(6)

sin3 2x

510

3.964

Trigonometric Functions



π/2

1.

xe−p tan x

0



π/2

2. 

π/2

3.8

1. 2. 3.966



∞

p − cos x 1 dx = 4 cos x cot x 4 2

2

x

xe−p tan

2

xp

0

3.965

p sin x − cos x dx = − sin p si(p) − ci(p) cos p cos3 x

xe−p tan

0



π p

− 2 cos2 x 1 + 2p dx = 6 cos x cot x 8p



π p





∞

xe−βx sin ax2 sin βx dx =

β 4

[p > 0]

LI (396)(4)

[p > 0]

BI (396)(7)

[p > 0]

BI (396)(8)

 π |arg β| < , 4

π − β2a2 e 2a3 0   ∞ π − β2 β −βx 2 xe cos ax cos βx dx = e 2a 4 2a3 0

1.

3.964

[a > 0,

 xe−px cos 2x2 + px dx = 0

a>0



Re β > |Im β|]

ET I 84(17) ET 26(27)

[p > 0]

BI (361)(16)

[p > 0]

BI (361)(17)

[p > 0]

BI (361)(18)

0

  √  1 p π exp − p2 xe−px cos 2x2 − px dx = 8 4 0  ∞   x2 e−px sin 2x2 + px + cos 2x2 + px dx = 0 

2. 3.

∞

0

  √ π 1 2 − p2 exp − p2 16 4 0    ∞ 1  1 e 4a Γ(μ) μπ D −μ √ xμ−1 e−x cos x + ax2 dx = cos μ 4 a (2a) 2 0

 4. 5.3



∞

∞

6.6

  x2 e−px sin 2x2 − px − cos 2x2 − px dx =

 e xμ−1 e−x sin x + ax2 dx =

0

1 4a

Γ(μ) μπ D −μ sin μ 4 (2a) 2



[Re μ > 0,  1 √ a [Re μ > −1,

3.967



∞

1.

β2

e− x2 sin a2 x2

0

 2. 0

∞

e

2 −β x2

√ √  dx π −√2aβ e = sin 2aβ x2 2β √

cos a2 x2

dx π e = x2 2β

[Re β > 0,

a > 0]

a > 0]

BI (361)(19)

ET I 321(37)

ET I 319(18)

a > 0] ET I 75(30)a, BI(369)(3)a

√ − 2aβ

√  cos 2aβ

[Re β > 0,

a > 0] BI (369)(4), ET I 16(20)

3.972

Trigonometric and exponentials functions and powers



∞

3.

2 −βx2

x e

0



√ π

cos ax dx =  cos 4 4 (a2 + β 2 )3 2

a 3 arctan 2 β

511



[Re β > 0] 3.968  1.12

∞

e

−βx2

0



∞

2.12

2

π sin ax dx = − 8

e−βx cos ax4 dx =

0



4

π 8



ET I 14(3)a

  2  2   2   2 β β π β π β β J 14 cos + + Y 14 sin + a 8a 8a 8 8a 8a 8 

β J 14 a



 2

β 8a

 sin

ET I 75(34) [Re β > 0, a > 0]     π π β2 β2 β2 + − Y 14 cos + 8a 8 8a 8a 8



[Re β > 0,

a > 0]

3.969  √ ∞   π −p2 x4 +q2 x2 3 3 1. 2px cos 2pqx + q sin 2pqx dx = e 2 0  ∞   2 4 2 2 2. e−p x +q x 2px sin 2pqx3 − q cos 2pqx3 dx = 0

ET I 16(24)

BI (363)(7) BI (363)(8)

0

  3.971 Notation: In formulas 3.971 1 and 2, p ≥ 0, q ≥ 0, r = 4 a2 + p2 , s = 4 b2 + q 2 , A = arctan ap , and B = arctan qb .       ∞   dx 1 ∞ dx q  b q  b 2 2 2 2 exp −px − 2 sin ax + 2 = exp −px − 2 sin ax + 2 1. 2 x x x 2 −∞ x x x2 0 √ π exp [−2rs cos(A + B)] sin [A + 2rs sin(A + B)] = 2s 

BI (369)(16, 17)

∞

2. 0

      q  b q  b dx 1 ∞ dx 2 2 cos ax exp −px2 − 2 cos ax2 + 2 = exp −px − + 2 2 2 x x x 2 −∞ x x x2 √ π exp [−2rs cos(A + B)] cos [A + 2rs sin(A + B)] = 2s 

BI (369)(15, 18)

3.972  1.

2.

   dx exp −β γ 4 + x4 sin ax2  4 4 γ 0 + x  2   2  aπ γ γ I 1/4 = β 2 + a2 − β K 1/4 8 2  4 π Re β > 0, |arg γ| < , 4  ∞    dx exp −β γ 4 + x4 cos ax2  4 4 0 γ + x  2   2  aπ γ γ I −1/4 = β 2 + a2 − β K 1/4 8 2  4 π Re β > 0, |arg γ| < , 4 ∞

  β 2 + a2 + β  a>0 ET I 75(37)

  β 2 + a2 + β  a>0 ET I 17(28)

512

Trigonometric Functions

3.973  1. 0



∞

∞

2.

exp (p cos ax) sin (p sin ax)

π dx = (ep − 1) x 2

exp (p cos ax) sin (p sin ax + bx)

0



exp (p cos ax) cos (p sin ax + bx)

0

a > 0]

 x dx π = exp −cb + pe−ac 2 +x 2 [a > 0, b > 0,

c2

WH, FI II 725

c > 0,

exp (p cos x) sin (p sin x + nx)

0 ∞

5.

 dx π exp −cb + pe−ac = c2 + x2 2c [a > 0, b > 0,

π dx = ep x 2

exp (p cos x) sin (p sin x) cos nx

0

c > 0,

[p > 0]

∞ pn π π  pk dx = · + x n! 4 2 k! k=n+1

∞

6.

exp (p cos x) cos (p sin x) sin nx

0

LI (366)(3)

n−1 π  pk pn π dx = + x 2 k! n! 4 k=0

[p > 0] 3.974



p > 0]

BI (366)(2)

[p > 0] 

p > 0]

BI (372)(4) ∞

4. 

[p > 0,

BI (372)(3) ∞

3.



3.973

∞

1. 0

LI (366)(4)

 π ep − exp pe−ab dx exp (p cos ax) sin (p sin ax) cosec ax 2 = b + x2 2b sinh ab [a > 0, b > 0, p > 0]  π ep − exp pe−ab x dx [1 − exp (p cos ax) cos (p sin ax)] cosec ax 2 = b + x2 2 sinh ab

BI (391)(4)

∞

[a > 0, b > 0, p > 0]  π ep − exp pe−ab − ab dx exp (p cos ax) sin (p sin ax + ax) cosec ax 2 = b + x2 2b sinh ab

BI (391)(5)

∞

[a > 0, b > 0, p > 0]  ∞ π ep − exp pe−ab − ab x dx exp (p cos ax) cos (p sin ax + ax) cosec ax 2 = b + x2 2 sinh ab 0

BI (391)(6)

 2. 0

 3. 0

 4.



[a > 0, ∞

5.

exp (p cos ax) sin (p sin ax)

0

 6. 0

exp (p cos ax) cos (p sin ax)

p > 0]

BI (391)(7)

x dx π = [1 − exp (p cos ab) cos (p sin ab)] 2 −x 2

b2

[p > 0, ∞

b > 0,

a > 0]

BI (378)(1)

dx π exp (p cos ab) sin (p sin ab) = b2 − x2 2b [a > 0,

b > 0,

p > 0]

BI (378)(2)

3.981

Trigonometric and hyperbolic functions



∞

7.

exp (p cos ax) sin (p sin ax) tan ax

0



dx π · tanh ab exp pe−ab − ep = b2 + x2 2b [a > 0,



∞

8.

exp (p cos ax) sin (p sin ax) cot ax

0

b2

∞

9.

exp (p cos ax) sin (p sin ax) cosec ax

0 ∞

10.

[1 − exp (p cos ax) cos (p sin ax)] cosec ax

0

3.975





∞ sin

1.

b arctan γx

(γ 2 + x2 )

0

b > 0,

b > 0,

∞

2.

(1 +

0

 3.976

∞



1 + x2

β− 12

BI (372)(15)

p > 0]

BI (391)(12)

1 1 γ 1−b dx = ζ(b, γ) − b − −1 2 4γ 2(b − 1)

e2πx

sin (b arctan x) b

p > 0]



b 2

x2 ) 2

BI (372)(14)

x dx π = − exp (p cos ab) sin (p sin ab) cosec ab b2 − x2 2 [a > 0, b > 0, p > 0] BI (391)(13)

[Re b > 1,



p > 0]

dx π cosec ab [ep − exp (p cos ab) cos (p sin ab)] = b2 − x2 2b [a > 0,



b > 0,

 dx π = coth ab ep − exp pe−ab 2 +x 2b [a > 0,



513

dx 1 ζ(b) = − b e2πx + 1 2(b − 1) 2 2

Re γ > 0]

[Re b > 1]

WH, ET I 26(7) EH I 33(13)

e−p sin πβ Γ(β) 2pβ [Re β > 0, p > 0]

e−px cos [2px + (2β − 1) arctan x] dx =

0

WH

3.98–3.99 Combinations of trigonometric and hyperbolic functions 3.981



∞

1. 0



∞

2. 0



[Re β > 0,

a > 0]

0 ∞

4. 0

     sin ax π aπ i β + ai β − ai dx = − tanh − ψ −ψ cosh βx 2β 2β 2β 4β 4β [Re β > 0, a > 0] cos ax π aπ dx = sech cosh βx 2β 2β

[Re β > 0,

all real a]

 0

∞

BI (264)(14)

     sinh aπ π β + γ + ia β + γ − ia sinh βx i γ dx = ψ −ψ sin ax + βπ sinh γx 2γ cosh aπ 2γ 2γ 2γ γ + cos γ [|Re β| < Re γ,

5.

BI (264)(16)

GW (335)(12), ET I 88(1) ∞

3. 

sin ax π aπ dx = tanh sinh βx 2β 2β

cos ax

sin πβ π sinh βx γ dx = sinh γx 2γ cosh aπ + cos βπ γ γ

[|Re β| < Re γ]

a > 0]

ET I 88(5) BI (265)(7)

514

6.

7.

8.

9.

Trigonometric Functions



∞



∞

βπ aπ π sin 2γ sinh 2γ sinh βx dx = sin ax [|Re β| < Re γ, a > 0] BI (265)(2) βπ cosh γx γ cosh aπ 0 γ + cos γ ⎡         ∞ 1 ⎣ 3γ − β − ia 3γ + β − ia sinh βx 3γ − β + ia dx = +ψ −ψ cos ax ψ cosh γx 4γ 4γ 4γ 4γ 0 ⎤   2π sin πβ 3γ + β + ia γ ⎦ + −ψ πβ 4γ cos γ + cosh πa γ

[|Re β| < Re γ,



∞

sin ax

cos ax

0



π/2

11.11

βπ aπ π cos 2γ cosh 2γ cosh βx dx = βπ cosh γx γ cosh aπ γ + cos γ

cos2m x cosh βx dx =

0



π/2

2.

ET I 31(13)

a > 0]

[|Re β| < Re γ,

all real a]

(2m + 1)! cosh (β 2 + 12 ) (β 2 + 32 ) . . . [β 2 + (2m + 1)2 ]

∞

γ

3.12 0

∞

π sin2 x cos ax dx = 4 sinh2 x



BI (265)(6)

WA 620a

πβ 2

cos ax aπ dx = [Re β > 0, aπ 2 0 cosh βx 2β 2 sinh 2β   βπ βπ aπ aπ  ∞ π a sin cosh − β cos sinh 2γ 2γ 2γ 2γ sinh βx   dx = sin ax 2 βπ aπ cosh γx 0 γ 2 cosh − cos 

ET I 88(6)

(2m)! sinh πβ 2 β (β 2 + 22 ) . . . [β 2 + (2m)2 ]

cos2m+1 x cosh βx dx =

0



[|Re β| < Re γ,

[β = 0]

12.11

1.

a > 0]

πa γ

sinh π cosh βx dx = · [|Re β| < Re γ, a > 0] BI (265)(4) πβ sinh γx 2γ cosh πa 0 γ + cos γ ⎡        ∞ i ⎣ cosh βx 3γ + β − ai 3γ − β + ia 3γ + β + ia dx = −ψ +ψ sin ax ψ cosh γx 4γ 4γ 4γ 4γ 0 ⎤   2πi sinh πa 3γ − β − ai γ ⎦ − −ψ βπ 4γ cosh aπ + cos γ γ

10.

3.982

3.982

WA 620a

a > 0]

BI (264)(16)

γ

[|Re β| < 2 Re γ, a > 0]  a−2 = I(a) + 1 − e−π(a−2)

2a a+2 − 1 − e−πa 1 − e−π(a+2)  1 I(0) = (π coth π − 1) , 2

I(±2) =

ET I 88(9)

1 π + (coth 2π − coth π) 4 2



3.984

Trigonometric and hyperbolic functions

3.983 

∞

1.6 0

 2. 3.12 4.

5.

  a c π sin arccosh β b cos ax dx = √ b cosh βx + c β c2 − b2 sinh aπ β  a π sinh β arccos cb = √ β b2 − c2 sinh aπ β

515

[c > b > 0] [b > |c| > 0] [Re β > 0,

a > 0]

GW (335)(13a)

aγ β

∞



cos ax dx π sinh = π Re β < Im βγ, a > 0 BI (267)(3) aπ β sin γ sinh β 0 cosh βx + cos γ  ∞ cos ax dx sin ab = −π coth aπ [a > 0, b > 0] PV ET I 30(8) sinh b 0 cosh x − cosh b π  ∞ cos ax dx 2 ET I 30(9) [a > 0]   =   2 2 0 1 + 2 cosh πx 1 + 2 cosh πa 3 3          β β a a  ∞ π sin (π − δ) sinh (π + δ) − sin (π + δ) sinh (π − δ) γ γ γ γ sin ax sinh βx   dx = 2πβ 2πa cosh γx + cos δ 0 γ sin δ cosh − cos γ

[π Re γ > |Re γδ|, 

∞

6.12 0

γ

|Re β| < Re γ,

a > 0]

BI (267)(2)

         β β a a π cos (π − b) cosh (π + b) − cos (π + b) cosh (π − b) γ γ γ γ cos ax cosh βx   dx = 2πβ 2πa cosh γx + cos b γ sin b cosh γ − cos γ [|Re β| < Re γ, 0 < b < π, a = 0] BI (267)(6)



∞

7.12 0

ai cos ax dx aπ Q ν (β)  ν+1 = Γ(ν + 1 − ai)e  Γ(ν + 1) β + β 2 − 1 cosh x [Re ν > −1,

|arg(β ± 1)| < π,

a > 0]

ET I 30(10)

3.984 1.

6



∞

lim

c↑1 0



2.6

lim

c↑1 0  ∞

3.8 0



∞

4. 0

 5.

0

∞

∞

sin ax sinh cx cosh ab dx = π cosh x + cos b sinh aπ

[|b| ≤ π,

cos ax cosh cx sinh ab dx = −π cot b cosh x + cos b sinh aπ

[0 < |b| < π,

a real] a real]

BI (267)(1) BI (267)(5)

sin ax sinh x2 π sinh aβ dx = cosh x + cos β 2 sin β2 cosh aπ

[Re β < π,

π cos aγ cos ax cosh β2 x β dx = cosh βx + cosh γ 2β cosh γ2 cosh aπ β

   π Re β > Im βγ 

ET I 31(16)

sin ax sinh βx aπ dx = 2 cosh 2βx + cos 2ax 4 (a + β 2 )

[a > 0,

BI (267)(7)

a > 0]

Re β > 0]

ET I 80(10)

516

Trigonometric Functions



∞

6. 0



∞

7.8 0

cos ax cosh βx βπ dx = cosh 2βx + cos 2ax 4 (a2 + β 2 )

3.985

[Re β > 0,

a > 0]

BI (267)(8)

sinh2μ−1 x cosh2 −2ν+1 x 1  dx = B(μ, ν − μ) 2 F 1 (, μ; ν; β) 2 cosh2 x − β sinh2 x [Re ν > Re μ > 0]

3.985  1.

∞

0

2ν−2 cos ax dx = Γ ν cosh βx β Γ(ν)



ai ν + 2 2β

   ai ν Γ − 2 2β

[Re β > 0,

Re ν > 0,

EH I 115(12)

a > 0] ET I 30(5)

 2.

3.

3.986

n−1 1

∞



cos ax dx a2 4n−1 πa = + k2 aπ 2n 4β 2 βx 2(2n − 1)!β 2 sinh 0 cosh k=1  22β 2 2

 2 2 2 a + 4 β · · · a2 + (2n − 2)2 β 2 πa a + 2 β = aπ 2(2n − 1)!β 2n sinh 2β [n ≥ 2, a > 0]      ∞ n 2 1 a2 π22n−1 cos ax dx 2k − 1 = + 2n+1 2 βx (2n)!β cosh aπ k=1 4β 2 0 cosh 2β 

 2 π a + β 2 a2 + 32 β 2 · · · a2 + (2n − 1)2 β 2 = aπ 2(2n)!β 2n+1 cosh 2β [Re β > 0, n = 0, 1, . . . , all real a]



∞

1. 0



∞

2. 0



∞

3. 0



∞

4.3 0

γπ sinh βπ sin βx sin γx π 2δ sinh 2δ dx = · cosh δx δ cosh βδ π + cosh γδ π

∞

BI (264)(19)

π sinh πα sin αx cos βx γ   dx = απ sinh γx 2γ cosh γ + cosh βπ γ

[|Im(α + β)| < Re γ]

LI (264)(20)

γπ cosh βπ cos βx cos γx π 2δ cosh 2δ dx = · γπ cosh δx δ cosh βπ δ + cosh δ

[|Im(β + γ)| < Re δ]

BI (264)(21)

sin2 βx β−1 β coth β − 1 β + = dx = π (e2β − 1) 2π 2π sinh2 πx



0

∞

EH I 44(3)

sin ax (1 − tanh βx) dx =

π 1 − a 2β sinh aπ 2β

[Re β > 0]

ET I 88(4)a

sin ax (coth βx − 1) dx =

aπ 1 π coth − 2β 2β a

[Re β > 0]

ET I 88(3)

0

2.

ET I 30(4)

[|Im(β + γ)| < Re δ]

[|Im β| < π] 3.987  1.

ET I 30(3)

3.991

3.988

Trigonometric and hyperbolic functions



π/2

1. 0



π/2

0

∞

0



∞

1. 0



∞

2. 0



∞

3. 0

 4. 5.

ET I 37(66)

cos ax cosh (2b cos x) π√ √ dx = πb I a2 − 14 (b) I − a2 − 14 (b) 2 cos x [a > 0]

3.12 3.989

cos ax sinh (2b cos x) π√ √ dx = πb I a2 + 14 (b) I − a2 + 14 (b) cos x 2 [a > 0]

2. 

517

π P − 12 +ia (cos b) cos ax dx √ √ = cosh x + cos b 2 cosh aπ 2

[a > 0,

b > 0]

ET I 30(7)

[a > 0,

b > 0]

ET I 93(44)

[a > 0,

b > 0]

ET I 93(45)

2

sin a πx sin bx π πb2 πb dx = sin 2 cosech sinh ax 2a 4a 2a 2

ET I 37(67)

2

2

πb cos a πx sin bx π cosh πb a − cos 4a2 dx = sinh ax 2a sinh πb 2a a2 π

2

sin xπ cos ax π cos 4 − √2 dx = cosh x 2 cosh aπ 2 1

ET I 36(54)

2

2 a π 1 cos xπ cos ax π sin 4 + √2 dx = · cosh x 2 cosh aπ 0 2   ∞ ∞ 2   2   sin πax cos bx dx = − exp − k + 12 b sin k + 12 πa cosh πx 0 k=0      2  ∞ k + 12 π b k + 12 π b2 1  sin − + exp − +√ a 4 4πa a a

∞

ET I 36(55)

k=0

ET I 36(56) [a > 0, b > 0]        ∞ 2 ∞  cos πax2 cos bx 1 1 dx = b cos k + (−1)k exp − k + πa cosh πx 2 2 0 k=0      2  ∞ k + 12 π b k + 12 π b2 1  cos − + exp − +√ a a 4 4πa a



 6.

k=0

3.991 1. 2.11

[a > 0,

  a π a2 1 + sin πx sin ax coth πx dx = tanh sin 2 2 4 4π 0     ∞ a a2 1 π 1 − cos + cos πx2 sin ax coth πx dx = tanh 2 2 4 4π 0



∞

2

b > 0]

ET I 36(57)

ET I 93(42) ET I 93(43)

518

3.992

Trigonometric Functions

3.992

  π a2 cos 12 − 4π √ sin πx2 cos ax   dx = − 3 + 1. 2 4 cosh √a3 − 2 0 1 + 2 cosh √ πx 3   π a2  ∞ sin 2 − 12 4π cos πx cos ax   dx = 1 − 2. a √ 2 4 cosh − 2 0 3 1 + 2 cosh √ πx 3  2   2  √  ∞ sin x + cos x π sin a2 + cos a2 12 √ √ cos(2ax) dx = 3.993 cosh ( πx) 2 cosh ( πa) 0 3.994  ∞ sin (2a cosh x) cos bx π√  √ 1. aπ J 1 + ib (a) Y 1 − ib (a) + J 1 − ib (a) Y dx = − 4 2 4 2 4 2 4 cosh x 0 



∞

∞

ET I 93(47) [a > 0, b > 0]   cos (2a sinh x) sin bx i√ √ πa I − 1 − ib (a) K − 1 + ib (a) − I − 1 + ib (a) K − 1 − ib (a) dx = − 4 2 4 2 4 2 4 2 2 sinh x

∞

[a > 0, b > 0] √   sin (2a sinh x) cos bx πa √ I 1 − ib (a) K 1 + ib (a) + I 1 + ib (a) K 1 − ib (a) dx = 4 2 4 2 4 2 4 2 2 sinh x

∞

ET I 37(64) [a > 0, b > 0] √   cos (2a sinh x) cos bx πa √ I − 1 − ib (a) K − 1 + ib (a) + I − 1 + ib (a) K − 1 − ib (a) dx = 4 2 4 2 4 2 4 2 2 sinh x

4. 0

5. 0

6. 0

 7.

[a > 0, ∞

sin (a cosh x) sin (a sinh x)

0

3.995



1. 0

2.

π/2

π dx = sin a sinh x 2

b > 0]

[a > 0]

ET I 93(48)

ET I 37(65) BI (264)(22)

 sin 2a cos2 x cosh (a sin 2x) 2ac π sin dx = 2 2 2 2 2bc b+c b cos x + c sin x

π/2 cos 2a cos2 x cosh (a sin 2x) 2ac π cos dx = 2 2 2 2 2bc b +c b cos x + c sin x 0



 (a)

ET I 37(63) [a > 0, b > 0]   sin (2a sinh x) sin bx i√ √ πa I 1 − ib (a) K − 1 + ib (a) − I 1 + ib (a) K 1 − ib (a) dx = − 4 2 4 2 4 2 4 2 2 sinh x

0



1 ib 4+ 2

∞

3.



ET I 37(58)

ET I 37(62) [a > 0, b > 0]   cos (2a cosh x) cos bx π√ √ aπ J − 1 + ib (a) Y − 1 − ib (a) + J − 1 − ib (a) Y − 1 + ib (a) dx = − 4 2 4 2 4 2 4 2 4 cosh x

0



ET I 37(61)

∞

2. 

ET I 37(60)



[b > 0,

c > 0]

BI (273)(9)

[b > 0,

c > 0]

BI (273)(10)

3.997

3.996

Trigonometric and hyperbolic functions



∞

1. 0

 2. 3. 4. 5. 3.997

∞

sin (a sinh x) sinh bx dx = sin

bπ K b (a) 2

[|Re b| < 1,

bπ K b (a) [|Re b| < 1, 2 0  π/2   π cos (a sin x) cosh (b cos x) dx = J 0 a2 − b 2 2 0  ∞  π sin a cosh x − 12 bπ cosh bx dx = J b (a) [|Re b| < 1, 2 0  ∞  π cos a cosh x − 12 bπ cosh bx dx = − Y b (a) [|Re b| < 1, 2 0 

cos (a sinh x) cosh bx dx = cos

π/2

1. 0

sinν x sinh (b cos x) dx =

519

a > 0]

EH II 82(26)

a > 0]

WA 202(13)

MO 40

a > 0]

WA 199(12)

a > 0]

WA 199(13)

 √   ν2  π 2 ν +1 L ν2 (b) Γ 2 b 2

[Re ν > −1] ν     π √ 2 2 ν +1 I ν2 (b) sinν x cosh (b cos x) dx = π Γ b 2 0

EH II 38(53)

[Re ν > −1]

WH

 2.



π/2

3. 0

 4. 0

π/2

∞ √  dx (−1)k √ √ = 2π 2k + 1 cosh (tan x) cos x sin 2x k=0

BI (276)(13)

∞

tanq x sin kλ dx Γ(q)  (−1)k−1 = cosh (tan x) + cos λ sin 2x sin λ kq k=1

[q > 0]

BI (275)(20)

520

4.111

4.11–4.12 Combinations involving trigonometric and hyperbolic functions and powers 4.111



∞

1. 0

 0

∞

3. 0



∞

4. 0

 5.

6.

7.

[Re b > 0]

(cf. 3.981 1) GW (336)(17a)

∞

2. 

sin ax 2m π ∂ 2m  aπ  x dx = (−1)m tanh sinh bx 2b ∂a2m 2b 2m+1

cos ax 2m+1 π ∂ x dx = (−1)m sinh bx 2b ∂a2m+1

 aπ  tanh 2b

sin ax 2m+1 π ∂ 2m+1 x dx = (−1)m+1 cosh bx 2b ∂a2m+1

cos ax 2m π ∂ 2m x dx = (−1)m cosh bx 2b ∂a2m





1 cosh aπ 2b

1 cosh aπ 2b

[Re b > 0]

(cf. 3.981 1) GW (336)(17b)

 [Re b > 0]

(cf. 3.981 3) GW (336)(18b)

 [Re b > 0]

(cf. 3.981 3) GW (336)(18a)

∞

aπ b

sinh π2 sin 2ax dx = 2 · cosh bx 4b cosh2 aπ 0 b  ∞ 2 π cos 2ax 1 dx = 2 · x aπ sinh bx 4b 0 cosh2 b  ∞  sin ax dx πa  π = 2 arctan exp − 2b 2 0 cosh bx x x

[Re b > 0,

a > 0]

BI (364)(6)a

[Re b > 0,

a > 0]

BI (364)(1)a

[Re b > 0,

a > 0]

BI (387)(1), ET I 89(13), LI (298)(17)

4.112 1.

2. 4.113



∞

 cos ax 2b3 dx = πx cosh3 ab 0 cosh 2b  ∞  cos ax  6b4 x x2 + 4b2 πx dx = cosh4 ab 0 sinh 2b



1. 0

∞



x2 + b2

[Re b > 0,

a > 0]

ET I 32(19)

[Re b > 0,

a > 0]

ET I 32(20)

sin ax dx 1 πe−aβ =− 2 − 2 2 sinh πx x + β 2β β sin πβ   1 + 2 2 F 1 1, −β; 1 − β; −e−a + 2β ∞  (−1)k e−ak πe−aβ 1 − − = 2β 2 2β sin πβ k2 − β 2 k=1

[Re β > 0,

2F 1



1, β; 1 + β; −e−a

β = 0, 1, 2, . . . ,



a > 0]

ET I 90(18)

4.113

2.

3. 4.

5.

6.

7.

8.

9.11

10.11

Trigonometric and hyperbolic functions and powers



∞



∞

m−1  sin ax dx 1  (−1)k e−ka (−1)m e−ma  (−1)m ae−ma + + ln 1 + e−a = 2 + m2 sinh πx x 2m 2m m − k 2m 0 k=1

1 dm−1 (1 + z)m−1 ln(1 + z) + 2m! dz m−1 z z=e−a [a > 0] ET I 89(17)  ∞  ∞   sin ax dx sin ax dx a 1 a = = − cosh a + sinh a ln 2 cosh GW (336)(21b) 2 2 sinh πx 1 + x 2 sinh πx 1 + x 2 2 0 −∞   ∞ dx dx sin ax 1 ∞ sin ax π GW (336)(21a) π 1 + x2 = 2 π 1 + x2 = 2 sinh a − cosh a arctan (sinh a) 0 sinh x −∞ sinh x 2 2    √ √   ∞ √ dx sin ax 2 2 cosh a + 2 π −a sinh a √ + 2 cosh a arctan = − √ e + √ ln π 2 2 sinh a 2 2 2 cosh a − 2 0 sinh x 1 + x 4 [a > 0] LI (389)(1)  √     ∞ √ sin ax x dx π −a sinh a 1 2 cosh a + 2 √ √ √ √ = + 2 cosh a arctan e ln − π 2 2 2 2 cosh a − 2 2 sinh a 0 cosh x 1 + x 4 [a > 0] BI (388)(1)  ∞   cos ax x dx 1 a = − + e−a + cosh a ln 1 + e−a 2 sinh πx 1 + x 2 2 0 [a > 0] BI (389)(14), ET I 32(24)  ∞  −a  π −a cos ax x dx + e −1 π 1 + x2 = 2 sinh a arctan e 2 0 sinh x 2 [a > 0] BI (389)(11)  ∞ ∞  (−1)k e−(k+1/2)a cos ax x dx πe−aβ − =  2 2 2 2β cos(βπ) 0 cosh πx x + β k + 12 − β 2 k=0

0

12. 13.

m −aβ



(−1) e aβ + cos ax dx 2 =  2 1 cosh πx x2 + m + 2β 2

1 2

[Re β > 0,

 −

∞  k=0

a > 0]

ET I 32(26)

k −(k+1/2)a

(−1) e  2 k + 12 − β 2

[Re β > 0, a > 0] ET I 32(25)  a  a 

cos ax dx a = 2 cosh − ea arctan e− 2 + e−a arctan e 2 2 2 0 cosh πx 1 + x [a > 0] ET I 32(21)  ∞   cos ax dx [a > 0] = ae−a + cosh a ln 1 + e−2a BI (388)(6) π 2 cosh x 1 + x 0 2 √    ∞ cos ax dx π −a 2 sinh a 1 cosh a 2 cosh a + 2 √ =√ e + √ arctan √ − √ ln π 2 2 2 2 sinh a 2 2 cosh a − 2 0 cosh 4 x 1 + x BI (388)(5) [a > 0] 

11.

521

∞

522

4.114

4.114



∞

1. 0



∞

2. 0

4.115



∞

1. 0



b > 0]

cos ax sinh βx e−ak sin kβ π e−ab sin bβ  dx = + (−1)k 2 2 x + b sinh πx 2b sin bπ k 2 − b2 k=1 [0 < Re β < π,

∞

a > 0,

b > 0] BI (389)(22)

∞

  cos ax sinh βx 1 1 dx = e−a (a sin β − β cos β) + cosh a sin β ln 1 + 2e−a cos β + e−2a x2 + 1 sinh πx 2 2 sin β − sinh a cos β arctan a e + cos β [|Re β| < π, a > 0, b > 0] BI (389)(20)a

∞

cos ax sinh βx dx = x2 + 1 sinh π2 x

5. 0

6. 0

0

a > 0,

∞

0

7.

∞

−ak x sin ax sinh βx sin kβ π e−ab sin bβ  k ke dx = + (−1) 2 2 2 x + b sinh πx 2 sin bπ k − b2 k=1 [0 < Re β < π,

x sin ax sinh βx dx x2 + 1 sinh π x 2   π cosh a + sin β cos β 1 = e−a sin β + cos β sinh a ln − sin β cosh a arctan 2 2 cosh a − sin β sinh a

|Re β| < π2 , a > 0 BI (389)(8)

4.



ET I 33(34)

∞

0



[|Re β| < Re γ]

BI (387)(6)a



x sin ax sinh βx 1 1 dx = e−a (a sin β − β cos β) − sinh a sin β ln 1 + 2e−a cos β + e−2a 2 x + 1 sinh πx 2 2 sin β + cosh a cos β arctan a e + cos β [|Re β| < π, a > 0] LI (389)(10)

3.



a > 0]

∞

0



βπ aπ cos ax sinh βx 1 cosh 2γ + sin 2γ dx = ln βπ x cosh γx 2 cosh aπ 2γ − sin 2γ

[|Re β| < Re γ,

BI (389)(23)

2.



  sin ax sinh βx βπ aπ dx = arctan tan tanh x sinh γx 2γ 2γ

∞

π −a 2e

sin β −



cosh a + sin β cos β 1 cosh a cos β ln + sinh a sin β arctan 2 cosh a − sin β sinh a

π |Re β| < 2 , a > 0, b > 0 

BI (389)(18)

 sin ax sinh βx a β a β  β dx = e− 2 a sin − β cos − sinh sin ln 1 + 2e−a cos β + e−2a 1 cosh πx 2 2 2 2 x2 + β sin β a 4 + cosh cos arctan 2 2 1 + e−a cos β [|Re β| < π, a > 0] ET I 91(26)

4.117

Trigonometric and hyperbolic functions and powers



∞

8. 0



 sin ax cosh βx 1 1 dx = − e−a (a cos β + β sin β) + sinh a cos β ln 1 + 2e−a cos β + e−2a 2 x + 1 sinh πx 2 2 sin β + cosh a sin β arctan a e + cos β [|Re β| < π, a > 0] ET I 91(25), LI (389)(9)

∞

π sin ax cosh βx cosh a + sin β cos β 1 dx = − e−a cos β + sinh a sin β ln + cosh a cos β arctan x2 + 1 sinh π x 2 2 cosh a − sin β sinh a 2

BI (389)(7) |Re β| < π2 , a > 0

∞

x cos ax cosh βx ke−ak cos kβ π e−ab cos bβ  (−1)k dx = + 2 2 x + b sinh πx 2 sin bπ k 2 − b2

10. 0

11. 0



∞

13. 0

∞

14. 0

4.116  1.6 0



∞

∞

2. 0

4.117  1. 0

∞

[|Re β| < π, ∞

0





k=1

12.



a > 0] BI (389)(21)

0



∞

sin ax cosh γx e−ak cos kγ 1 π e−aβ cos βγ  dx = + − (−1)k−1 2 2 2 2 x + β sinh πx 2β 2β sin βπ k − β2 k=1 [0 ≤ Re β, |Re γ| < π,

∞

9.



523

∞

a > 0]

BI (389)(24)

x cos ax cosh βx 1 dx = e−a (a cos β + β sin β) x2 + 1 sinh πx 2

1 1 − + cosh a cos β ln 1 + 2e−a cos β + e−2a 2 2 sin β + sinh a sin β arctan a e + cos β [|Re β| < π, a > 0]

BI (389)(19)

π x cos ax cosh βx cosh a + sin β 1 dx = −1 + e−a cos β + cosh a sin β ln x2 + 1 sinh π x 2 2 cosh a − sin β cos β 2 + sinh a cos β arctan sinh a

|Re β| < π2 , a > 0

BI (389)(17)

−2a

e cos ax cosh βx sin 2β dx = ae−a cos β + βe−a sin β + sinh a sin β arctan x2 + 1 cosh π2 x 1 + e−2a cos 2β   1 + cosh a cos β ln 1 + 2e−2a cos 2β + e−4a 2

|Re β| < π2 , a > 0 ET I 34(37) x cos 2ax tanh x dx cos ax tanh βx

aπ dx = ln coth x 4β

the integral is divergent

BI (364)(2)

[Re β > 0,

BI (387)(8)

a > 0]

sin ax πx dx = a cosh a − sinh a ln (2 sinh a) tanh 1 + x2 2 [a > 0]

BI (388)(3)

524

4.118

 2. 3. 4. 5.12

6.12

∞

sin ax π πx a dx = − ea + sinh a ln coth + 2 cosh a arctan (ea ) tanh 2 1 + x 4 2 2 0  ∞   sin ax a [a > 0] coth πx dx = e−a − sinh a ln 1 − e−a 2 2 0 1+x  ∞ sin ax a π [a > 0] coth x dx = sinh a ln coth 2 1 + x 2 2 0  ∞   x cos ax π tanh x dx = −ae−a + sinh a ln 1 − e−2a 2 1+x 2 0 

0

∞

8. 0



BI (389)(6)

BI (388)(7)

x cos ax π π a tanh x dx = ea + cosh a ln coth + 2 sinh a arctan (ea ) 2 1+x 4 2 2 [a > 0]

∞

7. 

BI (389)(5)

[a > 0] ∞

0



BI (388)(4)

∞

9. 0

  x cos ax a 1 coth πx dx = − e−a − − cosh a ln 1 − e−a 1 + x2 2 2 x cos ax a π coth x dx = −1 + cosh a ln coth 2 1+x 2 2

BI (388)(8) BI (389)(15)a, ET I 33(31)a

[a > 0]

  x cos ax π π a coth x dx = −2 + e−a + cosh a ln coth + 2 sinh a arctan e−a 2 1+x 4 2 2 [a > 0]



∞



4.122 1.6

2.

γπ sinh cos βx sin γx dx 2δ = arctan βπ cosh δx x 0 cosh 2δ  ∞ dx 1 cosh 2aπ + cos βπ cosh βx = ln sin2 ax sinh x x 4 1 + cos βπ 0 

∞

BI (389)(13)



x sin ax 1 1 1 π πa coth πa − 1 dx = 2 1 2 2 2 sinh πa cosh x 2  0 ∞ pπ 1 − cos px dx = ln cosh 4.119 sinh qx x 2q 0 4.121 aπ bπ  ∞ exp − exp sin ax − sin bx dx 2β 2β = 2 arctan 1. (a + b)π cosh βx x 0 1 + exp 2β bπ  ∞ cosh cos ax − cos bx dx 2β = ln 2. aπ sinh βx x 0 cosh 2β 4.1188

BI (389)(12)

ET I 89(14) BI (387)(2)a

[Re β > 0]

GW (336)(19b)

[Re β > 0]

GW (336)(19a)

[Re δ > |Im(β + γ)|]

[|Re β| < 1]

ET I 93(46)a

BI (387)(7)

4.125

Trigonometric and hyperbolic functions and powers

4.123  1. 0



∞

∞

2. 0



∞

3. 0



∞

4. 0



∞

5. 0

sin x x dx 1 1 = arctan − 2 2 cosh ax + cos x x − π a a

BI (390)(1)

sin x x dx a 1 = − arctan cosh ax − cos x x2 − π 2 1 + a2 a

BI (390)(2)

sin 2x x dx 1 1 + 2a2 1 = − arctan cosh 2ax − cos 2x x2 − π 2 2a 1 + a2 a

BI (390)(4)

cosh ax sin x x dx −1 = cosh 2ax − cos 2x x2 − π 2 2a (1 + a2 )

LI (390)(3)

cos ax dx πe−aγ = 2 2 cosh πx + cos πβ x + γ 2γ (cos γπ + cos βπ)  ∞   e−(2k+1−β)a e−(2k+1+β)a 1 − 2 + sinh βπ γ 2 − (2k + 1 − β)2 γ − (2k + 1 + β)2 k=0



∞

6. 0



∞

7. 0

525

[0 < Re β < 1,

Re γ > 0,

a > 0]

 Γ(p) a   (−1)k sin ax sinh bx p arctan xp−1 dx = p sin cos 2ax + cosh 2bx b (2k + 1)p (a2 + b2 ) 2 k=0 ∞

sin ax2



πx sin πx 1 ∂ ϑ1 (z | q) 2 sinh 2 x dx = cos πx + cosh πx 4 ∂z

[p > 0]

BI (364)(8)

z=0,q=e−2a

[a > 0] 4.124 1. 2. 4.125

ET I 93(49)

  √   cos px cosh q 1 − x2 π √ dx = J 0 p2 − q 2 2 1 − x2 0    ∞  u dx π cos ax cosh β (u2 − x2 ) √ = J0  2 u2 − x2 a2 − β 2 u





1

∞

1. 0



ET I 33(27)

∞

2. 0

MO (40)

ET I 34(38)



(−1)n−1 a2n−1 π a2 dx = 1+ sinh (a sin x) cos (a cos x) sin x sin 2nx x (2n − 1)! 8 2n(2n + 1) n−1 2(n−1)

a (−1) dx = cosh (a sin x) cos (a cos x) sin x cos(2n − 1)x x [2(n − 1)]!



LI (367)(14) 2

π a 1− 8 2n(2n − 1)

LI (367)(15)

 3. 0

∞

sinh (a sin x) cos (a cos x) cos x cos 2nx

dx π = x 2

∞  k=n+1

k 2k+1

n 2n+1

(−1) a (−1) a 3π + (2k + 1)! (2n + 1)! 8

(−1)n−1 a2n−1 π + (2n − 1)! 8 LI (367)(21)

526

4.126

4.126



∞

1. 0

 0

sin (a cos bx) cosh (a sin bx)

c2

0

cos (a cos bx) sinh (a sin bx)

BI (381)(2)

dx π cos (a cos bc) sinh (a sin bc) = 2 −x 2c [b > 0,

∞

3. 

x dx π = [cos (a cos bc) cosh (a sin bc) − 1] c2 − x2 2 [b > 0]

∞

2. 

sin (a cos bx) sinh (a sin bx)

c > 0]

BI (381)(1)

x dx π = [a cos bc − sin (a cos bc) cosh (a sin bc)] c2 − x2 2 [b > 0]

∞

4. 0

cos (a cos bx) cosh (a sin bx)

c2

BI (381)(4)

dx π = − sin (a cos bc) sinh (a sin bc) 2 −x 2c [b > 0]

BI (381)(3)

4.13 Combinations of trigonometric and hyperbolic functions and exponentials 4.131 

∞

1. 0

   ⎫  ⎧ β−νγ−ai β−νγ+ai ⎨ ⎬ Γ Γ 2γ 2γ i Γ(ν + 1)  −   sin ax sinhν γxe−βx dx = − ν+2 2 γ ⎩ Γ β+νγ−ai + 1 Γ β+γν+ai +1 ⎭ 2γ 2γ [Re ν > −2,

2.12



∞

0



∞

0



∞

4. 0

4.132



1. 0

∞

|Re(γν)| < Re β]

ET I 91(30)a

   ⎫  ⎧ ⎨ Γ β−νγ−ai ⎬ Γ β−νγ+ai 2γ 2γ Γ(ν + 1)  +   cos ax sinhν γxe−βx dx = ν+2 2 γ ⎩ Γ β+γν−ai + 1 Γ β+νγ+ai +1 ⎭ 2γ 2γ [Re ν > −1,

3.

Re γ > 0,

e−βx

e−x

Re γ > 0,

∞  2a sin ax dx = 2 + [β + (2k − 1)γ]2 sinh γx a k=1     β + γ + ia β + γ − ia 1 ψ −ψ = 2γi 2γ 2γ

|Re(γν)| < Re β]

ET I 34(40)a

BI (264)(9)a

[Re β > |Re γ|]

sin ax π aπ 1 dx = coth − sinh x 2 2 a

sinh 2πa sin ax sinh bx a π γ dx = − + eγx − 1 2 (a2 + b2 ) 2γ cosh 2πa − cos 2πb γ   γ   b a b a i ψ +i +1 −ψ −i +1 + 2γ γ γ γ γ [Re γ > |Re b|,

ET I 91(28)

ET I 91(29)

a > 0]

ET I 92(33)

4.136

Trigonometric and hyperbolic functions and exponentials



∞

2. 0

2πa

sinh γ sin ax cosh bx a π dx = − + 2πb γx 2 2 e −1 2 (a + b ) 2γ cosh 2πa γ − cos γ [Re γ > |Re b|]



∞

3. 0

a π sin ax cosh bx dx = − eγx + 1 2 (a2 + b2 ) γ

∞

4. 0

BI (265)(5)a, ET I 92(34)

bπ sinh aπ γ cos γ 2bπ cosh 2aπ γ − cos γ

[Re γ > |Re b|] 

527

ET I 92(35)

2πb γ

sin cos ax sinh bx b π dx = − 2bπ eγx − 1 2 (a2 + b2 ) 2γ cosh 2aπ γ − cos γ [Re γ > |Re b|]



∞

5. 0

πb πa cos ax sinh bx b π sin γ cosh γ dx = − + 2bπ eγx + 1 2 (a2 + b2 ) γ cosh 2aπ γ − cos γ

[Re γ > |Re b|] 4.133 1.11



∞

0

2.11



∞

0

4.134  1.12

∞

0



∞

2. 0

4.135 1.

12



∞

0

2.12



∞

0

1.12

 0

∞

ET I 34(39)

 2 

 √ x dx = πγ exp γ b2 − a2 sin(2abγ) sin ax sinh bx exp − 4γ  cos ax cosh bx exp −

e

−βx2

2

x 4γ

[Re γ > 0]



 (cosh x + cos x) dx =

2

e−βx (cosh x − cos x) dx =



[Re γ > 0]

ET I 35(41)

π 1 cosh β 4β

[Re β > 0]

ME 24

π 1 sinh β 4β

[Re β > 0]

ME 24

 2

sin ax cosh 2γxe

−βx

2

1 dx = 2

4

π2 exp a2 + β 2

 2

cos ax2 cosh 2γxe−βx dx =

1 2

ET I 92(37)



 √ πγ exp γ b2 − a2 cos(2abγ)

dx =

4

π2 exp a2 + β 2





βγ 2 a2 + β 2



 sin

a aγ 2 1 + arctan a2 + β 2 2 β



[Re β > 0] LI (268)(7)    a aγ 2 βγ 2 1 cos + arctan a2 + β 2 a2 + β 2 2 β [Re β > 0]

4.136

LI (265)(8)

LI (268)(8)

√    4 2π 1 1 cosh sinh x2 + sin x2 e−βx dx = √ I 14 8β 8β 4 β



[Re β > 0]

ME 24

528

2.

12

4.137





[Re β > 0] √     4 2π 1 1 cosh cosh x2 + cos x2 e−βx dx = √ I − 14 8β 8β 4 β

ME 24

∞

[Re β > 0] √     4 1 2π 1 sinh cosh x2 − cos x2 e−βx dx = √ I − 14 8β 8β 4 β

ME 24

∞

[Re β > 0]

ME 24

2

3.12

0

4.12

 0

4.137



∞

1. 0



[Re β > 0]     π 1 1 −π 2 2 −βx4 sin − cos 2x sinh 2x e dx =  J1 4 β 4 128β 2 4 β

MI 32

[Re β > 0]     1 π 1 sin + β β 4

MI 32

∞

[Re β > 0]

MI 32

4. 0

∞

1.

4

cos 2x2 cosh 2x2 e−βx dx =  4



2

2

2

π 128β 2

2

sin 2x cosh 2x + cos 2x sinh 2x

0



    1 1 cos β β

[Re β > 0]       −βx4 1 1 π 2 2 2 2 cos cos 2x cosh 2x + sin 2x sinh 2x e dx =  J − 14 4 2 β β 32β

MI 32

∞

[Re β > 0]       −βx4 π 1 1 2 2 2 2 sin cos 2x cosh 2x − sin 2x sinh 2x e dx =  J − 14 4 2 β β 32β

MI 32

∞

0

0

π dx =  J1 4 32β 2 4

MI 32

3.

4.

e

−βx4

[Re β > 0]       −βx4 1 1 π 2 2 2 2 sin sin 2x cosh 2x − cos 2x sinh 2x e dx =  J 14 4 2 β β 32β

0





J − 14

∞

2. 

    1 π 1 cos + β β 4

∞

0



4 π sin 2x2 sinh 2x2 e−βx dx =  J − 14 4 128β 2

MI 32

3.

4.138

e

−βx4

[Re β > 0]     π 1 1 π 2 2 −βx4  1 cos − sin 2x cosh 2x e dx = 4 J β 4 128β 2 4 β

0





∞

2. 

2

sinh x − sin x

0



√   2π 1 1 sinh dx = √ I 14 8β 8β 4 β

∞

[Re β > 0]

MI 32

4.144

Trigonometric and hyperbolic functions, exponentials, and powers

529

4.14 Combinations of trigonometric and hyperbolic functions, exponentials, and powers 4.141



∞

1. 0



∞

2. 0



∞

3. 0



∞

4. 0

4.142



∞

1. 0

 2. 3.

∞

2

xe−βx cosh x sin x dx =

2

xe−βx sinh x cos x dx =

1 4 1 4

2



1 4

2

x2 e−βx cosh x cos x dx =

x2 e−βx sinh x sin x dx =



1 4

π β3

[Re β > 0]

MI 32

[Re β > 0]   π 1 1 1 − sin cos β3 2β β 2β

MI 32

[Re β > 0]   π 1 1 1 + cos sin β3 2β β 2β

MI 32

[Re β > 0]

MI 32

[Re β > 0]

ME 24

  π 1 1 − sin cos β3 2β 2β





2

xe−βx (sinh x + sin x) dx =

  1 1 + sin cos 2β 2β

1 2

 

π 1 cosh 3 β 4β

π 1 [Re β > 0] sinh 3 β 4β 0     ∞ 2 1 1 1 1 π + sinh cosh x2 e−βx (cosh x + cos x) dx = 2 β3 4β 2β 4β 0 

∞

4. 0

xe

−βx2

1 (sinh x − sin x) dx = 2

2

x2 e−βx (cosh x − cos x) dx =

1 2



π β3



[Re β > 0]  1 1 1 + cosh sinh 4β 2β 4β [Re β > 0]

4.143



∞

1. 0



∞

2. 0

 4.144

0

2

xe−βx (cosh x sin x + sinh x cos x) dx =

2

xe−βx (cosh x sin x − sinh x cos x) dx =

∞

2

e−x sinh x2 cos ax

dx = x2



π − a2 e 8 2

1 2β 1 2β





ME 24

ME 24

ME 24

π 1 cos β 2β [Re β > 0]

MI 32

π 1 sin β 2β

[Re β > 0]   a πa 1−Φ √ − 4 8 [a > 0]

MI 32

ET I 35(44)

530

4.145

Logarithmic Functions



∞

1. 0



∞

2. 0

2

xe−βx cosh (2ax sin t) sin (2ax cos t) dx =

2

xe−βx sinh (2ax sin t) cos (2ax cos t) dx =

a 2 a 2





4.145

    2 π a2 a cos 2t cos t − sin 2t exp − β3 β β [Re β > 0] BI (363)(5)     2 2 π a a cos 2t sin t − sin 2t exp − β3 β β [Re β > 0]

4.14610 1.

8



∞

0

2.8



∞

0



∞

3. 0



∞

4. 0

5.8



∞

0

6.8



∞

0

e

−βx2

1 sinh ax sin bx dx = 2

2

e−βx cosh ax cos bx dx =

2

xe−βx sinh ax cos ax dx =

a 4β

a 4β

2

x2 e−βx cosh ax sin ax dx =

2

π exp β



1 2

xe−βx cosh ax sin ax dx =

2



x2 e−βx cosh ax cos ax dx =

1 4

1 4



π exp β 







a2 − b 2 4β



2

a −b 4β



2

sin

BI (363)(6)

ab 2β [Re β > 0]

 cos

ab 2β

[Re β > 0]   2 2 π a a cos + sin β 2β 2β [Re β > 0]   2 2 π a a cos − sin β 2β 2β

[Re β > 0]   2 2 π a a2 a + cos sin β3 2β β 2β [Re β > 0]   2 2 π a a2 a − sin cos β3 2β β 2β [Re β > 0]

4.2–4.4 Logarithmic Functions 4.21 Logarithmic functions 4.211  1.

2.

∞

dx = −∞ 1 e ln x  u dx = li u ln x 0

BI (33)(9)

FI III 653, FI II 606

4.213

4.212 1.

7

Logarithmic functions



1

0



1

2. 0

3.7



1



1

1

6. e

7.

8.



BI (31)(5)

[a ≥ 0]

BI (31)(14)

[a > 0]

BI (31)(16)

= 1 + (1 − a)e−a Ei(a)

[a ≥ 0]

BI (31)(15)

= 1 + (1 + a)ea Ei(−a)

[a > 0]

BI (31)(17)

dx

1 = − + e−a Ei(a) a (a + ln x) dx 2

ln x dx ln x dx ln x dx (1 + ln x)

1 7

[a > 0]

(a − ln x)

0



dx = −ea Ei(−a) a − ln x

(a + ln x)2

0



BI (31)(4)

(a − ln x)

0



[a > 0]

1

4. 5.8

dx = e−a Ei(a) a + ln x

2

0

1

0

531

2

2

=

=

1 + ea Ei(−a) a

e −1 2

BI (33)(10)

n−1  dx 1 1 e−a Ei(a) − (n − k − 1)!ak−n n = (n − 1)! (n − 1)! (a + ln x) k=1

[a ≥ 0] 

1

9. 0

BI (31))(22)

n−1 dx (−1)n a (−1)n−1  e = Ei(−a) + (n − k − 1)!(−a)k−n (a − ln x)n (n − 1)! (n − 1)! k=1

[a > 0, 

(ln x)

n odd]

BI (31)(23)

m

dx it is convenient to make the substitution x = e−t . n l [an + (ln x) ] Results 4.212 3, 4.212 5, and 4.212 8 [for n > 1] and 4.213 6, 4.213 8 below are divergent but may be considered to be valid if defined as follows:  n−1

 a  a f (z) dz f (z) d 1 PV dz n = (n − 1)! dz0 0 (z − z0 ) 0 z − z0 where a > z0 > 0, n = 1, 2, 3, . . . and PV indicates the Cauchy principal value. In integrals of the form

4.213



1

1.

a2 + (ln x)

0

2.7



dx

1

dx a2 − (ln x)

0

2

2

=

1 [ci(a) sin a − si(a) cos a] a

[a > 0]

=

1 −a e Ei(a) − ea Ei(−a) 2a

[a > 0] ,

BI (31)(6)

(cf. 4.212 1 and 2) BI (31)(8)

 3.

1

0 a2

ln x dx + (ln x)

2

= ci(a) cos a + si(a) sin a

[a > 0]

BI (31)(7)

532

4.7

Logarithmic Functions



1

ln x dx

0 a2

− (ln x)

2

=−

1 −a e Ei(a) + ea Ei(−a) 2

4.214

[a > 0] ,

(cf. 4.212 1 and 2) BI (31)(9)



1

5. 0

6.8



[a > 0] 1

0



1

7. 0

8.8





ln x dx a4

=−

1 a e Ei(−a) − e−a Ei(a) − 2 ci(a) sin a + 2 si(a) cos a 4a3

− (ln x)

4

=−

1

2

(ln x) dx a4 − (ln x)

0

4

=−

3

(ln x) dx a4

0

− (ln x)

4

=−



1

1. 0

 2. 3.

1

(− ln x)

μ−1

BI (31)(12)

1 a e Ei(−a) + e−a Ei(a) + 2 ci(a) cos a + 2 si(a) sin a 4 [a > 0]

4.215

BI (31)(11)

1 a e Ei(−a) − e−a Ei(a) + 2 ci(a) sin a − 2 si(a) cos a 4a [a > 0]

1

BI (31)(10)

1 a e Ei(−a) + e−a Ei(a) − 2 ci(a) cos a − 2 si(a) sin a 2 4a [a > 0]

3.

4.7

− (ln x)

4

BI (31)(19)

is divergent

[a > 0]

0



dx

1

2.



is divergent

ln x dx  2 2 a2 − (ln x)

a4

0

LI (31)(18)

ln x dx 1 1 [ci(a) sin a − si(a) cos a] − 2  2 = 2a 2a 2 a2 + (ln x)

1

1. 

dx  2 2 a2 − (ln x)

[a > 0] 1

0

4.214

dx 1 1  2 = 3 [ci(a) sin a − si(a) cos a] − 2 [ci(a) cos a + si(a) sin a] 2a 2a 2 a2 + (ln x)

dx = Γ(μ)

dx π cosec μπ μ = Γ(μ) (− ln x) 0 √  1 √ π − ln x dx = 2 0

BI (31)(13)

[Re μ > 0]

FI II 778

[Re μ < 1]

BI (31)(1)

BI (32)(1)

4.222

Logarithms of more complicated arguments



1

4. 0

4.216 1.

2.∗



533

√ dx √ = π − ln x

BI (32)(3)

1/e

dx  = K 0 (1) 2 0 (ln x) − 1 √  1/e dx π √ = e − ln x − 1 0

GW (32)(2)

4.22 Logarithms of more complicated arguments 4.221



1

1. 0



1

2. 0

 3.

ln x ln(1 − x) dx = 2 −

π2 6

BI (30)(7)

ln x ln(1 + x) dx = 2 −

π2 − 2 ln 2 12

BI (30)(8)

∞

1

ln 0

4.222



∞

ln 0



3. 4.

[a < 1]

BI (31)(3)

k=1

1. 2.

 ln(1 + k) 1 − ax dx =− ak 1 − a ln x k

∞

a2 + x2 dx = (a − b)π b2 + x2

[a > 0,

b > 0]

a2 + x2 aa dx = π(b − a) + π ln [a > 0, b > 0] b2 + x2 bb 0    ∞ b2 [b > 0] ln x ln 1 + 2 dx = πb (ln b − 1) x 0

   ∞   1 + ab b2 2 2 ln(1 + ab) − b ln 1 + a x ln 1 + 2 dx = 2π x a 0 ln x ln

[a > 0, b > 0]   ∞   b2 ln a2 + x2 ln 1 + 2 dx = 2π [(a + b) ln(a + b) − a ln a − b] x 0

 5.



0

BI (33)(2)

BI (33)(3)

[a > 0, b > 0]     b2 a2 ln 1 + 2 ln 1 + 2 dx = 2π [(a + b) ln(a + b) − a ln a − b ln b] x x

BI (33)(4)

[a > 0, b > 0]

    2 b 1 + ab 1 ln(1 + ab) − b ln b ln a2 + 2 ln 1 + 2 dx = 2π x x a

BI (33)(5)

∞

0

7.

BI (33)(1)

∞

6. 

GW (322)(20)

[a > 0,

b > 0]

BI (33)(7)

534

8.

12

Logarithmic Functions



∞

0

b −x

ln (1 + ax) x e

dx = b!e +

1/a

4.223



∞

1. 0



∞

2. 0



∞

3. 0

4.224



1. 0

 2. 3. 4.

u

     b−1 b! (−1)b−m−1 1/a 1 1 + Ei − e Ei − a (b − m)! ab−m a m=0

b−m  k=1

4.223

(k − 1)! (−a)b−m−k

[b > 0,

an integer]

  π2 ln 1 + e−x dx = 12

BI (256)(10)

  π2 ln 1 − e−x dx = − 6

BI (256)(11)

  t2 π2 − ln 1 + 2e−x cos t + e−2x dx = 6 2

ln sin x dx = L

π/4

[|t| < π]

BI (256)(18)

 π  −u −L 2 2

π

LO III 186(15)

1 π ln 2 − G 4 2 0  π  π/2 1 π ln sin x dx = ln sin x dx = − ln 2 2 0 2 0  u ln cos x dx = − L(u) ln sin x dx = −

BI (285)(1)

FI II 629,643 LO III 184(10)

0



π/4

5. 0

 6. 7. 8. 9.8 10.

π/2

ln cos x dx = −

1 π ln 2 + G 4 2

π ln 2 2 0

 π/2 π π2 2 2 (ln 2) + (ln sin x) dx = 2 12 0

 π/2 π π2 2 2 (ln 2) + (ln cos x) dx = 2 12 0 √  π a + a2 − b 2 ln (a + b cos x) dx = π ln 2 0  π ln (1 ± sin x) dx = −π ln 2 ± 4G

BI (286)(1)

ln cos x dx = −

BI 306(1)

BI (305)(19)

BI (306)(14)

[a ≥ |b| > 0]

GW (322)(15) GW (322)(16a)

0

11.7



0

π/2

∞ k  k  b (−1)n+1 π a ln + 2G + 2 2 2 k n=1 2n − 1 k=1 π = − ln 2 + 2G 2

ln (1 + a sin x) dx =

[a > 0] [a = 1]

b=

1−a 1+a

4.226

12.12

Logarithms of more complicated arguments



π

0

 ln (1 + a cos x) dx = π ln = 2π ln

14.11

 √ 1 − a2 2

2

a ≤1 2

a ≥1

|a| 2

  ⎧ √ 2 ⎪  π ⎨2π ln 1 + 1 − a 2 ln2 (1 + a cos x) dx = ⎪ 0 ⎩ a2 π ln 4

12 (1)12

13.12

1+

535

BI (330)(1)

for a2 ≤ 1

for a2 ≥ 1  2k+1  π/2 ∞ 2    2k (k!) 2a 2 ln 1 + 2a sin x + a dx = (2k + 1) · (2k + 1)!! 1 + a2 0 k=0 2

a ≤1  nπ   ln a2 − 2ab cos x + b2 dx = 2nπ ln [max (|a|, |b|)]

BI (308)(24)

0

15.8

 0

4.225 1. 2. 3.

4.



[ab > 0] nπ

  ln 1 − 2a cos x + a2 dx = 0 = nπ ln a2

π/4

FI II 142, 163, 688

2

a ≤1 2

a ≥1

1 π ln 2 − G 8 2 0  π/2  π/4 1 π 1 ln (cos x + sin x) dx = ln (cos x + sin x) dx = − ln 2 + G 2 0 8 2 0 √  2π 2 2 1+ 1−a −b ln (1 + a sin x + b cos x) dx = 2π ln 2 0

2 a + b2 < 1  2π   2

ln 1 + a2 + b2 + 2a sin x + 2b cos x dx = 0 a + b2 ≤ 1 0  2

 a + b2 ≥ 1 = 2π ln a2 + b2 ln (cos x − sin x) dx = −

GW (322)(9b)

GW (322)(9a)

BI (332)(2)

BI (322)(3)

4.226 1.12

 0

π/2

ln2 (a2 − sin2 x) dx = −2π ln 2 = 2π ln

a+

√ a2 − 1 = 2π (arccosh a − ln 2) 2

a2 ≤ 1



[a > 1] FI II 644, 687

536

Logarithmic Functions

4.227

  π/2      1 π  ln 1 + a sin2 x dx = ln 1 + a sin2 x dx = ln 1 + a cos2 x dx 2 0 0 0 √   1+ 1+a 1 π  2 ln 1 + a cos x dx = π ln = 2 0 2 [a ≥ −1] BI (308)(15), GW(322)(12)    u   1 α sin α − π2 ln 2 ln 1 − sin2 α sin2 x dx = (π − 2θ) ln cot + 2u ln 2 2 0   + L(θ + u) − L(θ − u) + L π2 − 2u

cot θ = cos α tan u; −π ≤ α ≤ π, − π2 ≤ u ≤ π2 LO III 287

2.

3.



π/2



π/2

4. 0

    

α β 1 2 β 2 α 4 2 cos + cos + sin cos ln 1 − cos x sin α − sin β sin x dx = π ln 2 2 2 2 2

2



2

2

2

[α > β > 0] LO III 283   u     sin2 x dx = −u ln sin2 α − L π2 − α + u + L π2 − α − u ln 1 − 2 sin α 0 π

− 2 ≤ u ≤ π2 , |sin u| ≤ |sin α|

 5.

LO III 287



π/2

6. 0

  1 ln a2 cos2 x + b2 sin2 x dx = 2

 0

π

  a+b ln a2 cos2 x + b2 sin2 x dx = π ln 2 [a > 0,



π/2

7. 0

4.227  1. 0



u

ln tan x dx = L(u) + L

π/4

2. 0



π/2

3. 0

4.

7



t 1 + sin π−t 1 + sin t cos2 x 2 dx = π ln = π ln cot ln t 1 − sin t cos2 x 4 cos 2  π |t| < 2

π/4

0

 ln tan x dx = −

π/4

ln (a tan x) dx = n

π/2

GW (322)(13)

LO III 283

 π −u −L 2 2

LO III 186(16)

ln tan x dx = −G

BI (286)(11)

π

π ln a 2

[a > 0] n

(ln tan x) dx = n!(−1)

∞  k=0

=

b > 0]

BI (307)(2)

(−1)k (2k + 1)n+1

1  π n+1 |En | 2 2

[n even] BI (286)(21)

5.7

 0

π/2

(ln tan x)

2n

dx = 2(2n)!

∞  k=0

k

(−1) = (2k + 1)2n+1

 π 2n+1 2

|E2n |

BI (307)(15)

4.228

Logarithms of more complicated arguments



π/2

6. 0



π/4

7. 0



π/4

8. 0



π/4

9. 0



π/2

10. 0



π/4

11. 0

12.11



π/2

0



π/4

13. 0

 14. 15. 16.11

π/4

(ln tan x)

2n+1

537

dx = 0

BI (307)(14)

π3 16

BI (286)(16)

5 5 π 64

BI (286)(19)

(ln tan x)2 dx = 4

(ln tan x) dx =

ln (1 + tan x) dx =

π ln 2 8

BI (287)(1)

ln (1 + tan x) dx =

π ln 2 + G 4

BI (308)(9)

ln (1 − tan x) dx =

π ln 2 − G 8

BI (287)(2)

2

(ln (1 − tan x)) dx = ln (1 + cot x) dx =

π ln 2 − 2G 2

BI (308)(10)

π ln 2 + G 8

BI (287)(3)

π ln 2 8 0   π/4 1 π/2 π ln (tan x + cot x) dx = ln (tan x + cot x) dx = ln 2 2 0 2 0  π/2  π/4 1 π 2 2 (ln (cot x − tan x)) dx = (ln (cot x − tan x)) dx = ln 2 2 2 0 0 ln (cot x − 1) dx =

BI (287)(4)

BI (287)(5), BI (308)(11)

BI (287)(6), BI (308)(12)



π/2

17. 0

  1 ln a2 + b2 tan2 x dx = 2

 0

π

  ln a2 + b2 tan2 x dx = π ln(a + b) [a > 0,

4.228 1. 2.

b > 0]

GW (322)(17)

      π t π−t 2 2 −2L ln sin t sin x + 1 − cos t sin x dx = ln 2 − 2 L LO III 290 2 2 2 0  u    π  1 1 − t − ϕ ln cos t + L(u + ϕ) − L(u − ϕ) − L(ϕ) ln cos x + cos2 x − cos2 t dx = − 2 2 2 0

sin u π cos ϕ = ; 0≤u≤t≤ sin t 2



π/2



LO III 290

 3. 0

t

   π  − t ln cos t ln cos x + cos2 x − cos2 t dx = − 2

LO III 285

538

Logarithmic Functions



u

4. 0

 5.

   t t sin u + sin t cos x sin2 u − sin2 x  dx = π ln tan sin u + tan2 sin2 u + 1 ln 2 2 2 2 sin u − sin t cos x sin u − sin x

π/4

6. 0



[t > 0,

√  ∞ π (−1)k  ln cot x dx = 2 (2k + 1)3 k=0

u > 0]

LO III 283

π/4 √

0



4.229

π/4

7. 0

√

BI (297)(9)

∞ √  dx (−1)k √ = π 2k + 1 ln cot x k=0

BI (304)(24)

 √   √ √ 1 1 π/2 √ π ln tan x + cot x dx = ln tan x + cot x dx = ln 2 + G 2 0 8 2 BI (287)(7), BI (308)(22)



π/4

8.

ln2

0

√  √ 1 cot x − tan x dx = 2



π/2

ln2

0

√  √ cot x − tan x dx =

π 4

ln 2 − G BI (287)(8), BI (308)(23)

4.229



1

1. 0

2.11

 0 1

3. 0



1

0

5.7

1

PV 

4.11

ln (− ln x) dx = −C dx = PV ln (− ln x)

FI II 807



∞ −u

0

e du ≈ −0.154479 ln u

BI (31)(2)

√ dx = − (C + 2 ln 2) π ln (− ln x) √ − ln x ln (− ln x) (− ln x)μ−1 dx = ψ(μ) Γ(μ)

BI (32)(4)

[Re μ > 0]

BI (30)(10)

If the integrand contains (ln ln x1 ), it is convenient to make the substitution ln x1 = u so that x = e−u .  1 ln (a + ln x) dx = ln a − e−a Ei(a) [a > 0] BI (30)(5) 0



1

6. 0



ln (a − ln x) dx = ln a − ea Ei(−a)

π/2

7. π/4

π ln ln tan x dx = ln 2

[a > 0]

    Γ 34 √   2π Γ 14

BI (30)(6)

BI (308)(28)

4.23 Combinations of logarithms and rational functions 4.231



1

1. 0

 2.

0

1

ln x π2 dx = − 1+x 12

FI II 483a

ln x π2 dx = − 1−x 6

FI II 714

4.231

Logarithms and rational functions



1

3. 0



1

4. 0

5.11



x ln x π2 dx = 1 − 1−x 6

BI (108)(7)

1+x π2 ln x dx = 1 − 1−x 3

BI (108)(9)

∞

0



1

6. 0

7.

7



∞

0

∞

9. 0



∞

10. 0



11. 12. 13. 14. 15. 16.

17.

18. 19.∗ 20.∗

[0 < a]

BI (139)(1)

BI (111)(1)

 √   Γ n − 12 π a 1 dx 2 ln − C − ψ n − ln x 2 = n 4(n − 1)!a2n−1 b 2b 2 (a + b2 x2 ) [a > 0,

∞

8. 

ln x dx ln a = (x + a)2 a

ln x dx = − ln 2 (1 + x)2

0



539

ln x dx a π ln = a2 + b2 x2 2ab b

[ab > 0]

ln px π ln pq dx = q 2 + x2 2q

[p > 0,

ln x dx π2 =− 2 2 −b x 4ab

a2

b > 0]

LI (139)(3) BI (135)(6)

q > 0]

BI (135)(4)

[ab > 0]

a

ln x dx π ln a G − [a > 0] = 2 + a2 x 4a a 0  1  ∞ ln x ln x dx = − dx = −G 2 2 0 1+x 1 1+x  1 ln x dx π2 =− 2 8 0 1−x  1 x ln x π2 dx = − 2 48 0 1+x  1 x ln x π2 dx = − 2 24 0 1−x  1 n 1 − x2n+2 (n + 1)π 2  n − k + 1 + ln x dx = − 2 8 (2k − 1)2 (1 − x2 ) 0 k=1  1 n 1 + (−1)n xn+1 (n + 1)π 2  n−k+1 − ln x dx = − (−1)k (1 + x)2 12 k2 0 k=1  1 n 1 − xn+1 (n + 1)π 2  n − k + 1 + ln x dx = − (1 − x)2 6 k2 0 k=1  1 x ln x π2 dx = −1 + 2 0 1+x  1 (1 − x) ln x π2 dx = 1 − 1+x 6 0

GW (324)(7b) FI II 482, 614 BI (108)(11) GW (324)(7b)

BI (111)(5)

BI (111)(2)

BI (111)(3)

540

4.232 1.

12

Logarithmic Functions



υ

u



ln x dx log2 u − log2 v = (x + u)(x + υ) 2(u − v)

∞

2. 0

2

4.232

[u > 0, 2

ln x dx (ln β) − (ln γ) = (x + β)(x + γ) 2(β − γ)

v > 0]

[|arg β| < π,

BI (145)(32)

|arg γ| < π] ET II 218(24)



∞

3. 0

4.233 1.

3

2.3 3.11 4.3 5.

[a > 0]



∞

1

∞

3. ∞

4.

0



1 + x2 1 − x2

2

(1 + x2 )

0



2 x2 )

(1 − x2 )2

0

5.

x2 )2

x ln x dx (1 +

0



ln x dx (1 +

2. 

BI (140)(10)

1

1

7.

ln x dx π 2 + (ln a) = x+ax−1 2(a + 1)

  ln x dx 1 2 2π 2  −ψ = −0.7813024129 . . . = 2 9 3 3 0 1+x+x    1 ln x dx 1 1 2π 2  −ψ = −1.17195361934 . . . = 2 1 − x + x 3 3 3 0    1 1 7π 2 x ln x dx 1  − ψ = −0.15766014917 . . . = − 2 1 + x + x 9 6 3 0    1 x ln x dx 1 1 5π 2  − ψ = −0.3118211319 . . . = 2 6 6 3 0 1−x+x  ∞ ln x dx t ln a [a > 0, = 2 + 2xa cos t + a2 x a sin t 0 

4.234  1.11

6.

2

1

LI (113)(1)

LI (113)(2)

LI (113)(2)

LI (113)(4)

0 < t < π]

G π − 2 8

BI (144)(18)a

1 = − ln 2 4

BI (111)(4)

=

ln x dx = 0 ln x dx = −

BI (142)(2)a

π 2

BI (142)(1)a

x2 ln x dx π2 √  =−  2 4 (1 − x ) (1 + x ) 16 2 + 2

BI (112)(21)

∞

ln x dx bπ a = ln [ab > 0] 2 2 2 2 2 + b x ) (1 + x ) 2a (b − a ) b 0    ∞ ln x 1 dx π ln a + b ln b · = 2 2 1 + b2 x2 2 (1 − a2 b2 ) a 0 x +a

BI (317)(16)a

(a2

 8. 0

GW (324)(13c)

[a > 0, ∞

(a2

x2 ln x dx aπ b = ln + b2 x2 ) (1 + x2 ) 2b (b2 − a2 ) a

[ab > 0]

b > 0]

LI (140)(12) LI (140)(12), BI (317)(15)a

4.241

4.235 1. 2. 3.11 4. 4.236

Logarithms and algebraic functions



∞

541

π (1 − x)xn−2 π2 [n > 1] dx = − 2 tan2 2n 1 − x 4n 2n 0        ∞ 1 − x2 xm−1 π 2 sin m+1 π sin πn n ln x dx = − 2 2  mπ  2  m+2  1 − x2n 4n sin 2n sin 0 2n π   n−3  ∞   2 2 1−x x π π [n > 2] ln x dx = − 2 tan2 2n 1−x 4n n 0  1 xm−1 + xn−m−1 π2   [n > m] ln x dx = − 1 − xn n2 sin2 m 0 nπ



1

ln x



1. 0



1

2. 0

x ln x 1 + (p − 1) ln x + 1−x (1 − x)2

x ln x 1 + 1 − x (1 − x)2



BI (135)(10)

LI (135)(12)

BI (135)(11) BI (108)(15)

xp−1 dx = −1 + ψ  (p) [p > 0]

dx =

BI (111)(6)a, GW (326)(13)

π2 −1 6

GW (326)(13a)

4.24 Combinations of logarithms and algebraic functions 4.241 1.

2.

3.

 2n  (2n − 1)!! π  (−1)k−1 x2n ln x √ · − ln 2 dx = (2n)!! 2 k 1 − x2 0 k=1    1 2n+1 2n+1  (−1)k (2n)!! x ln x √ ln 2 + dx = (2n + 1)!! k 1 − x2 0 k=1    1 2n  1 (2n − 1)!! π  (−1)k−1 2n 2 · − − ln 2 x 1 − x ln x dx = (2n + 2)!! 2 k 2n + 2 0



1

BI (118)(5)a

BI (118)(5)a

k=1

 4. 0

5.12

6. 7. 8.

1

x2n+1

 1 − x2 ln x dx =

(2n)!! (2n + 3)!!

 ln 2 +

2n+1  k=1

(−1)k 1 − k 2n + 3

   n 1  1 (2n − 1)!! 2n−1 π 2 ln 2 + ln x · (1 − x2 ) dx = − 4 · (2n)!! k 0 k=1  √ 12 ln x dx 1 π √ = − ln 2 − G 2 4 2 1−x 0  1 ln x dx π √ = − ln 2 2 2 1−x 0  ∞ ln x dx √ = 1 − ln 2 2 x2 − 1 1 x



LI (117)(4), GW (324)(53a)

BI (117)(5), GW (324)(53b)



BI (117)(3)

BI (145)(1)

FI II 614, 643 BI (144)(17)

542

9. 10.

Logarithmic Functions

 1 π π 1 − x2 ln x dx = − − ln 2 8 4 0  1  x 1 − x2 ln x dx = 13 ln 2 − 49 0



1

11. 0

4.242

4.242



BI (117)(1), GW (324)(53c)

BI (117)(2)

√   2 ln x dx 2π 1  Γ =− 2 8 4 x (1 − x )

∞

1. 0

ln x dx 1  K = 2 2 2 2 2a (a + x ) (x + b )

GW (324)(54a)

√  a2 − b 2 ln ab a

[a > b > 0]     b ln x dx a b π 1  K √ ln ab − K √ = √ 2 2 a2 + b 2 a2 + b 2 a2 + b 2 (a2 + x2 ) (b2 − x2 ) 0

BY (800.04)

 2.



∞

3. b

[a > 0, b > 0] BY (800.02)     ln x dx b a π 1  K √ ln ab + K √ = √ 2 2 2 2 2 2 2 2 2 2 2 a +b a +b a + b2 (x + a ) (x − b )

[a > 0, b > 0]  √   b π 1 ln x dx a2 − b 2 b  K ln ab − K = 2a a 2 a (a2 − x2 ) (b2 − x2 ) 0 

 4.

[a > b > 0]  √ a 1 ln x dx a2 − b 2  ln ab = K 2a a (a2 − x2 ) (x2 − b2 ) b    √   ∞ ln x dx a2 − b 2 b π 1  K ln ab + K = 2a a 2 a (x2 − a2 ) (x2 − b2 ) a

BY (800.06)

BY (800.01)

 5.

6.

[a > b > 0]  4.243 4.244 1.

2. 3.

1

0

x ln x π √ dx = − ln 2 4 8 1−x

  3 ln x dx 1 1  =− Γ 8 3 3 2 0 x (1 − x2 )    1 ln x dx π π √ √ √ ln 3 + = − 3 3 3 3 3 1 − x3 0    1 x ln x dx π π  √ − ln 3 = √ 3 3 3 3 0 3 (1 − x3 )2



BY (800.03)

BY (800.05) GW (324)(56b)

1

GW (324)(54b)

BI (118)(7)

BI (118)(8)

4.252

4.245 1.

2.

Logarithms and powers

543

  2n (2n − 1)!! π  (−1)k−1 x4n+1 ln x √ · − ln 2 dx = (2n)!! 8 k 1 − x4 0 k=1    1 4n+3 2n+1  (−1)k x ln x (2n)!! √ ln 2 + dx = 4 · (2n + 1)!! k 1 − x4 0



1

GW (324)(56a)

GW (324)(56c)

k=1

 4.246

1

 n  1 (2n − 1)!! π 2 ln 2 + ln x dx = − (2n)!! 4 k

GW (324)(55)

k=1

 1 1 , ln x 2n 2n √ dx = − π n 2 1 − x2n 8n sin 2n   1 1 , πB ln x dx 2n 2n  = − π n xn−1 (1 − x2 ) 8 sin 2n 



1

0

2.6

 1 2 n− 2

1−x

0

4.247 1.6





1

0

πB

[n > 1]

GW (324)(54c)a

GW (324)(54)

4.25 Combinations of logarithms and powers 4.251



∞

1. 0



∞

2. 0

3.

10



1

0



1

4. 0

5.

11



1

0

6.11

 0

4.252



1. 0

xμ−1 ln x πβ μ−1 dx = (ln β − π cot μπ) β+x sin μπ   xμ−1 ln x π dx = πaμ−1 cot μπ ln a − a−x sin2 μπ

[|arg β| < π,

0 < Re μ < 1] BI (135)(1)

[a > 0,

0 < Re μ < 1]

ET I 314(5)

xμ−1 ln x dx = β  (μ) x+1

[Re μ > 0]

GW (324)(6), ET I 314(3)

xμ−1 ln x dx = − ψ (μ) = − ζ(2, μ) 1−x

[Re μ > 0]

BI (108)(8)

2n

ln x

π 2  (−1)k−1 x2n dx = − + 1+x 12 k2

BI (108)(4)

2n−1  (−1)k π2 x2n−1 dx = + 1+x 12 k2

BI (108)(5)

k=1

1

ln x

k=1

∞

μ−1 

 xμ−1 ln x π dx = β ln β − γ μ−1 ln γ − π cot μπ β μ−1 − γ μ−1 (x + β)(x + γ) (γ − β) sin μπ [|arg β| < π, |arg γ| < π, 0 < Re μ < 2, μ = 1] BI (140)(9)a, ET 314(6)

544

Logarithmic Functions



∞

2. 0

 3. 4.6

4.253



π xμ−1 ln x dx = π − β μ−1 (sin μπ ln β − π cos μπ) 2 (x + β)(x − 1) (β + 1) sin μπ [|arg β| < π, 0 < Re μ < 2,

μ = 1] BI (140)(11)

∞

xp−1 ln x pπ π2 cosec2 [0 < p < 2] dx = − 2 1 − x 4 2 0    ∞ μ−1 x ln x 1 (1 − μ)aμ−2 π ln a − π cot μπ + dx = (x + a)2 sin μπ μ−1 0 [|arg a| < π,

(see also 4.254 2)

0 < Re μ < 2,

(μ = 1)]

GW (324)(13b)

4.253 1.8



1

0

ν−1

xμ−1 (1 − xr )

ln x dx =

μ  1 μ   μ , ν ψ − ψ + ν B r2 r r r [Re μ > 0,

Re ν > 0,

r > 0]

GW (324)(3b)a, BI (107)(5)a



1

2. 0



xp−1 π ln x dx = − cosec pπ (1 − x)p+1 p

∞

3. u

4.11



∞

0



∞

5. 1

6.

7



∞

0

7.7



∞

0

[0 < p < 1]

(x − u)μ−1 ln x dx = uμ−λ B(λ − μ, μ) [ln u + ψ(λ) − ψ(λ − μ)] xλ  ln x

a2

x + x2

p

[0 < Re μ < Re λ] dx ln a  p p  = pB , x 2a 2 2

(x − 1)p−1 ln x dx = ln x

ln x

π cosec πp p

(a +

1 x)n+ 2

=

2 (2n −

1 1)an− 2



1. 0



1

xp−1 ln x 1 dx = − 2 ψ  q 1−x q

∞

2. 0

 3. 0

∞

  p q

xp−1 ln x π2 dx = − pπ 1 − xq q 2 sin2 q ln x dx = − 1 xp

xq

π2 p−1 π q 2 sin2 q

ET II 203(18)

p > 0]

BI (140)(6)

[−1 < p < 0]

dx 1 = (ln a − C − ψ(μ)) (a + x)μ+1 μaμ dx

[a > 0,

 ln a + 2 ln 2 − 2

[Re μ > 0, n−1  k=1

1 2k − 1

[p > 0,

BI (289)(12)a

a = 0,

|arg a| < π] NT 68(7)



[|arg a| < π, 4.254

bi (319)(10)a

n = 1, 2, . . .]

q > 0]

BI (142)(5)

GW (324)(5)

[0 < p < q]

BI (135)(8)

[p < 1,

BI (140)(2)

p + q > 1]

4.257

4.3

5.

6. 4.255

Logarithms and powers

  p q 0 pπ  ∞ p−1 cos x ln x π2 q dx = − 2 pπ q 2 1+x q sin 0 q  1 q−1 2 π x ln x dx = − 2 2q 8q 0 1−x 

1

xp−1 ln x 1 dx = 2 β  q 1+x q

545

[p > 0,

q > 0]

[0 < p < q]

BI (135)(7)

[q > 0]

BI (108)(12)



  2 π sin 2p 1 − x2 xp−2 π   [p > 1] 1. ln x dx = − π 1 + x2p 2p 0 cos2 2p      2  1 1 + x2 xp−2 π π 2 [p > 1] 2. ln x dx = − sec 2p 1−x 2p 2p 0  ∞  pπ  1 − xp π2 tan2 [p < 1] 3. ln x dx = 2 1−x 4 2 0    1  μ  xμ−1 dx μ+m 1 1 μ m , ψ −ψ 4.256 ln  = 2B x n n n n n n 0 (1 − xn )n−m [Re μ > 0] 4.257   x  ∞ xν ln dx π γ ν ln βγ + π (β ν − γ ν ) cot νπ β = 1. sin νπ(γ − β) 0 (x + β)(x + γ) [|arg β| < π, 



1

∞

2.

ln 0

 3.

4.

∞

x q

x ln q

 

q 2p

xp + x2p

xp q 2p + x2p

 r

5. 0

BI (108)(13)

BI (108)(14) BI (140)(3)

LI (118)(12)

|arg γ|< π,

| Re ν| < 1] ET II 219(30)

dx =0 x

dx =0 + x2 0   2 2  ∞ 4π ln a + (ln a) dx x = ln x ln a (x − 1)(x − a) 6(a − 1) 0 

GW (324)(7)

q2

[q > 0]

BI (140)(4)a

[q > 0]

BI (140)(4)a

[a > 0]

(for a = 1 see 4.261 5) BI (141)(5)

∞

ln x ln

xp dx π 2 [(ap + 1) ln a − 2π (ap − 1) cot pπ] x = a (x − 1)(x − a) (a − 1) sin2 pπ 2

p < 1, a > 0

BI (141)(6)

546

Logarithmic Functions

4.261

4.26-4.27 Combinations involving powers of the logarithm and other powers 4.261 1.

7

2. 3.

4. 5. 6. 7. 8.11 9. 10. 11.7

  t π 2 − t2 dx [0 ≤ t ≤ π] (ln x) = 1 + 2x cos t + x2 6 sin t 0   1 (ln x)2 dx 1 ∞ (ln x)2 dx 10π 3 √ = = 2 2 0 x2 − x + 1 81 3 0 x −x+1   1 2 2 (ln x) dx 1 ∞ (ln x) dx 8π 3 √ = = 2 2 0 x2 + x + 1 81 3 0 x +x+1    ∞ π 2 + (ln a)2 ln a dx 2 = [a > 0] (ln x) (x − 1)(x + a) 3(1 + a) 0  ∞ 2 dx 2 (ln x) = π2 2 (1 − x) 3 0  1 dx π3 2 (ln x) = 2 1+x 16 0 √   1 2 2 1 ∞ 3 2 3 2 1+x 2 1+x π (ln x) dx = (ln x) dx = 1 + x4 2 0 1 + x4 64 0 √  1 8 3π 3 + 351 ζ(3) 2 1−x (ln x) dx = 1 − x6 486 0

 1 dx π π2 (ln 2)2 + (ln x)2 √ = 2 12 1 − x2 0    ∞ 2 3 μ−1 π 2 − sin μπ 2 x dx = [0 < Re μ < 1] (ln x) 1+x sin3 μπ 0    1 ∞ n n   (−1)n+k (−1)k 3 2 x dx n =2 ζ(3) + 2 (ln x) = (−1) 1+x (k + 1)3 2 k3 0 

1

2

k=n

12.7



1

0

13.11

 0

 14. 0

2

(ln x)

∞

1

2

(ln x)



n



(ln x)

GW (324)(16b)

BI (141)(1) BI (139)(4)

BI (109)(3)

BI (109)(5), BI (135)(13)

BI (118)(13)

ET I 315(10)

[n = 0, 1, . . .]

BI (109)(1)

[n = 0, 1, . . .]

BI (109)(2)

k=1

∞

n

k=n

k=1

  x2n dx 1 1 7 ζ(3) − 2 = 2 = 2 3 1−x (2k + 1) 4 (2k − 1)3 [n = 0, 1, . . .]

∞

GW (324)(16c)

k=1

  1 xn dx 1 =2 = 2 ζ(3) − 1−x (k + 1)3 k3 k=n

BI (113)(7)

2

x2

BI (109)(4)

" xp−1 dx π sin(1 − p)t ! 2 = π − t2 + 2π cot pπ [π cot pπ + t cot(1 − p)t] + 2x cos t + 1 sin t sin pπ [0 < t < π, 0 < p < 2, p = 1] GW (324)(17)

4.262

Powers and logarithms

⎧ ⎫  2n 2 2n ⎨ ⎬ 2 k k   π dx (−1) (−1) (2n − 1)!! 2 x π + + ln 2 (ln x) √ = + ⎭ 2 · (2n)!! ⎩ 12 k2 k 1 − x2 0 k=1 k=1 ⎫ ⎧ 2n+1 2  1 2n+1 ⎬ 2n+1  (−1)k  (−1)k dx (2n)!! ⎨ π 2 2 x − + ln 2 (ln x) √ = + − ⎭ (2n + 1)!! ⎩ 12 k2 k 1 − x2 0 

15.

16.

547

1

2n

k=1

GW (324)(60a)

k=1

GW (324)(60b)

17.7



1

0



1

0 1

19. 0

20.7



2

(ln x)

2

(ln x)

1 − xn+1 dx = 2(n + 1) ζ(3) − 2 (1 − x)2

n  k=1

1

1

21. 0

n−k+1 k3



1

1. 0



1

2. 0

 3. 4.

n

2

(ln x)

1 − x2n+2 2

(1 − x2 )

2

 n−k+1 7 (n + 1) ζ(3) − 2 4 (2k − 1)3 k=1

q−1

(ln x) xp−1 (1 − xr )

3

(ln x)

3

(ln x)

∞

[n = 0, 1, . . .] LI (111)(9)          2   p p 1  p  p ,q ψ −ψ +q + ψ −ψ +q dx = 3 B r r r r r r p

1

0

7 4 dx =− π 1+x 120 dx π4 =− 1−x 15

1

0

 0

3

r > 0]

GW (324)(8a)

BI (109)(9)

BI (109)(11)

 2 2 π 2 + (ln a)

[a > 0]

BI (141)(2)

[n = 1, 2, . . .]

BI (109)(10)

3

(ln x)

[n = 1, 2, . . .]

BI (109)(12)

[n = 1, 2, . . .]

BI (109)(14)

n−1

 xn dx 1 π4 =− +6 1−x 15 (k + 1)4 k=0

6.

7.

(ln x)

q > 0,

k=0

5. 

LI (111)(7)

n

dx =

dx = (x + a)(x − 1) 4(a + 1) 0   1 n−1 n  (−1)k 7π 4 3 x dx = (−1)n+1 −6 (ln x) 1+x 120 (k + 1)4 0 

ET I 315(11) LI (111)(8)

 1 + (−1)n xn+1 n−k+1 3 dx = (n + 1) ζ(3) − 2 (−1)k−1 2 (1 + x) 2 k3

[p > 0, 4.262

Re ν > 0]

k=1

0



$ 2 2 (ln x) xμ−1 (1 − x)ν−1 dx = B(μ, ν) [ψ(μ) − ψ(ν + μ)] + ψ  (μ) − ψ  (μ + ν) [Re μ > 0,

18. 

#

n−1

 x2n dx 1 π4 + 6 (ln x) = − 1 − x2 16 (2k + 1)4 3

k=0

1

(ln x)3

n

 n−k+1 1 − xn+1 (n + 1)π 4 +6 dx = − 2 (1 − x) 15 k4 k=1

BI (111)(11)

548

Logarithmic Functions



1

8. 0



3

(ln x)

n

 1 + (−1)n xn+1 n−k+1 7(n + 1)π 4 +6 dx = − (−1)k−1 2 (1 + x) 120 k4

1

0

3

(ln x)

1 − x2n+2 2

(1 − x2 )

n

dx = −

4.263 1.8

∞

0



(ln x)

4

dx = (x − 1)(x + a)

 n−k+1 (n + 1)π 4 +6 16 (2k − 1)4

1

0

(ln x)4

dx 5π 5 = 1 + x2 64

   ln a π 2 + (ln a)2 7π 2 + 3 (ln a)2 15(1 + a)

BI (109)(17)

   1 t π 2 − t2 7π 2 − 3t2 dx 4 (ln x) = 1 + 2x cos t + x2 30 sin t 0



1

1. 0



1

2. 0

 0

dx 31π 6 =− 1+x 252

BI (109)(20)

(ln x)5

dx 8π 6 =− 1−x 63

BI (109)(21)

5

 2   π 2 + (ln a)2 3π 2 + (ln a)2 dx 5 = (ln x) (x − 1)(x + a) 6(1 + a) [a > 0]

 4.266

1

0



1

1. 0



1

2. 0

4.267



1

1. 0

 2.

0

1

BI (113)(8)

(ln x)

∞

3.

4.265

BI (141)(3)



[|t| < π] 4.264

BI (111)(12)

k=1

[a > 0]

2. 3.

BI (111)(10)

k=1

9.



4.263

BI (141)(4)

dx 61π 7 = 1 + x2 256

BI (109)(25)

(ln x)

dx 127π 8 =− 1+x 240

BI (109)(28)

(ln x)7

dx 8π 8 =− 1−x 15

BI (109)(29)

6

(ln x)

7

1 − x dx 2 = ln 1 + x ln x π

BI (127)(3)

(1 − x)2 dx π = ln 1 + x2 ln x 4

BI (128)(2)

4.267

3.8

Powers and logarithms



1

0

 4. 5. 6.11 7. 8. 9. 10.

12. 13.

(1 − x)2 dx · mx 2 ln x +x 1 + 2x cos n !  n+k+1 "2  k+2   k    n−1  Γ Γ 2n Γ 2n kmπ 1 k  mπ  ln !  2n"2  (−1) sin =    k+1 n+k n sin n k=1 Γ 2n Γ 2n Γ n+k+2 2n !  n−k+1 "2  k+2   k    12 (n−1)   Γ Γ n Γ n 1 kmπ k n  mπ  ln !  = (−1) sin "   n−k+2   2 n sin n Γ k+1 Γ Γ n−k k=1 n n n [m < n]

1

14. 0

 15. 0

[m + n is even] BI (130)(3) BI (130)(16) BI (130)(17)

BI (123)(2)

GW (326)(10)

q > 0]

FI II 647

q > 0]

FI II 186

[0 < p < 1] 1

dx p+r = ln ln x r+q 0   1 p ∞  q  p+k x −x dx n+k−1 = ak ln n x ln x (1 − ax) q+k k 0 k=0  1 p+q+1 dx = ln (xp − 1) (xq − 1) ln x (p + 1)(q + 1) 0 

[m + n is odd]

1

1−x 1 ln 2 dx · =− · 2 ln x 2 0 1+x 1+x √  1 2 1−x x 2 2 dx · = ln · 2 ln x π 0 1+x 1+x  1 ∞   dx p = ln(1 + k) [p ≥ 1] (1 − x)p (−1)k ln x k 0 k=1   1 dx 1 − xp −p = ln Γ(p + 1) 1 − x ln x 0  1 p−1 x − xq−1 p dx = ln [p > 0, ln x q 0      1 p−1 Γ q2 Γ p+1 x − xq−1 dx 2     · = ln [p > 0, ln x 1+x Γ p2 Γ q+1 0 2   1 p−1  x − x−p 1 ∞ xp−1 − x−p pπ  dx = dx = ln tan 2 0 (1 + x) ln x 2 0 (1 + x) ln x 

11.

549

(xp − xq ) xr−1

xp − xq 1 + x2n+1 · 1+x x ln x

[r > 0,

p > 0,

q > 0]

p > 0,

q > 0,

a2 < 1

[p > −1,

q > −1,

LI (123)(5)



    p+1   q  + n + 1 Γ q+1 Γ Γ 2 2 +n Γ 2 2       dx = ln   p+1 q+1 p q +n+1 Γ +n Γ Γ Γ 2 2 2 2

xp − xq 1 − xr Γ(q + 1) Γ(p + r + 1) · dx = ln 1−x ln x Γ(p + 1) Γ(q + r + 1) [p > −1, q > −1,

q > 0]

p + r > −1,

BI (130)(15)

p + q > −1]

p

[p > 0, 1

FI II 816

q + r > −1]

GW (324)(19b)

BI (127)(7)

GW (324)(23)

550

Logarithmic Functions

  q Γ p+r Γ 2r xp−1 − xq−1 2r p dx = ln   r 0 (1 + x ) ln x Γ Γ q+r 2r 2r  1 2p−2q q−1 1−x x dx qπ = ln tan 2p 1+x ln x 4p 0  ∞ p−1  q−1 pπ qπ  x −x dx = ln tan cot r (1 + x ) ln x 2r 2r 0 

16.

17. 18.

1

4.267

[p > 0,

q > 0,

[0 < q < p] [0 < p < r,

0 < q < r] GW (324)(22), BI (143)(2)

pπ ⎞  ∞ p−1 sin x − xq−1 r ⎠ dx = ln ⎝ (1 − xr ) ln x sin qπ 0 r

20.

21.

23.

24. 25.

0 < q < r]

BI (143)(4)

1

4(2n+1)

22.

[0 < p < r,

  q Γ p+2 xp−1 − xq−1 1 − x2 2n Γ 2n dx = ln  q+2   p  [p > 0, q > 0] · 1 − x2n ln x Γ 2n Γ 2n 0         q+2 p+4n+2 q  1 p−1 Γ p+4n+4 4(2n+1) Γ 4(2n+1) Γ 4(2n+1) Γ 4(2n+1) x − xq−1 1 + x2        dx = ln  2(2n+1) ln x p+2 p 0 1+x Γ q+4n+2 Γ Γ q+4n+4 Γ 

GW (324)(21)

BI (128)(6)

⎛

19.

r > 0]

4(2n+1)

[p > 0,

4(2n+1)

q > 0]

BI (128)(7)

 x −x 1+x pπ (p + 2)π qπ (q + 2)π dx = ln tan tan cot cot 2(2n+1) ln x 4(2n + 1) 4(2n + 1) 4(2n + 1) 4(2n + 1) 0 1+x [0 < p < 4n, 0 < q < 4n] BI (143)(5) pπ  ∞ p−1 · sin (q+2)π sin 2n x − xq−1 1 − x2 2n dx = ln 2n (p + 2)π qπ 1−x ln x 0 · sin sin 2n 2n [0 < p < 2n, 0 < q < 2n] BI (143)(6)  1 r−1 dx (p + q + r)r x = ln [p > 0, q > 0, r > 0] (1 − xp ) (1 − xq ) BI (123)(8) ln x (p + r)(q + r) 0  1 Γ(p + r) Γ(q + r) xr−1 dx = ln (1 − xp ) (1 − xq ) (1 − x) ln x Γ(p + q + r) Γ(r) 0 [r > 0, r + p > 0, r + q > 0, r + p + q > 0] FI II 815a 

∞



1

26. 0

p−1

q−1



4(2n+1)

BI (128)(11)

2

(p + q + 1)(q + r + 1)(r + p + 1) dx = ln ln x (p + q + r + 1)(p + 1)(q + 1)(r + 1) r > −1, p + q > −1, p + r > −1, q + r > −1,

(1 − xp ) (1 − xq ) (1 − xr ) [p > −1,

q > −1,

p + q + r > −1] GW (324)(19c)

 27. 0

1

Γ(p + 1) Γ(q + 1) Γ(r + 1) Γ(p + q + r + 1) dx = ln (1 − x) ln x Γ(p + q + 1) Γ(p + r + 1) Γ(q + r + 1) r > −1, p + q > −1, p + r > −1, q + r > −1, p + q + r > −1]

(1 − xp ) (1 − xq ) (1 − xr ) [p > −1,

q > −1,

FI II 815

4.267

Powers and logarithms



1

28. 0

(1 − xp ) (1 − xq ) (1 − xr )

551

(p + q + s)(p + r + s)(q + r + s)s xs−1 dx = ln ln x (p + s)(q + s)(r + s)(p + q + r + s) [p > 0, q > 0, r > 0,

   q+s  1 Γ p+s Γ r xs−1 dx p q r  = ln  s   p+q+s (1 − x ) (1 − x ) r (1 − x ) ln x Γ r Γ 0 r

s > 0] BI (123)(10)

 29.

[p > 0,

q > 0,

r > 0,

s > 0] GW (324)(23a)



∞

30. 0

31.12



1

0



1

32. 0

33. 34. 35.

36.

(1 − xp ) (1 − xq )

s−1

x dx =2 p+q+2s (1 − x ) ln x

(1 − xp ) (1 − xq ) (1 − xr )



1

s−1

x dx (1 − xp ) (1 − xq ) p+q+2s (1 − x ) ln x 0 (p + s)π sπ cosec = 2 ln sin p + q + 2s p + q + 2s [s > 0, s + p > 0, s + p + q > 0] GW (324)(23b)a

Γ(p + s) Γ(q + s) Γ(r + s) Γ(p + q + r + s) xs−1 dx = ln (1 − x) ln x Γ(p + q + s) Γ(p + r + s) Γ(q + r + s) Γ(s) [p > 0, q > 0, r > 0, s > 0] ∗ BI (127)(11)

   q+s   r+s   p+q+r+s  Γ p+s Γ t Γ t Γ xs−1 dx t   q+r+s   p+r+s  t  s   = ln (1 − x ) (1 − x ) (1 − x ) (1 − xt ) ln x Γ Γ Γ t Γ p+q+s t t t [p > 0, q > 0, r > 0, s > 0, t > 0] ∗ GW (324)(23b) p





1



1

q

xp − xp+q −q 1−x

r



dx Γ(p + q + 1) = ln [p > −1, p + q > −1] BI (127)(19) ln x Γ(p + 1) 0   1 μ dx x −x − x(μ − 1) = ln Γ(μ) [Re μ > 0] WH, BI (127)(18) x−1 x ln x 0   1 (1 − xp ) (1 − xq ) dx 1−x− = − ln {B(p, q)} 1 − x x ln x 0 BI (130)(18) [p > 0, q > 0]   1  p−1 xpq−1 1 dx x 1 − + = q ln p − q q) 1 − x 1 − x x(1 − x) x (1 − x ln x 0

37. 0

 38. 0

1

xpq−1 xq−1 p − 1 p−1 p − 1 p−1 − x − x − 1 − x 1 − xp 1 − xp 2

(1 − xp ) (1 − xq ) − (1 − x)2 dx = ln B(p, q) x(1 − x) ln x



[p > 0]

BI (130)(20)

  1−p 1 dx = ln(2π) + pq − ln p ln x 2 2 [p > 0,

q > 0]

BI (130)(22)

[p > 0,

q > 0]

GW (324)(24)

∗ In 4.267.31 the restrictions can be somewhat weakened by writing, for example, s > 0, p + s > 0, q + s > 0, r + s > 0, p + q + s > 0, p + r + s > 0, q + r + s > 0, p + q + r + s > 0, in 4.267 31 and 32.

552

39.6

Logarithmic Functions



1

0

n

 dx = ln x

n

(xp − 1)

k=0



n n−k



4.268

(−1)n−k ln(pk + 1) [n > 0,

pn > −1] GW (324)(19d), BI (123)(12)a

40.6



1

0



1

41. 0

p n

(1 − x ) dx = 1 − x ln x (xp − 1)n xq−1

n 

(−1)k−1 ln Γ[(n − k)p + 1]

[n > 1,

pn > −1]

[n > 0,

q > 0,

BI (127)(12)

k=0

n  dx = ln[q + (n − k)p] (−1)k ln x k n

k=0

pn > −q] BI (123)(12)

42.6



1

0

(1 − xp )n xq−1

n

 dx = (−1)k−1 ln Γ[(n − k)p + q] (1 − x) ln x k=0 [n > 1,

q > 0,

pn > −q] BI (127)(13)

43.10



1

0

n

m

(xp − 1) (xq − 1)

  n m m xr−1 dx  j n = ln[r + (m − k)q + (n − j)p] (−1) (−1)k ln x j k j=0 [n ≥ 0,

4.268



1

1.

(xp − xq ) (1 − xr ) 2

(ln x)

0

m ≥ 0,



1

0



1

3. 0

2

(xp − xq )

r > 0,

pn + qm + r > 0]

BI (123)(16)

dx = (p + 1) ln(p + 1) − (q + 1) ln(q + 1) −(p + r + 1) ln(p + r + 1) + (q + r + 1) ln(q + r + 1) [p > −1,

2.

k=0

n + m > 0,

dx (ln x)

2

q > −1,

p + r > −1,

q + r > −1]

GW (324)(26)

= (2p + 1) ln(2p + 1) + (2q + 1) ln(2q + 1) − 2(p + q + 1) ln(p + q + 1) p > − 21 ,

(1 − xp ) (1 − xq ) (1 − xr )

q > − 12



GW (324)(26a)

dx (ln x)

2

= (p + q + 1) ln(p + q + 1) + (q + r + 1) ln(q + r + 1) + (p + r + 1) ln(p + r + 1) −(p + 1) ln(p + 1) − (q + 1) ln(q + 1) − (r + 1) ln(r + 1) − (p + q + r) ln(p + q + r) [p > −1,

q > −1,

r > −1,

p + q > −1,

p + r > −1,

q + r > −1,

p + q + r > 0] BI (124)(4)

 4. 0

1

  dx 1 k n (pk + q)2 ln(pk + q) (1 − xp )n xq−1 = (−1) 2 k (ln x)2 k=0  q q > 0, p > − n n 

BI (124)(14)

4.269

Powers and logarithms



1

5. 0

⎛

553

⎞     m m dx n p n q m r−1 j k ⎝ (−1) ⎠ (1 − x ) (1 − x ) x (−1) 2 = j k (ln x) j=0 k=0 n 

×[(m − k)q + (n − j)p + r] ln[(m − k)q + (n − j)p + r] [r > 0, 

1

6. 0



mq + r > 0,

(q − r)xp−1 + (r − p)xq−1 + (p − q)xr−1



np + r > 0,

mq + np + r > 0]

BI (124)(8)

dx (ln x)

2

= (q − r)p ln p + (r − p)q ln q + (p − q)r ln r 

1

7. 0

4.269



1.

3. 4. 5.



q > 0,

− ln x

√  ∞ π (−1)k dx  = 2 1+x 2 (2k + 1)3 k=0

1

[|t| < π] 1

6. 0



BI (124)(9)

BI (115)(33)

∞ √  dx (−1)k √ √ π =   2 2k + 1 − ln x 1 + x 0 k=0   1 1 π (− ln x)xp−1 dx = [p > 0] 2 p3 0  1 p−1  x π √ [p > 0] dx = p − ln x 0  1 n √  sin t − xn sin[(n + 1)t] + xn+1 sin nt sin kt dx √ √ · π = 2 1 − 2x cos t + x k − ln x 0 k=1



r > 0]

xq−1 xr−1 xp−1 + + (p − q)(p − r)(p − s) (q − p)(q − q)(r − s)

− r)(q − s) (r − p)(r dx p2 ln p q 2 ln q 1 xs−1 + = + (s − p)(s − q)(s − r) (ln x)2 2 (p − q)(p − r)(p − s) (q − p)(q − r)(q − s)

s2 ln s r2 ln r + + (r − p)(r − q)(r − s) (s − p)(s − q)(s − r) [p > 0, q > 0, r > 0, s > 0] BI (124)(16)

1√

0

2.11

[p > 0,

v

7. u

BI (133)(2)

GW (324)(1c)

BI (133)(1)

BI (133)(5)

 cos kt √ n−1 cos t − x − xn−1 cos nt + xn cos[(n − 1)t] dx √ √ · π = 2 1 − 2x cos t + x − ln x k k=1 dx  =π x v x · ln ln u x

[|t| < π]

BI (133)(6)

[uv > 0]

BI (145)(37)

554

4.271

Logarithmic Functions



1

1. 0



1

2. 0



1

3. 0



1

4. 0

 5.

6. 7. 8. 9. 10.

22n − 1 dx = · (2n)! ζ(2n + 1) 1+x 22n

2n−1

(ln x)

2n−1

(ln x)

p−1

(ln x)

BI (110)(1)

1 − 22n−1 2n dx = π |B2n | 1+x 2n

[n = 1, 2, . . .]

BI (110)(2)

1 dx = − 22n−2 π 2n |B2n | 1−x n

[n = 1, 2, . . .]

BI (110)(5), GW(324)(9a)

dx = ei(p−1)π Γ(p) ζ(p) 1−x

[p > 1]

GW (324)(9b)

∞  dx (−1)k n = (−1) n! 1 + x2 (2k + 1)n+1 0 k=0   1 dx dx 1 ∞ π 2n+1 2n 2n (ln x) = (ln x) = |E2n | 1 + x2 2 0 1 + x2 22n+2 0  ∞ (ln x)2n+1 dx = 0 [|b| < 2] 2 0 1 + bx + x  1 dx 22n+1 − 1 (ln x)2n = · (2n)! ζ(2n + 1) [n = 1, 2, . . .] 1 − x2 22n+1 0  ∞ dx 2n (ln x) =0 1 − x2 0   1 dx dx 1 ∞ 1 − 22n 2n 2n−1 2n−1 π |B2n | (ln x) = (ln x) = 1 − x2 2 0 1 − x2 4n 0



1

1

11. 0



1

12. 0

 13.

0

14.6

2n

(ln x)

4.271

 0

1

n

(ln x)

2n−1

(ln x)

2n

(ln x)

1 + x2 2

(1 − x2 )

2n+1

(ln x)

1 x dx = − π 2n |B2n | 1 − x2 4n dx =

22n − 1 2n π |B2n | 2

16.3

GW (324)(10)a

BI (135)(2) BI (110)(12) BI (312)(7)a

[n = 1, 2, . . .]

BI (290)(17)a, BI(312)(6)a

[n = 1, 2, . . .]

BI (290)(19)a

[n = 1, 2, . . .]

BI (296)(17)a

∞

 cos 2akπ (cos 2aπ − x) dx = −(2n + 1)! 2 1 − 2x cos 2aπ + x k 2n+2 k=1

∞

[a is not an integer]

x dx d n ν−2 sin(ν − 1)t (ln x) 2 = −π cosec t n a a + 2ax cos t + x2 dν sin νπ [a > 0, 0 < Re ν < 2, ν−1

  1 p−1 p 1 n x (n) (ln x) dx = − n+1 ψ q 1 − x q q 0    1 xp−1 p 1 (ln x)n dx = n+1 β (n) q 1 + x q q 0

LI (113)(10)

n

0 < |t| < π] ET I 315(12)

 15.

BI (110)(11)

[p > 0,

q > 0]

GW (324)(9)

[p > 0,

q > 0]

GW (324)(10)

4.272

4.272

Powers and logarithms



1

1. 0



1

0



∞

 [− ln x]q−1 dx sin kt = cosec t Γ(q) (−1)k−1 q 2 1 + 2x cos t + x k

1

0

(− ln x)

q−1

μ

[− ln x]



 ∞ 1  t (1 + x) dx k−1 cos k − 2 t = sec · Γ(q) (−1) 1 + 2x cos t + x2 2 kq k=1

|t| < π, q < 12

1

4. 0



∞

1

0 1

7. 0

8.



1

0

 9. 10.

k=1

Re μ > 0,

Re ν > 0,

−π < t < π]

BI (140)(14)a

∞

 pk−1 cos kλ cos λ − px xq−1 dx = Γ(r) 2 2 1 + p x − 2px cos λ (q + k − 1)r [r > 0,

6.

12

(− ln x)r−1

∞

1



LI (130)(5)

k=1

5. 

LI (130)(1)

Γ(μ + 1)  xν−1 dx ak sin kt = 1 − 2ax cos t + x2 a2 a sin t (ν + k − 1)μ+1 [a > 0,



[|t| < π, q < 1]

k=1

2.

3.9

555

(ln x)

p

dx = Γ(1 + p) x2

(− ln x)μ−1 xν−1 dx = (− ln x)

n− 12

(− ln x)n−1

ν−1

x

q > 0]

[p > −1]

1 Γ(μ) νμ

[Re μ > 0,

(2n − 1)!! dx = (2ν)n



π ν

BI (113)(11) BI (149)(1)

Re ν > 0]

BI (107)(3)

[Re ν > 0]

BI (107)(2)

[Re ν > 0]

BI (110)(4)

∞

 (−1)k xν−1 dx = (n − 1)! 1+x (ν + k)n k=0

1

xν−1 dx = (n − 1)! ζ(n, ν) [Re ν > 0] BI (110)(7) 1−x 0    1 n  (−1)k n(n − 1) . . . (n − k + 1) nx μ−1 xa−1 dx = Γ(μ) (− ln x) (x − 1)n a + x−1 (a + n − k)μ−1 k! 0 (− ln x)n−1

k=0



[Re μ > 0] 1

11. 0



1

12. 0

 13. 0

(− ln x) (− ln x)

n−1

μ−1

1 − xm dx = (n − 1)! 1−x

m  k=1

1 kn

LI (110)(10) LI (110)(9)

 ν  xν−1 dx 1 1 = Γ(μ) = μ Γ(μ) ζ μ, 2 μ 1−x (ν + 2k) 2 2 ∞

k=0

1

xq − x−q (− ln x)p dx = Γ(p + 1) 1 − x2

∞   k=1

[Re μ > 0,

Re ν > 0]  1 1 − (2k + q − 1)p+1 (2k − q − 1)p+1

p > −1, q 2 < 1

BI (110)(13)

LI (326)(12)a

556

Logarithmic Functions



1

14. 0

(− ln x)

r−1

∞

 xp−1 dx s = Γ(r) q (1 + x ) k=0



4.273

 −s 1 k (p + kq)r [p > 0,

q > 0,

r > 0,

0 < s < r + 2] GW (324)(11)



1

15. 0

1

0

1 (p + kq)n+1

(− ln x)n (1 − xq )m xp−1 dx = n!

m   k=0

m k

1

17. 0



1

0



1

19. 0



(− ln x)

p−1

1

(− ln x)2− n

(− ln x)

1 xq−1 dx = p Γ(p) 1 − axq aq

2n−1

1 xp − x−p q−1 x dx = 2n 1 − xq p

ln

4.274 

1



1. 0



 (− ln x)q−1 − xp−1 (1 − x)q−1 dx =

1

2. 0

 x−

q > 0]

BI (107)(7)

ak [p > 0, q > 0, kp k=1     p−1  1 1  −3+ 1 q−1 n − q −3+ n p x dx = Γ 3 − −x n



v

BI (107)(6)

∞ 

∞  k=n



2pπ q

k

 v p+q−1 x p−1  v q−1 dx = B(p, q) ln ln u x x u √ u1/e √ q qπ x dx  = √ q e x − (1 + ln x) 0

4.273

q > 0]

(−1)k (p + kq)n+1 [p > 0,



4.275

k

k=0

16.

18.

m

m

[p > 0,



12

n

(− ln x) (1 + xq ) xp−1 dx = n!

m  

1 1 − ln x

q

[q > p > 0]

[p > 0,

q > 0,

[q > 0]

LI (110)(8)

BI (133)(4)

|B2k | 2k · (2k − 2n)!  q p< 2

LI (110)(16)

uv > 0]

BI (145)(36) BI (145)(4)

Γ(q) [Γ(p + q) − Γ(p)] Γ(p + q) [p > 0,

dx = − ψ(q) x ln x

a < 1]

q > 0]

[q > 0]

BI (107)(8) BI (126)(5)

4.28 Combinations of rational functions of ln x and powers 4.281



1



1. 0

 2.

1

1 1 + dx = C ln x 1 − x

∞

x2

dx 1 = li(p) (ln p − ln x) p

BI (127)(15) LA 281(30)

Rational functions of ln x and powers

4.282

 3. 4. 5. 6. 7.

1

xp−1 dx = ±e∓pq Ei (±pq) [p > 0, q > 0] 0 q ± ln x

 1 xμ−1 1 + dx = − ψ(μ) [Re μ > 0] ln x 1 − x 0

 1 p−1 xq−1 x + dx = ln p − ψ(q) [p > 0, q > 0] ln x 1−x 0

 1 ln 2 dx 1 1 = + 2 1−x 2x ln x ln x 2 0

 1 1 ln 2π 1 (1 − x) (1 + q ln x) + x ln x q−1 dx = − q − ln Γ(q) + q− + x 2 2 (1 − x) ln x 2 2 0 [q > 0]

4.282  1.12 2. 3. 4. 5. 6. 7. 8. 9.10 10. 11.

12.

557

LI (144)(11,12) WH

BI (127)(17)

LI (130)(19)

BI (128)(15)

1

ln x dx 1 1 · = − C 2 + ln x 1 − x 4π 4 2 0    1  1 2a + π π dx 1 β a > − · = 2 1 + x2 2 2a 4π 2 0 a + (ln x)  1 1 dx 4−π = 2 2 2 4π 0 π + (ln x) 1 + x    1 ln x dx 1 1 − ln 2 · = 2 1 − x2 2 2 2 0 π + (ln x)  1  a  ln x π x dx 1π + ln + ψ [a > 0] · = 2 1 − x2 2 2 2a a π 0 a + (ln x)    1 ln x x dx 1 1 − C · = 2 1 − x2 2 2 2 0 π + (ln x)  1 1 dx ln 2 2 · 1 + x2 = 4π 2 0 π + 4 (ln x)  1 ln x dx 2−π 2 · 1 − x2 = 2 16 0 π + 4 (ln x)  1 √  dx 1  1 √ π + 2 ln · = 2 − 1 2 1 + x2 8π 2 0 π 2 + 16 (ln x)  1 √  1 1 dx π ln x √ √ + + ln · = − 2 − 1 2 1 − x2 32 2 16 16 2 0 π 2 + 16 (ln x)  2k−2  1 ∞ ln x dx π2  2π = − |B |  2 2k 4 1 − x a a 2 0 k=1 a2 + (ln x)  1 ∞  π 2k−2 ln x x dx π2  = − |B |  2 2k 1 − x2 4a4 a 2 0 k=1 a2 + (ln x)

BI (129)(1) BI (129)(9)

BI (129)(6)

BI (129)(10)

BI (129)(14)

BI (129)(13)

BI (129)(7)

BI (129)(11)

BI (129)(8)

BI (129)(12)

BI (129)(4)

BI (129)(16)

558

Logarithmic Functions



1

13. 0

4.283 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

∞ xp − x−p sin kpπ dx 2π  = (−1)k−1 x2 − 1 q 2 + (ln x)2 q 2q + kπ

4.283

2

p 0] 2 − ln x dx = q ln q − q x (ln x) 0

 1 a dx 1 = ln + C [a > 0, q > 0] x+ a ln x − 1 x ln x q 0

p−1    1 1+x 1 ln 2π x 1 + dx = − ln Γ(p) + p − ln p − p + ln x 2(1 − x) ln x 2 2 0





1

1

11. 0



1

12. 0

 13. 0

BI (132)(13)a

k=1

1





1 + 1−x

p−1−



1 1 − 2 ln x



xp−1

dx = ln x



[p > 0]  1 ln 2π − p ln p + p − 2 2 [p > 0]

 −

1 2

(ln x)

+

BI (132)(17)a BI (127)(20) BI (127)(23)

GW (326)(8a)

BI (128)(14) BI (127)(22) GW (326)(11a)

BI (126)(2)

BI (126)(8)

GW (326)(9)

BI (127)(25)

3 (p − 2)xp − (p − 1)xp−1 dx = − ψ(p) + p − (1 − x)2 2

[p > 0]      

 2p−1  dx 1 1 1 1 3 x + 1− x = − p (ln p − 1) p− −1 2 2 ln x ln x 2

GW (326)(8)

[p > 0] BI (132)(23)a 



1 pxpq−1 1 ln 2π 1 xp−1 − xr−1 rxrq−1 dx + = (p − r) − q − ln Γ(q) + q− − 2 ln x 1 − xp 1 − xr ln x 2 2 0

 14.

[q > 0]

BI (132)(13)

4.291

4.284

Logarithmic functions and powers



1



1.

x (ln x)

0



xq − 1

1



2.

q x (ln x)

2

−

q2 3 q2 dx = ln q − q 2 2 ln x 2 4 [q > 0]

xq − 1 x (ln x)

0

3

−

559

4

−

q x (ln x)

3

−

q2 2x (ln x)

2

−

BI (126)(3)

q3 11 q3 dx = ln q − q 3 6 ln x 6 36 [q > 0]

 4.285

1

0

BI (126)(4)

n−1  xp−1 dx pn−1 −pq 1 e = Ei(pq) − (n − k − 1)!(pq)k−1 n n−1 (n − 1)! (n − 1)!q (q + ln x) k=1



xa (ln x)n dx

[p > 0,

q < 0]

BI (125)(21)

, we should make the substitution x = et or x = e−t and m l [b ± (ln x) ] then seek the resulting integrals in 3.351–3.356. In integrals of the form

4.29–4.32 Combinations of logarithmic functions of more complicated arguments and powers 4.291



1

1. 0

 2.

0

 3. 4.

5. 6. 7.7 8. 9. 10.

1

ln(1 + x) π2 dx = x 12

FI II 483

ln(1 − x) π2 dx = − x 6

FI II 714

1/2

ln(1 − x) 1 π2 2 dx = (ln 2) − x 2 12 0  1   1 x dx π2 = (ln 2)2 − ln 1 − 2 x 2 12 0 1 + x  1 ln 2 2 dx = 1 (ln 2)2 − π 1−x 2 12 0  1 ln(1 + x) 1 2 dx = (ln 2) 1+x 2 0  ∞  a  ln(1 + ax) ln u du π  2 ln 1 + a − dx = 2 2 1 + x 4 0 0 1+u  1 ln(1 + x) π dx = ln 2 2 1+x 8 0  ∞ ln(1 + x) π dx = ln 2 + G 2 1 + x 4 0  1 ln(1 − x) π dx = ln 2 − G 2 1+x 8 0

BI (145)(2)

BI (114)(18)

BI (115)(1)

BI (114)(14)a

[a > 0]

GI II (2209)

FI II 157 BI (136)(1)

BI (114)(17)

560

Logarithmic Functions



∞

11. 1



1

12. 0



∞

0 1

14. 0

ln(x − 1) π dx = ln 2 1 + x2 8

BI (144)(4)

ln(1 + x) π2 1 2 dx = − (ln 2) x(1 + x) 12 2

13. 

4.291

BI (144)(4)

ln(1 + x) π2 dx = x(1 + x) 6

ln(1 + x) a+b 2 ln 2 1 ln + 2 dx = (ax + b)2 a(a − b) b b − a2 1 = 2 (1 − ln 2) 2a

BI (141)(9)a

[a = b,

ab > 0]

[a = b] LI (114)(5)a

15. 16. 17. 18. 19. 20.

a  ∞ ln ln(1 + x) b [ab > 0] dx = 2 (ax + b) a(a − b) 0  1 √ dx 1 ln(a + x) = √ arccot a ln[(1 + a)a] [a > 0] 2 a+x 2 a 0  ∞ dx a ln a − b ln b [a > 0, b > 0, ln(a + x) = 2 (b + x) b(a − b) 0  a   ln(1 + ax) 1 dx = arctan a ln 1 + a2 2 1+x 2 0  1 √ 1 ln(1 + ax) dx = √ arctan a ln(1 + a) [a > 0] 2 1 + ax 2 a 0

 1 ln(ax + b) 1 1 (a + b) ln(a + b) − b ln b − a ln 2 dx = 2 a−b 2 0 (1 + x) 

21. 22.

[a > 0, ∞

ln(ax + b) 1 [a ln a − b ln b] [a > 0, dx = 2 (1 + x) a−b 0    ∞ aπ a2 x dx 1 ln b + + ln(a + x) = ln a 2 2 (a2 + b2 ) 2b b2 (b2 + x2 ) 0 

[a > 0, 1

23. 0

 24.

0

1

b > 0,

BI (139)(5)

BI (114)(20)

a = b]

LI (139)(6) GI II (2195)

BI (114)(21)

a = b]

BI (114)(22)

b > 0]

BI (139)(8)

b > 0]

BI (139)(9)

ln(1 + x)

23 1 + x2 1 dx = − ln 2 + (1 + x)4 3 72

ln(1 + x)

π    1 + x2 dx 1 ln 1 + a2 − 2 arctan a · ln a · = 2 2 2 2 2 a +x 1+a x 2a (1 + a ) 2

LI (114)(12)

[a > 0]

LI (114)(11)

4.292

Logarithmic functions and powers



1

25. 0



ln(1 + x)

∞

26. 0

 27. 0





 dx a+b 1 1 1 1 − x2 1 ln(a + b) − ln b − ln a = (ax + b)2 (bx + a)2 a2 − b 2  a−b ab a b 4 ln 2 + 2 b − a2

a > 0, b > 0, a2 = b2 LI (114)(13)

ln(1 + x)

b 1 − x2 dx 1 ln · = 2 2 2 2 (ax + b) (bx + a) ab (a − b ) a [a > 0,

1

ln(1 + ax)

∞

28. 0

2

1−x (1 +

x2 )2

dx =

a a 1 (1 + a) 1 π ln(1 + a) − · ln 2 − · 2 2 2 1+a 2 1+a 4 1 + a2

  bπ b b2 − x2 1 − a ln ln(a + x) dx = 2 a2 + b 2 a 2 (b2 + x2 )

[a > −1]

[a > 0,   ∞ a2 aπ x dx 1 2 + 2 ln a ln b − ln (a − x) 2 = a2 + b 2 2b b (b2 + x2 ) 0

b > 0]

BI (139)(12)

[a > 0,

b > 0]

BI (139)(10)

∞

ln2 (a − x)

b2 − x2 (b2 + x2 )2

dx =

2 a2 + b 2

 a ln

a bπ − b 2







1

1. 0



1

2. 0

3.

BI (114)(23)

BI (139)(11)

0

12

LI (139)(14) 2

b > 0]

29.

4.292

b > 0]

2

[a > 0,



30.

561

ln (1 ± x) π √ dx = − ln 2 ± 2G 2 2 1−x

GW (325)(20)

x ln (1 ± x) π √ dx = −1 ± 2 1 − x2

GW (325)(22c)

√ ln |1 + bx| 1 + 1 − a2 b 2 √ dx = π ln a2 − x2 −a 2  a|b| = π ln 2



a



1

4. 0



1 |b| ≤ |a| 1 |b| ≥ |a|

√ x ln(1 + ax) 1 − a2 π 1 − 1 − a2 √ + arcsin a dx = −1 + · 2 a 1 − x2 √ a    a2 − 1 π + ln a + a2 − 1 = −1 + 2a a √

BI (145)(16, 17)a, GW (325)(21e)

[|a| ≤ 1] [a ≥ 1] GW (325)(22)

562

5.12

Logarithmic Functions



1

0

4.293

ln |1 + ax| π2 1 1 2 √ − (arccos a) dx = arcsin a (π − arcsin a) = 2 2 8 2 x 1−x    1 π2 + ln2 a + a2 − 1 = 8 2   3 2 1 2 = − π + ln −a + a2 − 1 8 2

[|a| ≤ 1] [a ≥ 1] [a ≤ −1] BI (120)(4), GW (325)(21a)

4.293



1

1. 0

2.6



1



xμ−1 ln(1 + x) dx =

∞

∞

3. 0

 4.

5. 6.11 7.

1 [ln 2 − β(μ + 1)] μ

[Re μ > −1]

BI (106)(4)a ET I 315(17)

xμ−1 ln(1 + x) dx =

−1 [β(−μ) + ln 2] μ

[Re μ < 0]

xμ−1 ln(1 + x) dx =

π μ sin μπ

[−1 < Re μ < 0]

GW (325)(3)a

2n

1  (−1)k−1 2n k 0 k=1   1 2n+1  (−1)k 1 2n ln 4 + x ln(1 + x) dx = 2n + 1 k 0 k=1   1 n 1 (−1)n · 4 π  (−1)k 2 ln 2 + − xn− 2 ln(1 + x) dx = 2n + 1 2n + 1 4 2k + 1 0 k=0  ∞ π [−1 < Re μ < 0] xμ−1 ln |1 − x| dx = cot(μπ) μ 0 1

x2n−1 ln(1 + x) dx =

GW (325)(2b)

GW (325)(2c)

GW (325)(2f)

BI (134)(4), ET I 315(18)



1

8. 0

9.7



xμ−1 ln(1 − x) dx = −

∞

1



∞

10. 0

11.11



1

12. 0

 13.

0

[Re μ > −1]

ET I 316(19)

[Re μ < 0]

ET I 316(20)

1 [π cot(μπ) + ψ(μ + 1) + C] μ

xμ−1 ln(1 + γx) dx =

π μγ μ sin μπ

[−1 < Re μ < 0,

|arg γ| < π] BI (134)(3)

∞

0



xμ−1 ln(x − 1) dx =

1 1 [ψ(μ + 1) − ψ(1)] = − [ψ(μ + 1) + C] μ μ

1

μ−1

x

ln(1 + x) π dx = − [C + ψ(1 − μ)] 1+x sin μπ

[−1 < Re μ < 1]

ln(1 + x) 2μ − 1 ln 2 + dx = − (1 + x)μ+1 2μ μ 2 μ μ2 xμ−1 ln(1 − x) dx = B(μ, ν) [ψ(ν) − ψ(μ + ν)] (1 − x)1−ν

ET I 316(21)

BI (114)(6)

[Re μ > 0,

Re ν > 0]

ET I 316(122)

4.295

Logarithmic functions and powers



∞

14. 0

563

xμ−1 ln(γ + x) dx = γ μ−ν B(μ, ν − μ) [ψ(ν) − ψ(ν − μ) + ln γ] (γ + x)ν [0 < Re μ < Re ν]

4.294



1

1. 0



1

2. 0



1

0 1

0 1

0 1

0 1

0

n−1

ln(1 + x)

ln(1 − x)

ln(1 − x)

∞

1

0 1

0 1

1

12. 0

1.12

0

BI (114)(16)

2π cot pπ p+1

n



1

1 ln 1 − x2 2n

(ln x) ln



j n  11 1 − xn dx = − 1−x j k j=1

[ln(1 − x)] (1 − x)r dx = (−1)n

0

4.295

BI (114)(10)

k=1

BI (114)(15)

[−2 < p < −1]

[ln(1 + x)]n (1 + x)r dx = (−1)n−1

0



BI (114)(9)

k=1

j n  (−1)j  1 1 − (−1)n xn dx = 1−x j k j=1

ln2 (1 − x)xp dx =

11.

13.6

j n−1 2n  1  (−1)j  (−1)k−1 1 − x2n dx = 2 ln 2 · + 1−x 2k + 1 i=1 j k

BI (134)(13)a

n

 (−1)k n! (ln 2)n−k n! r+1 + 2 (r + 1)n+1 (n − k)!(r + 1)k+1

LI (106)(34)a

k=0

10.



BI (114)(8)

k=1

k=1

9.



j

j n 2n+1   (−1)j  1 (−1)k−1 1 − x2n+1 dx = 2 ln 2 + ln(1 + x) 1−x 2k + 1 j k j=1

0



2n

 1  1  (−1)k−1 1 − x2n dx = 2 ln 2 · − ln(1 + x) 1+x 2k + 1 j=1 j k

8. 

BI (114)(7)

k=1

k=1

7. 

j n 2n+1   1 1 (−1)k−1 1 + x2n+1 dx = 2 ln 2 − 1+x 2k + 1 j k j=1

k=0

6. 

BI (114)(2)

k=0

5. 

[0 < p < 1]

k=0

4. 

ln(1 + x)

π (p − 1)xp−1 − px−p dx = 2 ln 2 − x sin pπ

k=0

3. 

ln(1 + x)

ET I 316(23)

∞

1 x

n

x2q−1 dx =

n! (r + 1)n+1

n! ζ(n + 1, q + 1) 2

[r > −1]

BI (106)(35)a

[−1 < q < 0]

BI (311)(15)a

 dx  π 2n+2 =− |B2n+2 | ln 1 − x2 x 2(n + 1)(2n + 1)

m

∞    ln 1 − x2 dx = −

Γ(m + 1) n(2n + 1)m+1 n=1

√ √  ln(ax2 + b) dx = π ln a + b 1 + x2

BI (309)(5)a

[m + 1 > 0,

[Re a > 0,

n + 1 > 0]

Re b > 0]

ET II 218(27)

564

Logarithmic Functions

 2. 3. 4. 5. 6. 7.

8. 0

10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

GW (325)(2g) GW (325)(4c)

[a > 0]

BI (319)(6)a

BI (114)(24) BI (114)(5)

[a > 0,

b > 0,

c > 0,

g > 0] BI (136)(11-14)a

∞

 ln a2 + b2 x2 

bc dx π = − arctan c2 − g 2 x2 cg ag

   ∞ ln 1 + p2 x2 − ln 1 + q 2 x2 dx = π(p − q) x2 0  1 1 + a2 x2 dx 2 ln = − (arctan a) 1 + a2 1 − x2 0  1  dx  π2 =− ln 1 − x2 x 12 0  ∞   dx ln2 1 − x2 2 = 0 x 0  1  dx  π ln 1 − x2 = ln 2 − G 2 1 + x 4 0  ∞   dx π ln x2 − 1 = ln 2 + G 2 1+x 4 1  ∞    dx π  ln2 a2 − x2 2 = ln a2 + b2 2 b +x b 0  ∞ 2 2   b −x 2bπ ln2 a2 − x2 2 dx = − a2 + b2 2 2 (b + x ) 0

 1   1 π2 1 dx 2 2 = − (ln 2) ln 1 + x x (1 + x2 ) 2 12 2 0  ∞   π2 dx = ln 1 + x2 2 x (1 + x ) 12 0  1   dx ln cos2 t + x2 sin2 t = −t2 1 − x2 0 

9.

1

 dx  π ln 1 + x2 = − ln 2 2 x 2 0  ∞   dx ln 1 + x2 2 = π x 0  ∞    dx 2a  π + ln a ln 1 + x2 = (a + x)2 1 + a2 2a 0  1  dx  π ln 1 + x2 = ln 2 − G 2 1 + x 2 0  ∞  dx  π ln 1 + x2 = ln 2 + G 2 1 + x 2 1  ∞   ag + bc dx π ln ln a2 + b2 x2 2 = 2 2 c +g x cg g 0 

4.295

[a > 0,

b > 0,

c > 0,

g > 0] BI (136)(15)a



[p > 0,

q > 0]

FI II 645

BI (115)(2)

BI (142)(9)a

GW (325)(17) BI (144)(6)

[b > 0]

BI (136)(16)

[b > 0]

BI (136)(20)

BI (114)(25) BI (141)(9)

BI (114)(27)a

4.295

Logarithmic functions and powers



∞

20. 0



2

2 2

ln a + b x



b2 2 ln b dx + = (c + gx)2 cg a2 g 2 + b 2 c2



565

 c c a2 g a a π + 2 ln + 2 2 ln b g g b c b [a > 0, b > 0, c > 0, g > 0] BI (139)(16)a



1

21. 0

22.

11



  ln a2 + b2 x2

dx (c + gx)2

b2 a cb2 − ga2 a2 + b2 2a 2 c c+g ln a + 2 2 arccot + 2 ln = − 2 ln c(c + g) a g + b 2 c2 b b b (c + g) a2 g c [a > 0, b > 0, c > 0, g > 0] BI (114)(28)a

∞

0



∞

23. 0



∞

24. 0



∞

25. 0



∞

26. 0



1

27. 0



1

28. 0

30.12

  ln 1 + a2 x2 dx π = 2 2 b2 + c2 x2 d2 + g 2 x 2 b g − c2 d2 a > 0, b > 0,



  ln 1 + a2 x2 x2 dx π = 2 2 2 2 2 2 2 2 b +c x d + g x b g − c2 d2 a > 0, b > 0,





2

2 2

ln a + b x



π 2 = 2c3 g 2 2 2 (c + g x ) dx

1

p > 0]

FI II 745a, BI (318)(1)a, BI (318)(4)a

    ad c ab g ln 1 + − ln 1 + d g b c

2 2 c > 0, d > 0, g > 0, b g = c2 d2

BI (141)(10)

    ab d ad b ln 1 + − ln 1 + c c g g

c > 0, d > 0, g > 0, b2 g 2 = c2 d2

BI (141)(11)

  ag + bc bc ln − g ag + bc [a > 0, b > 0,

    2 ag + bc π bc x dx 2 2 ln ln a + b x + 2 = 2cg 3 g ag + bc (c2 + g 2 x2 ) [a > 0, b > 0,

 ln 1 + ax2

c > 0,

g > 0] GW (325)(18a)

2



√ √   1+ 1+a 11− 1+a π 2 √ ln + 1 − x dx = 2 2 21+ 1+a

 ln 1 + a − ax2



[a > 0] √ √   1+ 1+a 11− 1+a π 2 √ ln − 1 − x dx = 2 2 21+ 1+a

√   dx 1 + 1 − a2 2 2 ln 1 − a x √ = π ln 2 1 − x2 0  1    dx π 2 ln 1 − a2 x2 √ = − arccos |a| − 2 x 1 − x2 0

 29.

    ∞  2 ln 1 + p2 x2 ln p + x2 q + pr π ln dx = dx = 2 2 2 2 2 2 r +q x q +r x qr r 0 [qr > 0,

c > 0,

g > 0] GW (325)(18b)

BI (117)(6)

[a > 0]

BI (117)(7)

2

a −2,

b > 0,

a = b]

BI (119)(27)

BI (119)(3)

BI (119)(7)

BI (145)(15)

q > −2]

[a ≥ −1]

BI (120)(14)

BI (106)(15) GW (325)(21b)

[Re μ > −2]

BI (106)(12)

[−2 < Re μ < 0] BI (311)(4)a, ET I 315(15)

 μπ   1 − μ   μπ   2 − μ  x  dx π = ln 2 + sin β − cos β ln 1 + x2 1+x sin μπ 2 2 2 2 μ−1



ET I 316(25)

4.296  1. 2. 3. 4.

1

 dx  π2 t2 = − ln 1 + 2x cos t + x2 x 6 2 0  ∞  dx    ln a2 − 2ax cos t + x2 = π ln 1 + 2a|sin t| + a2 2 1 + x −∞  ∞   2π cos μt [|t| < π, −1 < Re μ < 0] ln 1 + 2x cos t + x2 xμ−1 dx = μ sin μπ 0  2    ∞  2 a − b2 cos t x dx x + 2ax cos t + a2 1 2 π ln = − πt + π arctan x2 − 2ax cos t + a2 x2 + b2 2 (a2 + b2 ) sin t + 2ab 0 [a > 0,

b > 0,

0 < t < π]

BI (114)(34) BI (145)(28) ET I 316(27)

4.298

4.297

Logarithmic functions and powers



1

1. 0





dx a+b ax + b 1 (a + b) ln − a ln a − b ln b ln = bx + a (1 + x)2 a−b 2 [a > 0,

∞

2.

ln 0



1

3.

ln 0

 4.

dx ax + b =0 bx + a (1 + x)2

b > 0]

[ab > 0]

1 − x dx π = ln 2 2 x 1+x 8

BI (115)(16) BI (139)(23)

BI (115)(5)

1

1 + x dx =G BI (115)(17) 1 − x 1 + x2 0 2  ∞  dx π2 1+x = ln BI (141)(13) 1−x x (1 + x2 ) 2 0  v 1  v 2 v + x dx = ln ln [uv > 0] BI (145)(33) u+x x 2 u u  ∞ b ln(1 + ax) − a ln(1 + bx) b [a > 0, b > 0] dx = ab ln FI II 647 2 x a 0  1 dx 1 + ax √ ln = π arcsin a [|a| ≤ 1] GW (325)(21c), BI (122)(2) 1 − ax x 1 − x2 0   v  u dx 1 + ax π   ln = F arcsin av, 1 − ax v v (x2 − u2 ) (v 2 − x2 ) u ln

5.11 6. 7. 8. 9.

10.8 11.∗

4.298 1.

567

12

3.12

5.12

[|av| < 1] BI (145)(35) ) ) 2 )a + y) π )  dy [0 < a ≤ 1] PV ln )) = ) 2 a−y y 1−y 2 0   b   x2 dx c+x π 2  2  c − (c − a2 ) (c2 − b2 ) + πb [F (φ, k) − E(φ, k)] ln = c−x c (x2 − a2 ) (b2 − x2 ) a

b a φ = arcsin , k = , 0 < a < b < c KM(4.51) c b 

1

 

π 1 π 1 + x2 x2p−1 dx = ln 2 + cos(pπ)β(p) + sin(pπ)β −p − ln x 1+x sin(2pπ) 2 sin(2pπ) 0

1 0 0]

BI (138)(6, 7, 9, 10)a

18.

19.



+ n−1  (−1)k 1 1 + x 2n 1 nπ x dx = (−1) + ln 2 − +2 ln x 2n + 1 2 2n + 1 2n − 2k − 1 0 k=0 * +  1 n−1  (−1)k 1 + x2 2n−1 1 1 n+1 n+1 x (−1) + (−1) ln dx = ln 2 + ln 2 − x 2n 2n k 0 k=1  1 1 + x2 dx π ln = ln 2 2 x 1 + x 2 0  ∞ 2 dx 1+x ln = π ln 2 x 1 + x2 0  ∞ 1 + x2 dx ln =0 x 1 − x2 0  1 1 − x2 dx π ln = ln 2 2 x 1 + x 4 0  ∞ 2 dx 1+x 3π ln 2 ln = 2 x+1 1+x 8 1  1 1 + x2 dx 3π ln = ln 2 − G 2 x + 1 1 + x 8 0  ∞ 1 + x2 dx 3π ln 2 + G ln = 2 x − 1 1 + x 8 1  1 1 + x2 dx 3π ln 2 ln = 2 1 − x 1 + x 8 0  ∞ 1 + x2 x dx π2 ln = x2 1 + x2 12 0  ∞ 2 2 2 ag + bc a +b x π dx ln [a > 0, b > 0, c > 0, ln = 2 2 2 2 x c +g x cg c 0

 6.

1 + x2 x

4.298

dx ag a2 + b2 x2 1 arctan = x2 c2 − g 2 x2 cg bc

[a > 0, b > 0, c > 0, g > 0] BI (138)(8, 11)a

∞

2

2

x dx 1+x π = (ln 4 − 1) 2 2 2 x 4 (1 + x ) 0    1 2 1−x ln2 1 − x2 dx = π x2 0   1 1 ∞ 1 + 2x cos t + x2 dx t2 1 + 2x cos t + x2 dx = =− ln ln 2 2 (1 + x) x 2 0 (1 + x) x 2 0 ln

[|t| < π]

BI (139)(21)

FI II 643a

BI (115)(23), BI (134)(15)

4.313

Logarithmic functions and powers

 22. 23. 4.299 1. 2.

∞

1 + 2x cos t + x2 p−1 2π (1 − cos pt) x dx = − 2 (1 + x) p sin pπ 0    1 2 dx 1 + x sin t π−t √ ln = π ln cot 1 − x2 sin t 1 − x2 4 0 ln

3.



1

0

  1−a x 1 + ax2 2 2

1.

(1 −

2 ax2 )

3. 4. 5. 6.8 4.312

  π cosec πn ln (1 + xn ) dx = xn n−1 0  ∞   dx 2π ln 1 + x3 = √ ln 3 2 1−x+x 3 0  ∞   π2 dx π ln 1 + x3 = √ ln 3 − 3 1+x 9 3 0  ∞  x dx  π2 π √ ln 3 + ln 1 + x3 = 1 + x3 9 3 0  ∞   1−x 2 ln 1 + x3 dx = − π 2 3 1 + x 9 0 √ )  ∞) 3 ) ) )1 − x ) dx = − π 3 ) 3 a ) x3 6a2 0 



∞

∞

1.

0



1. 0

∞

BI (134)(14)

BI (115)(25)

[a > 0] [a > 0,

BI (115)(26)

Re μ > 0]

BI (134)(16)

[n = 2, 3, . . .] LI (136)(8) LI (136)(6) LI (136)(7) BI (136)(9)

π2 1 + x3 dx π √ ln 3 + = x3 1 + x3 9 3

BI (138)(12)

ln

1 + x3 x dx π π2 = √ ln 3 − 3 3 x 1+x 9 3

BI (138)(13)

∞

2. 4.313

GW (325)(21d)

ln 0



BI (134)(17)

√ dx 1 = √ arctan a ln(1 + a) 2 1 + ax a

  2 1 (x + 1) x + a2 μ−1 π (aμ − 1) ln x dx = (x + a)2 μ sin μπ 0

2.

[|t| < π]

[a > 0]



11

|t| < π]

∞

ln

4.311

[0 < |p| < 1,

  (x + 1) x + a2 dx 2 = (ln a) ln [a > 0] 2 (x + a) x 0    1 √ (1 − ax) 1 + ax2 dx 1 ln = √ arctan a ln(1 + a) 2 2 2 1 + ax 2 a (1 − ax ) 0





4.

569

 dx  ln x ln 1 + a2 x2 = πa (1 − ln a) x2

[a > 0]

BI (134)(18)

570

Logarithmic Functions



∞

2. 0



∞

3. 0



∞

4. 0



∞

5. 0



∞

6. 0



∞

7. 0

4.314 1.11



1

0





  dx ln a2 + b2 x2 2 = 2π x



 b b ln(b + ac) − ln b − c ln c c+ a a [a > 0, b > 0, c > 0]

[a > 0,

a + bc > 0] b > 0]

1 + a2 x2 dx bb = π(a − b) + π ln a 2 2 2 1+b x x a

[a > 0,

ln x ln

b a2 + 2bx + x2 dx = 2π ln a arcsin a2 − 2bx + x2 x a

[a ≥ |b|]

BI (134)(25)

(ln a)2 x ln x − x − a dx = (x + a)2 x 2(a − 1)

[a > 0]

BI (141)(7)

[a > 0]

LI (141)(8)

ln(1 + x)

ln2 (1 − x)

π 2 + (ln a)2 x ln x − x − a dx = (x + a)2 x 1+a

1

∞



1 (q − 1)x 1 + − (1 + x)2 x + 1 (1 + x)q

x ln x + 1 − x 2

1

1. 0



1

2. 0



1

3. 0

 4.

0

BI (134)(24)

 xp−1 − xq−1 ak p + k dx = (−1)k+1 ln ln x k q+k

[|a| < 1,

ln(1 + x)dx = ln

1

ln(1 + x) (ln x)

n−1

ln(1 + x) (ln x)

2n

ln(1 − x) (ln x)

n−1

ln(1 − x) (ln x)

2n

p > 0,

q > 0]

BI (123)(18)

dx = ln Γ(q) x ln(1 + x)

4 π

x (ln x)      1 ln 1 − x2 dx q+π π q π  = − ln Γ  + ln 2q + ln − 1 2 q π 2q π 0 x q 2 + (ln x)



BI (134)(22, 23)a

ln x ln

[q > 0] 4.315

BI (134)(20, 21)a

    a b2 dx a + bc ln(a + bc) − ln a − c ln 1 + c2 x2 ln a2 + 2 = 2π x x2 b b

[q > 0]

0

4.

ln 1 + c x

ln(1 + ax)

∞

0

3.

2 2

k=1

2. 



4.314

  dx 1 n−1 = (−1) (n − 1)! 1 − n ζ(n + 1) x 2

dx 22n+1 − 1 = π 2n+2 |B2n+2 | x (2n + 1)(2n + 2) dx = (−1)n (n − 1)! ζ(n + 1) x

dx 22n =− π 2n+2 |B2n+2 | x (n + 1)(2n + 1)

BI (143)(7) BI (126)(12)

LI (327)(12)a

BI (116)(3)

BI (116)(1)

BI (116)(4)

BI (116)(2)

4.318

4.316

Logarithmic functions and powers



1

1. 0



[p > −1, 1

0

1. 2.

3.

4.

5. 6. 7. 8. 9.12 10.

11.10 4.318

∞

 ak 1 dx = − p+1 Γ(p + 1) x r k p+2 k=1

2. 4.317

p

ln (1 − axr ) (− ln x)

571

  dx = −2 Γ(p + 1) ln 1 − 2ax cos t + a2 x2 (− ln x)p x

∞  k=1

a < 1,

r > 0]

a cos kt k p+2

√ dx 1 + x2 + a √ ln √ = π arcsin a [|a| < 1] 2 1 + x − a 1 + x2 0 √  1 √ 1 − a2 x2 − x 1 − a2 dx 1 2 = (arcsin a) ln 1 − x x 2 0 v−t √  1 cos dx 1 + cos t 1 − x2 2 √ = π cot t ln 2 x2 + tan2 v v + t 1 − cos t 1 − x 0 sin 2   √  1 π2 x + 1 − x2 x dx 2 √ ln = 2 x − 1 − x2 1 − x2 0  1 # $ √ √ π dx 1 ln 1 + kx + 1 − kx  = ln(4k) K(k) + K (k  ) 2 ) (1 − k 2 x2 ) 4 8 (1 − x 0  1 # $ √ √ 3 dx 1 ln 1 + kx − 1 − kx  = ln(4k) K(k) + π K (k  ) 2 2 2 4 8 (1 − x ) (1 − k x ) 0  1 # $  π 1 dx ln 1 + 1 − k 2 x2  = ln k K(k) + K (k  ) 2 2 2 2 4 (1 − x ) (1 − k x ) 0  1 # $  3 dx 1 ln 1 − 1 − k 2 x2  = ln k K(k) − π K (k  ) 2 2 2 2 4 (1 − x ) (1 − k x ) 0 √  1 2

1 + p 1 − x2 dx √ ln = π arcsin p p 0] b > −1, a

−px

ln 0

[a > 0]



dx a p a + be = ln ln −qx a + be x a+b q

BI (354)(6)

BI (354)(7)

pq > 0 FI II 635, BI (354)(1)

4.321  1.

∞

−∞  ∞

2. −∞

x ln cosh x dx = 0 ln cosh x

BI (358)(2)a

dx =0 1 − x2

BI (138)(20)a

4.322   π 1 π π2 x ln sin x dx = x ln cos2 x dx = − ln 2 1.11 2 0 2 0    ∞ 2 −2a ln sin (ax) 1−e [a > 0] 2.12 dx = π ln 2 1 + x 2 0    ∞ ln cos2 (ax) 1 + e−2a [a > 0] 3.12 dx = π ln 1 + x2 2 0  ∞ ln sin2 ax π2 + aπ [a > 0, b > 0] 4. dx = − b2 − x2 2b 0  ∞ ln cos2 ax 5.11 dx = ∞ b2 − x2 0  ∞ ln cos2 x 6. dx = −π x2 0   π/4 ∞  π μ ζ(2k) 2  1 μ−1 7 ln 2 + − 7. ln sin xx dx = − 2k−1 (μ + 2k) 2μ 4 μ 4 0

BI (432)(1, 2) FI II 643 GW (338)(28b) GW (338)(28a)

BI (418)(1) BI (418)(2) FI II 686

k=1

[Re μ > 0]

LI (425)(1)

4.325

8.7

Logarithmic functions and powers



π/2

0



ln sin xx

 ∞  ζ(2k) 1  π μ 1 −2 dx = − μ 2 μ 4k (μ + 2k) k=1

π/2

9. 0



μ−1

573



ln (1 − cos x) xμ−1 dx = −

[Re μ > 0]

LI (430)(1)

ζ(2k) 1  π μ 2  − μ 2 μ 42k−1 (μ + 2k) ∞

k=1

[Re μ > 0] ∞

10. 0

  ln 1 ± 2p cos βx + p2

q2

 π  dx = ln 1 ± pe−βq 2 +x q  π  = ln p ± e−βq q



p2 < 1 p2 > 1

LI (430)(2)



FI II 718a

4.323 1.

11



π

0



x ln tan2 x dx = 0

BI (432)(3)

∞

ln tan2 ax π dx = ln tanh ab 2 + x2 b b 0 2  ∞  dx π2 1 + tan x = ln 1 − tan x x 2 0

2. 3. 4.324





∞

ln

1. 0



∞

2.

ln 0



1 + sin x 1 − sin x

2

[a > 0,

b > 0]

GW (338)(28c)

GW (338)(26)

dx = π2 x

GW (338)(25)

q2 1 + 2a cos px + a2 dx = ln(1 + a) ln 2 1 + 2a cos qx + a2 x p 2  q 1 ln 2 = ln 1 + a p

[−1 < a ≤ 1] [a < −1 or a ≥ 1] GW (338)(27)

∞

3. 0

  ln a2 sin2 px + b2 cos2 px

dx π = [ln (a sinh cp + b cosh cp) − cp] c2 + x2 c [a > 0, b > 0, c > 0,

p > 0] GW (338)(29)

4.325 1.

12



1

0

 2.

3.

∞

ln (− ln x)

 ln k 1 dx 2 = −C ln 2 + = − (ln 2) (−1)k 1+x k 2

GW (325)(25a)

k=2

∞

 (−1)k dx e−ikλ (C + ln k) = GW (325)(26) x + eiλ k 0 k=1  

 1  ∞  1 1 π dx dx 1 ψ + ln 2π = ln − C ln (− ln x) = ln ln x = 2 2 (1 + x) (1 + x) 2 2 2 2 0 1 1

ln (− ln x)

BI (147)(7)

574

Logarithmic Functions

√   2π Γ 34 dx π   ln ln x = ln 1 + x2 2 Γ 14 0 1 √    1  ∞ 3 2π Γ 23 dx dx π   √ ln ln (− ln x) = ln ln x = 1 + x + x2 1 + x + x2 Γ 13 3 0 1    1  ∞ 1 dx dx 2π 5 ln 2π − ln Γ ln (− ln x) = ln ln x = √ 2 2 1 − x + x 1 − x + x 6 6 3 0 1 

4. 5. 6.

1

dx ln (− ln x) = 1 + x2



∞

BI (148)(1)

BI (148)(2)

BI (148)(5)

  t 1 t/π + (2π) Γ 1 ∞ dx dx π 2 2π   ln ln (− ln x) = ln ln x = 2 2 t 1 1 + 2x cos t + x 1 + 2x cos t + x 2 sin t 0 1 Γ − 2 2π

 7.

4.326



BI (147)(9)



1

8. 0



ln (− ln x) xμ−1 dx = −

∞

9. 1

ln ln x

1+

x2

1 (C + ln μ) μ

[Re μ > 0]

BI (147)(1)

xn−2 dx + x4 + · · · + x2n−2   n−1 π π π kπ Γ n+k k−1 2n tan ln 2π + ln  k  = (−1) sin 2n 2n n n Γ 2n k=1 n−1   2 π π π kπ Γ n−k k−1 n   = tan ln π + ln (−1) sin 2n 2n n n Γ nk k=1

[n is even]

[n is odd] BI (148)(4)



1

10. 0

dx √ = ln (− ln x) 2 (1 + x ) − ln x =



1

11. 0



1

12. 0

4.326



1

1. 0

 2.

0

 1

∞

ln ln x

∞ √  (−1)k+1 √ [ln(2k + 1) + 2 ln 2 + C] π 2k + 1 k=0

xμ−1 dx ln (− ln x) √ = − (C + ln 4μ) − ln x ln (− ln x) (− ln x)μ−1 xν−1 dx =

ln (a − ln x) xμ−1 dx =

1/e

dx √ (1 + x2 ) ln x



BI (147)(4)

π μ

[Re μ > 0]

BI (147)(3)

1 Γ(μ) [ψ(μ) − ln(ν)] νμ

1 [ln a − eaμ Ei(−aμ)] μ

 2μ−1  x 1 1 dx = − [Ei(−μ)]2 ln 2 ln − 1 x ln x 2

[Re μ > 0,

Re ν > 0]

[Re μ > 0,

a > 0]

[Re μ > 0]

BI (147)(2)

BI (107)(23)

BI (145)(5)

4.335

4.327

Logarithms and exponentials



1

1. 0



1

2. 0





2

2

ln a + (ln x)





ln a2 + 4 (ln x)

∞

3. 0

2

  2 Γ 2a+3π dx 4π   + = π ln 1 + x2 Γ 2a+π 4π 

  2 Γ a+3π dx 4π  +  = π ln 1 + x2 Γ a+π 4π

π 2

575

ln π2

a > − π2

π 2

BI (147)(10)

ln π

[a > −π]   2 2 ln a2 + (ln x) xμ−1 dx = [− cos aμ ci(aμ) − sin aμ si(aμ) + ln a] μ [a > 0,

BI (147)(16)a

Re μ > 0]

GW (325)(28)

If the integrand contains a logarithm whose argument also contains a logarithm, for example, if the integrand contains ln ln x1 , it is useful to make the substitution ln x = t and then seek the transformed integral in the tables.

4.33–4.34 Combinations of logarithms and exponentials 4.331



∞

1. 0

 2. 3. 4.332 1. 2.12

∞

e−μx ln x dx = −

1 (C + ln μ) μ

1 Ei(−μ) μ 1   1 1 1 eμx − 1 μx dx e ln x dx = − μ 0 x 0 e−μx ln x dx = −

  ln x dx 2π 5 1 = √ ln 2π − ln Γ x −x −1 6 3 6 0 e +e     ∞ 2 √ Γ 3 3 ln x dx π   2π = √ ln x −x +1 Γ 13 3 0 e +e



∞

[Re μ > 0]

BI (256)(2)

[Re μ > 0]

BI (260)(5)

[μ = 0]

(cf. 4.325 6)

BI (257)(6)

(cf. 4.325 5)

BI (257)(7)a, LI (260)(3)

 2 π 1 [Re μ > 0] e−μx ln x dx = − (C + ln 4μ) 4 μ 0   ∞ ∞ ln x dx C + ln 4k 1 π kπ √ 4.334 = (−1)k sin 2 2 x −x 2 3 3 +1+e k 0 e 

4.333

4.335 1. 2. 3.7

GW (324)(81a)

∞

BI (256)(8), FI II 807a BI (357)(13)

k=1



1 π2 2 + (C + ln μ) [Re μ > 0] μ 6 0

√  ∞ π π2 2 2 −x2 (C + 2 ln 2) + e (ln x) dx = 8 2 0

 ∞ 2 1 π 3 3 (C + ln μ) + (C + ln μ) − ψ  (1) e−μx (ln x) dx = − μ 2 0



∞

2

e−μx (ln x) dx =

ET I 149(13) FI II 808 MI 26

576

Logarithmic Functions

4.336 1.

7

 PV 

∞

2.

2. 3. 4.7 5.12

∞ −x

e dx ≈ −0.154479567 ln x

e−μx dx π 2 + (ln x)

0

4.337  1.

0

∞

4.336

2

BI (260)(9)

= ν  (μ) − eμ

[Re μ > 0]

MI 26

1 ln β − eμβ Ei(−βμ) [|arg β| < π, Re μ > 0] BI (256)(3) μ 0    ∞ 1 βμ μ −μx [|arg β| < π, Re μ > 0] e ln(1 + βx)dx = − e Ei − ET I 148(4) μ β 0  ∞

1 ln a − e−aμ Ei(aμ) e−μx ln |a − x| dx = [a > 0, Re μ > 0] BI (256)(4) μ 0 ) )  ∞ ) β )

) dx = 1 e−βμ Ei(βμ) e−μx ln )) [Re μ > 0] MI 26 ) β−x μ 0  ∞ ln(1 + ax)xn e−x dx 0   n−μ n−μ−k     n−1   n! (−1)n−μ−1 1/a 1 1 1 1/a + n!e e Ei − (k − 1)! − Ei = n−μ (n − μ)! a a a a μ=0 e−μx ln(β + x)dx =

k=1

4.338  1.

∞

  2 e−μx ln β 2 + x2 dx = [ln β − ci(βμ) cos(βμ) − si(βμ) sin(βμ)] μ 0 [Re β > 0, Re μ > 0]  ∞  

2 2 ln β − eβμ Ei(−βμ) − eβμ Ei(βμ) 2. e−μx ln2 x2 − β 2 dx = μ 0 [Im β > 0, Re μ > 0] ) )  ∞ )x + 1)

) dx = 1 e−μ (ln 2μ + γ) − eμ Ei(−2μ) 4.339 e−μx ln )) ) x−1 μ 0 [Re μ > 0] √ √  ∞ x + ai + x − ai π √ [H0 (aμ) − Y 0 (aμ)] dx = 4.341 e−μx ln 4μ 2a 0 [a > 0, Re μ > 0] 4.342 

∞ 1 1 −2nx + ln 2 − 2 β(2n + 1) 1. e ln (sinh x) dx = 2n n 0  

 ∞ μ 1 1 β − [Re μ > 0] 2. e−μx ln (cosh x) dx = μ 2 μ 0  ∞  μ  μ 1 1 −μx 11 ln − − ψ 3. e [ln (sinh x) − ln x] dx = μ 2 μ 2 0  4.343

0

[Re μ > 0] π

 

eμ cos x ln 2μ sin2 x + C dx = −π K 0 (μ)

BI (256)(6)

BI (256)(5)

MI 27

ET I 149(20)

BI (256)(17) ET I 165(32)

ET I 165(33) WA 95(16)

4.354

Logarithms, exponentials, and powers

577

4.35–4.36 Combinations of logarithms, exponentials, and powers 4.351



1

1. 0



1

2. 0



e

∞

0 ∞

2. 0



∞

3. 0



∞

4. 0

4.353  1. 0



∞

∞

2. 0

 3. 0

BI (352)(2)

∞ −μx

1



BI (352)(1)

  1 eμx μx2 + 2x ln x dx = 2 [(1 − μ)eμ − 1] μ

3. 4.352  1.

1−e e

(1 − x)e−x ln x dx =

1

ln x 1 dx = eμ [Ei(−μ)]2 1+x 2

xν−1 e−μx ln x dx =

xn e−μx ln x dx =

[Re μ > 0]

1 Γ(ν) [ψ(ν) − ln μ] μν n!



μn+1

[Re μ > 0,

1+

1.6



ET I 148(7)

[Re μ > 0]

ET I 148(10)

xμ−1 e−x ln x dx = Γ (μ)

[Re μ > 0]

GW (324)(83a)

(x − ν)xν−1 e−x ln x dx = Γ(ν)

[Re ν > 0]

GW (324)(84)

[Re μ > 0]

BI (357)(2)

   1 1 (2n − 1)!! π xn− 2 e−μx ln x dx = μx − n − 2 (2μ)n μ n  k=0

0

BI (353)(3), ET I 315(10)a

[Re μ > 0] 

 √ (2n − 1)!! 1 1 1 1 + + · · · + − C − ln 4μ 2 1 + xn− 2 e−μx ln x dx = π 1 3 5 2n − 1 2n μn+ 2

(μx + n + 1)xn eμx ln x dx = eμ

∞

Re ν > 0]

1 1 1 + + · · · + − C − ln μ 2 3 n

(−1)k−1

n! n! + (−1)n n+1 (n − k)!μk+1 μ [μ = 0]

4.354

NT 32(10)

GW (324)(82)

∞

 (−1)k−1 xν−1 ln x dx = Γ(ν) [ψ(ν) − ln k] x e +1 kν k=1 1 2 = − (ln 2) 2

[Re ν > 0] [for ν = 1] GW (324)(86a)

2.7

 0

∞

xν−1 ln x 2

(ex + 1)

dx = Γ(ν)

∞  k=2

(−1)k (k − 1) [ψ(ν) − ln k] kν [Re ν > 1]

GW (324)(86b)

578

Logarithmic Functions



∞

3.

(x − ν)ex − ν 2

(ex + 1)

0



∞

4.

2

(ex + 1) ∞

5. 0

4.355



∞

1. 0



∞

2. 0

 3. 4.

x2n−1 ln x dx =

xν−1 ln x n Γ(ν) n dx = (−1) (ex + 1) (n − 1)!

2

x2 e−μx ln x dx = 

2

∞  (−1)k−1

kν

k=1

(x − 2n)ex − 2n

0



xν−1 ln x dx = Γ(ν)



x μx − νx − 1 e

[Re ν > 0]

GW (324)(87a)

[n = 1, 2, . . .]

GW (324)(87b)

22n−1 − 1 2n π |B2n | 2n

∞  k=n

(−1)k (k − 1)! [ψ(ν) − ln k] (k − n)!k ν

1 (2 − ln 4μ − C) 8μ −μx2 +2νx

4.355



π μ

ν 1 + ln x dx = 4μ 4μ



[Re ν > 0]

GW (324)(86c)

[Re μ > 0]

BI (357)(1)a

π exp μ



ν2 μ



 1+Φ

[Re μ > 0] ∞



 2 (n − 1)! μx2 − n x2n−1 e−μx ln x dx = [Re μ > 0] 4μn 0   ∞   2n −μx2 (2n − 1)!! π 2 2μx − 2n − 1 x e ln x dx = 2(2μ)n μ 0 [Re μ > 0]

4.356 1. 2.



ν √ μ



BI (358)(1) BI (353)(4)

BI (353)(5)

  x a  dx + ln x = 2 ln a K 0 (2μ) exp −μ [a > 0, Re μ > 0] GW (324)(91) a x x 0    ∞

1 b ln x 2ax2 − (2n + 1)x − 2b xn− 2 dx exp −ax − x 0  n  ∞ π −2√ab  (n + k)! b 2 e =2  √ k a a k=0 (n − k)!(2k)!! 2 ab ∞

[a > 0, b > 0] BI (357)(4)   ∞

dx b ln x 2ax2 + (2n − 1)x − 2b n+ 3 exp −ax − x x 2 0 ∞ an π √  (n + k − 1)! 2 e−2 ab =2  √ k b a k=0 (n − k − 1)!(2k)!! 2 ab

 3.

For n = 12 :  ∞  √   ax2 − b a ln x 4. exp −ax − dx = 2 K 0 2 ab 2 x x 0

[a > 0,

b > 0]

BI (357)(11)

[a > 0,

b > 0]

GW (324)(92c)

4.357

Logarithms, exponentials, and powers

For n = 0:     ∞ π −2√ab 2ax2 − x − 2b b √ ln x dx = 2 e 5. exp −ax − x x x a 0 [a > 0,

579

b > 0] BI (357)(7), GW(324)(92a)

For n = −1: √     ∞ 2ax2 − 3x − 2b 1 + 2 ab π −2√ab b √ ln x dx = e exp −ax − 6. x x a a 0 [a > 0, b > 0] 7.9

8.9

9.12

10.12

[a > 0, b > 0]  

b 3 ln x 2ax2 − (2n + 1)x − 2b xn− 2 dx exp −ax − x 0  (2n+1)/4  √  b =4 K n+ 12 2 ab a   n2  n π −2√ab  (n + k)! b e =2  √ k a a k=0 (n − k)!(2k)!! 2 ab [n = 0, 1, . . . , a > 0, b > 0]    ∞  

dx b ln ax2 − b cos (c ln x) + cx sin (c ln x) 2 exp −ax − x x 0  √     = 2 cos c ln b/a K ic 2 ab 

∞



∞

0

11.12

LI (357)(6), GW (324)(92b)

    ∞  √  b b ln x a − 2 dx = K 0 2 ab exp −ax − x x 0





∞

0

[a > 0, b > 0]    

dx b ln x ax2 − b sin (c ln x) − cx cos (c ln x) 2 exp −ax − x x  √     = 2 sin c ln b/a K ic 2 ab

xc ln x a −

b c − 2 x x

[a > 0, b > 0]   c/2   √  b b dx = 2 exp −ax − K c 2 ab x a [a > 0,

4.357



1. 0

√   1 + ax2 − x4 1 + x4 2a3 π √ ln x exp − dx = − 2ax2 x2 2ae

√   ∞ 2a3 π x4 + ax2 − 1 1 + x4 √ ln x exp − dx = 2ax2 x4 2ae 0

 2.

∞

b > 0]

[a > 0]

BI (357)(8)

[a > 0]

BI (357)(9)

580

Logarithmic Functions



∞

3. 0

4.358

√   x4 + 3ax − 1 1 + x4 (1 + a) 2a3 π √ ln x exp − dx = 2ax2 x6 2ae [a > 0]

4.358 1.6



∞

1

2.12



3.9

∞

0

4.12



∞

0

5.3





∞

1. 0

2.11





1

∞

∞

0

 0

 2.

1

[Re μ > 0, Re ν > 0] # $ Γ(ν) xν−1 e−μx (ln x)3 dx = ν [ψ(ν) − ln μ]3 + 3 ζ(2, ν) [ψ(ν) − ln μ] − 2 ζ(3, ν) μ Γ(ν) μν

xν−1 e−μx (ln x)4 dx =

Re ν > 0]



[Re μ > 0,

e−μx

Re ν > 0]

MI 26

MI 26

[ψ(ν) − ln μ]4 + 6 ζ(2, ν) [ψ(ν) − ln μ]2

 2 −8 ζ(3, ν) [ψ(ν) − ln μ] + 3 [ζ(2, ν)] + 6 ζ(4, ν)

xν−1 e−μx (ln x)n dx =

n

∂ ∂ν n

!

" μ−ν Γ(ν)

Re ν > 0]

[n = 0, 1, 2, . . .]

xp−1 − xq−1 1 dx = [λ(μ, p − 1) − λ(μ, q − 1)] ln x μ p > 0,

q > 0]

MI 27

∞

 μk p + k xp−1 − xq−1 dx = ln ln x k! q+k

(x + 1)e−μx π 2 + (ln x)

2.

1.

Re μ > 0,

[p > 0,

q > 0]

BI (352)(9)

k=0

0

4.362

[m = 0, 1, . . . ,

$ Γ(ν) # xν−1 e−μx (ln x)2 dx = ν [ψ(ν) − ln μ]2 + ζ(2, ν − 1) μ

eμx

1. 

" ∂ m ! −ν μ Γ(ν, μ) m ∂ν

[Re μ > 0,

0

4.361

dx =

[Re μ > 0, ∞

0

4.359

m

MI 26 ∞

0



xν−1 e−μx (ln x)

BI (357)(10)

1

2

dx = ν  (μ) − ν  (μ)

e−μx dx  = eμ − ν(μ)  2 2 x π + (ln x)

[Re μ > 0]

MI 27

[Re μ > 0]

MI 27

xex ln(1 − x) dx = 1 − e

∞

e−μx ln(2x − 1)

1   μ 2 dx = Ei − x 2 2

BI (352)(5)a

[Re μ > 0]

ET I 148(8)

4.369

4.363

Logarithms, exponentials, and powers



∞

1. 0



1

2. 0

4.364



μ(x + a) ln(x + a) − 2 dx x+a  ∞ 1 μ(x − a) ln2 (x − a) − 4 2 dx = (ln a) = e−μx ln2 (a − x) 4 0 x−a [Re μ > 0, a > 0] BI (354)(4, 5) 2

x(1 − x)(2 − x)e−(1−x) ln(1 − x) dx =

∞

1. 0



e−μx ln(a + x)

∞

2. 0

581

e−μx ln[(x + a)(x + b)]

e−μx ln(x + a + b)



1−e 4e

BI (352)(4)

dx = e(a+b)μ {Ei(−aμ) Ei(−bμ) − ln(ab) Ei[−(a + b)μ]} x+a+b [a > 0, b > 0, Re μ > 0] BI (354)(11)

1 1 + x+a x+b

 dx = (1 + ln a ln b) ln(a + b) + e−(a+b)μ {Ei(−αμ) Ei(−bμ) + (1 − ln(ab)) Ei[−(a + b)μ]} [a > 0,

 4.365 4.366 1. 2. 3.

0



e−x −

x (1 +

x)p+1

ln(1 + x)

dx = ln p x

[p > 0]

  x2 dx = [ci(aμ)]2 + [si(aμ)]2 e−μx ln 1 + 2 [Re μ > 0] a x 0 ) )  ∞ ) x2 )) dx −μx ) = Ei(aμ) Ei(−aμ) [Re μ > 0] e ln )1 − 2 ) a x 0 ) )  ∞ ) 1 + x2 ) 2 ) dx = 1 [cosh μ sinh(iμ) − sinh μ cosh(iμ)] xe−μx ln )) 2 1−x ) μ 0



Re μ > 0]

BI (354)(12)

BI (354)(15)

∞

NT 32(11)a ME 18

MI 27 [Re μ > 0] (cf. 4.339)   ∞ 2 eβμ x + x2 + 2β √ K 0 (βμ) dx = xe−μx ln [|arg β| < π, Re μ > 0] ET I 149(19) 4μ 2β   0 2u  2   2  x2 4u2 − x2 2 dx π −2u2 μ  π √ e Y 2iu 2iu μ e−μx ln = μ − (C − ln 2) J 0 0 u4 2 2 4u2 − x2 0 ET I 149(21)a [Re μ > 0]

4.367 4.368 4.369

∞

b > 0,



1. 0

∞

xν−1 e−μx [ψ(ν) − ln x] dx =

Γ(ν) ln μ μν

[Re ν > 0]

ET I 149(12)

582

Logarithmic Functions



∞

2. 0

xn e−μx

#

ln x −

1 2

2 ψ(n + 1) −

1 2

$ ψ  (n + 1) dx

4.371

*

+

2 1 1  ln μ − ψ(n + 1) + ψ (n + 1) = n+1 μ 2 2 MI 26 [Re μ > 0] n!

4.37 Combinations of logarithms and hyperbolic functions 4.371



∞

1. 0

ln x dx = π ln cosh x

√   2π Γ 34 1 Γ 4

LI (260)(1)a

 π+t  ∞ (2π) Γ 2

ln x dx π 2π   = ln t < π2 π−t sin t 0 cosh x + cos t Γ 2π    ∞ ln x dx 1 + ln π = ln π − 2 ln 2 − C =ψ 2 2 0 cosh x  ∞ 2 ln x 7 ζ(3) 4 dx = − 3 (2 ln 2 + C − ln π) − 3 π 2 0 cosh x

 ∞ ln x 1 ζ(3) 8 ζ(5) 6 dx = − 15 (2 ln 2 + C − ln π) − 15 35 π 2 + 93 π 4 0 cosh x

 ∞ ln x 1 ζ(3) 16 ζ(5) ζ(7) 8 dx = − 35 (2 ln 2 + C − ln π) − 315 686 π 2 + 2604 π 4 + 5715 π 6 0 cosh x 

t/π

2.

3. 4.∗ 5.∗ 6.∗ 4.372



∞

1. 1

  n−1 π mπ π sinh mx kmπ Γ n+k k−1 2n dx = tan ln 2π + ln  k  ln x (−1) sin sinh nx 2n 2n n n Γ 2n k=1    n−1  2 mπ π  π kmπ Γ n−k k−1 n   tan ln π + ln = (−1) sin 2n 2n n n Γ nk

BI (257)(7)a

BI (257)(4)a GC GC GC

[m + n is odd] [m + n is even]

k=1

 2. 1

BI (148)(3)a ∞

ln x

cosh mx dx cosh nx   n π ln 2π π (2k − 1)mπ Γ 2n+2k−1 k−1 4n   ln = + (−1) cos 2n cos mπ n 2n Γ 2k−1 2n 4n k=1    n−1 2  π ln π π  (2k − 1)mπ Γ 2n−2k+1 k−1 2n   ln = + (−1) cos 2n cos mπ n 2n Γ 2k−1 2n 2n

[m + n is odd] [m + n is even]

k=1

BI (148)(6)a

4.376

4.373

Logarithms and hyperbolic functions



∞

1. 0



∞

     2 Γ 2ab+3π ln a2 + x2 π 2b 4π  − ln  dx = 2 ln cosh bx b π Γ 2ab+π 4π

 b > 0,

a>−

π . 2b

  ln 1 + x2

dx 4 πx = 2 ln cosh π 0 2  2    a+5   ∞   2 sinh 3 πx π 6 Γ a+4 6  Γ 6  dx = 2 sin ln  a+1 ln a + x2 sinh πx 3 Γ 6 Γ a+2 0 6

2. 3.



∞

4. 0

  ln 1 + x2



a π 2 dx ln + −ψ = a π 2a sinh2 ax



π+a π

6. 4.374

[a > −1] . 



∞

2.

BI (258)(3)

BI (258)(2)

  ln cos2 t + e−2x sin2 t

[a > 0, 4.375 11



∞

ln cosh 0



∞

2. 0

4.376



∞

1. 0



∞

0

3. 0

BI (259)(10)a

LI (259)(14)

BI (259)(11)

π dx = ln 2 cosh x 2

BI (259)(16)

∞

√  (−1)k+1 ln x √ √ dx = 2 π {ln(2k + 1) + 2 ln 2 + C} x cosh x 2k + 1

BI (147)(4)

k=0

2.



a + b > 0]

π x dx = G − ln 2 2 cosh x 4

ln coth x

BI (258)(12)

BI (258)(5)

dx = −2t2 sinh x 0

 ∞  dx  a+b 2 ln(a + b) − a ln a − b ln 2 = ln a + be−2x (b − a) 2 cosh2 x 0

1.

BI (258)(11)a

BI (258)(1)a

[a > 0] π   ∞  cosh 2 x  2π − 4  π  dx = ln 1 + x2 π 0 x sinh2 2 π  √  ∞  x cosh √   16 8 2 √ 4π  dx = 4 2 − + ln ln 1 + x2 2+1 π π 0 x sinh2 4

5.

1.

583

∞

ln x

∞  (−1)k+1 (μ + 1) cosh x − x sinh x μ x dx = 2 Γ(μ + 1) (2k + 1)μ+1 cosh2 x k=0

[Re μ > −1]   (n + 1) cosh x − x sinh x n (−1) (n) 1 x dx = ln x β 2 2n 2 cosh x n

BI (356)(10)

584

Logarithmic Functions



∞

4.

ln 2x 0

 5. 6.

7.

8.6

9.6

4.377

n sinh 2ax − ax 2n−1 1  π 2n x dx = − |B2n | n a sinh2 ax [n = 1, 2, . . .]

∞

ax cosh ax − (2n + 1) sinh ax 2n 22n+1 − 1 x dx = 2 (2n)! ζ(2n + 1) 2 (2a)2n+1 sinh ax 0  ∞  π 2n ax cosh ax − 2n sinh ax 2n−1 22n−1 − 1 |B x ln x dx = | 2n 2n a sinh2 ax 0 [n = 1, 2, . . . , a > 0]  ∞   (2n + 1) cosh ax − ax sinh ax 2n π 2n+1 x dx = − ln |E2n | 2 2a cosh ax 0 ln x

BI (356)(9)a BI (356)(14)

BI (356)(15)

[a > 0] BI (356)(11) ⎧2   π 2n  2n−1 ⎪ −1 |B2n | n = 1, 2, . . . ⎪  ∞ ⎨a 2 2a 2ax sinh ax − (2n + 1) cosh ax 2n x dx = 1 ln x ⎪ cosh3 ax 0 ⎪ n=0 ⎩ a 

∞

0



∞

10. 0



[a > 0]   2ax cosh ax − (2n + 1) sinh ax 2n 1 π 2n x dx = ln x |B2n | 3 a a sinh ax

ln x

x

[a > 0,

BI (356)(2)

n = 1, 2, . . .]

BI (356)(6)a

t   2 − 6 cos2 2 2 x2 dx = π − t t 3 sin t (cosh x + cos t)2

x sinh x − 6 sinh2

[0 < t < π]

BI (356)(16)a

∞

 cosh πx + πx sinh πx dx  ln 1 + x2 =4−π x2 cosh2 πx 0  ∞  cosh πx + πx sinh πx dx  12. ln 1 + 4x2 = 4 ln 2 x2 cosh2 πx 0    ∞ ax − n 1 − e−2ax 2n−1 1  π 2n x 4.377 ln 2x dx = |B2n | 2n a sinh2 ax 0 [n = 1, 2, . . .] 11.

BI (356)(12) BI (356)(13)

LI (356)(8)a

4.38–4.41 Logarithms and trigonometric functions 4.381 1. 2. 3.



1

1 ln x sin ax dx = − [C + ln a − ci(a)] a 0  1 π 1 ln x cos ax dx = − si(a) + a 2 0  2π 1 ln x sin nx dx = − [C + ln(2nπ) − ci(2nπ)] n 0

[a > 0]

GW (338)(2a)

[a > 0]

BI (284)(2) GW (338)(1a)

4.384

Logarithms and trigonometric functions



2π

4. 0

4.382 1.12 2.10

ln x cos nx dx = −

585

π 1 si(2nπ) + n 2

GW (338)(1b)

) ) )x + a) ) sin bx dx = π sin ab [a > 0, b > 0] ln )) ) x − a b 0 )  ∞ )  # $  )x + a) ) cos bx dx = 2 cos(ab) si(ab) + π − sin(ab) ci(ab) ) ln ) x − a) b 2 0 



∞

∞

3.

ln 0





∞

4.

ln 0



 a2 + x2 π  −bc −ac e cos cx dx = − e b2 + x2 c x2 + x + a2 2π exp −b sin bx dx = 2 2 x −x+a b

 a2 −

1 4

[a > 0,

b > 0]

[a > 0,

b > 0,

ET I 77(11)

ET I 18(9)

c > 0] FI III 648a, BI (337)(5)

 sin

b 2

[b > 0] ∞

5.

ln 0

(x + β)2 + γ 2 2π −γb e sin bx dx = sin βb (x − β)2 + γ 2 b

ET I 77(12)

[Re γ > 0,

|Im β| ≤ Re γ,

b > 0] ET I 77(13)

4.383 1.

2. 4.384



∞

π   πb 0 2b sinh β    ∞   β π πb ln 1 − e−βx cos bx dx = 2 − coth 2b 2b β 0



1

1. 0

2.

7

  β ln 1 + e−βx cos bx dx = 2 − 2b



1

0

[Re β > 0,

b > 0]

ET I 18(13)

[Re β > 0,

b > 0]

ET I 18(14)

ln (sin πx) sin 2nπx dx = 0

GW (338)(3a)

 ln (sin πx) sin(2n + 1)πx dx = 2

1/2

ln (sin πx) sin(2n + 1)πx dx  n  1 1 2 ln 2 − −2 = (2n + 1)π 2n + 1 2k − 1 0

k=1

GW (338)(3b)

3.6

 0

1

 ln (sin πx) cos 2nπx dx = 2

1/2

0

= − ln 2 1 =− 2n

ln (sin πx) cos 2nπx dx [n = 0] [n > 0] GW (338)(3c)

586

Logarithmic Functions



1

4. 0



ln (sin πx) cos(2n + 1)πx dx = 0

π/2

5. 0



π/2

6. 0

7.

8. 9. 10. 11. 12. 13.



π

14. 0



π

15. 0



ln sin x sin x dx = ln 2 − 1

BI (305)(4)

ln sin x cos x dx = −1

BI (305)(5)

π

0



LI (305)(6)

LI (330)(8)

BI (305)(7)

BI (305)(8)

BI (305)(9)

BI (305)(11)

BI (305)(16, 17)

0

ln (1 + p cos x) dx = π arcsin p cos x ln sin x

dx π 1 − a2 = ln 1 − 2a cos x + a2 1 − a2 2 a2 − 1 π ln = 2 a −1 2a2

2

p 0 4n ln sin x cos 2nx dx = π ⎪ 0 ⎩− ln 2, for n = 0 2  π π cos 2mn ln sin x cos[2m(x − n)] dx = − 2m 0  π/2 π ln sin x sin2 x dx = (1 − ln 4) 8 0  π/2 π ln sin x cos2 x dx = − (1 + ln 4) 8 0  π/2 1 ln sin x sin x cos2 x dx = (ln 8 − 4) 9 0  π/2 π2 ln sin x tan x dx = − 24 0  π/2  π/2 ln sin 2x sin x dx = ln sin 2x cos x dx = 2 (ln 2 − 1) 0

17.

4.384

π

ln sin bx

dx π 1 − a2b = ln 2 2 1 − 2a cos x + a 1−a 2

2

a 0, a > b] 2 2 a + b cos x 2 a −b a + a2 − b 2 0  π/2  π/2 dx dx ln sin x ln cos x 2 = 2 (a sin x ± b cos x) 0 0  (a cos x ± b sin x) a bπ 1 ∓a ln − = b (a2 + b2 ) b 2 [a > 0, b > 0] π

BI (331)(6)

BI (319)(1,6)a

588

Logarithmic Functions

 3.

π/2

4.



π/2

ln cos x dx b π ln = 2ab a + b b2 sin2 x + a2 cos2 x BI (317)(4, 10) [a > 0, b > 0]  π/2  π/2 sin 2x dx sin 2x dx ln sin x  ln cos x  2 = 2 2 2 2 0 0 a sin x + b cos x b sin x + a cos2 x a 1 ln = 2b(b − a) b [a > 0, b > 0] BI (319)(3, 7), LI (319)(3) 0



π/2

5. 0

ln sin x dx = a2 sin2 x + b2 cos2 x

ln sin x 

0

a2 sin2 x − b2 cos2 x 2 dx = a2 sin2 x + b2 cos2 x =

4.386



π/2

1. 0



π/2

2. 0



π/2

3. 0



π/2

4. 0

4.387 1.



3.

π/2

ln sin x 



sin x 2

1 + sin x

sin3 x ln sin x  dx = 1 + sin2 x ln sin x 

0



dx 1−

π/2

dx =

k 2 sin2

π/2

0

 0

π/2

ln cos x 

π 2b(a + b)

a2 cos2 x − b2 sin2 x 2 dx a2 cos2 x + b2 sin2 x [a > 0,

b > 0]

cos x ln cos x π √ dx = − ln 2 2 8 1 + cos x

ln 2 − 1 cos3 x ln cos x √ dx = 4 1 + cos2 x

π 1 = − K(k) ln k − K (k  ) 2 4 x



BI (322)Zsurround1, 6

BI (322)(2, 7)

BI (322)(3)

ln cos x dx k π 1  = K(k) ln − K (k  ) 2 2 k 4 1 − k 2 sin x

μ

LI (319)(2, 8)

BI (322)(9)

π/2

ln cos x cosμ x sinν x dx       μ+1 ν +1 μ+1 μ+ν +2 1 , ψ −ψ = B 4 2 2 2 2 [Re μ > −1, Re ν > −1] GW (338)(6c) μ √      π/2 πΓ μ+1 μ μ−1 2   −ψ ψ ln sin x sin x dx = μ+1 2 2 0 4Γ 2 0

2.

4.386

ν

ln sin x sin x cos x dx =

0

[Re μ > 0] ν  √      π/2 πΓ ν +1 ν  2  ψ −ψ ln sin x cosν−1 x dx = ν +1 2 2 0 4Γ 2

GW (338)(6a)

[Re ν > 0]

GW (338)(6b)

4.388

Logarithms and trigonometric functions

 2n  (−1)k+1 π (2n − 1)!! − ln 2 ln sin x sin2n x dx = (2n)!! 2 k 0 k=1 2n+1  π/2  (−1)k (2n)!! 2n+1 + ln 2 ln sin x sin x dx = (2n + 1)!! k 0 k=1  n  π/2 (2n − 1)!! π  1 2n + ln 4 ln sin x cos x dx = − (2n)!! 4 k 0 k=1 (2n − 1)!! π [C + ψ(n + 1) + ln 4] =− (2n)!! 4 

4.

5.

6.

589

π/2

FI II 811

BI (305)(13)

BI (305)(14)



π/2

7. 0

n 

(2n)!! 1 (2n + 1)!! 2k + 1 k=0     3 1 (2n)!! ψ n+ −ψ =− 2(2n + 1)!! 2 2

ln sin x cos2n+1 x dx = −

GW (338)(7b)



π/2

8. 0



π/2

9. 0

10.12



π/2

0

ln cos x sin2n x dx = −

ln cos x cos2n x dx = −

(2n − 1)!! π {C + 2 ln 2 + ψ(n + 1)} 2n+1 · n! 2 (2n − 1)!! π 2n n! 2

 ln 2 +

k=1



ln cos x cos2n−1 x dx =

2n  (−1)k

2n−1 (n − 1)! ln 2 + (2n − 1)!!



BI (306)(8)

k BI (306)(10)

2n−1  k=1

(−1)k k BI (306)(9)

4.388 1.

2.

3.

4.

5.

 n−1 π  (−1)k 1 1 sin2n x ln 2 + (−1)n + ln sin x 2n+2 dx = cos x 2n + 1 2 4 2n − 2k − 1 0 k=0   π/4 n−1  (−1)k 1 sin2n−1 x − ln 2 + (−1)n ln 2 + ln sin x 2n+1 dx = cos x 4n n−k 0 k=1   π/4 n π  (−1)k−1 1 1 sin2n x − ln 2 + (−1)n+1 + ln cos x 2n+2 dx = cos x 2n + 1 2 4 2n − 2k + 1 0 k=0   π/4 n−1  (−1)k 1 sin2n−1 x − ln 2 + (−1)n ln 2 + ln cos x 2n+1 dx = cos x 4n n−k 0 k=0  π/2 π pπ sinp−1 x [0 < p < 2] ln sin x p+1 dx = − cosec cos x 2p 2 0



π/4

BI (288)(1)

LI (288)(2)

BI (288)(10)

BI (288)(11)

BI (310)(4)

590

Logarithmic Functions



π/2

6. 0

4.389



π

1. 0



π/4

0 π/4

3. 0



π/2

4. 0

5.3



π 2

−π 2

 6. 0

4.391



π/4

1. 0



tan

π/4

2.

ln cos x cos

p−1



π/4

0

4.12



π/4

0

4.392



π/4

1. 0

 2. 0

1 β(μ) 4(1 − μ)

π/2

0

x sin px sin x dx =

n

(ln cos 2x) cosp−1 2x tan x dx =

n

(ln sin 2x) sinp−1 2x tan

[Re μ > 0]

ln cos x cosμ−1 x sin x dx = − [Re μ > 0]

BI (286)(2)

1 μ2 BI (306)(11)

π 2p+2



π/4



[p > −1] 1 C + ψ(p) − − 2 ln 2 p [p > 0]

n

(ln sin 2x) sinp−1 2x tan

0

BI (306)(12)

 1 − x dx = β (n) (p) 4 2

π

[p > 0]

BI (286)(10), BI (285)(18)

π

(ln cos 2x)2n−1 tan x dx = (ln cos 2x)2n tan x dx =

BI (285)(2)

π [C + ψ(p + 1) − 2 ln 2] 2p+1

0

3.



BI (330)(9)

1 {C + ψ(n + 2) + ln 2} 4(n + 1)

ln cos x cosμ−1 2x tan 2x dx =

ln sin x sinμ−1 x cos x dx =

BI (310)(3)

(2n − 1)!! π (2n)!! 4n + 2

ln sin x cosn 2x sin 2x dx = −

ln cos x cosp x cos px dx =

π/2

2

p 1

π 1−a ln tan bx dx = ln 2 2 1 − 2a cos 2x + a 1−a 1 + ab

5. 

2

a 0]

BI (331)(24)

ln tan x

arcsin a cos 2x dx =− (π + arcsin a) 1 − a sin 2x 4a

2

a ≤1

BI (291)(2,3)

ln tan x

π cos 2x dx = − arcsin a 2 2 4a 1 − a sin 2x

2 a 0]

GW (338)(19)

[a > 0]

GW (338)(19)

4.421

4.416

Logarithms, trigonometric functions, and powers



π/2 cos x ln

   1 + sin2 β − cos2 β tan2 α sin2 x

1.

1 − sin2 α cos2 x

0

2.

3.

4.7

597

dx

= cosec 2α {(2α + 2γ − π) ln cos β + 2 L(α) − 2 L(γ) + L(α + γ) − L(α − γ)}

sin α π ; 0 0]

ET I 17(3), NT 27(11)a

4. 

b > 0]

ET I 76(5), NT 27(10)a

3.



FI II 810a

∞

ln ax sin bx

x dx π = {− si(bc) sin bc + cos bc [ln ac − ci(bc)]} x2 − c2 2 [a > 0, b > 0,

c > 0]

BI (422)(5)

dx π {sin bc [ci(bc) − ln ac] − cos bc si(bc)} = x2 − c2 2c [a > 0, b > 0,

c > 0]

BI (422)(6)

ln ax cos bx

598

4.422

Logarithmic Functions



∞

1. 0



∞

2. 0

4.423



∞

1. 0



∞

2. 0

 0



∞

1. 0

2.12



∞

0

4.425



∞

1. 0

2.12 3.



π μπ  Γ(μ) μπ  ψ(μ) − ln a + cot sin aμ 2 2 2

[a > 0, |Re μ| < 1]  π Γ(μ) μπ μπ  ψ(μ) − ln a − tan ln(cos ax)xμ−1 dx = μ cos a 2 2 2

a cos ax − cos bx dx = ln ln x x b ln x

  1 C + ln ab 2

∞

ln x

BI (411)(5)

[a > 0,

0 < Re μ < 1]

[a > 0,

b > 0]

GW (338)(21a)

b > 0]

GW (338)(21b)

BI (411)(6)

cos ax − cos bx π dx = [(a − b) (C − 1) + a ln a − b ln b] 2 x 2 [a > 0,

∞

3. 4.424

ln(sin ax)xμ−1 dx =

4.422

2

sin ax aπ (C + ln 2a − 1) dx = − 2 x 2

(ln x)2 sin ax

[a > 0]

GW (338)(20b)

π π dx π3 = C2 + + πC ln a + (ln a)2 x 2 24 2

[a > 0] ET I 77(9), FI II 810a  Γ(μ) μπ  μπ ψ (μ) + ψ 2 (μ) + π ψ(μ) cot − 2 ψ(μ) ln a (ln x)2 (sin ax) xμ−1 dx = μ sin a 2 2  μπ 2 + (ln a) − π 2 − π ln a cot 2 [a > 0, 0 < Re μ < 1] ET I 77(10)

ln(1 + x) cos ax 

b+x b−x



$ 1# dx 2 2 = [si(a)] + [ci(a)] x 2

dx = −2π si(ab) x 0  ∞  a   dx = −π Ei − ln 1 + b2 x2 sin ax x b 0 ln2

cos ax

[a > 0]

ET I 18(8)

[a > 0,

b > 0]

[a > 0,

b > 0]

ET I 18(11)

GW (338)(24), ET I 77(14)

 4. 0

4.426 1.11

 0

1

  1 pπ dx π = 2+ coth ln 1 − x2 cos (p ln x) x 2p 2p 2

∞

x ln

LI (309)(1)a

b2 + x2 π sin ax dx = 2 (1 + ac)e−ac − (1 + ab)e−ab 2 2 c +x a [b ≥ 0,

c ≥ 0,

a > 0]

GW (338)(23)

4.431

Logarithms, trigonometric functions, and powers



∞

2.

ln 0

  ap   ap  b2 x2 + p2 dx = π Ei − − Ei − sin ax c2 x2 + p2 x c b [b > 0,

599

c > 0,

p > 0,

a > 0] ET I 77(15)

 4.427 4.428 1.12

∞

0



∞

0

 2.

3.

 sin ax   π π ln x + β 2 + x2  dx = K 0 (aβ) + ln(β) [I 0 (aβ) − L(aβ)] 2 2 2 2 β +x [Re β > 0, a > 0] ⎛

ln cos2 ax

cos bx dx = π ⎝b ln 2 − a + x2

b  2a 



n=1

(−1)n

⎞ (b − 2an) ⎠ n [a > 0,

[a > 0, ∞

0

ln cos2 ax

x2

0



1

∞

0 ∞

2. 0



b > 0]

∞

3. 0

b > 0]

4. 0

ET I 22(31)

ln (2 ± 2 cos x)

ln (2 ± 2 cos x)

BI (326)(2)a

sin bx x dx = − π sinh(bc) ln (1 ± e−c ) x2 + c2 [b > 0,

c > 0]

ET I 22(32)

[b > 0,

c > 0]

ET I 22(32)

  cos bx π dx = cosh(bc) ln 1 ± e−c 2 2 x +c c

b  sin bx  π  (−a)k dx = − [1 + sign(b − k)] ln 1 + 2a cos x + a2 x 2 k k=1

[0 < a < 1, 

ET I 82(36)

(1 + x)x π sin (ln x) dx = ln x 4

1. 

ET I 22(30)

    cos bx dx = −π ln 1 + e−2a cosh b + b + e−b π ln 2 − aπ 2 (1 + x ) [a > 0,

 4.431

ET I 22(29)

∞

4.

4.429

b > 0]

    cos bx π ln 4 cos2 ax 2 dx = cosh(bc) ln 1 + e−2ac 2 x +c c 0  π a < b < 2a < c  ∞     sin bx dx = π ln 1 + e−2a sinh b − π ln 2 1 − e−b ln cos2 ax 2) x (1 + x 0 

ET I 77(16)

∞

b > 0]

ET I 82(25)

b  cos bx  π  ak π  −c sinh[c(b − k)] cosh(bc) + ln 1 − 2a cos x + a dx = ln 1 − ae x2 + c2 c c k



2

k=1

[|a| < 1,

b > 0,

c > 0]

ET I 22(33)

600

4.432

Logarithmic Functions



∞

1. 0

  dx sin x = ln 1 − k 2 sin2 x  2 2 1 − k sin x x



∞

0

4.432

  dx sin x = ln k  K(k) ln 1 − k 2 cos2 x √ 1 − k 2 cos2 x x BI ((412, 414))(4)



π/2

2. 0

  sin x cos x x dx ln 1 − k 2 sin2 x  1 − k 2 sin2 x   " 1 ! = 2 πk  (1 − ln k  ) + 2 − k 2 K(k) − (4 − ln k  ) E(k) k

BI (426)(3)



π/2

3. 0



BI (426)(6) ∞

4. 0



 $  sin x cos x dx 1 # 2 = 2 2 − k 2 − k  ln k  K(k) − (2 − ln k  ) E(k) ln 1 − k 2 sin2 x  k 1 − k 2 sin2 x x 

BI (412)(5) ∞

5. 0



    sin x cos x " 1 ! ln 1 − k 2 cos2 x √ x dx = 2 −π − 2 − k 2 K(k) + (4 − ln k  ) E(k) 2 2 k 1 − k cos x

∞

6. 0

   sin x cos x dx " 1 ! = 2 k 2 − 2 + ln k  K(k) + (2 − ln k  ) E(k) ln 1 − k 2 cos2 x √ 2 2 x k 1 − k cos x 

BI (414)(5) ∞

    dx dx sin x sin x = ln 1 ± k sin2 x  ln 1 ± k cos2 x √ 2 2 1 − k cos x x 1 − k 2 sin2 x x 0 ∞   dx tan x 2 ln 1 ± k sin x  = 2 2 1 − k sin x x 0 ∞   dx tan x = ln 1 ± k cos2 x √ 2 2 1 − k cos x x 0 ∞   dx tan x 2 ln 1 ± k sin 2x  = 2 1 − k 2 sin 2x x 0 ∞   dx tan x 2 2 = ln 1 ± k cos 2x √ 2 cos2 2x x 1 − k 0 π 1 2 (1 ± k) K(k) − K (k  ) = ln √ 2 8 k BI (413)(1–6), BI (415)(1–6)



∞

   " 1 ! dx sin3 x = 2 k 2 − 2 + ln k  K(k) + (2 − ln k  ) E(k) ln 1 − k 2 sin2 x  2 2 x k 1 − k sin x

∞

 $   dx 1 # sin x 2 = 2 2 − k 2 − k  ln k  K(k) − (2 − ln k  ) E(k) ln 1 − k 2 cos2 x √ k 1 − k 2 cos2 x x

∞

 $   sin x cos x dx 1 # 2 = 2 2 − k 2 − k  ln k  K(k) − (2 − ln k  ) E(k) ln 1 − k 2 sin2 x  k 1 − k 2 sin2 x x

7. 0

BI (412)(6)

 8. 0

 9. 0

3

BI (414)(6)a 2

BI (412)(7)

4.432

Logarithms, trigonometric functions, and powers



∞

10. 0



∞

11. 0

601

   sin x cos2 x dx " 1 ! = 2 k 2 − 2 + ln k  K(k) + (2 − ln k  ) E(k) ln 1 − k 2 cos2 x √ 2 2 k 1 − k cos x x   dx tan x = ln 1 − k 2 sin2 x  2 2 1 − k sin x x

BI (414)(7)



∞

0

  dx tan x = ln k  K(k) ln 1 − k 2 cos2 x √ 1 − k 2 cos2 x x BI ((412, 414))(9)



∞

   sin x tan x dx " 1 ! = 2 k 2 − 2 + ln k  K(k) + (2 − ln k  ) E(k) ln 1 − k 2 sin2 x  2 k 1 − k 2 sin x x

∞

 $   sin x tan x dx 1 # 2 = 2 2 − k 2 − k  ln k  K(k) − (2 − ln k  ) E(k) ln 1 − k 2 cos2 x √ k 1 − k 2 cos2 x x

12. 0

2

BI (412)(8)

 13. 0

14.12



∞

0

2

  ln 1 − k 2 sin2 x 



BI (414)(8) ∞

  dx sin x ln 1 − k 2 cos2 x  3 x 0 (1 − k 2 cos2 x)  " 1 ! = 2 k 2 − 2 K(k) + (2 + ln k  ) E(k)  k

sin x

dx =  3 x 1 − k 2 sin2 x

BI ((412, 414))(13)



π/2

15. 0

$   1 # π sin x cos x ln 1 − k 2 sin2 x  x dx = 2 (1 + ln k  )  − (2 + ln k  ) K(k)  3 k k 1 − k 2 sin2 x BI (426)(9)

 16.

17.

π/2

  sin x cos x 1 ln 1 − k 2 cos2 x  x dx = 2 {−π + (2 + ln k  ) K(k)} BI (426)(15) k 3 0 (1 − k 2 cos2 x)  ∞  ∞     dx dx sin x cos x sin3 x 2 2  = ln 1 − k 2 sin2 x  ln 1 − k cos x 3 x x 3 2 0 0 (1 − k 2 cos2 x) 1 − k 2 sin x  " 1 ! = 2 2 − k 2 + ln k  K(k) − (2 + ln k  ) E(k) k

BI (412)(14), BI(414)(15)

 18. 0

∞

  ln 1 − k 2 sin2 x 

sin3 x

dx 3 x 1 − k 2 sin x  = 2

∞

  dx sin x cos x ln 1 − k 2 cos2 x  x 3 0 (1 − k 2 cos2 x)  $  1 # 2 = (2 + ln k  ) E(k) − 2 − k 2 + k  ln k  K(k) 2 k2 k

BI (412)(15), BI(414)(14)

602

Logarithmic Functions



∞

19. 0



4.432

∞

  dx sin2 x tan x ln 1 − k 2 cos2 x  x 0 (1 − k 2 cos2 x)3  " 1 ! = 2 2 − k 2 + ln k  K(k) − (2 + ln k  ) E(k) k

  dx sin x cos2 x = ln 1 − k 2 sin2 x   3 x 1 − k 2 sin2 x

BI (412)(16), BI(414)(17)



∞

20. 0

2

  dx sin x tan x ln 1 − k 2 sin2 x   3 x 1 − k 2 sin2 x  =

∞

  dx sin x cos2 x ln 1 − k 2 cos2 x  x 3 0 (1 − k 2 cos2 x)  $  1 # 2 = 2  2 (2 + ln k  ) E(k) − 2 − k 2 + k  ln k  K(k) k k



∞

21. 0





 ln 1 − k 2 sin2 x 



BI (412)(17), BI(414)(16)

∞



 dx dx tan x tan x 2 2  = ln 1 − k cos x  3 x x 3 0 (1 − k 2 cos2 x) 1 − k 2 sin2 x  " 1 ! =  2 k 2 − 2 K(k) + (2 + ln k  ) E(k) k

BI ((412, 414))(18)

∞

22. 0

 ∞     dx dx 2 2 2 2 = ln 1 − k sin x 1 − k sin x sin x ln 1 − k 2 cos2 x 1 − k 2 cos2 x sin x x x 0   2  = 2 − k K(k) − (2 − ln k ) E(k)

BI ((412, 414))(1)



π/2

23. 0

  ln 1 − k 2 sin2 x 1 − k 2 sin2 x sin x cos x · x dx $   1 # 3  2 4 2  2  3πk K (k) − (2 − k (1 − 3 ln k ) + 22k + 6k − 3k ln k )(14 − 6 ln k )E (k) = 27k 2 BI (426)(1)



π/2

24. 0

 25. 0

  ln 1 − k 2 cos2 x 1 − k 2 cos2 x sin x cos x · x dx #   $   1 2 4 2  2  −3π − 22k K(k) + 2 − k (14 − 6 ln k + 6k − 3k ln k ) E(k) = 27k 2 BI (426)(2)

∞

 ∞     dx dx = ln 1 − k 2 sin2 x 1 − k 2 sin2 x tan x ln 1 − k 2 cos2 x 1 − k 2 cos2 x tan x x x 0   2  = 2 − k K(k) − (2 − ln k ) E(k)

((412,414))(2)

4.521

Arcsines, arccosines, and powers



∞

26. 0

603

 ∞    dx dx sin x tan x = ln sin x + k cos x √ ln sin2 x + k  cos2 x √ 2 2 2 2 x 1 − k cos x 1 − k cos x x 0 ∞  2  dx tan x ln sin 2x + k  cos2 2x √ = 2 2 1 − k cos 2x x 0 ⎡ √ 3 ⎤ k ⎥ 1 ⎢2 = ln ⎣ ⎦ K(k) 2 1 + k 

2



2

BI (415)(19–21)

4.44 Combinations of logarithms, trigonometric functions, and exponentials 4.441 1.12



∞

0



∞

2. 0

 4.442

e−qx sin px ln x dx =



 p  2 p 1 2 − pC − ln p q arctan + q p2 + q 2 q 2

e−qx cos px ln x dx = −

p2

1 + q2



p > 0]

 q  2 p ln p + q 2 + p arctan + qC 2 q

π/2 −p tan x

e

0

ln cos x dx 1 1 2 2 = − [ci(p)] + [si(p)] sin x cos x 2 2

[q > 0,

BI (467)(1)

[q > 0]

BI (467)(2)

[Re p > 0]

NT 32(11)

4.5 Inverse Trigonometric Functions 4.51 Inverse trigonometric functions 

4.511

0

 4.512

∞

π

0

π arccot px arccot qx dx = 2



    1 p 1 q ln 1 + + ln 1 + p q q p [p > 0, q > 0]

arctan (cos x) dx = 0

BI (77)(8) BI (345)(1)

4.52 Combinations of arcsines, arccosines, and powers 4.521



1

1. 0



1

2. 0

4.

FI II 614, 623

arccos x π dx = ∓ ln 2 + 2G 1±x 2

BI (231)(7, 8)

√ 2 1+q x π √ ln dx = 1 + qx2 2q 1 + 1 + q 0   1 x π 1 + 1 − p2 arcsin x dx = 2 ln  1 − p2 x2 2p 2 1 − p2 0 

3.

arcsin x π dx = ln 2 x 2

1

arcsin x

[q > −1]

BI (231)(1)

2

p −1]

BI (235)(10)

[q > −1]

BI (234)(2)

[q > −1]

BI (234)(4)



   1 3 2 2 π + k  K(k) − 2 1 + k  E(k) x 1 − k 2 x2 arccos x dx = 2 9k 2 0

 1    3 1 3 2 2 x 1 − k 2 x2 arcsin x dx = 2 − πk  − k  K(k) + 2 1 + k  E(k) 9k 2 0

 1    1 3 2 2 2  2 2 π + k K(k) − 2 1 + k E(k) x k + k x arcsin x dx = 2 9k 2 0  1  x arcsin x 1  π √ dx = 2 − k  + E(k) k 2 1 − k 2 x2 0  1  1 π x arccos x √ dx = 2 − E(k) k 2 1 − k 2 x2 0  1  x arcsin x 1 π  − E(k) dx = 2 k 2 0 k  2 + k 2 x2  1  x arccos x 1  π  dx = 2 − k  + E(k) k 2 0 k  2 + k 2 x2  1 ∞ x arcsin x dx 2  sin[(2k + 1)λ] √ = 2 2 2 sin λ (2k + 1)2 0 (x − cos λ) 1 − x k=0  1 x arcsin kx π  dx = − ln k  2 ) (1 − k 2 x2 ) 2k (1 − x 0  1 x arccos kx π  ln(1 + k) dx = 2 2 2 2k (1 − x ) (1 − k x ) 0



1



π 2n n! 1 − 2n + 1 2 (2n + 1)!! 0

 1 (2n − 1)!! π 1− x2n−1 arcsin x dx = 4n 2n n! 0  1 2n n! x2n arccos x dx = (2n + 1)(2n + 1)!! 0



4.522

1

x2n arcsin x dx =

BI (236)(9)

BI (236)(1)

BI(236)(5)

BI (237)(1)

BI (240)(1)

BI (238)(1)

BI (241)(1)

BI (243)(11)

BI (239)(1)

BI (242)(1)

BI (229)(1)

BI (229)(2)

BI (229)(4)

4.531

Arctangents, arccotangents, and powers



1

4. 0



x2n−1 arccos x dx =

1

5.



n



n− 12

1 − x2

−1



1

6.

1 − x2

−1

4.524



1

1. 0



1

2. 0

(arcsin x)

π (2n − 1)!! 4n 2n n!

arccos x dx = π

x2

BI (229)(5)

2n n! (2n + 1)!!

arccos x dx =

2

605

BI (254)(2)

π 2 (2n − 1)!! 2 2n n!

BI (254)(3)

dx √ = π ln 2 1 − x2 dx

2

(arccos x) √

1 − x2

BI (243)(13)

3 = π ln 2

BI (244)(9)

4.53–4.54 Combinations of arctangents, arccotangents, and powers 4.531



1

1. 0



arctan x dx = x

∞

0



1

BI (235)(11)

∞

0



1

0

 0

7. 0



arctan qx

arccot qx



0

10. 11.12

q dx 1 (1 + p)2 q2 − p arctan q = ln + 2 2 2 2 (1 + px) 2p +q 1+q (1 + p) (p2 + q 2 )

∞

∞

BI (243)(7)

2

q dx 1 1+q p 1 arccot q = ln + 2 arctan q + 2 2 2 2 2 (1 + px) 2p +q (1 + p) p +q 1+p

arctan x π 1 dx = ln 2 + G 2 x (1 + x ) 8 2

9. 

BI (248)(2)

[p > −1] 1

8. 0

arctan x dx = −G. 1 − x2

[p > −1] 1

6. 

FI II 482, BI (253)(8)

π arctan x dx = − ln 2 + G x(1 + x) 8

4. 5.

arccot x dx = G x

BI (248)(6, 7)

0



1

∞

arccot x π dx = ± ln 2 + G 1±x 4

2. 3.12



BI (234)(10) BI (235)(12)

x arctan x π2 dx = 4 1+x 16

BI (248)(3)

x arctan x π dx = − ln 2 1 − x4 8

BI (248)(4)

∞

x arccot x π dx = ln 2 4 1 − x 8 0  ∞  ∞ arctan x arccot x √ √ dx = dx = 2G 2 1 + x2 0 x 1+x 0

BI (248)(12) BI (251)(3, 10)

606

Inverse Trigonometric Functions

 12.

13. 4.532

√  arctan x π  √ dx = ln 1 + 2 2 2 0 x 1−x ⎤ ⎡  1 x arctan x dx π 1 ⎣ π  ⎦   = k2 F 4 , k −     2 2 2 2   0 (1 + x ) 1 + k x 2 2 1+k 

1

1

1. 0

 2. 3. 4. 5. 4.533

xp arctan x dx =

∞

p  π 1 −β +1 2(p + 1) 2 2

pπ π cosec 2(p + 1) 2 0  1    π p 1 +β +1 xp arccot x dx = 2(p + 1) 2 2 0  ∞ pπ π cosec xp arccot x dx = − 2(p + 1) 2 0 √   ∞ 2q dx π3 Γ(q) xp   = 2q+2 arctan x 2p 1 + x x 2 p Γ q + 12 0 

∞

1. 0



xp arctan x dx =

(1 − x arccot x) dx =

1

π

FI II 694

BI (294)(14)

[p > −2]

BI (229)(7)

[−1 > p > −2]

BI (246)(1)

[p > −1]

BI (229)(8)

[−1 < p < 0]

BI (246)(2)

[q > 0]

π 4

 dx π − arctan x = − ln 2 + G 4 1−x 8 0  1  1 + x dx 1 π π − arctan x 3. = ln 2 + G 2 4 1 − x 1 + x 8 2 0   1 dx 1 π x arccot x − arctan x 4. = − ln 2 2 x 1 − x 4 0  ∞  ∞ dx x dx π2 2 2 √ + 4G 4.534 (arctan x) = (arccot x) √ =− 4 x2 1 + x2 1 + x2 0 0 4.535  1   arctan px 1 dx = 2 arctan p ln 1 + p2 1. 2 2p 0 1+p x

 1   arccot px 1 π 1 dx = + arccot p ln 1 + p2 [p > 0] 2. 2 2 p 4 2 0 1+p x  ∞  π  q arctan qx pq [p > 0, q > 0] ln pq − 3. dx = − (p + x)2 1 + p2 q 2 2 0    ∞ π q arccot qx [p > 0, q > 0] ln pq + 4. dx = (p + x)2 1 + p2 q 2 2pq 0  ∞ x arccot px π 1 + pq [p > 0, q > 0] 5. dx = ln 2 + x2 q 2 pq 0 2.

4.532

BI (250)(10)

BI (246)(3)

BI (232)(2)

BI (235)(25)

BI (232)(1)

BI (251)(9, 17)

BI (231)(19)

BI (231)(24) BI (249)(1) BI (249)(8) BI (248)(9)

4.537

Arctangents, arccotangents, and powers



∞

6. 0

 7. 8. 9. 10.12 11. 12. 13.

14.9

9



∞

∞

 0



∞

3. 0

2.

BI (248)(10)





FI II 745 BI (250)(6) BI (250)(3) BI (250)(6) BI (252)(12)a

BI (252)(20)a

BI (244)(11)

for α, β, γ real sign(α)

[α, β, γ real;

β = γ]

[β = γ]

  π  dx 1 π  1 + 1 + q2 arctan qx arcsin x 2 = qπ ln  + ln q + 1 + q 2 − − arctan q x 2 2 2 1 + q2 BI (230)(7)

∞

2.

1.

⎧ ⎪ ⎨

π 1 + |α/γ| ln x arctan (α/x) dx 2 − γ2 β 1 + |α/β| = πα 2 2 2 2 ⎪ −∞ (x + β ) (x + γ ) ⎩ 2 2β (|β| + |α|)



0

8

q > 0]

∞

1.

4.537

[p > 0,

π arctan px dx = ln(1 + p) [p ≥ 0] 2 2 0 x (1 + x )  ∞  π  arctan px dx = ln 1 + p2 [p ≥ 0] 2 4 0 x (1 − x )  ∞ π dx = 2 ln(1 + pq) arctan qx [p > 0, q ≥ 0] 2 2 x (p + x ) 2p 0  ∞ π p2 + q 2 dx = ln arctan qx [q ≥ 0] 2 2 x (1 − p x ) 4 p2 0  ∞ x arctan qx πq [p > 0, q ≥ 0] 2 dx = 4p(1 + pq) 2 2 (p + x ) 0  ∞ x arccot qx π [p > 0, q ≥ 0] 2 dx = 4p2 (1 + pq) 2 2 (p + x ) 0  1   π  arctan qx √ dx = ln q + 1 + q 2 2 2 0 x 1−x   ⎧ π 1 + |αβ| ⎪  ∞ ⎨ sign(α) for β = γ ln x arctan(αx) dx β2 − γ2 1 + |αγ| = πα 2 2 2 2 ⎪ −∞ (x + β ) (x + γ ) ⎩ for β = γ 2|β| (1 + |αβ|)

15.

4.536

π 1 + p2 q 2 x arccot px dx ln = x2 − q 2 4 p2 q 2

607

π p arctan px − arctan qx dx = ln x 2 q

[p > 0,

q > 0]

FI II 635

arctan px arctan qx π (p + q)p+q dx = ln 2 x 2 pp q q

[p > 0,

q > 0]

FI II 745

    π π − 4λ π + 4λ dx = ln cos cosec arctan 1− 1 − x2 cos2 λ 2 cos λ 8 8 0  1  dx      1 arctan p 1 − x2 = π ln p + 1 + p2 2 1−x 2 0 

1



x2



[p > 0]

BI (245)(9)

BI (245)(10)

608

3.12

Inverse Trigonometric Functions



1

0



1

4. 0



1

5. 0

4.538  1.

  arctan tan λ 1 − k 2 x2 

   arctan tan λ 1 − k 2 x2





 1 − x2 π  2 E (λ, k) − k dx = F (λ, k) 1 − k 2 x2 2k 2    π − 2 cot λ 1 − 1 − k 2 sin2 λ 2k BI (245)(12)

   1 − k 2 x2 π π 2 sin2 λ E (λ, k) − cot λ 1 − dx = 1 − k 1 − x2 2 2 BI (245)(11)

√  arctan tan λ 1 − k 2 x2 π  dx = F (λ, k) 2 2 2 2 (1 − x ) (1 − k x ) 

BI (245)(13)

 ∞ dx dx arctan x = arctan x3 2 1+x 1 + x2  0 0 ∞ ∞ π2 dx dx 3 = arccot x2 = arccot x = 1 + x2 1 + x2 8 0 0  1   2 √ 1−x π 2.12 arctan x2 dx = 2−1 2 x 2 0  ∞  ∞ dx π2 n dx [n is an integer] 3.∗ arctan xn = arccot x = 1 + x2 1 + x2 8 0 0  ∞     xs−1 arctan ae−x dx = 2−s−1 Γ(s)a Φ −a2 , s + 1, 12 4.539 0   ∞  x dx p sin qx π  4.541 arctan = ln 1 + pe−q [p > −eq ] 2 1 + p cos qx 1 + x 2 0 ∞

4.538

2

BI (252)(10, 11) BI (252)(18, 19) BI (244)(10)a

ET I 222(47) BI (341)(14)a

4.55 Combinations of inverse trigonometric functions and exponentials 4.551 1.9



1

0



1

2. 0

3.9



(arcsin x) e−bx dx =

x (arcsin x) e−bx dx =

∞

0

4.9



∞

0

4.552

π πe−b [I 0 (b) − L0 (b)] − 2b 2b

ET I 160(1)

π 1 [L0 (b) − I 0 (b) + b L1 (b) − b I 1 (b)] + 2b2 b

ET I 161(2)

 x  −bx 1 e arctan dx = [ci(ab) sin(ab) − si(ab) cos(ab)] a b

[Re b > 0]     x −bx 1 π e arccot dx = − ci(ab) sin(ab) + si(ab) cos(ab) a b 2

ET I 161(3)

[Re b > 0]

ET I 161(4)

x  

 ∞ arctan 1 1 1 q dx = ln Γ(q) − q − ln q + q − ln 2π e2πx − 1 2 2 2 0 [q > 0]

WH

4.574

Inverse and direct trigonometric functions

 4.553

∞

0



2 arccot x − e−px π



dx = C + ln p x

609

[p > 0]

NT 66(12)

4.56 A combination of the arctangent and a hyperbolic function 

∞

1 arctan e−x dx = 2q 2 cosh px −∞

4.561

√ Π(x) π 3 Γ(q)   dx = 2q 4p Γ q + 12 −∞ cosh px [q > 0]



∞

LI (282)(10)

4.57 Combinations of inverse and direct trigonometric functions  4.571 4.572 4.573 1. 2.

3.12

π/2

π sin x dx = − ln k  arcsin (k sin x)  2 2k 0 1 − k 2 sin x  ∞ 2 arccot x − cos px dx = C + ln p π 0



BI (344)(2)

[p > 0]

 p π  1 − e− q [p > 0, 2p 0      ∞ p 1 p p −p q q e Ei − e Ei − arccot qx cos px dx = 2p q q 0 

∞

[p > 0, ∞

0



arccot qx sin px dx =

∞

4. 0

arccot rx

1±q sin px dx π ln =± p 1 ± 2q cos px + q 2 2pq 1 ± qe− r q±1 π ln =± p 2pq q ± e− r



 arctan

0

2.

7



∞

arctan 0



arctan 0

 4.

2a x

 sin(bx) dx =

π −ab sinh(ab) e b

BI (347)(2)a

r > 0,

p>0

2 q > 1,

r > 0,

p>0



BI (347)(10)

q > 0,

[Re a > 0,

[a > 0,

r > 0]

b > 0]

BI (347)(9)

ET I 87(8)

b > 0]

ET I 29(7)

√ π 2ax 2 2 sin(bx) dx = e−b a +c sinh(ab) 2 +c b

x2 

∞

arctan 0

q > 0]

1 −ab a cos(bx) dx = e Ei(ab) − eab Ei(−ab) x 2b

∞

3.

BI (347)(1)a

  1 π tan x dx r tanh = arccot px 2 ln 1 + 2r2 q p q cos2 x + r2 sin2 x

∞

1.

q > 0]

2 q < 1,

[p > 0, 4.574

NT 66(12)

2 x2

 cos(bx) dx =

π −b e sin b b

[b > 0]

ET I 87(9)

[b > 0]

ET I 29(8)

610

4.575

Inverse Trigonometric Functions



π

arctan

1. 0

3.

4.576 1. 2. 4.577

2

p q 2 , lim f (x) = 0  2.



2π

dω 0

0

x→+∞

∞

f  [p cosh x + (q cos ω + r sin ω) sinh x] sinh x dx

LO III 389

   2π sign p f sign p p2 − q 2 − r2 = − p2 − q 2 − r 2

2 2 2 p >q +r , lim f (x) = 0 LO III 390 x→+∞

4.622

Double and triple integrals with constant limits

 π

π

3. 0

0

615

   dx dy 2π sign p  p − q cos x 2 − q 2 − r2  + r cot y = − f f sign p p sin x sin y sin x sin2 y p2 − q 2 − r2

p2 > q 2 + r 2 , lim f (x) = 0 x→+∞



∞

4.

LO III 280

 dx

−∞

∞

−∞

f  (p cosh x cosh y + q sinh x cosh y + r sinh y) cosh y dy    2π sign p = − f sign p p2 − q 2 − r2 p2 − q 2 − r 2

2 2 2 p >q +r , lim f (x) = 0 LO III 390 x→+∞

 5.

 π  ∞  ∞   2 dx f (p cosh x + q cos ω sinh x) sinh x sin ω dω = 2 f sign p p2 − q 2 cosh x sinh2 x dx 0 0 0

lim f (x) = 0 LO III 391

 6.

 ∞ dx

0



2π

x→+∞

π

dω

0

0

f [p cosh x + (q cos ω + r sin ω) sin θ sinh x] sinh2 x sin θ dθ  ∞    =4 f sign p p2 − q 2 − r2 cosh x sinh2 x dx 0

2 2 2 p >q +r , lim f (x) = 0 LO III 390 x→+∞

 7.

∞

 dx

0



2π

dω

0

0

π

f {p cosh x + [(q cos ω + r sin ω) sin θ + s cosh θ] sinh x} sinh2 x sin θ dθ  ∞    = 4π f sign p p2 − q 2 − r2 − s2 cosh x sinh2 x dx 0

p 2 > q 2 + r 2 + s2 , lim f (x) = 0 LO III 391 x→+∞

4.621 1. 2. 3. 4.622

 1 − k 2 sin2 x sin2 y π dx dy = √ 2 2 1 − k sin y 2 1 − k2 0 0   π/2  π/2 cos y 1 − k 2 sin2 x sin2 y dx dy = K(k) 1 − k 2 sin2 y 0 0  π/2  π/2 sin α sin y dx dy πα  = 2 2 2 2 1 − sin α sin x sin y 0 0



π/2  π/2

 π π

π

1. 0

0

 π π

0 π

2. 0

0

0

sin y

dx dy dz = 4π K 2 1 − cos x cos y cos z

√  2 2

 √ dx dy dz π = 3π K 2 sin 3 − cos y cos z − cos x cos z − cos x cos y 12

LO I 252(90)

LO I 252(91)

LO I 253

MO 137

MO 137

616

Multiple Integrals

 π π

π

3. 0

0

0

4.623

  √ √  √ √ √  √  dx dy dz = 4π 18 + 12 2 − 10 3 − 7 6 K 2 2 − 3 3− 2 3 − cos x − cos y − cos z MO 137

4.6233



∞ ∞

0 π 

0 2π

  π ϕ a2 x2 + b2 y 2 dx dy = 2ab

 0

∞

  ϕ x2 x dx

f (α cos θ + β sin θ cos ψ + γ sin θ sin ψ) sin θ dθ dψ  1  π f (R cos p) sin p dp = 2π f (Rt) dt = 2π −1 0   R = α2 + β 2 + γ 2  a  b     −3/2 dx dy x2 + y 2 + 1 P l 1/ x2 + y 2 + 1 4.6258 pl (a, b) =

4.624

0

0

0

0

Then, for even and odd subscripts:    l+k l−1 (−1)l−k−1 22k 2l+2k  l+k l−k−1 ab 1   √ • p2l (a, b) = 2l 2 2 2k l(2l + 1)2 a + b + 1 k=0 (2k + 1) k     2j k  j 1 1 1 ×(2l + 2k + 1) + 22j (a2 + b2 + 1)j (a2 + 1)k−j+1 (b2 + 1)k−j+1 j=0     l  (−1)l+k l l+k+1 2l + 2k + 1 1 • p2l+1 (a, b) = 2l+1 2 (2l + 1) 22k k k l+k k=0 * b a a b 1 1 √ √ × arctan−1 √ + arctan−1 √ k k 2+1 2+1 2+1 2+1 2 2 b b a a (b + 1) (a + 1)  ⎫ k ⎬ 2j−1  2 1 1 1  · + +ab 2j (a2 + b2 + 1)j (a2 + 1)k−j+1 (b2 + 1)k−j+1 ⎭ j=1 j j 4.626∗



1

1.

 dx

0



1

 dx

0

1

0



0

1

dy 0

3. 4.

dy 0

2.



1

1

 dx  dx

1

GS

(xy)u−1 (− ln xy)s = Γ(s + 2)Φ(z, s + 2, u) 1 − xyz [z ∈ C − [1, ∞], Re(s) > −2] or [z = 1, Re(s) > −1]

GS

dy

−x ln xy π2 = G − 1 + x2 y 2 48

GS

dy

−x ln xy π2 = 2 2 1−x y 12

GS

0 1 0

xu−1 y v−1 Φ(z, s + 1, v) − Φ(z, s + 1, u) (− ln xy)s = Γ(s + 1) 1 − xyz u−v [z ∈ C − [1, ∞], Re(s) > −2] or [z = 1, Re(s) > −1]

4.632

Multiple integrals



1

5. 0



1

6. 0



1

7. 0



1

8. 0



1

9. 0



1

10. 0



1

11. 0



1

12. 0



1

13. 0



1

14. 0



1

15.

 dx  dx  dx  dx  dx  dx  dx  dx  dx  dx  dx

0

1

dy

1 =G 1 + x2 y 2

dy

− ln(1 − z) −1 = (1 − xyz) ln xy z

dy

1 = ζ(2) 1 − xy

GS

dy

− ln xy = 2ζ(3) 1 − xy

GS

dy

1 u xu−1 y v−1 = ln − ln xy u−v v

GS

dy

1 (xy)u−1 = − ln xy u

GS

dy

π −x = ln (1 + xy) ln xy 2

GS

0 1 0 1 0 1 0 1 0 1 0 1 0 1 0

617

GS

−x = ln dy (1 + x2 y 2 ) ln xy

1

dy 0 1

−1 (1 +

x2 y 2 ) ln xy

=

[z = 1]

 √  2π  2 Γ 34

GS

GS

π 4

GS

dy

y π = √ 1 − x3 y 3 3 3

GS

dy

1 π2 = 2 2 1−x y 8

GS

0 1 0

4.63–4.64 Multiple integrals

 x 1 4.631 dtn−1 dtn−2 . . . f (t) dt = (x − t)n−1 f (t) dt, (n − 1)! p p p p where f (t) is continuous on the interval [p, q] and p ≤ x ≤ q. 

4.632

x



 

tn−1

dx1 dx2 · · · dxn =

x1 ≥0,x2 ≥0,...,xn ≥0 x1 +x2 +···+xn ≤h

FI II 692

hn n! [the volume of an n-dimensional simplex]

√ πn  Rn  ··· dx1 dx2 · · · dxn = n + 1 Γ x21 +x22 +···+x2n ≤R2 2  

2.

t1

 ···

1.



FI III 472



[the volume of an n-dimensional sphere] FI III 473

618

Multiple Integrals

 



4.633

π (n+1)/2   [n > 1] n+1 Γ 2 2 2 x1 +x2 +···+xn ≤1 2

Half-area of the surface of an (n + 1)-dimensional sphere x21 + x22 + · · · + x2n+1 = 1    8 ··· 4.634 x1p1 −1 x2p2 −1 · · · xnpn −1 dx1 dx2 . . . dxn 4.633

···



x1 q1

4.635

dx1 dx2 · · · dxn  = 1 − x21 − x22 − · · · − x2n

α x1 ≥0,x  2α≥0,...,xn ≥0 α 1 2 x n n + q2 +···+( x ≤1 qn )

x1 q1



 ···



      p2 pn p1 Γ . . . Γ Γ q1p1 q2p2 . . . qnpn α1 α2 αn   = p1 p2 pn α1 α2 . . . αn Γ + + ··· + +1 α1 α2 αn [αi > 0, pi > 0, qi > 0, i = 1, 2, . . . , n] FI III 477

2

 

1.8

f

α x1 ≥0,x  2α≥0,...,xn ≥0 α 1 2 x n n + q2 +···+( x ≥1 qn )

FI III 474

x1 q1

α1

 +

x2 q2

α2

 + ···+

xn qn

αn

2

×x1p1 −1 x2p2 −1 · · · xpnn −1 dx1 dx2 · · · dxn       p2 pn p1  ∞ Γ . . . Γ Γ p1 p pn q1p1 q2p2 . . . qnpn α1 α2 αn + 2 +···+ α −1 n   = f (x)x α1 α2 dx p1 p2 pn α1 α2 · · · αn 1 Γ + + ···+ α1 α2 αn

2.8

under the assumption that the integral on the right converges absolutely.  α1  α2  αn    x1 x2 xn ··· f + + ···+ q1 q2 qn 

x1 q1

FI III 487

≥0,··· ,xn ≥0 α x1 ≥0,x  2α 1 2 x n αn ≤1 + q2 +···+( x qn ) 2

×x1p1 −1 x2p2 −1 · · · xpnn −1 dx1 dx2 · · · dxn       p2 pn p1  1 Γ ···Γ p1 p2 pn Γ p1 p2 pn q q . . . qn α α2 αn  1 = 1 2 f (x)x α1 + α2 +···+ αn −1 dx p1 p2 pn α1 α2 . . . αn 0 Γ + + ···+ α1 α2 αn under the assumptions that the one-dimensional integral on the right converges absolutely and that the numbers qi , αi , and pi are positive. FI III 479

4.636

3.

Multiple integrals

In particular,    ···

x1p1 −1 x2p2 −1 . . . xpnn −1 e−q(x1 +x2 +···+xn ) dx1 dx2 . . . dxn

x1 ≥0,x2 ≥0,...,xn ≥0 x1 +x2 +···+xn ≤1

=

8

 

 ···

4.

x1 ≥0,x2 ≥0,··· ,xn ≥0 α α n x1 1 +x2 2 +···+xα n ≤1

4.636 1.

8

 

 ···

x1 ≥0,x2 ≥0,...,xn ≥0 α α α x1 1 +x2 2 +···+xn n ≥1

2.8

 

619

 Γ (p1 ) Γ (p2 ) . . . Γ (pn ) 1 p1 +p2 +···+pn −1 −qx x e dx Γ (p1 + p2 + · · · + pn ) 0 [n > 0, pi > 0]

x1p1 −1 x2p2 −1 . . . xnpn −1 α2 αn μ dx1 dx2 . . . dxn 1 (1 − xα 1 − x2 − · · · − xn )       p2 pn p1 Γ ...Γ Γ Γ(1 − μ) α1 α2 αn   = p1 p2 pn α1 α2 . . . αn Γ 1−μ+ + + ··· + α1 α2 αn [pi > 0, μ < 1] FI III 480

x1p1 −1 x2p2 −1 . . . xpnn −1 α2 αn μ dx1 dx2 . . . dxn 1 (xα 1 + x2 + · · · + xn )      p2 pn p1 Γ ...Γ Γ 1 α1 α2 αn     = p1 p2 pn p1 p2 pn α1 α2 . . . αn μ − − − ··· − Γ + + ···+ α1 α2 αn α1 α2

αn p1 p2 pn pi > 0, μ > + + ···+ FI III 488 α1 α2 αn 

 ···

x1 ≥0,x2 ≥0,··· ,xn ≥0 α α n x1 1 +x2 2 +···+xα n ≤1

x1p1 −1 x2p2 −1 · · · xpnn −1 α2 αn μ dx1 dx2 . . . dxn 1 (xα 1 + x2 + · · · + xn )

     p2 pn p1 Γ ...Γ 1 α1 α2 αn     = p1 p2 pn p1 p2 pn α1 α2 . . . αn + + ···+ −μ Γ + + ···+ α1 α2 α1 α2 αn αn p1 p2 pn μ< + + ···+ FI III 480 α1 α2 αn  α2 αn 1 1 − xα 1 − x2 − · · · − xn x1p1 −1 x2p2 −1 . . . xpnn −1 dx1 dx2 . . . dxn α1 α2 n 1 + x1 + x2 + · · · + xα n 

Γ

3.

12

 

 ···

x1 ≥0,x2 ≥0,...,xn ≥0 α α α x1 1 +x2 2 +···+xn n ≤1

 =

√ Γ π 2

p1 α1

⎤       ⎡ m m+1 p2 pn Γ Γ ...Γ Γ ⎥ 1 ⎢ α2 αn ⎢  2  −  2 ⎥ , ⎣ m+1 m+2 ⎦ α1 α2 . . . αn Γ(m) Γ Γ 2 2

620

Multiple Integrals

p1 p2 pn + + ··· + . α1 α2 αn    ··· f (x1 + x2 + · · · + xn )

4.637

where m = 4.63712

x1 ≥0, x2 ≥0,...,xn ≥0 x1 +x2 +···+xn ≤1

=

FI III 480

x1p1 −1 x2p2 −1 . . . xpnn −1 dx1 dx2 . . . dxn (q1 x1 + q2 x2 + · · · + qn xn + r)p1 +p2 +···+pn

Γ (p1 ) Γ (p2 ) . . . Γ (pn ) Γ (p1 + p2 + . . . pn )



∞

0

f (x)

xp1 +p2 +···+pn −1 p p dx, (q1 x + r) (q2 x + r) 2 . . . (qn x + r) n [qi ≥ 0, r > 0] p1

where f (x) is continuous on the interval (0, 1). 4.638    ∞ p1 −1 p2 −1 ∞ ∞ x1 x2 · · · xpnn −1 e−(q1 x1 +q2 x2 +···+qn xn ) 1.12 ... dx1 dx2 . . . dxn (r0 + r1 x1 + r2 x2 + · · · + rn xn )s 0 0 0 e−r0 x xs−1 dx Γ (p1 ) Γ (p2 ) . . . Γ (pn ) ∞ = p1 p2 pn Γ(s) 0 (q1 + r1 x) (q2 + r2 x) . . . (qn + rn x) where pi , qi , ri , and s are positive. This result is also valid for r0 = 0 provided p1 +p2 +· · ·+pn > s. 

∞ ∞

2. 0

3.12



∞ ∞

0

4.639

0

0

 ...

 

xp11 −1 x2p2 −1 . . . xnpn −1 dx1 dx2 . . . dxn (r0 + r1 x1 + r2 x2 + · · · + rn xn )s Γ (p1 ) Γ (p2 ) . . . Γ (pn ) Γ (sp1 p2 − · · · − pn ) = r1p1 r2p2 · · · rnpn r0s−p1 −p2 −···−pn Γ(s) [pi > 0, ri > 0, s > 0]

∞

x1p1 −1 x2p2 −1 . . . xpnn −1 q q q s dx1 dx2 . . . dxn [1 + (r1 x1 ) 1 + (r2 x2 ) 2+ ·  · · +(rn xn ) n ]     p2 pn p1 p2 pn p1 Γ ...Γ Γ s− − − ···− Γ q1 q2 qn q1 q2 qn = p1 p2 pn q1 q2 . . . qn r1 r2 . . . rn Γ(s) [pi > 0, qi > 0, ri > 0, s > 0]

0

 ...

∞

0

 (p1 x1 + p2 x2 + · · · + pn xn )

···

1.

x21 +x22 +···+x2n ≤1

 

dx1 dx2 . . . dxn

√  m πn (2m − 1)!!  p21 + p22 + · · · + p2n  = n 2m +m+1 Γ 2 FI III 482

 ···

2.

2m

(p1 x1 + p2 x2 + · · · + pn xn )

2m+1

dx1 dx2 . . . dxn = 0

FI III 483

x21 +x22 +···+x2n ≤1

4.641 1.11

 

 ···

ep1 x1 +p2 x2 +···+pn xn dx1 dx2 . . . dxn

x21 +x22 +···+x2n ≤1

=

∞ √  πn k=0

k! Γ

n 2

1 +k+1

 

p21 + p22 + · · · + p2n 4

k

FI III 483

4.645

Multiple integrals

 

2.12

 ···

ep1 x1 +p2 x2 +···+p2n x2n dx1 dx2 . . . dx2n =

621

(2π)n I n

  p21 + p22 + · · · + p22n n

(p21 + p22 + · · · + p22n )

x21 +x22 +···+x22n ≤1

FI III 483a

√    2 π n R n−1 4.642 ··· f x21 + x22 + · · · + x2n dx1 dx2 . . . dxn =  n  x f (x) dx, 0 Γ x21 +x22 +···+x2n ≤R2 2 where f (x) is a function that is continuous on the interval (0, R).  1 1  1 p −1 p −1 p −1 4.643 ... f (x1 , x2 , · · · , xn ) (1 − x1 ) 1 (1 − x2 ) 2 . . . (1 − xn ) n  

0

0



FI III 485

0

×xp21 xp31 +p2 · · · xpn1 +p2 +···+pn−1 dx1 dx2 . . . dxn  Γ (p1 ) Γ (p2 ) . . . Γ (pn ) 1 = f (x)(1 − x)p1 +p2 +···+pn −1 dx Γ (p1 + p2 + · · · + pn ) 0 under the assumption that the integral on the right converges absolutely. n−1 2   34  5 dx1 dx2 . . . dxn−1 ··· 4.644 f (p1 x1 + p2 x2 + · · · + pn xn ) |xn | x21 +x22 +···+x2n =1    dx1 dx2 · · · dxn−1 =2 ··· f (p1 x1 + p2 x2 + · · · + pn xn )  1 − x21 − x22 − · · · − x2n−1 x21 +x22 +···+x2n−1 ≤1 √   π  2 π n−1 2 2 2  =  f p1 + p2 + · · · + pn cos x sinn−2 x dx n−1 0 Γ 2 #  $  where f (x) is continuous on the interval − p21 + p22 + · · · + p2n , p21 + p22 + · · · + p2n

FI III 488

[n ≥ 3]

FI III 489

4.645 Suppose that two functions f (x1 , x2 , . . . , xn ) and g (x1 , x2 , . . . , xn ) are continuous in a closed bounded region D and that the smallest and greatest values of the function g in D are m and M respectively. Let ϕ(u) denote a function that is continuous for m ≤ u ≤M. We denote by ψ(u) the integral    1. ψ(u) = ··· f (x1 , x2 , . . . , xn ) dx1 dx2 . . . dxn , m≤g(x1 ,x2 ,...,xn )≤u

2.

over that portion of the region D on which the inequality m ≤ g (x1 , x2 , . . . , xn ) ≤ u is satisfied. Then    ··· f (x1 , x2 , . . . , xn ) ϕ [g (x1 , x2 , . . . , xn )] dx1 dx2 . . . dxn m≤g(x1 ,x2 ,...,xn )≤M

 = (S)

M m

 ϕ(u)d ψ(u) = (R)

M m

ϕ(u)

d ψ(u) du du

where the middle integral must be understood in the sense of Stieltjes. If the derivative dψ du exists and is continuous, the Riemann integral on the right exists.  M 6 +∞ M may be +∞ in formulas 4.645 2, in which case m should be understood to mean lim . M→+∞

m

622

Multiple Integrals

 

4.6468

 ···

x1 ≥0,x2 ≥0,...,xn ≥0 x1 +x2 +···+xn ≤1

4.646

x1p1 −1 x2p2 −1 . . . xpnn −1 r dx1 dx2 . . . dxn (q1 x1 + q2 x2 + · · · + qn xn )

 ∞ xr−1 dx Γ (p1 ) Γ (p2 ) . . . Γ (pn ) = p1 Γ (p1 + p2 + · · · + pn − r + 1) Γ(r) 0 (1 + q1 x) (1 + q2 x)p2 · · · (1 + qn x)pn [pi > 0, qi > 0, p1 + p2 + · · · + pn > r > 0] FI III 493 + *    p1 x1 + p2 x2 + · · · + pn xn  dx1 dx2 . . . dxn 4.647 ··· exp x21 + x22 + · · · + x2n 2 2 2 0≤x1 +x2 +···+xn ≤1 √   2 πn 2 2 2 n = I 2 −1 p1 + p2 + · · · + pn n −1 n (p21 + p22 + · · · + p2n ) 4 2 4.64812

 0

∞ ∞ 0

 ···

0

∞

 exp − x1 + x2 + · · · + xn +

n+1

λ x1 x2 . . . xn 1

FI III 495

 −1

2

−1

n

× x1n+1 x2n+1 . . . xnn+1 1 n (2π) 2 e−(n+1)λ =√ n+1

−1

dx1 dx2 · · · dxn

FI III 496

Table of Integrals, Series, and Products. http://dx.doi.org/10.1016/B978-0-12-384933-5.00005-9 c 2015 Elsevier Inc. All rights reserved. Copyright 

5 Indefinite Integrals of Special Functions 5.1 Elliptic Integrals and Functions Notation: k  =

√ 1 − k 2 (cf. 8.1).

5.11 Complete elliptic integrals 5.111



1.  2.

K(k)k 2p+3 dk =

E(k)k 2p+3 dk =

1 (2p + 3)2

    2 4(p + 1)2 K(k)k 2p+1 dk + k 2p+2 E(k) − (2p + 3) K(k)k  BY (610.04)

⎧ ⎨

1 4(p + 1)2 4p2 + 16p + 15 ⎩



E(k)k 2p+1 dk

⎫ ⎬   2 2 − E(k)k 2p+2 (2p + 3)k  − 2 − k 2p+2 k  K(k) ⎭ BY (611.04)

5.112



1.

2.6

⎤ ⎡ ∞ 2  [(2j)!] k 2j ⎦ πk ⎣ K(k) dk = 1+ 4 4j 2 j=1 (2j + 1)2 (j!) ⎡

 E(k) dk = 

3.

∞ 

2

2j

BY (610.00)

⎤

[(2j)!] k πk ⎣ ⎦ 1− 2 − 1) 24j (j!)4 2 (4j j=1 2

K(k)k dk = E(k) − k  K(k)

BY (611.00)

BY (610.01)

 4. 5.

  1  2 1 + k 2 E(k) − k  K(k) 3     1  2 4 + k 2 E(k) − k  4 + 3k 2 K(k) K(k)k 3 dk = 9 E(k)k dk =

623

BY (611.01) BY (610.02)

624

Elliptic Integrals and Functions



   1  2 4 + k 2 + 9k 4 E(k) − k  4 + 3k 2 K(k) BY 611.02) 45     1  2 64 + 16k 2 + 9k 4 E(k) − k  64 + 48k 2 + 45k 4 K(k) K(k)k 5 dk = BY (610.03) 225     1  2 64 + 16k 2 + 9k 4 + 225k 6 E(k) − k  64 + 48k 2 + 45k 4 K(k) E(k)k 5 dk = 1575

6. 7. 8.

 9.  10.  11.  12.  13. 14.∗ 15.

5.113

∗

5.113

 



BY (611.03)

K(k) E(k) dk = − k2 k   E(k) 1 2 k dk = K(k) − 2 E(k) k2 k E(k) dk = k K(k) k 2   E(k) 1   2 2 2 k dk = − 2 E(k) + k K(k) k4 9k 3 k E(k) dk = K(k) − E(k) k 2   K(k) 1  2 dk = − 3 1 + 4k 2 E(k) + 2k  K(k) 4 k 9k

 2.  3.  4.  5.  6. 

BY (612.05) BY (612.02) BY (612.01) BY (612.03) BY (612.04)

(1 ± k  )3 E (k) ∓ k  K (k) K (k) dk = k3 k 1 ± k dk = − E(k) k   dk 2 2 = 2 E(k) − k  K(k) E(k) − k  K(k) k   dk  2 = −k  K(k) 1 + k 2 K(k) − E(k) k  dk 1 2 E(k) − k  K(k) [K(k) − E(k)] 2 = k k   dk 1 2 E(k) − k  K(k) 2  2 = [K(k) − E(k)] k k k   dk  E(k) 2 1 + k 2 E(k) − k  K(k) 4 =  kk k 2 [K(k) − E(k)]

1.

5.114

E(k)k 3 dk =



k K(k) dk

1 2 =  2 k K(k) − E(k) E(k) − k K(k) 2

5.115  π   π 2   1. Π , r2 , k k dk = k 2 − r2 Π , r , k − K(k) + E(k) 2 2   π   π 2   , r2 , k k dk = k 2 K(k) − k 2 − r2 Π ,r ,k 2. K(k) − Π 2 2

BY (612.06) BY (612.09) BY (612.12) BY (612.07)

BY (612.13)

BY (612.11)

BY (612.14) BY (612.15)

5.124

Elliptic integrals

  3.

625

π   π 2   2 E(k) 2 2 , r ,r ,k Π + Π , k k dk = k − r 2 2 k 2

BY (612.16)

5.12 Elliptic integrals 

2  F (x, k) dx π [F (x, k)]  0 k 2

 

BY (630.13) x

    Π x, α2 , k cos x dx = sin x Π x, α2 , k − f − f0

#     $  2 1 − α2 α2 − k 2 + 1 − α2 sin2 x 2k 2 − α2 − α2 k 2 1  f=  arctan  2 (1 − α2 ) (α2 − k 2 ) 2α2 (1 − α2 ) (α2 − k 2 ) cos x 1 − k2 sin2 x  for 1 − α2 α2 − k 2 > 0; #        2 2 α2 − 1 α2 − k 2 + 1 − α2 sin x α2 + α2 k 2 − 2k 2 1 =  ln 1 − α2 sin2 x 2 (α2 − 1) (α2 − k 2 ) $   2α2 (α2 − 1) (α2 − k 2 ) cos x 1 − k 2 sin2 x + 1 − α2 sin2 x    for 1 − α2 α2 − k 2 < 0, f0 is the value of f at x = 0 BY (630.23) where

Integration with respect to the modulus    2 1 − k 2 sin2 x − 1 cot x BY (613.01) 5.126 F (x, k)k dk = E (x, k) − k  F (x, k) +        1  2 1 + k 2 E (x, k) − k  F (x, k) + 5.127 E (x, k)k dk = 1 − k 2 sin2 x − 1 cot x BY (613.02) 3           5.128 Π x, r2 , k k dk = k 2 − r2 Π x, r2 , k − F (x, k) + E (x, k) + 1 − k 2 sin2 x − 1 cot x BY (613.03)

5.13 Jacobian elliptic functions 5.131 1.



⎡    1 ⎣ snm+1 u cn u dn u + (m + 2) 1 + k 2 snm+2 u du snm u du = m+1 ⎤  − (m + 3)k 2 snm+4 u du⎦ SI 259, PE(567)

628

Elliptic Integrals and Functions

 2.

⎡ cn m u du =

1 ⎣ − cn m+1 u snu dn u (m + 1)k  2   + (m + 2) 1 − 2k 2

 3.

5.132



cn m+2 u du + (m + 3)k 2



⎤ cn m+4 u du⎦ PE (568)

⎡ dn m u du =

1 ⎣k 2 dn m+1 u snu cn u (m + 1)k  2 

+ (m + 2) 2 − k

 2



⎤  dn m+2 u du − (m + 3) dn m+4 u du⎦ PE (569)

%

By using formulas 5.131, we can reduce the integrals (for m = 1) dn m u du to the integrals 5.132, 5.133 and 5.134.

5.132 1.

2. 3.

%

snm u du,

%

cn m u du, and



sn u du = ln sn u cn u + dn u dn u − cn u = ln sn u  1 du k  sn u + dn u =  ln cn u k cn u  1 du k  sn u − cn u =  arctan  dn u k k sn u + cn u 1 cn u =  arccos k dn u 1 cn u + ik  sn u =  ln ik dn u 1 k  sn u =  arcsin k dn u

5.133  1 1. sn u du = ln (dn u − k cn u) k dn u − k 2 cn u 1 = arccosh 2 k  1−k  dn u − cn u 1 = arcsinh k k 1 − k2 1 = − ln (dn u + k cn u) k  1 2. cn u du = arccos (dn u) k i = ln (dn u − ik sn u) k 1 = arcsin (k sn u) k

H 87(164) SI 266(4) SI 266(5) H 88(166) JA SI 266(6) JA

H 87(161) JA JA SI 365(1) H 87(162) SI 265(2)a, ZH 87(162) JA

5.137

Jacobian elliptic functions

629

 dn u du = arcsin (sn u)

3.

= am u = i ln (cn u − i sn u) 5.134 1. 2. 3.



1 [u − E (am u, k)] k2   1  2 cn 2 u du = 2 E (am u, k) − k  u k  dn 2 u du = E (am u, k) sn2 u du =

5.135  1 dn u + k  sn u du =  ln 1. cn u k cn u 1 dn u + k  =  ln 2k dn u − k   i sn u ik  − k cn u du =  ln 2. dn u kk dn u 1 k cn u = arccot  kk k  1 − dn u cn u du = ln 3. sn u sn u 1 1 − dn u = ln 2 1 + dn u  1 1 − k sn u cn u du = − ln 4. dn u k dn u 1 + k sn u 1 ln = 2k 1 − k sn u  1 1 + sn u dn u du = ln 5. cn u 2 1 − sn u 1 + sn u = ln cn u  1 1 − cn u dn u du = ln 6. sn u 2 1 + cn u

H 87(163) SI 266(3), ZH 87(163)

PE (564) PE (565) PE (566)

SI 266(7) H 88(167) SI 266(8)

SI 266(10) H 88(168) SI 266(9)

H 88(172) JA H 87(170)

5.136  1 1. snu cn u du = − 2 dn u k  2. snu dn u du = − cn u  3. cn u dn u du = sn u 5.137 1.



1 dn u sn u du = 2 2 cn u k  cn u

H 88(173)

630

Elliptic Integrals and Functions

 2.  3.  4.  5.  6.

5.138

1 cn u sn u du = −  2 dn 2 u k dn u dn u cn u du = − sn2 u sn u sn u cn u du = dn 2 u dn u cn u dn u du = − 2 sn u sn u sn u dn u du = cn 2 u cn u

H 88(175) H 88(174) H 88(177) H 88(176) H 88(178)

5.138  cn u sn u 1. du = ln sn u dn u dn u  1 sn u dn u du = 2 ln 2.  cn u dn u cn u k  sn u dn u du = ln 3. sn u cn u cn u

H 88(183) H 88(182) H 88(184)

5.139  cn u dn u du = ln sn u 1.11 sn u  1 sn u dn u du = ln 2. cn u cn u  1 sn u cn u du = − 2 ln dn u 3. dn u k

H 88(179) H 88(180) H 88(181)

5.14 Weierstrass elliptic functions The invariants g1 and g2 used below are defined in 8.161. 5.141  ℘(u) du = − ζ(u)

1.  2. 3. 4.8

5.12

1  1 ℘ (u) + g2 u 6 12  1  3 1 ℘ (u) − g2 ζ(u) + g3 u ℘3 (u) du = 120 20 10    1 σ(u − v) du =  2u ζ(v) + ln ℘(u) − ℘(v) ℘ (v) σ(u + v) 

℘2 (u) du =

H 120(192) H 120(193)

[℘(v) = e1 , e2 , e3 ]

  αu αδ − βγ σ(u + v) α℘(u) + β du = − 2  ln − 2u ζ(v) γ℘(u) + δ γ γ ℘ (v) σ(u − v) where v = (℘ )−1

(see 8.162) H 120(194)



−δ γ

 H 120(195)

5.221

The exponential integral function and powers

631

5.2 The Exponential Integral Function 5.21 The exponential integral function 

5.211



∞

Ei(−βx) Ei(−γx) dx = x

1 1 + β γ



Ei[−(β + γ)x]

−x Ei(−βx) Ei(−γx) −

e−γx e−βx Ei(−γx) − Ei(−βx) β γ [Re(β + γ) > 0]

NT 53(2)

5.22 Combinations of the exponential integral function and powers 5.221



∞

1. x



   n−1 (−1)n Ei[−a(x + b)] e−ab  (−1)n−k−1 ∞ e−ax Ei[−a(x + b)] 1 + dx = n − dx k+1 xn+1 x bn n n bn−k x x k=0

∞

2. x

Ei[−a(x + b)] dx = x2



1 1 + x b

[a > 0,

 Ei[−a(x + b)] −

4.

NT 52(3)

b > 0]

NT 52(4)

e−ab Ei(−ax) b [a > 0,

 3.

b > 0]

1 x2 xe−ax Ei(−ax) + 2 e−ax + [a > 0] 2 2a 2a  ∞ n!e−ax  (ax)k xn+1 Ei(−ax) + xn Ei(−ax) dx = n+1 (n + 1)an+1 k! x Ei(−ax) dx =

k=0

 5.  6.

 7.

 8. 0

 9.

0

[a > 0] x Ei(−ax)e−bx dx =

1 1 x 1 e−(a+b)x Ei [−(a + b)x] − 2 Ei(−ax)e−bx − Ei(−ax)e−bx − 2 b b b b(a + b)

[a > 0,   2 Ei(−ax)e−ax − Ei(−2ax) Ei2 (−ax) dx = x Ei2 (−ax) + a

x Ei2 (−ax) dx =

x2 2 Ei (−ax) + 2



1 x + a2 a



b > 0]

[a > 0] Ei(−ax)e−ax −

1 1 Ei(−2ax) + 2 e−2ax a2 a

[a > 0] u

Ei(−ax) dx = u Ei(−au) +

∞

e

−au

a

−1

[a > 0]

 2   x  x a + b2 a2 b2 ab Ei − dx = ln(a + b) − ln a − ln b − x Ei − a b 2 2 2 2 [a > 0,

b > 0]

632

The Sine Integral and the Cosine Integral



∞

10. 0

5.231

   x  x   2  3 ab  2 3 3 3 2 Ei − dx = a + b ln(a + b) − a ln a − b ln b − a − ab + b x Ei − a b 3 a+b 2

[a > 0,

b > 0]

5.23 Combinations of the exponential integral and the exponential 5.231



x

1. 0

 2.

0

x

ex Ei(−x) dx = − ln x − C + ex Ei(−x) e−βx Ei(−αx) dx = −

1 β



   β e−βx Ei(−αx) + ln 1 + − Ei[−(α + β)x] α

ET II 308(11) ET II 308(12)

5.3 The Sine Integral and the Cosine Integral 5.31

 cos αx ci(βx) dx =

1. 

sin αx ci(βx) dx = −

2. 5.32

 cos αx si(βx) dx =

1. 



1.



cos αx si(βx) si(αx + βx) − si(αx − βx) + α 2α

NT 49(1) NT 49(2)

NT 49(3) NT 49(4)

1 [si(αx + βx) + si(αx − βx)] ci(αx) ci(βx) dx = x ci(αx) ci(βx) + 2α 1 1 1 [si(αx + βx) + si(βx − αx)] − sin αx ci(βx) − sin βx ci(αx) + 2β α β NT 53(5)

1 [si(αx + βx) + si(αx − βx)] 2β 1 1 1 [si(αx + βx) + si(βx + αx)] + cos αx si(βx) + cos βx si(αx) − 2α α β

si(αx) si(βx) dx = x si(αx) si(βx) −

2.

 3.

cos αx ci(βx) ci(αx + βx) + ci(αx − βx) + α 2α

sin αx si(βx) ci(αx + βx) − ci(αx − βx) + α 2α

sin αx si(βx) dx = −

2. 5.33

sin αx ci(βx) si(αx + βx) + si(αx − βx) − α 2α

NT 54(6)

1 cos αx ci(βx) α    1 1 1 1 1 − sin βx si(αx) − + ci(αx + βx) − − ci(αx − βx) β 2α 2β 2α 2β

si(αx) ci(βx) dx = x si(αx) ci(βx) +

NT 54(10)

5.54

Combinations of the exponential integral and the exponential

5.34



∞

1. x



∞

2. x

si[a(x + b)]

ci[a(x + b)]

dx = x2 dx = x2





1 1 + x b 1 1 + x b

 si[a(x + b)] −

cos ab si(ax) + sin ab ci(ax) b [a > 0,

 ci[a(x + b)] +

633

b > 0]

NT 52(6)

sin ab si(ax) − cos ab ci(ax) b [a > 0,

b > 0]

NT 52(5)

5.4 The Probability Integral and Fresnel Integrals 5.41

11

5.42 5.43



2

2

e−α x Φ(αx) dx = x Φ(αx) + √ α π  cos2 αx2 √ S (αx) dx = x S (αx) + α 2π  sin2 αx2 C (αx) dx = x C (αx) − √ α 2π

NT 12(20)a NT 12(22)a NT 12(21)a

5.5 Bessel Functions Notation: Z and Z denote any of J, N , H (1) , H (2) . In formulae 5.52–5.56, Z p (x) and Zp (x) are arbitrary Bessel functions of the first, second, or third kinds.  ∞  5.51 J p (x) dx = 2 J p+2k+1 (x) JA, MO 30 5.52

k=0



1. 2.12 5.5310



xp+1 Z p (x) dx = xp+1 Z p+1 (x)

WA 132(1)

x−p+1 Z p (x) dx = −x−p+1 Z p−1 (x)

WA 132(2)

 

  p2 − q 2 Z p (αx) Zq (βx) dx α2 − β 2 x − x



= αx Z p+1 (αx) Zq (βx) − βx Z p (αx) Zq+1 (βx) − (p − q) Z p (αx) Zq (βx) = βx Z p (αx) Zq−1 (βx) − αx Z p−1 (αx) Zq (βx) + (p − q) Z p (αx) Zq (βx) JA, MO 30, WA 134(7)

5.54 1.

10



 2.

αx Z p+1 (αx) Zp (βx) − βx Z p (αx) Zp+1 (βx) α2 − β 2 βx Z p (αx) Zp−1 (βx) − αx Z p−1 (αx) Zp (βx) = α2 − β 2

x Z p (αx) Zp (βx) dx =

WA 134(8) 2

x [Z p (αx)] dx =

' x2 & 2 [Z p (αx)] − Z p−1 (αx) Z p+1 (αx) 2

WA 135(11)

634

Hypergeometric Functions

 x Z p (ax) Zp (ax) dx =

3. 5.55

10



5.55

 x4  2 Z p (ax) Zp (ax) − Z p−1 (ax) Zp+1 (ax) − Z p+1 (ax) Zp−1 (ax) 4

Z p (αx) Zq+1 (αx) − Z p+1 (αx) Zq (αx) Z p (αx) Zq (αx) 1 Z p (αx) Zq (αx) dx = αx + x p2 − q 2 p+q Z p−1 (αx) Zq (αx) − Z p (αx) Zq−1 (αx) Z p (αx) Zq (αx) = αx − p2 − q 2 p+q WA 135(13)

5.56



1.

Z 1 (x) dx = − Z 0 (x)

JA

x Z 0 (x) dx = x Z 1 (x)

JA

 2.

5.6 Orthogonal Polynomials  1. 

2. 3. 4.

  1 Tn+1 (x) Tn−1 (x) Tn (x) dx = − 2 n+1 n−1

[n ≥ 2]

Tn+1 (x) n+1   Un−1 (x) Tn (x) √ dx = − 1 − x2 2 n 1−x √     1 − x2 Un+1 (x) Un−1 (x) − 1 − x2 Un (x) dx = 2 n+2 n Un (x) dx =

[n ≥ 1] [n ≥ 1] [n ≥ 1]

5.7 Hypergeometric Functions  F (α, β; γ, x) dx =

1. 

2. 3. 4.

[α = 1,

β = 1,

γ = 1]

xβ−1 F (α, β − 1; γ, x) [β = 1] β−1  xγ F (α, β; γ + 1, x) xγ−1 F (α, β; γ, x) dx = γ  γ−1 (1 − x)β−1 F (α, β − 1; γ − 1, x) (1 − x)β−2 F (α, β; γ, x) dx = (α − γ + 1)(β − 1) 

5.

γ−1 F (α − 1, β − 1; γ − 1, x) (α − 1)(β − 1)

xβ−2 F (α, β; γ, x) dx =

[β = 1, xγ−1 (1 − x)β−γ−1 F (α, β; γ, x) dx =

1 γ x (1 − x)β−γ F (α + 1, β; γ + 1, x) γ

γ − α = 1,

γ = 1]

5.71

Combinations of the exponential integral and the exponential

5.71*  1. 

2. 3. 4. 5. 6.

1 F1 (α; γ; x) dx

=

γ−1 1 F1 (α − 1; γ − 1; x) α−1

xα−1 1 F1 (α − 1; γ; x) α−1  xγ xγ−1 1 F1 (α; γ; x) dx = 1 F1 (α; γ + 1; x) γ  1 Ψ(α − 1, γ − 1; x) Ψ(α, γ; x) dx = − α−1  xα−1 xα−2 Ψ(α, γ; x) dx = − Ψ(α − 1, γ; x) (α − 1)(γ − α)  xγ Ψ(α, γ + 1; x) xγ−1 Ψ(α, γ; x) dx = γ−α xα−2 1 F1 (α; γ; x) dx =

[α = 1, [α = 1]

[α = 1] [α = 1]

γ = 1]

635

Table of Integrals, Series, and Products. http://dx.doi.org/10.1016/B978-0-12-384933-5.00006-0 c 2015 Elsevier Inc. All rights reserved. Copyright 

6–7 Definite Integrals of Special Functions 6.1 Elliptic Integrals and Functions Notation: k  =

√ 1 − k 2 (cf. 8.1).

6.11 Forms containing F (x, k) 

6.111 6.112

π/2

F (x, k) cot x dx =

0



π/2

1. 0



π/2

2. π/2

3.

F (x, k)

2 1 sin x cos x π √ − K(k) ln K(k  ) dx = 4k 1 − k sin2 x (1 − k) k 16k

F (x, k)

1 sin x cos x dx = − 2 ln k  K(k) 2k 1 − k 2 sin2 x

0

6.113



π/2

1.

F (x, k  )

0



π/2

2. 0

BI (350)(1)

√ (1 + k) k π 1 sin x cos x K(k) ln + K(k  ) dx = F (x, k) 4k 2 16k 1 + k sin2 x

0



π 1 K(k  ) + ln k K(k) 4 2

F (x, k)

2 1 sin x cos x dx √ K(k  ) ln = 4(1 − k) (1 + k) k cos2 x + k sin2 x

dx sin x cos x · 1 − k 2 sin2 t sin2 x 1 − k 2 sin2 x =−

BI (350)(6)

BI (350)(7)

BI (350)(2)a, BY(802.12)a

BI (350)(5)

  π 1  K(k) arctan (k F (t, k) tan t) − k 2 sin t cos t 2

BI (350)(12)

 6.114

6.115



dx 1 K(k) K 1 − tan2 u cot2 v =    2 cos u sin v u sin2 x − sin2 u sin2 v − sin2 x  2 k = 1 − cot2 u · cot2 v BI (351)(9) √  1 (1 + k) k π x dx 1 K(k) ln + K(k  ) F (arcsin x, k) = 2 1 + kx 4k 2 16k 0 BI (466)(1) (cf. 6.112 2) v

F (x, k) 

637

638

Elliptic Integrals and Functions

6.116*

This and similar formulas can be obtained from formulas 6.111–6.113 by means of the substitution x = arcsin t.  π/2 sin x cos x  6.116* dx F (x, k) 2 0 (1 + k 2 sinh μ sin2 x) 1 − k 2 sin2 x

−1 π K (k) arctanh(k  tanh μ) − F (φ, k  ) = 2 k sinh μ cosh μ 2    k  = 1 − k 2 , φ = arcsin(tanh μ), 0 < tanh μ < 1, 0 < k < 1 KM (4.48)  6.117* 0

 6.118* 0

π/2

sin x cos x  dx (1 − k 2 cosh ν sin2 x) 1 − k 2 sin2 x

  1 tanh ν π  = 2 K (k)arctanh F (φ, k − ) k sinh ν cosh ν k 2  

  tanh ν   2 , 0 < tanh ν < k < 1 k = 1 − k , φ = arcsin k

π/2

F (x, k)

2

KM (4.49)

sin x cos x  dx (1 − ψ sin2 x) 1 − k 2 sin2 x

  1 tan ψ π = 2 K (k)arctan F (β, k) − k sin ψ cos ψ k 2 F (x, k)

k2

cos2

  k = 1 − k2 ,

β = arctan

tan ψ k

 ,

0 −1]

DLMF (19.28.3)

[a, b, c > 0]

DLMF (19.28.5)

642

Elliptic Integrals and Functions



1

RD (a, b, x2 c + (1 − x2 )d) dx = RJ (a, b, c, d)

5.

6.161

[a, b, c, d > 0]

DLMF (19.28.6)

[a, b, c > 0]

DLMF (19.28.7)

[a, b, c, d > 0]

DLMF (19.28.8)

0



∞

6.

RJ (a, b, c, x2 ) dx =

0



∞

7. 0

3π RF (ab, ac, bc) 2

6 RJ (ax, b, c, xd) dx = √ RC (d, a)RF (0, b, c) d

6.16 The theta function 6.161



∞

1. 0



∞

2. 0



∞

3. 0



      s xs−1 ϑ2 0 | ix2 dx = 2s 1 − 2−s π − 2 Γ 12 s ζ(s)      s xs−1 ϑ3 0 | ix2 − 1 dx = π − 2 Γ 12 s ζ(s)

[Re s > 2]

ET I 339(20)

[Re s > 2]

ET I 339(21)

       1 xs−1 1 − ϑ4 0 | ix2 dx = 1 − 21−s π − 2 s Γ 12 s ζ(s) [Re s > 2]

∞

4. 0

ET I 339(22)

           1 xs−1 ϑ4 0 | ix2 + ϑ2 0 | ix2 − ϑ3 0 | ix2 dx = − (2s − 1) 21−s − 1 π − 2 s Γ 12 s ζ(s) ET I 339(24)

6.162



∞

1.11

e−ax ϑ4

0



∞

2.

e−ax ϑ1

0



∞

3.11

e−ax ϑ2

0

 4.11 0

∞

e−ax ϑ3

bπ 2l

!  ! iπx  √  √  l ! dx = √ cosh b a cosech l a ! l2 a

[Re a > 0, !   √  √  bπ !! iπx l dx = − √ sinh b a sech l a ! 2 2l l a

|b| ≤ l]

ET I 224(1)a

[Re a > 0, !   √  √  (l + b)π !! iπx l dx = − √ sinh b a sech l a ! l2 2l a

|b| ≤ l]

ET I 224(2)a

[Re a > 0, !   √  √  (l + b)π !! iπx l dx = √ cosh b a cosech l a ! 2 2l l a

|b| ≤ l]

ET I 224(3)a

[Re a > 0,

|b| ≤ l]

ET I 224(4)a

6.165

6.16310

Generalized elliptic integrals



∞

1.12 0



√ √ √ 1  √  √  e−(a−μ)x ϑ3 (π μx |iπx ) dx = √ tanh a + μ + tanh a − μ 2 a [Re a > 0]

∞

2.10

ϑ3 (iπkx | iπx) e−(k

2

+l2 )x

0



643

dx =

ET I 224(7)a

sinh 2l l (cosh 2l − cos 2k)

∞

       1 ϑ4 0 | ie2x + ϑ2 0 | ie2x − ϑ3 0 | ie2x e 2 x cos(ax) dx 0

    1 1 1 1 1 +ia 22 − 1 1 − 2 2 −ia π − 4 − 2 ia Γ 14 + 12 ia ζ 12 + ia = 2 [a > 0] ET I 61(11)  ∞    1 6.165 e 2 x ϑ3 0 | ie2x − 1 cos(ax) dx 0        2 1 1 1 + a2 + 14 π − 2 ia− 4 Γ 12 ia + 14 ζ ia + 12 = 2 1 + 4a ET I 61(12) [a > 0] 6.16411

6.1710 Generalized elliptic integrals 1.

Set 

π

Ωj (k) ≡

 −(j+ 12 ) dφ, 1 − k 2 cos φ

0

j! (4m + 2j)! π αm (j) = m (64) (2j)! (2m + j)!

1 m!

2 ,

π λ= 2



(2j + 1)k 2 , 1 − k2

then

⎡

   π 1 1 2 −j ⎣ −1 (2j + 1) 1 + 1 − k αm (j)k 4m = Ωj (k) = erf λ + (2j + 1)k 2 2 2k 2 m=0  $



#

1 13 2 2 1 2 −λ2 −2 λe − (2j + 1) 16 + 2 + 4 1+ λ × erf λ − √ π 3 12 k k ⎤  $

#

2 2 4 2 λe−λ + . . .⎦ 1 + λ2 + λ4 × erf λ − √ π 3 15 

∞ "

while for large λ  lim Ωj (k) =

j→∞

 −j π k2 1 − k2 (2j + 1) # $ # $   1 4 13 1 1 + . . . × 1 + (2j + 1)−1 1 + 2 − (2j + 1)−2 1 + + 2 2k 3 16k 2 16k 4

644

The Exponential Integral Function and Functions Generated by It

2.

6.211

Set  Rμ (k, α, δ) =

π

cos2α−1 (θ/2) sin2δ−2α−1 (θ/2) dθ

, μ+ 1 [1 − k 2 cos θ] 2 0 < k < 1, Re δ > Re α > 0, Re μ > −1/2,   (−1)ν 2ν μ + 12 ν Γ(α) Γ (δ − α + ν) , Mν (μ, α, δ) = ν! Γ(δ + ν) with (λ)ν = Γ(λ + ν)/ Γ(λ), and   ν 2 μ + 12 ν Γ(α + ν) Γ (δ − α) , Wν (μ, α, δ) = ν! Γ(δ + ν) 0

then: • for small k; ∞ −(μ+ 12 ) "  2   ν  Rμ (k, α, δ) = 1 − k 2 k / 1 − k2 Mν (μ, α, δ) ν=0

∞  −(μ+ 12 ) "  2   ν = 1 + k2 k / 1 + k2 W ν (μ, α, δ), ν=0

• for k 2 close to 1; Rμ (k, α, δ)       2 α−δ  δ−α−μ− 12 = Γ(δ − α) Γ μ + α − δ + 12 Γ μ + 12 2k 1 − k2       μ+ 12  × Γ δ − α − μ − 12 Γ(α) Γ δ − μ − 12 2k 2    Re μ + α − δ + 12 not an integer     1 = 2μ+ 2 k 2μ+1 Γ μ + 12 Γ(1 − α) ×

∞ "      α−δ+μ−n+ 1  2 Γ (δ − α + n) Γ(1 − α + n) Γ α − δ + μ − n + 12 n! 2k 2 / 1 − k 2 n=0  α − δ + μ + 12 = m, with m a non-negative integer

6.2–6.3 The Exponential Integral Function and Functions Generated by It 6.21 The logarithm integral 

1

6.211 6.212

li(x) dx = − ln 2

BI (79)(5)

 1 x dx = 0 li x

BI (255)(1)

0



1. 0

1

6.214

The logarithm integral



1

2. 0



1 li(x)xp−1 dx = − ln(p + 1) p

1

3.

li(x) 0



∞

4.

li(x) 1

6.213 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

dx 1 = − ln(q − 1) q+1 x q

[p > −1]

BI (255)(2)

[q < 1]

BI (255)(3)

[q > 1]

BI (255)(4)

 1 π

1 sin (a ln x) dx = [a > 0] a ln a − 2 x 1+a 2 0  ∞ 

1 π 1 sin (a ln x) dx = − + a ln a [a > 0] li x 1 + a2 2 1  1  1 π

1 cos (a ln x) dx = − a [a > 0] ln a + li x 1 + a2 2 0  ∞  1 π

1 cos (a ln x) dx = a [a > 0] ln a − li x 1 + a2 2 1    1 ln 1 + a2 dx = [a > 0] li(x) sin (a ln x) BI(479)(1), x 2a 0  1 arctan a dx =− li(x) cos (a ln x) x a 0  1 π

dx 1 [a > 0] a ln a + li(x) sin (a ln x) 2 = 2 x 1+a 2 0  ∞

dx 1 π − a ln a [a > 0] li(x) sin (a ln x) 2 = x 1 + a2 2 1  1 π

dx 1 a [a > 0] ln a − li(x) cos (a ln x) 2 = x 1 + a2 2 0  ∞ π

dx 1 ln a + li(x) cos (a ln x) 2 = − a [a > 0] x 1 + a2 2 1 $ #  1 a  a 1 2 2 ln (1 + p) − p arctan li(x) sin (a ln x) xp−1 dx = 2 + a a + p2 2 1+p 0



1

li



1

12. 0

6.214

dx 1 = ln(1 − q) xq+1 q

645



1. 0

1

li(x) cos (a ln x) xp−1 dx = −

a2

1 + p2

# a arctan

 1 (− ln x)p−1 dx = −π cot pπ · Γ(p) li x

[p > 0] p  a + ln (1 + p)2 + a2 1+p 2

$

BI (475)(1) BI (475)(9)

BI (475)(2) BI (475)(10)

ET I 98(20)a

BI (479)(2)

BI (479)(3) BI (479)(13)

BI (479)(4) BI (479)(14)

BI (477)(1)

[p > 0]

BI (477)(2)

[0 < p < 1]

BI (340)(1)

646

The Exponential Integral Function and Functions Generated by It



∞

2. 1

6.215



1

1. 0



xp−1 dx = −2 li(x) √ − ln x

1

2.

li(x) 0

6.216

 1 π (ln x)p−1 dx = − Γ(p) li x sin pπ



1

1.12

xp+1

dx √ = −2 − ln x

1

2.

π √ arcsinh p = −2 p



[0 < p < 1]



BI (340)(9)

 π √ ln p+ p+1 p

π √ arcsin p p

[p > 0]

BI (444)(3)

[1 > p > 0]

BI (444)(4)

li(x)[− ln x]p−1

dx 1 = − Γ(p) x p

[0 < p < 1]

BI (444)(1)

li(x)[− ln x]p−1

dx π Γ(p) =− x2 sin pπ

[0 < p < 1]

BI (444)(2)

0





6.215

0

6.22–6.23 The exponential integral function 

6.221 6.222 6.223

p

1 − eαp NT 11(7) 0 

α ∞ ln q ln p 1 1 + ln(p + q) − − Ei(−px) Ei(−qx) dx = p q p q 0 [p > 0, q > 0] FI II 653, NT 53(3)  ∞ Γ(μ) Ei(−βx)xμ−1 dx = − μ [Re β ≥ 0, Re μ > 0] μβ 0 Ei(αx) dx = p Ei(αp) +

NT 55(7), ET I 325(10)

6.224



∞

1.

Ei(−βx)e 0

−μx

 μ 1 dx = − ln 1 + μ β

[Re(β + μ) ≥ 0,

= −1/β 

μ > 0]

[μ = 0] FI II 652, NT 48(8)

∞

2. 0

Ei(ax)e−μx dx = −

μ

1 ln −1 μ a



[a > 0,

Re μ > 0,

μ > a]

ET I 178(23)a, BI (283)(3)

6.225



∞

1.

  2 Ei −x2 e−μx dx = −



0

 2. 0

∞

  2 Ei −x2 epx dx = −



π √ arcsinh μ = − μ



 π √ ln μ+ 1+μ μ [Re μ > 0]

π √ arcsin p p

[1 > p > 0]

BI (283)(5), ET I 178(25)a NT 59(9)a

6.233

6.226 1. 2. 3. 4.

6.227 1. 2.

The exponential integral function

647



1 2 √ e−μx dx = − K 0 ( μ) Ei − [Re μ > 0] 4x μ 0  ∞ 2 a 2 √ e−μx dx = − K 0 (a μ) Ei [a > 0, Re μ > 0] 4x μ 0    ∞ π √ 1 −μx2 Ei (− μ) Ei − 2 e dx = [Re μ > 0] 4x μ 0    ∞ 2 1 π √ 1 √ √ √ [cos μ ci μ − sin μ si μ] Ei − 2 e−μx + 4x2 dx = 4x μ 0



∞



∞

1 1 − 2 ln(1 + μ) μ(μ + 1) μ 0   ∞  −ax ax Ei(ax) e Ei(−ax) e − dx = 0 x−b x+b 0 = π 2 e−ab Ei(−x)e−μx x dx =

MI 34 MI 34 MI 34

[Re μ > 0]

MI 34

[Re μ > 0]

MI 34

[a > 0,

b < 0]

[a > 0,

b > 0] ET II 253(1)a

6.228  1.

∞

Ei(−x)ex xν−1 dx = −

0



∞

2.

Ei(−βx)e 0

6.229 6.231

1. 2. 6.233

x

[0 < Re ν < 1]



μ Γ(ν) dx = − 2 F 1 1, ν; ν + 1; ν(β + μ)ν β+μ [|arg β| < π, Re(β + μ) > 0,

ET II 308(13)

Re ν > 0]





√ 1 1 dx √ √ √ √ 2 Ei − 2 exp −μx + 2 = 2 π (cos μ si μ − sin μ ci μ) 2 4x 4x x 0 [Re μ > 0]  ∞   −x  −μx 1 [a < 1, Re μ > 0] Ei(−a) − Ei −e e dx = γ(μ, a) μ − ln a 

6.232

−μx ν−1

π Γ(ν) sin νπ

∞



b2  ∞ ln 1 + 2 a Ei(−ax) sin bx dx = − 2b 0  ∞ b 1 Ei(−ax) cos bx dx = − arctan b a 0 

1. 0

ET II 308(14)

∞

Ei(−x)e−μx sin βx dx = −

1 β 2 + μ2

#

MI 34 MI 34

[a > 0,

b > 0]

BI (473)(1)a

[a > 0,

b > 0]

BI (473)(2)a

β  β ln (1 + μ)2 + β 2 − μ arctan 2 1+μ [Re μ > |Im β|]

$

BI (473)(7)a

648

The Exponential Integral Function and Functions Generated by It



∞

2.

Ei(−x)e

−μx

0

1 cos βx dx = − 2 β + μ2

#

6.234

μ  β ln (1 + μ)2 + β 2 + β arctan 2 1+μ

$

[Re μ > |Im β|] 

∞

6.234

BI (473)(8)a

Ei(−x) ln x dx = C + 1

NT 56(10)

0

6.24–6.26 The sine integral and cosine integral functions 6.241



∞

1.

si(px) si(qx) dx =

π 2p

[p ≥ q]

BI II 653, NT 54(8)

ci(px) ci(qx) dx =

π 2p

[p ≥ q]

FI II 653, NT 54(7)

0



∞

2. 0



∞

3. 0

 2 2

2 p − q2 p+q 1 1 ln ln si(px) ci(qx) dx = + 4q p−q 4p q4 1 = ln 2 q

[p = q] [p = q] FI II 653, NT 54(10, 12)

 6.242 6.243 1. 2. 6.244

∞

0



 ci(ax) 1 2 2 dx = − [si(aβ)] + [ci(aβ)] β+x 2

∞

si (a|x|) sign x dx = π ci (a|b|) −∞ x − b  ∞ ci (a|x|) dx = −π sign b · si (a|b|) −∞ x − b 

∞

1.8

si(px) 0



∞

2.8

si(px) 0

6.245

 

6.246



1. 0

∞

[a > 0,

b > 0]

ET II 253(3)

[a > 0]

ET II 253(2)

[p > 0,

q > 0]

BI (255)(6)

x dx π = − ci(pq) q 2 − x2 2

[p > 0,

q > 0]

BI (255)(6)

q2

dx π = Ei(−pq) 2 +x 2q

[p > 0,

q > 0]

BI (255)(7)

ci(px)

q2

dx π si(pq) = 2 −x 2q

[p > 0,

q > 0]

BI (255)(8)

[a > 0,

0 < Re μ < 1]

∞

0

ET II 224(1)

ci(px) 0

2.

|arg β| < π]

x dx π = Ei(−pq) 2 +x 2

q2

∞

1.

[a > 0,

si(ax)xμ−1 dx = −

Γ(μ) μπ sin μ μa 2

NT 56(9), ET I 325(12)a

6.252

The sine integral and cosine integral functions



∞

2.

ci(ax)xμ−1 dx = −

0

Γ(μ) μπ cos μaμ 2

[a > 0,

649

0 < Re μ < 1] NT 56(8), ET I 325(13)a

6.247



∞

μ 1 arctan μ β 0   ∞ 1 μ2 −μx ci(βx)e dx = − ln 1 + 2 μ β 0

1.

2. 6.248

si(βx)e−μx dx = −

[Re μ > 0]

NT 49(12), ET I 177(18)

[Re μ > 0]

NT 49(11), ET I 178(19)a

   1 π Φ −1 [Re μ > 0] 1. si(x)e x dx = √ 4μ 2 μ 0

   ∞ 2 1 1 π Ei − [Re μ > 0] 2. ci(x)e−μx dx = 4 μ 4μ 0 %  2  2  2 &  ∞  2  π  −μx 1 1 μ2 π μ e S − − si x + 6.249 dx = + C 2 μ 4 2 4 2 0 

∞

−μx2

8

6.251 1. 2. 6.252

 1 2 √ e−μx dx = kei (2 μ) x μ 0  ∞  1 2 √ e−μx dx = − ker (2 μ) ci x μ 0



∞

si



∞

1. 0

MI 34

MI 34

[Re μ > 0]

ME 26

[Re μ > 0]

MI 34

[Re μ > 0]

MI 34

 2 p > q2  2 p = q2  2 p < q2

π 2p π =− 4p

sin px si(qx) dx = −

=0

FI II 652, NT 50(8)



∞

2.6

cos px si(qx) dx = −

0

=

1 ln 4p

1 q

p+q p−q

2

 p = 0,

p2 = q 2



[p = 0] FI II 652, NT 50(10)

 3.

∞

sin px ci(qx) dx = −

0

=0

1 ln 4p

p2 −1 q2

2

 p = 0,

p2 = q 2



[p = 0] FI II 652, NT 50(9)

650

The Exponential Integral Function and Functions Generated by It



∞

 2 p > q2  2 p = q2  2 p < q2

π 2p π =− 4p

cos px ci(qx) dx = −

4. 0

6.253

=0

FI II 654, NT 50(7)

 6.253

∞

0



 m+1

π rm + r si(ax) sin bx dx = − 2 1 − 2r cos x + r2 4b(1  − r) (1 −mr ) m+1  π 2 + 2r − r − r =− 4b(1 − r) (1 − r2 ) πrm+1 =− 2b(1 − r2)  − r) (1 m+1 π 1+r−r =− 2b(1 − r) (1 − r2 )

[b = a − m] [b = a + m] [a − m − 1 < b < a − m] [a + m < b < a + m + 1] ET I 97(10)

6.254



∞

1. 0

  

1 1 1 dx = L2 − L2 − ci(x) sin x x 2 2 2 2



2.12

6.255

z

log(1 − t) dt and this in turn can t 0 be expressed as L2 (z) = Φ(z, 2, 1) in terms of the Lerch function defined in 9.550, with z real.  ∞ dx π a π cos bx = ln if a > b > 0 si(ax) + 2 x 2 b 0 =0 if a2 ≤ b2 ET I 41(11)

where L2 (x) is the Euler dilogarithm defined as L2 (z) = −



∞

1. −∞



[cos ax ci (a|x|) + sin (a|x|) si (a|x|)]

dx = −π [sign b cos ab si (a|b|) − sin ab ci (a|b|)] x−b [a > 0]

∞

2. −∞

[sin ax ci (a|x|) − sign x cos ax si (a|x|)]

dx = −π [sin (a|b|) si (a|b|) + cos ab ci (a|b|)] x−b [a > 0]

6.256



∞

1. 0

2.

∗



∞

 2 π si (x) + ci2 (x) cos ax dx = ln(1 + a) a [si(x) cos x − ci(x) sin x]2 dx =

0

3.∗



0

∞

si2 (x) cos(ax) dx =

[a > 0]

π 2

π log(1 + a) 2a

ET II 253(4)

[0 ≤ a ≤ 2]

ET II 253(5)

6.259

4.

∗

6.257 6.258

The sine integral and cosine integral functions



∞

π log(1 + a) 2a 0  ∞

√

π a sin bx dx = − J 0 2 ab si x 2b 0



ci2 (x) cos(ax) dx =

∞

si(ax) +

1. 0



651

[0 ≤ a ≤ 2] [b > 0]

ET I 42(18)

dx π sin bx 2 2 x + c2 ( π ' −bc e = [Ei(bc) − Ei(−ac)] + ebc [Ei(−ac) − Ei(−bc)] 4c π = e−bc [Ei(ac) − Ei(−ac)] 4c

[0 < b ≤ a,

c > 0]

[0 < a ≤ b,

c > 0] BI (460)(1)

∞

si(ax) +

2. 0

x dx π cos bx 2 2 x + c2 ( π ' −bc e =− [Ei(bc) − Ei(−ac)] + ebc [Ei(−bc) − Ei(−ac)] 4 π = e−bc [Ei(−ac) − Ei(ac)] 4

[0 < b ≤ a,

c > 0]

[0 < a ≤ b,

c > 0]

BI (460)(2, 5)

6.259



∞

1.

si(ax) sin bx

0



x2

∞

ci(ax) sin bx

0

0

4.∗

0

c > 0]

[0 < a ≤ b,

c > 0]

x dx π = − sinh(bc) Ei(−ac) x2 + c2 2 π π = − sinh(bc) Ei(−bc) + e−bc [Ei(−bc) + Ei(bc) 2 4 − Ei(−ac) − Ei(ac)]

[0 < b ≤ a,

c > 0]

[0 < a ≤ b,

c > 0]

BI (460)(3)a, ET I 97(15)a ∞

3.



[0 < b ≤ a,

ET I 96(8)

2.



dx π Ei(−ac) sinh(bc) = 2 +c 2c π −cb = e [Ei(−bc) + Ei(bc) − Ei(−ac) − Ei(ac)] 4c π + Ei(−bc) sinh(bc) 2c

ci(ax) cos bx

dx + c2 π = cosh bc Ei(−ac) 2c ( π ' −bc e [Ei(ac) + Ei(−ac) − Ei(bc)] + ebc Ei(−bc) = 4c

x2

[0 < b ≤ a,

c > 0]

[0 < a ≤ b,

c > 0]

BI (460)(4), ET I 41(15) ∞

[ci(x) sin x − Si(x) cos x] sin x

2 x dx 1 Ei(a)e−a − Ei(−a)ea = a2 + x2 8 [a real]

652

5.∗

The Exponential Integral Function and Functions Generated by It



∞

[ci(x) sin x − Si(x) cos x]

0

2

6.261

2 x dx π  π 3 e−|a| sinh(a) − Ei(a)e−a − Ei(−a)ea = 2 2 a +x 8a 8|a| [a real]

6.261



∞

1.

si(bx) cos (ax) e

−px

0

  1 2bp a p2 + (a + b)2 dx = − + p arctan 2 ln 2 (a2 + p2 ) 2 p2 + (a − b)2 b − a2 − p 2 [a > 0,



∞

2.

si(βx) cos (ax) e

−μx

∞

1.

ci(bx) sin (ax) e

−μx

0



∞

2. 0

3.

dx = −

2. 3. 6.264

ET I 40(9)

⎡ −1 ⎣ p ln ci(bx) cos (ax) e−px dx = 2 2 2 (a + p ) 2



b 2 + p 2 − a2

2

+ 4a2 p2

b4



ET I 98(16)a

⎤ 2ap ⎦ + a arctan 2 b + p 2 − a2

[a > 0, b > 0, Re p > 0]     (μ − ai)2 (μ + ai)2  ∞ ln 1 + − ln 1 + β2 β2 − ci(βx) cos (ax) e−μx dx = 4(μ + ai) 4(μ − ai) 0

π − μ ln μ [ci(x) cos x + si(x) sin x] e−μx dx = 2 1 + μ2 0 π  ∞ − μ + ln μ −μx [si(x) cos x − ci(x) sin x] e dx = 2 1 + μ2 0    ∞ ln 1 + μ2 [sin x − x ci(x)] e−μx dx = 2μ2 0

 1.

Re μ > |Im β|]

) *  2 2 μ + b2 − a2 + 4a2 μ2 2aμ 1 a μ arctan 2 dx = − ln 2 (a2 + μ2 ) μ + b 2 − a2 2 b4 [a > 0, b > 0, Re μ > 0]

[a > 0, 6.263

ET I 40(8)

μ − ai μ + ai arctan β β − 2(μ + ai) 2(μ − ai) [a > 0,



p > 0]

arctan

0

6.262

b > 0,



−

∞

∞

1.

si(x) ln x dx = C + 1

Re μ > |Im β|]

ET I 41(16)

ET I 41(17)

[Re μ > 0]

ME 26a, ET I 178(21)a

[Re μ > 0]

ME 26a, ET I 178(20)a

[Re μ > 0]

ME 26

NT 46(10)

0

 2.

0

∞

ci(x) ln x dx =

π 2

NT 56(11)

6.282

The probability integral

653

6.27 The hyperbolic sine integral and hyperbolic cosine integral functions 6.271  1.

∞

μ+1 1 1 ln = arccoth μ [Re μ > 1] 2μ μ − 1 μ 0  ∞   1 ln μ2 − 1 2.11 chi(x)e−μx dx = − [Re μ > 1] 2μ 0

   ∞ 2 1 1 π Ei [p > 0] 6.27211 chi(x)e−px dx = 4 p 4p 0 6.273  ∞ ln μ [Re μ > 0] [cosh x shi(x) − sinh x chi(x)] e−μx dx = 2 1.11 μ −1 0  ∞ μ ln μ 2.11 [cosh x chi(x) + sinh x shi(x)] e−μx dx = [Re μ > 2] 1 − μ2 0 

  ∞ 1 1 π 4μ 1 −μx2 11 e Ei − 6.274 [cosh x shi(x) − sinh x chi(x)] e dx = 4 μ 4μ 0 [Re μ > 0]  2   ∞ ln μ − 1 6.275 [x chi(x) − sinh x] e−μx dx = − [Re μ > 1] 2μ2 

  0 ∞ 1 1 π 1 −μx2 Ei − 6.276 [cosh x chi(x) + sinh x shi(x)] e x dx = exp 8 μ3 4μ 4μ 0 [Re μ > 0] 6.277    ∞ ln μ4 − 1 −μx [Re μ > 1] [chi(x) + ci(x)] e dx = − 1. 2μ 0  ∞ μ2 + 1 1 ln 2 [Re μ > 1] 2. [chi(x) − ci(x)] e−μx dx = 2μ μ − 1 0 shi(x)e−μx dx =

MI 34 MI 34

MI 35

MI 35 MI 35

MI 35 MI 35

MI 35

MI 34 MI 35

6.28–6.31 The probability integral 6.281 1.



∞

6

[1 − Φ(px)] x

2q−1

0

  Γ q + 12 dx = √ 2 πqp2q

[Re q > 0,

Re p > 0] NT 56(12), ET II 306(1)a

 1−α

 ∞  2b b 2α  b α 1 − Φ at ± α dt = √ K 1+α (2ab) ± K 1−α (2ab) e±2ab 2α 2α t π a 0

 2.6

[a > 0, 6.282



1. 0

∞

Φ(qt)e−pt dt =





2  p p 1 1−Φ exp p 2q 4q 2

b > 0,

 Re p > 0,

α = 0]

|arg q|
0,

Re β < Re α]

[Re p > 0,

Re(q + p) > 0]

ET II 307(5)

EH II 148(12)



∞ √ q 1 √ e−px dx = e−q p 1−Φ 2 x p 0

 6.284

6.283

 Re p > 0,

|arg q|
0]

s2 4a2



∞

1.

[1 − Φ(βx)] e

μ2 x2 ν−1

x

Γ dx =

0



∞

2. 0

) 1−Φ

√

2x 2

ν +1 2 √ πνβ ν



 s2 Ei − 2 4a  Re s > 0,

MI 37

|arg a|
Re μ2 ,

Re ν > 0

ET II 306(2)

* e

x2 2

ν

xν−1 dx = 2 2 −1 sec

νπ ν

Γ 2 2 [0 < Re ν < 1]

6.287  1.

∞

2

Φ(βx)e−μx x dx =

0



∞

2. 0

2

 Re μ > − Re β 2 ,

β  2μ μ + β 2

[1 − Φ(βx)] e−μx x dx =

1 2μ



β 1−  μ + β2



ET I 325(9)

Re μ > 0



ME 27a, ET I 176(4)

 Re μ > − Re β 2 ,

Re μ > 0



NT 49(14), ET I 177(9)

 3.12 0

∞





 

1 1 r2 r A B − α arctan + β arctan Q(rA) Q(rB) dr = exp − σ2 2σ 2 4 2π αB βA

1 1 1 = − α arctan π α  

4   ∞ 1 σ 2 A2 x 1 −t2 /2 Q(x) = √ 1 − erf √ , α= e dt = , 2 1 + σ 2 A2 2π x 2

 β=

B = A B=A

σ2 B 2 1 + σ2 B 2

BEA

6.295

4.

∗

The probability integral

0

  A2 + 2p A2 3 2ν−2 ABΓ(ν) F1 1, ν, 1; ν + ; 2 , r e Q(Ar)Q(Br)dr = πc(1 + 2ν) 2 A + B 2 + 2p A2 + B 2

*  B 2 + 2p B2 3 , +F1 1, ν, 1; ν + ; 2 LEI c = (A2 + B 2 )(A2 + B 2 + 2p)ν 2 A + B 2 + 2p A2 + B 2



∞



∞

6.288

2ν−1 −pr 2

2

Φ(iax)e−μx x dx =

0

6.289

655



∞

1.

Φ(βx)e(β

2

−μ2 )x2

ai  2μ μ − a2

x dx =

0



β 2μ (μ2 − β 2 )

 a > 0,

Re μ > Re a2

 Re μ2 > Re β 2 ,



MI 37a

|arg μ|
Re β 2 ,

√ ∞ √

2 b−a √ [Re μ > −a > 0, Φ b − ax e−(a+μ)x x dx = 2(μ + a) μ + b 0  

 ∞ 2 μ 2 i μ2 1 + eμ /4 Ei − Φ(ix)e−(μx+x ) x dx = √ π μ a 4 0 [Re μ > # $ 0]  ∞ 2 2 arctan μ 1 1 [1 − Φ(x)] e−μ x x2 dx = √ − 2 2 2 π μ3 μ (μ  + 1) 0 |arg μ| < π4 √  ∞  2 dx μ+1+1 1 = ln √ = arccoth μ + 1 Φ(x)e−μx x 2 μ+1−1 0 [Re μ > 0]

|arg μ|
0]  

2    b + 2ax e−bμ exp − μ2 − a2 x2 + ab x dx = 1−Φ 2x 2μ(μ + a)

3.

6.298



MI 37

MI 38a

 

2 2 β √ 1  exp [−2 (βγ + β μ)] e(γ −μ)x x dx = √ √ 1 − Φ γx + x 2 μ μ+γ

∞ #



2

6.296

ET I 177(12)a

MI 38 [a > 0, b > 0, Re μ > 0]   $ 



2 b − 2ax2 b + 2ax2 1 e−ab + 1 − Φ eab e−μx x dx = exp −b a2 + μ 2x 2x μ

MI 38 [a > 0, b > 0, Re μ > 0]  $

 ∞# 2 2 b − 2ax2 b + 2ax2 1 √ − eab Φ e−(μ−a )x x dx = 2 cosh ab − e−ab Φ exp (−b μ) 2 2x 2x μ−a 0 [a > 0, b > 0, Re μ > 0] MI 38  ∞    1 2  2 1 2 exp 2 a K ν a cosh(2νt) exp (a cosh t) [1 − Φ (a cosh t)] dt = 2 cos(νπ) 0 Re a > 0, − 12 < Re ν < 12



b2 1 1 − e− 4a2 [a > 0, b > 0] [1 − Φ(ax)] sin bx dx = b  √ √  0 ∞ 2 + a 2b 2b 1 b + a a √ + 2 arctan Φ(ax) sin bx2 dx = √ ln 2 b − a2 4 2πb b + a − a 2b 0 [a > 0, b > 0]

6.311 6.312

∞

6.313 

∞

1. 0

2. 0

∞

ET I 96(4)

ET I 96(3)

⎛ α ⎞ 12  − 12 1 √  1 α2 + β 2 2 − α sin(βx) 1 − Φ αx dx = − ⎝ 2 2 2 ⎠ β α +β 

[Re α > |Im β|] 

ET II 308(10)



⎛

√  cos(βx) 1 − Φ αx dx = ⎝

ET II 307(6)

⎞ 12

α  − 12 1 2 2 2 2 ⎠ α + β + α α2 + β 2 [Re α > |Im β|]

ET II 307(7)

6.321

6.314

Fresnel integrals



∞

1. 0

      a 1 1 dx = b−1 exp −(2ab) 2 cos (2ab) 2 sin(bx) 1 − Φ x 

[Re a > 0, b > 0]  

∞     1 1 a dx = −b−1 exp −(2ab) 2 sin (2ab) 2 cos(bx) 1 − Φ x 0 [Re a > 0,

6.315

ET II 307(8)



 2.

657



∞

1.

ν−1

x

0



∞

2.

xν−1 cos(βx) [1 − Φ(αx)] dx =



2F 2

[Re α > 0,

Re ν > −1]  β2 ν ν+1 1 ν , ; , + 1; − 2 2 2 2 2 4α

∞

[a > 0,



2  2 1 p p dx = Ei − 2 − Ei [Φ(ax) − Φ(bx)] cos px x 2 4b 4a2

4. 0



Γ 2 + 12 ν √ πναν

[Re α > 0, Re ν > 0]  



2 1 b2 1 b [1 − Φ(ax)] cos bx · x dx = 2 exp − 2 − 2 1 − exp − 2 2a 4a b 4a

0



1

∞

3.

∞

5. 0

 6.316



 √  1 1 x− 2 Φ a x sin bx dx = √ 2 2πb

∞

1

e2x

ET II 307(9)

 

 Γ 1 + 12 ν β 3 ν +3 β2 ν +1 ν , + 1; , ; − sin(βx) [1 − Φ(αx)] dx = √ F 2 2 2 2 2 2 4α2 π(ν + 1)αν+1

0



b > 0]

2



1−Φ

0

x √ 2

%

sin bx dx =

ET II 307(4)

b > 0]

ET I 40(5)

[a > 0, b > 0, p > 0] * ) √ *& √ a 2b b + a 2b + a2 √ + 2 arctan ln b − a2 b − a 2b + a2





)



π b2 e 2 1−Φ 2

[a > 0,  b √ 2 [b > 0]

b > 0]

6.3176 6.318

∞

2

ET I 40(6)

ET I 96(3)

ET I 96(5)

√ i π − b22 e 4a e−a x Φ(iax) sin bx dx = [b > 0] a 2 0 ∞

2 2 2 1 − e−p − √ (1 − Φ(p)) [1 − Φ(x)] si(2px) dx = πp π 0 [p > 0] 

ET II 307(3)

2

ET I 96(2)

NT 61(13)a

6.32 Fresnel integrals 6.321  1. 0

∞



 1 − S (px) x2q−1 dx = 2

√   2q + 1 π 2 Γ q + 12 sin 4 √ 2q 4 πqp

 0 < Re q < 32 ,

p>0



NT 56(14)a

658

The Exponential Integral Function and Functions Generated by It



∞



2. 0

6.322 1. 2. 6.323

2. 6.324 1. 2. 6.325

√

  2q + 1 π 2 Γ q + 12 cos 4 √ 4 πqp2q

#  p  p2 1 p2 cos −C + sin S (t)e 4 2 2 4 0 #    ∞

p2 1 p p2 1 cos −S − sin C (t)e−pt dt = p 4 2 2 4 0



∞

−pt

1 dt = p

 0 < Re q < 32 ,

p>0

p $ 1 −S 2 2  $ p

1 −C 2 2

∞

√

S t e−pt dx =



∞

∞

1. 0



∞

2. 0

NT 56(13)a

MO 173a MO 172a



 1 + sin p2 − cos p2 1 − S (x) sin 2px dx = 2 4p 0   ∞ 1 − sin p2 − cos p2 1 − C (x) sin 2px dx = 2 4p 0







12 p2 + 1 − p  2p p2 + 1 0 

12  ∞

2+1+p p √  C t e−pt dt = 2p p2 + 1 0

 1.

 1 − C (px) x2q−1 dx = 2

6.322

EF 122(58)a

EF 122(58)a



√ π −5 2 2 S (x) sin b x dx = b =0 2 2

√ π −5 2 2 C (x) cos b2 x2 dx = b =0

[p > 0]

NT 61(12)a

[p > 0]

NT 61(11)a

 0 < b2 < 1  2 b >1 ET I 98(21)a

 0 < b2 < 1  2 b >1 ET I 42(22)

6.326



∞



1. 0

 2. 0

∞

 π 1/2 1 1 + sin p2 − cos p2 − S (x) si(2px) dx = [S (p) + C (p) − 1] − 2 8 4p [p > 0]  π 1/2 1 1 − sin p2 − cos p2 − C (x) si(2px) dx = [S (p) − C (p)] − 2 8 4p [p > 0]

NT 61(15)a

NT 61(14)a

6.414

The gamma function

659

6.4 The Gamma Function and Functions Generated by It 6.41 The gamma function 

6.411

∞

12 −∞

 6.412

Γ(α + x) Γ(β − x) dx

i∞

12 −i∞

= −iπ21−α−β Γ(α + β)

[Re(α + β) < 1,

Im α > 0,

Im β > 0] ET II 297(3)

= iπ21−α−β Γ(α + β)

[Re(α + β) < 1,

Im α < 0,

Im β < 0] ET II 297(2)

=0

[Re(α + β) < 1,

(Im α) (Im β) < 0]

Γ(α + s) Γ(β + s) Γ(γ − s) Γ(δ − s) ds = 4πi

ET II 297(1)

Γ(α + γ) Γ(α + δ) Γ(β + γ) Γ(β + δ) Γ(α + β + γ + δ) [Re α, Re β, Re γ, Re δ > 0] ET II 302(32)

6.413



∞

1.

√

0

2.

6.414 1.

3.



5.

∞

Γ(α + x) dx = 0 −∞ Γ(β + x)

[a > 0,

b > 0]

ET II 302(27)

 0 < a < b − 12

ET II 302(28)

[Im α = 0,

Re(α − β) < −1] ET II 297(4)

∞

dx 2α+β−2 = [Re(α + β) > 1] Γ(α + β − 1) −∞ Γ(α + x) Γ(β − x)  ∞ Γ(γ + x) Γ(δ + x) dx = 0 −∞ Γ(α + x) Γ(β + x) [Re(α + β − γ − δ) > 1, Im γ, Im δ > 0] 

4.

    π Γ(a) Γ a + 12 Γ(b) Γ b + 12 Γ(a + b)   2 Γ a + b + 12

    √ !  ∞! 1 1 ! Γ(a + ix) !2 ! dx = π Γ(a) Γ a + 2 Γ b − a − 2 ! ! Γ(b + ix) ! 2 Γ(b) Γ b − 12 Γ(b − a) 0

 2.

2

|Γ(a + ix) Γ(b + ix)| dx =

ET II 297(5)

ET II 299(18)

∞

Γ(γ + x) Γ(δ + x) ±2π 2 i Γ(α + β − γ − δ − 1) dx = sin[π(γ − δ)] Γ(α − γ) Γ(α − δ) Γ(β − γ) Γ(β − δ) −∞ Γ(α + x) Γ(β + x)

[Re(α + β − γ − δ) > 1, Im γ < 0, Im δ < 0. In the numerator, we take the plus sign if Im γ > Im δ and the minus sign if Im γ < Im δ.] ET II 300(19)  1   ∞ π exp ± 2 π(δ − γ)i Γ(α − β − γ + x + 1) dx     = Γ(α + x) Γ(β − x) Γ(γ + x) Γ(β + γ − 1) Γ 12 (α + β) Γ 12 (γ − δ + 1) −∞ [Re(β + γ) > 1, δ = α − β − γ + 1, Im δ = 0. The sign is plus in the argument if the exponential for Im δ > 0 and minus for Im δ < 0]. ET II 300(20)

660

The Gamma Function and Functions Generated by It

 6.

6.415

∞

dx Γ(α + β + γ + δ − 3) = Γ(α + x) Γ(β − x) Γ(γ + x) Γ(δ − x) Γ(α + β − 1) Γ(β + γ − 1) Γ(γ + δ − 1) Γ(δ + α − 1) −∞ [Re(α + β + γ + δ) > 3]

6.415



−∞

1. −∞

R(x) dx Γ(α + x) Γ(β − x) Γ(γ + x) Γ(δ − x) =

 1 Γ(α + β + γ + δ − 3) R(t) dt Γ(α + β − 1) Γ(β + γ − 1) Γ(γ + δ − 1) Γ(δ + α − 1) 0 [Re(α + β + γ + δ) > 3, R(x + 1) = R(x)] ET II 301(24) 

 R(t) cos 12 π(2t + α − β) dt R(x) dx   = 0 γ+δ α+β Γ(α + x) Γ(β − x) Γ(γ + x) Γ(δ − x) −∞ Γ Γ(α + δ − 1) Γ 2 2 [α + δ = β + γ, Re(α + β + γ + δ) > 2, R(x + 1) = −R(x)]

 2.

ET II 300(21)

1

∞

ET II 301(25)

6.42 Combinations of the gamma function, the exponential, and powers 6.421



∞

1. −∞

Γ(α + x) Γ(β − x) exp [2(πn + θ)xi] dx = 2πi Γ(α + β)(2 cos θ)−α−β exp[(β − α)iθ]

) Re(α + β) < 1,  2.

3.

4.

− π2 < θ
0     n an integer, ηn (ξ) = sign 12 − n if 12 − n Im ξ < 0 ET II 298(7)

∞

eπicx dx =0 −∞ Γ(α + x) Γ(β − x) Γ(γ + kx) Γ(δ − kx) [Re(α + β + γ + δ) > 2, c and k are real,



∞



∞

|c| > |k| + 1]

ET II 301(26)

Γ(α + x) exp[(2πn + π − 2θ)xi] dx −∞ Γ(β + x)   (2 cos θ)β−α−1 = 2πi sign n + 12 exp[−(2πn + π − θ)αi + θi(β − 1)] Γ(β − α)    Re(β − α) > 0, − π2 < θ < π2 , n is an integer, n + 12 Im α < 0 ET II 298(8) Γ(α + x) exp[(2πn + π − 2θ)xi] dx = 0 −∞ Γ(β + x)  Re(β − α) > 0, − π2 < θ < π2 ,

n is an integer,



n+

1 2



Im α > 0



ET II 297(6)

6.422

6.422

The gamma function, the exponential, and powers



i∞

1. −i∞

    Γ(s − k − λ) Γ λ + μ − s + 12 Γ λ − μ − s + 12 z s ds 



γ+i∞

2. γ−i∞



γ+i∞

3. γ−i∞



−∞i



Γ(s) Γ −i∞

 6.

t−p 2

i∞

c+i∞

3 c−i∞



Re(k + λ) < 0,

Γ(−s) Γ(β + s)ts ds = 2πi Γ(β)(1 + t)−β

Γ

5.

   z − k − μ Γ 12 − k + μ z λ e 2 W k,μ (z) Re λ > |Re μ| − 12 , |arg z| < 32 π ET II 302(29)

= 2πi Γ

1 2

Γ(α + s) Γ(−s) Γ(1 − c − s)xs ds = 2πi Γ(α) Γ(α − c + 1)Φ(α, c; x)  − Re α < γ < min (0, 1 − Re c) , − 32 π < arg x < 32 π

∞i

4.

661

1



[0 > γ > Re(1 − β),

EH I 256(5)

|arg t| < π] EH I 256, BU 75

√ t−p−2 1 2 Γ(−t) 2 z t dt = 2πie 4 z Γ(−p) D p (z)  |arg z| < 34 π, p is not a positive integer

2ν +

1 4

  − s Γ 12 ν −

1 4

−s



z2 2

WH

s ds     1 1 1 3 2 = 2πi2 4 − 2 ν z − 2 e 4 z Γ 12 ν + 14 Γ 12 ν − 14 D ν (z)  |arg z| < 34 π, ν = 12 , − 21 , − 32 , . . . EH II 120

 1 −s  1    −1 Γ 2 ν + 12 s Γ 1 + 12 ν − 12 s ds = 4πi J ν (x) 2x [x > 0, − Re ν < c < 1]

−c+i∞

7. −c−i∞

 ν+2s 1 Γ(−ν − s) Γ(−s) − 21 iz ds = −2π 2 e 2 iνπ H (1) ν (z)  |arg(−iz)|
0] EH II 83(36) Γ(ν + s + 1) −i∞  i∞   5 Γ(−s) Γ(−2ν − s) Γ ν + s + 12 (−2iz)s ds = −π 2 e−i(z−νπ) sec(νπ)(2z)−ν H (1) ν (z) −i∞  3 |arg(−iz)| < 2 π, 2ν = ±1, ±3 . . . 

9. 10.

EH II 83(37)

662

The Gamma Function and Functions Generated by It



i∞



i∞

6.422

  5 Γ(−s) Γ(−2ν − s) Γ ν + s + 12 (2iz)s ds = π 2 ei(z−νπ) sec(νπ)(2z)−ν H (2) ν (z) −i∞  |arg(iz)| < 32 π, 2ν = ±1, ±3 . . .

11.

EH II 84(38)

12.

Γ(s) Γ −i∞

1 2

   3 3 1 − s − ν Γ 12 − s + ν (2z)s ds = 2 2 π 2 iz 2 ez sec(νπ) K ν (z)  |arg z| < 32 π, 2ν = ±1, ±3, . . . EH II 84(39)

 13.

− 12 +i∞

− 12 −i∞



Γ(−s) 2s x ds = 4π s Γ(1 + s)



∞

2x

J 0 (t) dt t

[x > 0]

MO 41

i∞

Γ(α + s) Γ(β + s) Γ(−s) Γ(α) Γ(β) (−z)s ds = 2πi F (α, β; γ; z) Γ(γ + s) Γ(γ) −i∞

14.

[For arg(−z) < π, the path of integration must separate the poles of the integrand at the points s = 0, 1, 2, 3, . . . from the poles s = −α − n and s = −β − n (for n = 0, 1, 2, . . . )]. 

δ+i∞

15. δ−i∞

Γ(α + s) Γ(−s) 2πi Γ(α) (−z)s ds = 1 F 1 (α; γ; z) Γ(γ + s) Γ(γ) − π2 < arg(−z) < π2 , 0 > δ > − Re α, γ = 0, 1, 2, . . .

)   *2  1

1  Γ 12 − s 1 z s ds = 2πiz 2 2π −1 K 0 4z 4 − Y 0 4z 4 Γ(s) −i∞

 16.

i∞

[z > 0]    1 Γ λ+μ−s+ Γ λ−μ−s+ 2 s z z ds = 2πiz λ e− 2 W k,μ (z) Γ(λ − k − s + 1) −i∞  Re λ > |Re μ| − 12 ,

 17.



i∞



ET II 303(33)

1 2



i∞





|arg z|
0, Re(λ + μ) > − 21 , |arg z|
0, j = 1, . . . , m

p + q < 2(m + n); Re ak < 1,

ET II 303(34)

6.433

6.423

Gamma functions and trigonometric functions



∞

1.

e−αx

0



∞

663

  dx = ν e−α Γ(1 + x)

MI 39, EH III 222(16)

  dx = eβα ν e−α , β MI 39, EH III 222(16) Γ(x + β + 1) 0  ∞   xm dx = μ e−α , m Γ(m + 1) 3. e−αx [Re m > −1] MI 39, EH III 222(17) Γ(x + 1) 0  ∞   xm dx = enα μ e−α , m, n Γ(m + 1) 4. e−αx MI 39, EH III 222(17) Γ(x + n + 1) 0 

α+β−2 θ   1  ∞ 2 cos R(x) exp[(2πn + θ)xi] dx 1 2 = exp θ(β − α)i 6.424 R(t) exp(2πnti) dt Γ(α + x) Γ(β − x) Γ(α + β − 1) 2 −∞ 0 [Re(α + β) > 1, −π < θ < π, n is an integer, R(x + 1) = R(x)] ET II 299(16) 2.

e−αx

6.43 Combinations of the gamma function and trigonometric functions 6.431

r p+q−2 r(q − p) 2 cos sin sin rx dx 2 2 = Γ(p + q − 1) −∞ Γ(p + x) Γ(q − x)

 1.12

∞

[|r| > π]

=0 [r is real;  2.

∞

cos rx dx = Γ(p + x) Γ(q − x) −∞

2 cos

r p+q−2 r(q − p) cos 2 2 Γ(p + q − 1) [r is real;



sin(mπx) dx =0 sin(πx) Γ(α + x) Γ(β − x) −∞

1.



MO 10a, ET II 298(9, 10)

[|r| < π]

Re(p + q) > 1]

MO 10a, ET II 299(13, 14)

∞

= 6.433

Re(p + q) > 1]

[|r| > π]

=0

6.432

[|r| < π]

∞

[m is an even integer]

2α+β−2 Γ(α + β − 1)

[m is an odd integer] [Re(α + β) > 1]

ET II 298(11, 12)

 sin π2 (β − α)  

γ+δ α+β Γ Γ(α + δ − 1) 2Γ 2 2 [α + δ = β + γ, Re(α + β + γ + δ) > 2]

sin πx dx = −∞ Γ(α + x) Γ(β − x) Γ(γ + x) Γ(δ − x)

ET II 300(22)

664

The Gamma Function and Functions Generated by It



6.441

 cos π2 (β − α)  

γ+δ α+β Γ Γ(α + δ − 1) 2Γ 2 2 [α + δ = β + γ, Re(α + β + γ + δ) > 2]

∞

cos πx dx = −∞ Γ(α + x) Γ(β − x) Γ(γ + x) Γ(δ − x)

2.

ET II 301(23)

6.44 The logarithm of the gamma function∗ 6.441



p+1

1. p



√ ln Γ(x) dx = ln 2π + p ln p − p 

1

2.

1

ln Γ(x) dx = 0



FI II 784

√ ln Γ(1 − x) dx = ln 2π

FI II 783

0

√ ln Γ(x + q) dx = ln 2π + q ln q − q

1

3.

[q ≥ 0]

NH 89(17), ET II 304(40)

0



√ z(z + 1) + z ln Γ(z + 1) − ln G(z + 1), ln Γ(x + 1) dx = z ln 2π − 2  ∞ # $

z z k z(z + 1) C z 2 + z2 2 − where G(z + 1) = (2π) exp − 1+ exp −z + 2 2 k 2k

z

4. 0

WH

k=1



n

5.

ln Γ(a + x) dx =

0

6.∗



n−1 " k=0

1

0



[a ≥ 0, n = 1, 2, . . .] ET II 304(41)

  √ √ √ C 4 π 1 ζ (2) ζ (2) ln2 Γ(x)dx = + + + C ln 2π + ln2 2π − C + 2 ln 2π 12 48 3 3 π2 2π 2

1

6.442

√ 1 (a + k) ln(a + k) − na + n ln 2π − n(n − 1) 2

2

2

exp(2πnxi) ln Γ(a + x) dx = (2πni)−1 [ln a − exp(−2πnai) Ei(2πnai)]

0

6.443



[a > 0,

n = ±1, ±2, . . .]

ET II 304(38)

1

1 NH 203(5), ET II 304(42) [ln(2πn) + C] 2πn 0



   1 π 1 1 1 1 ln + 2 1 + + ···+ + ln Γ(x) sin(2n + 1)πx dx = (2n + 1)π 2 3 2n − 1 2n + 1 0 ln Γ(x) sin 2πnx dx =

1. 2.

ET II 305(43)



1

3.

ln Γ(x) cos 2πnx dx = 0

 4.

8 0

1

1 4n

2 ln Γ(x) cos(2n + 1)πx dx = 2 π

NH 203(6), ET II 305(44)

)

∞

" ln k 1 (C + ln 2π) + 2 2 (2n + 1) 4k 2 − (2n + 1)2

* NH 203(6)

k=2

∗ Here, we are violating our usual order of presentation of the formulas in order to make it easier to examine the integrals involving the gamma function

6.456

The incomplete gamma function



1

5.12

665

sin(2πnx) ln Γ(a + x) dx = −(2πn)−1 [ln a − cos(2πna) ci(2πna) − sin(2πna) si(2πna)]

0

 6.12

[a > 0, 1

n = 1, 2, . . .]

ET II 304(36)

cos(2πnx) ln Γ(a + x) dx = −(2πn)−1 [− sin(2πna) ci(2πna) + cos(2πna) si(2πna)]

0

[a > 0,

n = 1, 2, . . .]

ET II 304(37)

6.45 The incomplete gamma function 6.451 1. 2. 6.452



∞

1 Γ(β)(1 + α)−β α 0    ∞ 1 1 e−αx Γ(β, x) dx = Γ(β) 1 − α (α + 1)β 0



∞

1. 0

e−αx γ(β, x) dx =

[β > 0]

MI 39

[β > 0]

MI 39



2 1 x2 e−μx γ ν, 2 dx = 2−ν−1 Γ(2ν)e(aμ) D −2ν (2aμ) 8a μ  |arg a|
− 12 ,

Re μ > 0



ET I 179(36)

 2. 6.453 6.454

∞



√   2 2 a dx = √ e(aμ) K 14 a2 μ2 μ 3 4

 1 x2 , 2 |arg a| < π4 , Re μ > 0 4 8a 0  ∞ a

 1 1 √ dx = 2a 2 ν μ 2 ν−1 K ν (2 μa) e−μx Γ ν, |arg a| < π2 , Re μ > 0 x 

2 0 ∞  √  α α −βx − 12 ν ν − 12 ν−1 D −ν √ e γ ν, α x dx = 2 α β Γ(ν) exp 8β 2β 0 [Re β > 0, Re ν > 0] e−μx γ

ET I 179(35)

ET I 179(32)

ET II 309(19), MI 39a

6.455



∞

1.

xμ−1 e−βx Γ(ν, αx) dx =

0



∞

2. 0

6.456 1. 2.

xμ−1 e−βx γ(ν, αx) dx =



β αν Γ(μ + ν) 1, μ + ν; μ + 1; 2F 1 μ(α + β)μ+ν α+β [Re(α + β) > 0, Re μ > 0, Re(μ + ν) > 0]

ET II 309(16)



α αν Γ(μ + ν) 1, μ + ν; ν + 1; F 2 1 ν(α + β)μ+ν α+β [Re(α + β) > 0, Re β > 0, Re(μ + ν) > 0]

ET II 308(15)



√ √ γ (2ν, α) 1 1 dx = π e−αx (4x)ν− 2 γ ν, 1 4x αν+ 2 0 

√ √  ∞ π Γ (2ν, α) 1 −αx ν− 12 dx = e (4x) Γ ν, 1 4x αν+ 2 0



∞

MI 39a MI 39a

666

6.457

The Gamma Function and Functions Generated by It

6.457

 √ √ γ (2ν + 1, α) 1 √ γ ν + 1, dx = π 1. e MI 39 1 x 4x αν+ 2 0 

√  ∞ √ Γ (2ν + 1, α) (4x)ν 1 dx = π 2. e−αx √ Γ ν + 1, MI 39 1 4x x αν+ 2 0



2  ∞       1 b b D 2ν−2 √ 6.458 x1−2ν exp αx2 sin(bx) Γ ν, αx2 dx = π 2 2−ν αν−1 Γ 32 − ν exp 8α 2α 0  |arg α| < 3π , 0 < Re ν < 1 2 

∞

ν −αx (4x)

6.46–6.47 The function ψ(x) 

6.461

ψ(t) dt = ln Γ(x)

1

1

ψ(α + x) dx = ln α 0 ∞ x−α [C + ψ(1 + x)] = −π cosec(πα) ζ(α) 0  1 e2πnxi ψ(α + x) dx = e−2πnαi Ei(2πnαi)

6.462 6.463 6.464 6.465

x

[α > 0]

ET II 305(1)

[1 < Re α < 2]

ET II 305(6)

[α > 0;

ET II 305(2)

n = ±i, ±2, . . .]

0



1

1.8 0

) * ∞ " ln k 2 C + ln 2π + 2 ψ(x) sin πx dx = − π 4k 2 − 1 k=2

(see 6.443 4)



1

2. 0

1 ψ(x) sin(2πnx) dx = − π 2



∞

6.466



[n = 1, 2, . . .]

NH 204 ET II 305(3)

 −1 [ψ(α + ix) − ψ(α − ix)] sin xy dx = iπe−αy 1 − e−y

0

6.467

ET II 309(18)

[α > 0,

y > 0]

ET I 96(1)

1

sin(2πnx) ψ(α + x) dx = − sin(2πnα) ci(2πnα) + cos(2πnα) si(2πnα)

1.12 0



[α ≥ 0;

n = 1, 2, . . .]

ET II 305(4)

1

cos(2πnx) ψ(α + x) dx = sin(2πnα) si(2πnα) + cos(2πnα) ci(2πnα)

2.12 0

[α > 0; 

1

6.468 6.469

0



1 ψ(x) sin2 πx dx = − [C + ln(2π)] 2

1

ψ(x) sin πx cos πx dx = −

1. 0

π 4

n = 1, 2, . . .]

ET II 305(5) NH 204

NH 204

6.511

Bessel functions



n 1 − n2 1 n−1 = ln 2 n+1

1

ψ(x) sin πx sin(nπx) dx =

2.8 0

667

[n is even] [n > 1 is odd] NH 204(8)a

6.471  ∞ x−α [ln x − ψ(1 + x)] dx = π cosec(πα) ζ(α) [0 < Re α < 1] 1. 0  ∞  2. x−α [ln(1 + x) − ψ(1 + x)] dx = π cosec(πα) ζ(α) − (α − 1)−1

ET II 306(7)

0



[0 < Re α < 1] ∞

[ψ(x + 1) − ln x] cos(2πxy) dx =

3. 0

6.472  1.

∞

ET II 306(8)

1 [ψ(y + 1) − ln y] 2

ET II 306(12)

  x−α (1 + x)−1 − ψ  (1 + x) dx = −πα cosec(πα) ζ(1 + α) − α−1

0



[|Re α| < 1] ∞

2.

ET II 306(9)

 x−α x−1 − ψ  (1 + x) dx = −πα cosec(πα) ζ(1 + α)

0

[−2 < Re α < 0]



∞

6.473

x−α ψ (n) (1 + x) dx = (−1)n−1

0

π Γ(α + n) ζ(α + n) Γ(α) sin πα [n = 1, 2, . . . ;

ET II 306(10)

0 < Re α < 1] ET II 306(11)

6.5–6.7 Bessel Functions 6.51 Bessel functions 6.511  ∞ 1 J ν (bx) dx = 1. b 0  ∞ νπ

1 2. Y ν (bx) dx = − tan b 2 0

[Re ν > −1,

b > 0]

[|Re ν| < 1,

b > 0]

ET II 22(3)

WA 432(7), ET II 96(1)



a

3. 0



a

4. 0

 5.

0

a

J ν (x) dx = 2

∞ "

J ν+2k+1 (a)

[Re ν > −1]

ET II 333(1)

k=0

J 12 (t) dt = 2 S

√  a

J − 12 (t) dt = 2 C

√  a

WA 599(4) WA 599(3)

668

Bessel Functions



a

6.

J 0 (x) dx = a J 0 (a) +

0



a

7.

6.512

πa [J 1 (a) H0 (a) − J 0 (a) H1 (a)] 2

J 1 (x) dx = 1 − J 0 (a)

[a > 0]

ET II 7(2)

[a > 0]

ET II 18(1)

0



∞

8. a



∞

9. a



b

10. a

 11.

J 0 (x) dx = 1 − a J 0 (a) +

J 1 (x) dx = J 0 (a)

Y ν (x) dx = 2

a

I ν (x) dx = 2

∞

∞

13. 0

1.11

[a > 0]

ET II 18(2)

[Y ν+2n+1 (b) − Y ν+2n+1 (a)]

ET II 339(46)

∞ "

(−1)n I ν+2n+1 (a)

[Re ν > −1]

K 0 (ax) =

π 2a

[a > 0]

K 20 (ax) =

π2 4a

[a > 0]

0

6.512

ET II 7(3)

ET II 364(1)

n=0

12. 

∞ "

[a > 0]

n=0

0



πa [J 0 (a) H1 (a) − J 1 (a) H0 (a)] 2

 μ+ν +1

  ∞ Γ b2 μ+ν+1 ν −μ+1 2 F

, ; ν + 1; 2 J μ (ax) J ν (bx) dx = bν a−ν−1 μ−ν +1 2 2 a 0 Γ(ν + 1) Γ 2 [a > 0, b > 0, Re(μ + ν) > −1, b < a.

For b > a, the positions of μ and ν should be reversed.] 

∞

2.7 0

 3.8 0

 β2 β ν−n−1 Γ(ν) F ν, −n; ν − n; 2 J ν+n (αt) J ν−n−1 (βt) dt = ν−n α n! Γ(ν − n) α n 1 = (−1) 2α =0 [Re(ν) > 0]

∞

β ν−1 αν 1 = 2β

J ν (αx) J ν−1 (βx) dx =

=0

ET II 48(6)

[0 < β < α] [0 < β = α] [0 < α < β] MO 50

[β < α] [β = α] [β > α] [Re ν > 0]

WA 444(8), KU (40)a

6.513

Bessel functions



∞

4.

ν −ν−1

J ν+2n+1 (ax) J ν (bx) dx = b a

0

P (ν,0) n

669

 2b2 1− 2 a

=0 

[Re ν > −1 − n,

0 < b < a]

[Re ν > −1 − n,

0 < a < b] ET II 47(5)

∞

5.

 Re ν > − 12 ,

1 2a

J ν+n (ax) Y ν−n (ax) dx = (−1)n+1

0

a > 0,

n = 0, 1, 2, . . . ET II 347(57)

 ∞ b2 b−1 ln 1 − 2 J 1 (bx) Y 0 (ax) dx = − π a 0  a ∞ " 2 J ν (x) J ν+1 (x) dx = [J ν+n+1 (a)] 

6. 7.

0



8.

9

9. 10. 6.513

ET II 21(31)

[Re ν > −1]

ET II 338(37)

n=0 ∞

1 δ(b − a) a 0

  ∞ b2 1 ln 1 + 2 K 0 (ax) J 1 (bx) = 2b a 0

  ∞ b2 1 K 0 (ax) I 1 (bx) = − ln 1 − 2 2b a 0

[n = 0, 1, . . .]

k J n (ka) J n (kb) dk =

JAC 110

[a > 0,

b > 0]

[a > 0,

b > 0]

 1 + ν + 2μ ∞ 2 

[J μ (ax)]2 J ν (bx) dx = a2μ b−2μ−1 1 + ν − 2μ 2 0 [Γ(μ + 1)] Γ 2 ⎡ ⎛

 1.

[0 < b < a]

Γ

1− ⎢ ⎜ 1 − ν + 2μ 1 + ν + 2μ ⎜ ×⎢ , ; μ + 1; ⎣F ⎝ 2 2 [Re ν + Re 2μ > −1, 

∞

2. 0

b−1 Γ [J μ (ax)] K ν (bx) dx = 2 2

2μ + ν + 1 2

 3. 0

∞



⎞⎤2 4a2 1 − 2 ⎟⎥ b ⎟⎥ ⎠⎦ 2

0 < 2a < b]

ET II 52(33)

 *2  ) 2μ − ν + 1 4a2 −μ Γ P 1 ν− 1 1+ 2 2 2 2 b [2 Re μ > |Re ν| − 1, Re b > 2|Im a|] 

ET II 138(18)

ν + 2μ + 1     eμπi Γ 4a2 4a2 2 −μ −μ  P 1 ν− 1

I μ (ax) K μ (ax) J ν (bx) dx = 1 + 2 Q 1 ν− 1 1+ 2 2 2 2 2 ν − 2μ + 1 b b bΓ 2 [Re a > 0, b > 0, Re ν > −1, Re(ν + 2μ) > −1] ET II 65(20)

670

Bessel Functions



∞

4. 0

νπ

π sec P μ1 ν− 1 J μ (ax) J −μ (ax) K ν (bx) dx = 2 2 2b 2

∞

5. 0



z

6.

   4a2 4a2 −μ 1 + 2 P 1 ν− 1 1+ 2 2 2 b b [|Re ν| < 1, Re b > 2|Im a|] ET II 138(21)

1 + ν + 2μ )  *2 e2μπi Γ 4a2 2 2 −μ 

Q 1 ν− 1 [K μ (ax)] J ν (bx) dx = 1+ 2 2 2 1 + ν − 2μ b bΓ 2    Re a > 0, b > 0, Re 12 ν ± μ > − 21

J μ (x) J ν (z − x) dx = 2

0









6.514

∞ "

(−1)k J μ+ν+2k+1 (z)

[Re μ > −1,

ET II 66(28)

Re ν > −1]

WA 414(2)

k=0 z

7.

J μ (x) J −μ (z − x) dx = sin z

[−1 < Re μ < 1]

WA 415(4)

J μ (x) J 1−μ (z − x) dx = J 0 (z) − cos(z)

[−1 < Re μ < 2]

WA 415(4)

0

 8.

z

0



∞

9.

1 b

 2 b = arcsin πb 2a

0

6.514



∞

1.

Jν

a

x

0



∞

2.

Jν

a

x

0



Jν

a

0

Yν

0

5. 0

√

J ν (bx) dx = b−1 J 2ν 2 ab

Y ν (bx) dx = b−1



 a > 0,

√ 2 √  Y 2ν 2 ab + K 2ν 2ab π  a > 0,

b > 0,

Re ν > − 12

ET II 57(9)

b > 0,

− 12 < Re ν
0, Re b > 0, |Re ν| < 52 ET II 141(31)

∞

4.



[2a > b > 0]

ET II 110(12) ∞

3.



[b > 2a > 0]

J 20 (ax) J 1 (bx) =

a

x

J ν (bx) dx = −

2b−1 π

 √

K 2ν 2 ab −

π 2

√  Y 2ν 2 ab  a > 0, b > 0,

|Re ν|
0,

b > 0,

|Re ν|
0, Re b > 0, |Re ν| < 52

0



ET II 143(37)

∞

7.

Kν

  

√

√  3νπ 3νπ Y ν (bx) dx = −2b−1 sin ker2ν 2 ab + cos kei2ν 2 ab x 2 2  Re a > 0, b > 0, |Re ν| < 12

a

0



ET II 113(28) ∞

8.

Kν

a

x

0

6.515



∞

1.

Jμ

a

0



∞

2. 0



671

∞

3. 0

x

√

K ν (bx) dx = πb−1 K 2ν 2 ab

Yμ

a

x

[Re a > 0,

Re b > 0]

√

√

K 0 (bx) dx = −2b−1 J 2μ 2 ab K 2μ 2 ab

[a > 0, Re b > 0]  a 2 1 √

√

1 Kμ K 0 (bx) dx = 2πb−1 K 2μ 2e 4 πi ab K 2μ 2e− 4 πi ab x

H (1) μ

 2

a x

H (2) μ

ET II 146(54)

 2

a x

ET II 143(42)

ET II 147(59) [Re a > 0, Re b > 0]

√ √

J 0 (bx) dx = 16π −2 b−1 cos μπ K 2μ 2eπi/4 a b K 2μ 2e−πi/4 a b  |arg a| < π4 , b > 0, |Re μ| < 14 ET II 17(36)

6.516



∞

1.

 √  J 2ν a x J ν (bx) dx = b−1 J ν

0



∞

2.

a2 4b

 √  J 2ν a x Y ν (bx) dx = −b−1 Hν

0

 3. 0

4.12

5.

∞



 √  π J 2ν a x K ν (bx) dx = b−1 I ν 2



 a > 0,

a 4b

2

a 4b

Re ν > − 12

ET II 58(16)



2

b > 0,

 a > 0,



− Lν

a 4b 2



b > 0,

Re ν > − 12



ET II 111(18)

 Re b > 0, Re ν > − 21 ET II 144(45)

2

2

2   ∞  √  a a a sec(πν) 2 cos(πν)Yν − Y−ν + H−ν Y 2ν a x J ν (bx) dx = 2b 4b 4b 4b 0 [a > 0, b > 0] MC  ∞  √  Y 2ν a x Y ν (bx) dx 0 

2

2

2  b−1 a a a sec(νπ) J −ν + cosec(νπ) H−ν − 2 cot(2νπ) Hν = 2 4b 4b 4b  a > 0, b > 0, |Re ν| < 12 ET II 111(19)

672

6.

7.

Bessel Functions

⎡

2

2 −1  √  a a πb ⎣ − cot(2νπ) Lν Y 2ν a x K ν (bx) dx = cosec(2νπ) L−ν 2 4b 4b 0 ⎤

2

2 sec(νπ) a a ⎦ − Kν − tan(νπ) I ν 4b π 4b  ET II 144(46) Re b > 0, |Re ν| < 12

2

2    ∞  √  a a 1 − Y −ν K 2ν a x J ν (bx) dx = πb−1 sec(νπ) H−ν 4 4b 4b 0  Re a > 0, b > 0, Re ν > − 21 

∞



∞

 √  K 2ν a x Y ν (bx) dx 

2

2

2  1 −1 a a a − cosec(νπ) H−ν + 2 cosec(2νπ) Hν = − πb sec(νπ) J −ν 4 4b 4b 4b  Re a > 0, b > 0, |Re ν| < 12 ET II 114(34)

∞

 √  K 2ν a x K ν (bx) dx =

∞

 √  πb−1 I 2ν a x K ν (bx) dx = 2

ET II 70(22)

8. 0

 9. 0

 10. 0

 6.517

z

0

J0 ∞

6.51812 6.519

6.517



z 2 − x2 dx = sin z

K 2ν (2z sinh x) dx =

0



π/2

1. π/2

2.

MO 48

 2  π Jν (z) + Yν2 (z) 8 cos νπ 2

 Re z > 0,

− 21 < Re ν
− 12

WH

J 2ν (2z sin x) dx =

π 2 J (z) 2 ν

 Re ν > − 12

WA 42(1)a

0



# 

2

2

2 $ π a a a Kν + L−ν − Lν 4b 2 sin(νπ) 4b 4b  1 Re b > 0, |Re ν| < 2 ET II 147(63)  2

2  a a Iν + Lν 4b 4b  Re b > 0, Re ν > − 21 ET II 147(60)

πb−1 4 cos(νπ)

0

6.52 Bessel functions combined with x and x2 6.521



1

β J ν−1 (β) J ν (α) − α J ν−1 (α) J ν (β) α2 − β 2 α J ν (β) J ν (α) − β J ν (α) J ν (β) = β 2 − α2

x J ν (αx) J ν (βx) dx =

1. 0

[α = β,

ν > −1]

[α = β,

ν > −1] WH

Bessel functions combined with x and x2

6.522

 2.

∞

10

x K ν (ax) J ν (bx) dx =

0

bν aν (b2 + a2 )

673

[Re a > 0,

b > 0,

Re ν > −1] ET II 63(2)

  ∞ π(ab)−ν a2ν − b2ν x K ν (ax) K ν (bx) dx = 2 sin(νπ) (a2 − b2 ) 0

 3.

[|Re ν| < 1,

Re(a + b) > 0] ET II 145(48)

 ν  a −1  λ x J ν (λx) K ν (μx) dx = μ2 + λ2 + λa J ν+1 (λa) K ν (μa) − μa J ν (λa) K ν+1 (μa) μ 0

 4.



[Re ν > −1] ∞

5.

x K 1 (ax) =

π 2a2

[a > 0]

x K 20 (ax) =

1 2a2

[a > 0]

0



∞

6. 0



∞

7.

b a (a2 + b2 )

x K 1 (ax) J 1 (bx) =

0



∞

8.

x K 0 (ax) I 0 (bx) =

0



∞

9.

x K 1 (ax) I 1 (bx) =

0



∞

10. 

∞



∞

12.

[a > b > 0]

b a (a2 − b2 )

[a > b > 0]

π 2a3

[a > 0]

x2 K 1 (ax) =

2 a3

[a > 0]

0

x2 K 0 (ax) J 1 (bx) =

0



∞

13.

x2 K 1 (ax) J 0 (bx) =

0



∞

14.

x2 K 0 (ax) I 1 (bx) =

0



∞

15.

x2 K 1 (ax) I 0 (bx) =

0

6.522

Notation: 1 = 

1.12 0

∞

b > 0]

1 − b2

a2

x2 K 0 (ax) =

0

11.

[a > 0,

2b

[a > 0,

(a2 + b2 )2 2a (a2

[a > b > 0]

2

[a > b > 0]

2

[a > b > 0]

2b (a2

− b2 ) 2a

(a2

b > 0]

2

+ b2 )

− b2 )

ET II 367(26)

    1  1  (b + c)2 + a2 − (b − c)2 + a2 , 2 = (b + c)2 + a2 + (b − c)2 + a2 2 2

    2 x [J μ (ax)] K ν (bx) dx = Γ μ + 12 ν + 1 Γ μ − 12 ν + 1 b−2  − 1  1  −μ  1   2 −2 2 2 −2 2 1 + 4a P 1 + 4a × 1 + 4a2 b−2 2 P −μ b b 1 1 ν ν−1 2

[Re b > 2|Im a|,

2

2 Re μ > |Re ν| − 2]

ET II 138(19)

674

Bessel Functions



∞

2. 0



∞

3.12 0



6.522

  2e2μπi Γ 1 + 12 ν + μ x [K μ (ax)] J ν (bx) dx =  1   b 4a2 + b2 2 Γ 12 ν − μ  



2 b−2 ) Q −μ 2 b−2 ) (1 + 4a (1 + 4a × Q −μ 1 1 2ν 2 ν−1    b > 0, Re a > 0, Re 12 ν ± μ > −1 2

ET II 66(27)a

ν ν −ν x K 0 (ax) J ν (bx) J ν (cx) dx = r1−1 r2−1 (r2 − r1 ) (r2 + r1 ) = ν 2 1 2 , 2 (2 − 1 )     2 2 2 2 r1 = a + (b − c) , r2 = a + (b + c) , c > 0, Re ν > −1, Re a > |Im b| ET II 63(6)

∞

4.10

− 1  x I 0 (ax) K 0 (bx) J 0 (cx) dx = a4 + b4 + c4 − 2a2 b2 + 2a2 c2 + 2b2 c2 2

0

[Re b > Re a,

5.10

c > 0]

ET II 16(27)

alternatively, with a and c interchanged  ∞ 1 x I 0 (cx) K 0 (bx) J 0 (ax) dx = 2 [Re b > Re c, a > 0]  − 21 0 2  ∞ − 1  x J 0 (ax) K 0 (bx) J 0 (cx) dx = a4 + b4 + c4 − 2a2 c2 + 2a2 b2 + 2b2 c2 2 0

6.

alternatively, with a and b interchanged  ∞ 1 x J 0 (bx) K 0 (ax) J 0 (cx) dx = 2 2 − 21 0  ∞ x J 0 (ax) Y 0 (ax) J 0 (bx) dx = 0

∞

0

∞

8. 0

 9.

0

c > 0]

∞

ET II 15(25)

[0 < b < 2a] [0 < 2a < b < ∞] ET II 15(21)

   − 1 3−ν 3+ν Γ μ+ b−2 1 + 4a2 b−2 2 x J μ (ax) J μ+1 (ax) K ν (bx) dx = Γ μ + 2 2     −μ ×P 1 ν− 1 1 + 4a2 b−2 P −μ−1 1 + 4a2 b−2 1 ν− 1

2

8

[Re a > |Im b|,

− 12

7.12



c > 0]

 = −2π −1 b−1 b2 − 4a2

0



[Re b > |Im a|,

2

[Re b > 2|Im a|,

2

2

2 Re μ > |Re ν| − 3]

ET II 138(20)

x K μ− 12 (ax) K μ+ 12 (ax) J ν (bx) dx     1  −μ− 12  1  2e2μπi Γ 12 ν + μ + 1 −μ+ 12  2 −2 2 2 −2 2 1 + 4a Q 1 + 4a Q b b =− 1 1 1 1 1    ν− 2 ν− 2 b Γ 12 ν − μ b2 + 4a2 2 2 2 b > 0, Re a > 0, Re ν > −1, |Re μ| < 1 + 12 Re ν ET II 67(29)a  − 1 x I 12 ν (ax) K 12 ν (ax) J ν (bx) dx = b−1 b2 + 4a2 2 [b > 0,

Re a > 0,

Re ν > −1] ET II 65(16)

Bessel functions combined with x and x2

6.522



∞

10. 0

x J 12 ν (ax) Y

1 2ν

(ax) J ν (bx) dx =0   = −2π −1 b−1 b2 − 4a2

− 12



[a > 0,

Re ν > −1,

0 < b < 2a]

[a > 0,

Re ν > −1,

2a < b < ∞] ET II 55(48)

∞

11.8 0

x J 12 (ν+n) (ax) J 12 (ν−n) (ax) J ν (bx) dx = 2π

−1 −1

b



4a − b 2

 1 2 −2

Tn

b 2a



=0  0

13.

∞

0

 14.

Re ν > −1,

0 < b < 2a]

[a > 0,

Re ν > −1,

2a < b]

  x I 12 (ν−μ) (ax) K 12 (ν+μ) (ax) J ν (bx) dx = 2−μ a−μ b−1 b2 + 4a2

− 12

[b > 0,

8

8

[a > 0,

ET II 52(32) ∞

12.



675

∞

Re a > 0,

Re ν > −1,

   1 μ b + b2 + 4a2 2

Re(ν − μ) > −2] μ

ν

ET II 66(23)

ν−μ

μ−ν

(cos ψ) (sin ϕ) (sin ψ) (cos ϕ)   x J μ (xa sin ϕ) K ν−μ (ax cos ϕ cos ψ) J ν (xa sin ψ) dx = 2 2 2 a 1 − sin ϕ sin ψ  π π ET II 64(10) a > 0, 0 < ϕ < , 0 < ψ < , Re μ > −1, Re ν > −1 2 2 x J μ (xa sin ϕ cos ψ) J ν−μ (ax) J ν (xa cos ϕ sin ψ) dx

0 μ

15.10

16.10

17.11

ν

−ν

−μ

−1

= −2π −1 a−2 sin(μπ) (sin ϕ) (sin ψ) (cos ϕ) (cos ψ) [cos(ϕ + ψ) cos(ϕ − ψ)]   π a > 0, 0 < ϕ < , 0 < ψ < 12 π, Re ν > −1 ET II 54(39) 2    ∞ 23ν (abc)ν Γ ν + 12 ν+1 x J ν (bx) K ν (ax) J ν (cx) dx = √ 2ν+1 π (22 − 21 ) 0 [Re a > |Im b|, c > 0]    ∞ 1 3ν ν 2 (abc) Γ ν + 2 xν+1 I ν (cx) K ν (bx) J ν (ax) dx = √ 2ν+1 π (22 − 21 ) 0 [Re b > |Im a| + |Im c|]  ∞ tν−μ−ρ+1 J μ (ct) J ν (bt) K ρ (at) dt 0   μ−ν+ρ−1  1 1+2ν−2ρ  2 1 − x2 22 − x2 x 21+ν−μ−ρ dx = μ ν ρ μ−ν c b a Γ (μ − ν + ρ) 0 (b2 − x2 )     1  1  1 = (b + c)2 + a2 − (b − c)2 + a2 , 2 = (b + c)2 + a2 + (b − c)2 + a2 2 2 [Re a > |Im b|, c > 0]

676

Bessel Functions

 18.

∞

11

6.523

tμ−ν+ρ+1 J μ (ct) J ν (bt) K ρ (at) dt

  ν−μ−ρ−1  1 1+2μ+2ρ  2 1 − x2 22 − x2 x 21+μ−ν+ρ aρ dx = μ ν ν−μ c b Γ (ν − μ − ρ) 0 (c2 − x2 )       1  1  1 = (b + c)2 + a2 − (b − c)2 + a2 , 2 = (b + c)2 + a2 + (b − c)2 + a2 2 2 [Re a > |Im b|, c > 0]

0



∞

6.523 0

 −1  2 −1  b  ln x 2π −1 K 0 (ax) − Y 0 (ax) K 0 (bx) dx = 2π −1 a2 + b2 + b − a2 a [Re b > |Im a|, Re(a + b) > 0] ET II 145(50)

6.524



∞

1. 0



 0 < a < b,  0 < b < a,

x J 2ν (ax) J ν (bx) Y ν (bx) dx = 0 = −(2πab)−1

∞

2.

2

x [J 0 (ax) K 0 (bx)] dx =

0

1 b −a π − arcsin 2 8ab 4ab b + a2 2

2

1.10

Re ν > − 21

ET II 352(14)

 [a > 0,

6.525

Re ν > − 21

b > 0]

ET II 373(9)

    1  1  (b + c)2 + a2 − (b − c)2 + a2 , 2 = (b + c)2 + a2 + (b − c)2 + a2 2 2  ∞ − 32   2 2  a + b2 + c2 − 4a2 c2 x2 J 1 (ax) K 0 (bx) J 0 (cx) dx = 2a a2 + b2 − c2 Notation: 1 =

0

[c > 0,

Re b ≥ |Im a|,

Re a > 0] ET II 15(26)

2.10

alternatively, with a and b interchanged    ∞ 2b a2 + b2 − c2 2 x J 1 (bx) K 0 (ax) J 0 (cx) dx = [Re a > |Im b|, Re b > 0, 3 (22 − 21 ) 0  ∞ − 32   2 2  a + b2 + c2 − 4a2 b2 x2 I 0 (ax) K 1 (bx) J 0 (cx) dx = 2b b2 + c2 − a2

c > 0]

0



∞

3.10

x2 I 0 (cx) K 0 (bx) J 0 (ax) dx =

  2b a2 + b2 − c2

0

6.526



1. 0

∞



x J 12 ν ax

2



−1

J ν (bx) dx = (2a)

3

(22 − 21 )

J 12 ν

b2 4a

[Re b > |Re a|,

c > 0]

[Re a > |Im b|,

c > 0]

ET II 16(28)

 [a > 0,

b > 0,

Re ν > −1]

ET II 56(1)

Bessel functions combined with x and x2

6.527



∞

2. 0

  x J 12 ν ax2 Y ν (bx) dx −1



= (4a)



∞

3. 0



∞

4.



x J 12 ν ax

xY

0



1 2ν

2



0

1 2ν

 − tan

∞

6. 0

∞

7. 0



 ax2 J ν (bx) dx = −(2a)−1 H 12 ν

xY

1 2ν

J 12 ν 2 [a > 0,

ET II 140(27)

 2

b 4a

[a > 0,

∞

8. 0

1. 2. 3.



Re b > 0,

Re ν > −1]

 ax2 K ν (bx) dx



x K 12 ν ax



2

x K 12 ν ax

2







2

2

2  νπ

νπ

π b b b cos H− 12 ν − sin J − 12 ν − H 12 ν 4a sin(νπ) 2 4a 2 4a 4a [a > 0, Re b > 0, |Re ν| < 1] ET II 141(28)



2

2  b b π I 12 ν − L 12 ν J ν (bx) dx = 4a 4a 4a [Re a > 0, π ⎣ Y ν (bx) dx = cosec(νπ) L− 12 ν 4a

  x K 12 ν ax2 K ν (bx) dx π = 8a

∞

# sec

b > 0,

Re ν > −1] ET II 68(9)

⎡

νπ

2

I 12 ν

b2 4a

b2 4a





− cot(νπ) L 12 ν

νπ

1 − sec K 12 ν π 2

[Re a > 0,

6.527

Re ν > −1]



− tan





2  νπ

b2 b + sec H− 12 ν 4a 2 4a b > 0, Re ν > −1] ET II 109(9)

νπ



2

2  b b π νπ H− 1 ν − Y − 12 ν K ν (bx) dx = 2 4a 4a 8a cos 2 [a > 0, Re b > 0,

=



Y

b2 4a

ET II 61(35) ∞

5.



677

νπ

2

K

1 2ν

b2 4a



    1 x2 J 2ν (2ax) J ν− 12 x2 dx = a J ν+ 12 a2 2 0  ∞     1 x2 J 2ν (2ax) J ν+ 12 x2 dx = a J ν− 12 a2 2 0  ∞     1 x2 J 2ν (2ax) Y ν+ 12 x2 dx = − a Hν− 12 a2 2 0

b > 0,

b2 4a



⎤  b2 ⎦ 4a

|Re ν| < 1]



ET II 112(25)



2 $ b2 b 1 + π cosec(νπ) L − L2ν 4a 4a [Re a > 0, |Re ν| < 1] ET II 146(52)

− 12 ν

 a > 0,

Re ν > − 12



[a > 0,

Re ν > −2]

ET II 355(35)

[a > 0,

Re ν > −2]

ET II 355(36)

ET II 355(33)

678

Bessel Functions



∞

6.528 0

6.529



∞

1. 0

 2.

x K 14 ν

x2 4



I 14 ν

x2 4



J ν (bx) dx = K 14 ν

b2 4

6.528



 b2 I 14 ν 4 [b > 0, ν > −1]

 √   √  2a 1 x J ν 2 ax K ν 2 ax J ν (bx) dx = b−2 e− b 2

[Re a > 0,

MO 183a

b > 0,

Re ν > −1] ET II 70(23)

a





x J λ (2x) I λ (2x) J μ 2 a2 − x2 I μ 2 a2 − x2 dx

0

=

a2λ+2μ+2 2 Γ(λ + 1) Γ(μ + 1) Γ(λ + μ + 2)  λ+μ+3 λ+μ+1 ; λ + 1, μ + 1, λ + μ + 1, ; −a4 × 1F 4 2 2 [Re λ > −1, Re μ > −1] ET II 376(31)

6.53–6.54 Combinations of Bessel functions and rational functions 6.531 1.

12

2.



∞

Y ν (bx) π dx = [Eν (ab) + Yν (ab) + 2 cot(πν)(Jν (ab) − Jν (ab))] x + a sin(πν) 0 [Re ν < 1, arg a = π, b > 0]  ∞   Y ν (bx) 2 dx = π cot(νπ) [Y ν (ab) + Eν (ab)] + Jν (ab) + 2 [cot(νπ)] [Jν (ab) − J ν (ab)] x−a 0 [b > 0, a > 0, |Re ν| < 1] 

ET II 98(9) ∞

3. 0

4.∗

 

ET II 128(5)

Jν (bx) π dx = (Jν (ab) − Jν (ab)) x+a sin(πν)

∞

Jν (bx) π dx = (Jν (ab) − Jν (ab)) + Eν (ab) x−a tan(πν)

0

6.532



1.12 0

  K ν (bx) 1 1 π 2 dx = [cosec(νπ)] I ν (ab) + I −ν (ab) − e− 2 iνπ Jν (iab) − e 2 iνπ J−ν (iab) x+a 2 [Re b > 0, |arg a| < π, |Re ν| < 1] 2

∞

0

5.∗

MC

∞

[b > 0,

|arg(a)| < π,

[b > 0,

a > 0,

Re(ν) > −1]



J ν (x) 3 − ν 3 + ν a2 1 πIν (a)   , ; 1; + dx = F 1 2 x2 + a2 ν2 − 1 2 2 4 2a cos πν 2 [Re a > 0,

Re(ν) > −1]

Re ν > −1]

6.535

Bessel functions and rational functions

 1(b).* PV 0



∞

2.12 0

∞

πν

Jν (x) π tan (Jν (a) − Jν (a)) + Eν (a) dx = x2 − a2 2a 2 ⎡

[Re a > 0,



∞

0

 4. 5. 6.

4.

6.535

|Re ν| < 1]

ET II 99(13)

ET II 101(21) ∞

x J 0 (ax) dx = K 0 (ak) x2 + k 2 0  ∞ Y 0 (ax) K 0 (ak) dx = − 2 + k2 x k 0  ∞ J 0 (ax) π [I 0 (ak) − L0 (ak)] dx = 2 + k2 x 2k 0

[a > 0,

Re k > 0]

WA 466(5)

[a > 0,

Re k > 0]

WA 466(6)

[a > 0,

Re k > 0]

WA 467(7)

[Re p > 0,

Re q > −1]

WA 415(3)

[Re p > 0,

Re q > 0]

WA 415(5)

[0 < b < a] [0 < a < b]

⎧

 b b2 1 1 ⎪ ⎪  ∞ , ; 2, 2 − 1 [0 < b < a] dx ⎨ 2a 2 F 1 2 2 a

2 = [J 0 (ax) − 1] J 1 (bx) a 2 ⎪ x 0 ⎪ −1 [0 < a < b] ⎩ E π b2  ∞ dx =0 [0 < a < b] [1 − J 0 (ax)] J 0 (bx) x 0 a [0 < b < a] = ln b 

6.534

Re a > 0,

νπ

νπ   Y ν (bx) π  J ν (ab) + tan tan [Jν (ab) − J ν (ab)] − Eν (ab) − Y ν (ab) dx = 2 2 x −a 2a 2 2 [b > 0, a > 0, |Re ν| < 1]

6.533  z J p+q (z) dx = J p (x) J q (z − x) 1. x p 0

  z J p (x) J q (z − x) 1 1 J p+q (z) dx = + 2. x z−x p q z 0  ∞  a dx b 3.11 [J 0 (ax) − 1] J 1 (bx) 2 = − 1 + 2 ln x 4 b 0 a2 =− 4b

3b.12

Re ν > −1]

νπ

Y ν (bx) 1 ⎣ 1 π tan I ν (ab) − K ν (ab) dx = − νπ 2 2 x +a 2a 2 a cos 2

⎤ νπ 

b sin 2 2 3−ν 3+ν a b ⎦ 2 , ; + 1 F 2 1; 1 − ν2 2 2 4 [b > 0,

3.

679

∞

x3 J 0 (x) 1 1 dx = K 0 (a) − π Y 0 (a) 4 − a4 x 2 4 0 ∞ x 2 [J ν (x)] dx = I ν (a) K ν (a) 2 2 0 x +a

[a > 0] [Re a > 0,

ET II 21(28)a

ET II 14(16) ET II 340(5)

Re ν > −1]

ET II 342(26)

680

Bessel Functions



∞

6.536 0

 6.537 6.538 1.

∞

0



∞

12 0



6.536

 b > 0,

x3 J 0 (bx) dx = ker(ab) x4 + a4

x J 0 (bx) 1 dx = − 2 kei(ab) x4 + a4 a 2

 b > 0,

|arg a|
0,

∞

|arg a| < 14 π

x−1 J ν+2n+1 (x) J ν+2m+1 (x) dx = 0

0

b > 0]

ET II 21(30)

[m = n with m, n integers, ν > −1]

= (4n + 2ν + 2)−1

[m = n,

ν > −1] EH II 64

6.539 1. 2.

  π Y ν (b) Y ν (a) − 2 2 J ν (b) J ν (a) a x [J ν (x)]    b dx J ν (b) π J ν (a) 2 = 2 Y (a) − Y (b) ν ν a x [Y ν (x)]



b

dx

[J ν (x) = 0

=

for x ∈ [a, b]]

ET II 338(41)

[Y ν (x) = 0 for x ∈ [a, b]] ET II 339(49)

 3. a

6.541



b

dx π J ν (a) Y ν (b) = ln x J ν (x) Y ν (x) 2 J ν (b) Y ν (a)

∞

1. 0

 2.8 0



x J ν (ax) J ν (bx)

 ET II 339(50)

dx = I ν (bc) K ν (ac) x2 + c2 = I ν (ac) K ν (bc)

[0 < b < a,

Re c > 0,

Re ν > −1]

[0 < a < b,

Re c > 0,

Re ν > −1] ET II 49(10)

∞

dx x1−2n J ν (ax) J ν (bx) 2 x + c2  2 2 p n−1−p  2 2 k *

ν n )

n−1 " " a c /4 b c /4 π 1 1 b I ν (bc) K ν (ac) − = − 2 c 2 a sin(πν) p=0 p! Γ(1 − ν + p) k! Γ(1 − ν + k) k=0

    *

ν n−1 n )

" a2 c2 /4 p n−1−p " b2 c2 /4 k 1 1 b I ν (bc) K ν (ac) − = − 2 c 2ν a p!(1 − ν)p k!(1 + ν)k p=0 [n = 1, 2, . . . ,

k=0

Re ν > n − 1,

Re c > 0,

[0 < b < a]

0 < b < a]

6.544

Bessel functions and rational functions

 3.

∞

8 0

681

1 c 2ρ−α xα−1 J (cx) J (cx) dx = μ ν ρ (x2 + z 2 ) 2 2   (μ + ν + α)/2 − ρ, 1 + 2ρ − α ×Γ (μ⎛− ν − α)/2 + ρ + 1, (μ + ν − α)/2 + ρ + 1, (ν − μ − α)/2 + ρ + 1 1−α α μ+ν +α μ−ν −α + ρ, 1 − + ρ, ρ; ρ + 1 − ,ρ+ 1 + , 2 2 2 2 ⎞ ν − μ − α 2 2 ⎠ z α−2ρ cz μ+ν μ+ν −α ,ρ + 1 + ;c z ρ+1+ , + 2 2 2 2 ⎛   1+μ+ν μ+ν ρ − (α + μ + ν)/2, (α + μ + ν) /2 ,1 + Γ 3F 4 ⎝ 2 2 ρ, μ + 1, ν + 1 ⎞ α+μ+ν α+μ+ν ;1 − ρ+ , μ + 1, ν + 1, μ + ν + 1; c2 z 2 ⎠ 2 2

× 3F 4 ⎝

4.∗

    Γ (a1 ) . . . Γ (ap ) a 1 , . . . , ap , c > 0, Re z > 0, Re(α + μ + ν) > 0; Re(α − 2ρ) > 1 = Γ b1 , . . . , bq Γ (b1 ) . . . Γ (bq )  ∞ ρ−1      x J (bx) cos 12 (ρ − μ + ν)π Jν (ax) + sin(ax) + sin 12 (ρ + μ − ν)π Yν (ax) 2 + k2 μ x 0 = −k ρ−2 Iμ (kb)Kν (ka) [|Re ν| − Re μ < Re ρ < 4,

6.543

∞

[Re r > 0, 6.544

WA 430(3)

ν J ν (ax) Y ν (bx) − J ν (bx) Y ν (ax) π b   dx = − [0 < b < a] ET II 352(16) 2 2 2 a 0 x [J ν (bx)] + [Y ν (bx)]   #   $  ∞ 1 1 x dx J μ (bx) cos (ν − μ)π J ν (ax) − sin (ν − μ)π Y ν (ax) = I μ (br) K ν (ar) 2 2 2 x + r2 0

 6.542

0 < b ≤ a]



∞

1.

Jν

0



∞

2.

Jν

0

 3. 0

∞

Jν

a

x

Yν



√ 

√  x dx 2 a 2 a 1 2 √ √ K = − − Y 2ν 2ν 2 b x a π b  b a > 0, b > 0,

√  x dx 2 a 1 Jν = J 2ν √ 2 x b x a b

a

√  x dx 2 a 1 iπ 1 1 iνπ 2 4 Kν + = e K 2ν √ e x b x2 a b

a

a ≥ b > 0,

Re μ > |Re ν| − 2]

|Re ν|
0,

b > 0,

Re ν > − 12

ET II 57(10)

√  1 − 1 iνπ 2 a − 1 iπ 2 4 e K 2ν √ e a b Re b > 0, a > 0, |Re ν| < 12

ET II 142(32)

682

Bessel Functions



∞

4.

Yν

0



Yν

0

Kν

0

1 2





√ 

√  x dx 2 a 1 iπ 2 a − 1 iπ 1 1 i(ν+1)π − 12 i(ν+1)π 2 4 4 √ √ Kν e + e = K K e e 2ν 2ν x b x2 a b b  Re b > 0, a > 0, |Re ν| < 12

a



√  √  x dx 1 2 a i 1 νπi − 12 νπi − 14 πi 2 a 2 4 πi √ √ Jν e e e = K K − e 2ν 2ν x b x2 a b b  Re a > 0, b > 0, |Re ν| < 52

a

ET II 70(19)

∞

7.

Kν

0



|Re ν|
0, b > 0,

a

ET II 62(38)

∞

5.12



6.551

∞

8. 0

Kν

  

√ 

√ 

x dx 3 2 a 2 a 2 3 Yν sin πν kei2ν √ πν ker2ν √ = − cos 2 x b x a 2 2 b  b Re a > 0, b > 0, |Re ν| < 52

a

ET II 113(29)

√  x dx 2 a π Kν = K 2ν √ x b x2 a b

a

[Re a > 0,

Re b > 0]

ET II 146(55)

6.55 Combinations of Bessel functions and algebraic functions 6.55110



1

x1/2 J ν (xy) dx =

1. 0



∞

2. 1

  Γ 3 + 1ν 2y −3/2  41 21  Γ 4 + 2ν   +y −1/2 ν − 12 J ν (y) S −1/2,ν−1 (y) − J ν−1 (y) S 1/2,ν (y)  y > 0, Re ν > − 32 √

   x1/2 J ν (xy) dx = y −1/2 J ν−1 (y) S 1/2,ν (y) + 12 − ν J ν (y) S −1/2,ν−1 (y) [y > 0]

6.552



∞

1. 0



J ν (xy)

dx (x2

Y ν (xy)

0

3. 0

+

a2 )1/2

= I ν/2

1

2 ay



K ν/2

1

2 ay



[Re a > 0,

ET II 22(2)

y > 0,

Re ν > −1]

ET II 23(11), WA 477(3), MO 44 ∞

2.



ET II 21(1)

dx (x2 +

1/2 a2 )

=−

          1 sec 12 νπ K ν/2 12 ay K ν/2 12 ay + π sin 12 νπ I ν/2 12 ay π [y > 0, Re a > 0, |Re ν| < 1] ET II 100(18)

∞

K ν (xy)

dx (x2 + a2 )1/2

      2   2  π sec 12 νπ = J ν/2 12 ay + Y ν/2 12 ay 8 [Re a > 0, Re y > 0, 2

|Re ν| < 1] ET II 128(6)

6.561

Bessel functions and powers



1

J ν (xy)

4. 0



dx (1 −

1

Y 0 (xy)

5. 0



∞

1



∞

(1 −

  2 π J ν/2 12 y 2

[y > 0,

=

    π J 0 12 y Y 0 12 y 2

[y > 0]

1/2 x2 )

dx 1/2

(x2 − 1)

Y ν (xy)

7.

=

dx

J ν (xy)

6.

1/2 x2 )

1

683

Re ν > −1]

ET II 102(26)a

    π J ν/2 12 y Y ν/2 12 y [y > 0] 2   2    2  π  J ν/2 12 y = − Y ν/2 12 y 4

=−

dx (x2 − 1)1/2

ET II 24(23)a

[y > 0] 

∞

6.553

−1/2

x

I ν (x) K ν (x) K μ (2x) dx =

0

Γ

1 4

ET II 24(22)a

ET II 102(27)

       + 12 μ Γ 14 − 12 μ Γ 14 + ν + 12 μ Γ 14 + ν − 12 μ     4 Γ 34 + ν + 12 μ Γ 34 + ν − 12 μ |Re μ| < 12 , 2 Re ν > |Re μ| − 12 ET II 372(2)

6.554



∞

1.

x J 0 (xy)

0



1

x J 0 (xy)

2. 0



∞

3. ∞

4.

x J 0 (xy)

 5.

∞

11

ET II 7(4)

[y > 0]

ET II 7(6)a

= a−1 e−ay

[y > 0,

Re a > 0]

 1 ν √ 2a  π  J ν (ak) K ν (ak) dx = 2ν (2k) Γ ν + 12  a > 0,

|arg k| >

+ a2 )3/2

ν+1/2

Re a > 0]

= y −1 cos y

1/2

− 1)

xν+1 J ν (ax) (x4 + 4k 4 )

0

= y −1 sin y

dx (x2

[y > 0,

ET II 7(5)a

dx (x2

= y −1 e−ay

[y > 0]

1/2

(1 − x2 )

x J 0 (xy)

0

1/2

+ x2 ) dx

1



dx (a2

ET II 7(7)a

π 4,

Re ν > − 12



WA 473(1)

 6.555 0

 6.556

∞

0

∞



a x1/2 J 2ν−1 ax1/2 Y ν (xy) dx = − 2 Hν−1 2y

 2

a 4y a > 0,

  a

a

1/2  dx π √ Y ν/2 J ν a x2 + 1 = − J ν/2 2 2 2 x2 + 1

y > 0,

Re ν > − 12



ET II 111(17)

[Re ν > −1,

a > 0]

MO 46

6.56–6.58 Combinations of Bessel functions and powers 6.561



1. 0

1

  1 xν J ν (ax) dx = 2ν−1 a−ν π 2 Γ ν + 12 [J ν (a) Hν−1 (a) − Hν (a) J ν−1 (a)]  Re ν > − 12

ET II 333(2)a

684

Bessel Functions



1

2. 0



1

3. 0



1

4. 0



1

5.

6.561

  1 xν Y ν (ax) dx = 2ν−1 a−ν π 2 Γ ν + 12 [Y ν (a) Hν−1 (a) − Hν (a) Y ν−1 (a)]  Re ν > − 12

ET II 338(43)a

  1 xν I ν (ax) dx = 2ν−1 a−ν π 2 Γ ν + 12 [I ν (a) Lν−1 (a) − Lν (a) I ν−1 (a)]  Re ν > − 12

ET II 364(2)a

  1 xν K ν (ax) dx = 2ν−1 a−ν π 2 Γ ν + 12 [K ν (a) Lν−1 (a) + Lν (a) K ν−1 (a)]  Re ν > − 12 xν+1 J ν (ax) dx = a−1 J ν+1 (a)

ET II 367(21)a

[Re ν > −1]

ET II 333(3)a

0



1

6.

xν+1 Y ν (ax) dx = a−1 Y ν+1 (a) + 2ν+1 a−ν−2 π −1 Γ(ν + 1)

0



1

7.

xν+1 I ν (ax) dx = a−1 I ν+1 (a)

[Re ν > −1]

ET II 339(44)a

[Re ν > −1]

ET II 365(3)a

[Re ν > −1]

ET II 367(22)a

0



1

8.

xν+1 K ν (ax) dx = 2ν a−ν−2 Γ(ν + 1) − a−1 K ν+1 (a)

0



1

x1−ν J ν (ax) dx =

9. 0



1

x1−ν Y ν (ax) dx =

10. 0



1

11.

a

ν−2

2ν−1 Γ(ν)

− a−1 J ν−1 (a)

aν−2 cot(νπ) − a−1 Y ν−1 (a) 2ν−1 Γ(ν)

x1−ν I ν (ax) dx = a−1 I ν−1 (a) −

0



1

12.

ET II 333(4)a

[Re ν < 1]

ET II 339(45)a

aν−2

ET II 365(4)a

2ν−1 Γ(ν)

x1−ν K ν (ax) dx = 2−ν aν−2 Γ(1 − ν) − a−1 K ν−1 (a)

0

 13.7

1

xμ J ν (ax) dx =

0

aμ+1 Γ

 15. 0

∞

[Re ν < 1]

ET II 367(23)a

2  + a−μ {(μ + ν − 1) J ν (a) S μ−1,ν−1 (a) − J ν−1 (a) S μ,ν (a)}  ν−μ+1 2

1 1  1 ∞ μ μ −μ−1 Γ  2 + 2 ν + 2 μ x J ν (ax) dx = 2 a Γ 12 + 12 ν − 12 μ 0

 14.

2μ Γ

 ν+μ+1 

[a > 0,

Re(μ + ν) > −1]

 − Re ν − 1 < Re μ < 12 ,

  1 −μ−1 Γ 12 + 12 ν + 12 μ μ μ   x Y ν (ax) dx = 2 cot 2 (ν + 1 − μ)π a Γ 12 + 12 ν − 12 μ |Re ν| − 1 < μ < 12 ,

ET II 22(8)a

a>0



EH II 49(19)

a>0

ET II 97(3)a

6.563

Bessel functions and powers



∞

16.

μ

x K ν (ax) dx = 2

μ−1 −μ−1

a

Γ

0



∞

17. 0



∞

18. 0

1

1+μ+ν 2

  1+μ−ν Γ 2 [Re (μ + 1 ± ν) > 0,

Γ Y ν (x) dx = ν−μ x

2

Re a > 0] EH II 51(27)

 1

 −1 < Re q < Re ν − 12

Γ q+ J ν (ax)  dx = ν−q q−ν+12  2 1 xν−q 2 a Γ ν − 2 q + 12 1

685

WA 428(1), KU 144(5)

+

 1 1    Γ 2 + 2 μ − ν sin 12 μ − ν π 2ν−μ π  |Re ν| < Re(1 + μ − ν) < 32

1 2μ

WA 430(5)





1

x2m+n+1/2 K n+1/2 (αx) dx =

19. 0

6.562



∞

1.

xμ Y ν (bx)

0



∞

2. 0



π 2

n " k=0

(n + k)! γ(2m + n − k + 1, α) k!(n − k)! α2m+n+3/2 2k

STR

'    dx = (2a)μ π −1 sin 12 π(μ − ν) Γ 12 (μ + ν + 1) Γ 12 (1 + μ − ν) S −μ,ν (ab) x+a     (  −2 cos 12 π (μ − ν) Γ 1 + 12 μ + 12 ν Γ 1 + 12 μ − 12 ν S −μ−1,ν (ab)  b > 0, |arg a| < π, Re (μ ± ν) > −1, Re μ < 32 ET II 98(8)

xν J ν (ax) πk ν dx = [H−ν (ak) − Y −ν (ak)] x+k 2 cos νπ

 1 − 2 < Re ν < 32 ,

a > 0,

|arg k| < π



WA 479(7) ∞

3. 0

dx x+a 

1 1 −μ μ − ν a2 b 2 μ+ν μ−2 ,1− ; Γ 2 (μ + ν) Γ 2 (μ − ν) b 1 F 2 1; 1 − =2 2 2 4  1 1 1−μ 3 − μ − ν 3 − μ + ν a2 b 2 μ−3 , ; 1; Γ 2 (μ − ν − 1) Γ 2 (μ + ν − 1) ab −2 F 1 2 2 2 4

xμ K ν (bx)

−πaμ cosec[π(μ − ν)] {K ν (ab) + π cos(μπ) cosec[π(ν + μ)] I ν (ab)} [Re b > 0,  6.563 0

∞

x−1 J ν (bx)

|arg a| < π,

Re μ > |Re ν| − 1]

ET II 127(4)

dx πa−μ−1 = (x + a)1+μ sin[( + 1) ⎧ + ν − μ)π] Γ(μ  ν+2m ∞ ⎨" m 1 (−1) 2 ab Γ( + ν + 2m) × ⎩ m! Γ(ν + m + 1) Γ ( + ν − μ + 2m) m=0 μ+1−+m ⎫ 1 ∞ 1 ⎬ " ab Γ(μ + m + 1) ( + ν − μ − m)π sin 2 1 1 2 − m! Γ 2 (μ + ν −  + m + 3) Γ 2 (μ − ν −  + m + 3) ⎭ m=0  ET II 23(10), WA 479 b > 0, |arg a| < π, Re( + ν) > 0, Re( − μ) < 52

686

Bessel Functions

6.564  1.

∞

ν+1

x

0



∞

2.

dx J ν (bx) √ = 2 x + a2

x1−ν J ν (bx) √

0

dx = + a2





x2

2 ν+ 1 a 2 K ν+ 12 (ab) πb

6.564

 Re a > 0,

b > 0,

−1 < Re ν
0,

b > 0,

Re ν > − 21



ET II 23(16)

6.565  1.

∞

−ν

x



x +a 2

1  2 −ν− 2

J ν (bx) dx = 2 a

0



∞

2.

 −ν− 12 xν+1 x2 + a2

0



∞

3.

√ ν−1 πb  J ν (bx) dx = ν ab  2 e Γ ν + 12

 −ν− 32 xν+1 x2 + a2 J ν (bx) dx =

0



J ν (bx)x +

μ+1 a2 )

dx =

2

b π   aeab Γ ν + 32 [Re a > 0,

b > 0,

Re ν > −1]

b a K ν−μ (ab) Γ(μ + 1)    −1 < Re ν < Re 2μ + 32 ,

2μ

a > 0,

b>0



MO 43

 μ x1−ν x2 + a2 Y ν (bx) dx =

0

ET II 24(18)

ν√

ν+1



ν−μ μ

0

7.

Re ν > − 21

∞

12



b > 0,

 μ xν+1 x2 + a2 Y ν (bx) dx   aμ+1 b−μ−ν−1 2 μ+1 ν π 2 = (ab) csc(π(μ + ν))[cot(πμ)I−μ−ν−1 (ab) + cot(πν))Iμ+ν+1 (ab) 2πΓ(−μ) 

a2 b 2 −2ν (ab)μ+1 Γ(−μ − 1)Γ(ν) 1F2 1; μ + 2; 1 − ν; 4 [b > 0, Re a > 0, −1 < Re ν < −2 Re μ] ET II 100(19)

0

6.

 Re a > 0,

∞

12



ν+1

(x2

0

5.

WA 477(4), ET II 23(17)

ET II 24(19) ∞

4.





 Γ(ν + 1) ab ab Iν Kν b Γ(2ν + 1)  2 2 Re a > 0, b > 0, Re ν > − 21

ν −2ν ν

∞

  2μ aμ−ν+1 2 cos((μ − ν)π)K Γ(μ + 1) cot(νπ)I (ab) + (ab) μ−ν+1 μ−ν+1 bμ+1 π 

a2 b 2 a2μ+2 bν cot(νπ) 1; ν + 1, μ + 2; F − ν+1 1 2 2 (μ + 1)Γ(ν 4  + 1) Re ν < 1, Re(ν − 2μ) > −3, arg a2 = π, b > 0 MC

 μ x1+ν x2 + a2 K ν (bx) dx = 2ν Γ(ν + 1)aν+μ+1 b−1−μ S μ−ν,μ+ν+1 (ab) [Re a > 0,

Re b > 0,

Re ν > −1] ET II 128(8)

6.567

Bessel functions and powers

 8.

∞

11

μ+1

(x2 + k 2 )

0

6.566



x−1 J ν (ax)

∞

1.

xμ Y ν (bx)

0



∞

2.

x2

xν+1 J ν (ax)

0

 0

2

   + 12 ν Γ μ + 1 − 12  − 12 ν

2ν+1 Γ(μ + 1) Γ(ν + 1)  a2 k 2 +ν +ν ; − μ, ν + 1; × 1F 2 2  2  4 a2μ+2− Γ 12 ν + 12  − μ − 1 

+ 1 1 22μ+3− Γ μ + 2 + ν −  2 2 

ν +  a2 k 2 ν − ,μ+ 2 − ; × 1 F 2 μ + 1; μ + 2 + 2 2 4  a > 0, − Re ν < Re  < 2 Re μ + 72 , Re k > 0

WA 477(1)

dx = 2μ−2 π −1 b1−μ + a2   π    (μ − ν + 1) Γ 12 μ + 12 ν − 12 Γ 12 μ − 12 ν − 12 × cos 2 

μ+1+ν μ + 1 − ν a2 b 2 × 1 F 2 1; 2 − ,2 − ; 2 2 4     π 1 μ−1 π − πa cosec (μ + ν + 1) cot (μ − ν + 1) I ν (ab) 2 2  π 2 −aμ−1 cosec (μ − ν + 1) K ν (ab) 2 b > 0, Re a > 0, |Re ν| − 1 < Re μ < 52 ET II 100(17)

x2

dx = bν K ν (ab) + b2

 a > 0,

Re b > 0,

−1 < Re ν
0,

Re b > 0,

Re ν > − 21



WA 468(9) ∞

4.

x−ν K ν (ax)

0



1

EH II 96(58) ∞

3.



dx =

aν k +ν−2μ−2 Γ

687

dx π [Hν (ab) − Y ν (ab)] = ν+1 x2 + b2 4b cos νπ  a > 0, 2

Re b > 0,

Re ν
0,

Re b > 0,

Re ν > − 25



WA 468(11)

6.567



1. 0

1

 μ xν+1 1 − x2 J ν (bx) dx = 2μ Γ(μ + 1)b−(μ+1) J ν+μ+1 (b) [b > 0,

Re ν > −1,

Re μ > −1] ET II 26(33)a

688

Bessel Functions



1

2.

 μ xν+1 1 − x2 Y ν (bx) dx

0

 = b−(μ+1) 2μ Γ(μ + 1) Y μ+ν+1 (b) + 2ν+1 π −1 Γ(ν + 1) S μ−ν,μ+ν+1 (b) [b > 0,

 3. 4.

6.567

Re μ > −1,

Re ν > −1]

 μ 21−ν S ν+μ,μ−ν+1 (b) [b > 0, Re μ > −1] x1−ν 1 − x2 J ν (bx) dx = bμ+1 Γ(ν) 0   1  μ x1−ν 1 − x2 Y ν (bx) dx = b−(μ+1) 21−ν π −1 cos(νπ) Γ (1 − ν)

ET II 103(35)a

1

ET II 25(31)a

 × s μ+ν,μ−ν+1 (b) − 2 cosec(νπ) Γ(μ + 1) J μ−ν+1 (b)

0

μ

[b > 0, 

1

x

1−ν

5.



 2 μ

1−x

K ν (bx) dx = 2

0

−ν−2 ν

−1

b (μ + 1)

Re μ > −1,

Γ(−ν) 1 F 2

Re ν < 1]

b2 1; ν + 1, μ + 2; 4

ET II 104(37)a



+π2μ−1 b−(μ+1) cosec (νπ) Γ(μ + 1) I μ−ν+1 (b)  6. 7.

8.

9.

10.

11.

12. 13.



[Re μ > −1,

Re ν < 1]

π Hν− 12 (b) [b > 0] 2b 0   1   π dx cosec(νπ) cos(νπ) J ν+ 12 (b) − H−ν− 12 (b) x1+ν Y ν (bx) √ = 2b 1 − x2 0 [b > 0, Re ν > −1]    1    π dx cot(νπ) Hν− 12 (b) − Y ν− 12 (b) − J ν− 12 (b) x1−ν Y ν (bx) √ = 2b 1 − x2 0 [b > 0, Re ν < 1]  2  1  ν− 12   √ b xν 1 − x2 J ν (bx) dx = 2ν−1 πb−ν Γ ν + 12 J ν 2 0  b > 0, Re ν > − 12  



 1   1 b b 1 ν 2 ν− 2 ν−1 √ −ν Jν Yν x 1−x Y ν (bx) dx = 2 πb Γ ν + 2 2 2 0  b > 0, Re ν > − 12  



 1  ν− 12 √ b b 1 Iν Kν xν 1 − x2 K ν (bx) dx = 2ν−1 πb−ν Γ ν + 2 2 2 0  Re ν > − 12   2

 1   1 √ −ν b 1 ν 2 ν− 2 −ν−1 Iν x 1−x I ν (bx) dx = 2 πb Γ ν + 2 2 0 

 1 ν−1 1   b 1 −ν− 2 − ν sin b xν+1 1 − x2 J ν (bx) dx = 2−ν √ Γ 2 π 0  b > 0, |Re ν| < 12 1

x1−ν J ν (bx) √

dx = 1 − x2

ET II 129(12)a ET II 24(24)a

ET II 102(28)a

ET II 102(30)a

ET II 24(25)a

ET II 102(31)a

ET II 129(10)a ET II 365(5)a

ET II 25(27)a

6.571

Bessel functions and powers

 







b b b b 1 Jν J −ν −Y ν Y −ν x x −1 Y ν (bx) dx = 2 πb Γ ν + 2 2 2 2 2 1  |Re ν| < 12 , b > 0 ET II 103(32)a   

 ∞ 2  ν− 12 b 2ν−1 1 Kν xν x2 − 1 K ν (bx) dx = √ b−ν Γ ν + 2 2 π 1  Re b > 0, Re ν > − 21 ET II 129(11)a  



 ∞  −ν− 12 √ b b 1 − ν Jν Yν x−ν x2 − 1 J ν (bx) dx = −2−ν−1 πbν Γ 2 2 2 1  b > 0, |Re ν| < 12 ET II 25(26)a 

 ∞  ν− 12 2−ν 1 + ν cos b x−ν+1 x2 − 1 J ν (bx) dx = √ b−ν−1 Γ 2 π 1  b > 0, |Re ν| < 12 ET II 25(28)  1 2k k! x(1 − x2 )k I0 (ax)dx = k+1 Ik+1 (a) PBM 2.15.2.6 a 0 

14.

15.

16.

17.12

18.∗

6.568



∞

∞

1.

ν



2

ν− 12

xν Y ν (bx)

0



xμ Y ν (bx)

0

 6.569

ν−2

dx π = aν−1 J ν (ab) x2 − a2 2

1

x2

0

 a > 0,

b > 0,

− 12 < Re ν
0, b > 0, |Re ν| − 1 < Re μ < 52

ET II (101)(25)

xλ (1 − x)μ−1 J ν (ax) dx =



−ν

π

0

1.

√

ET II 101(22) ∞

2.

6.571

689

∞



x2 + a2

 12

Γ(μ) Γ(1 + λ + ν)2−ν aν Γ(ν + 1) Γ(1 + λ + μ + ν)  λ+1+μ+ν λ+2+μ+ν a2 λ+1+ν λ+2+ν , ; ν + 1, , ;− × 2F 3 2 2 2 2 4 [Re μ > 0, Re(λ + ν) > −1] ET II 193(56)a



 μ ab ab dx K 12 (ν±μ) ± x J ν (bx) √ = aμ I 12 (ν∓μ) 2 2 x2 + a 2 Re a > 0, b > 0, Re ν > −1, Re μ < 32

ET II 26(38)

690

Bessel Functions



∞



∞



2.

μ 1 dx x2 + a2 2 − x Y ν (bx) √ 2 2 

x + a





 ab ab ab ab μ K 12 (μ−ν) − cosec(νπ) I 12 (μ−ν) K 12 (μ+ν) = a cot(νπ) I 12 (μ+ν) 2  2 2 2 Re a > 0, b > 0, Re μ > − 23 , |Re ν| < 1 ET II 104(40)

0

 3.



∞

1.

μ dx + x K ν (bx) √ 2 2 x +

a 





  ab ab ab ab π2 μ a cosec(νπ) J 12 (ν−μ) Y − 12 (ν+μ) − Y 12 (ν−μ) J − 12 (ν+μ) = 4 2 2 2 2 [Re a > 0, Re b > 0] ET II 130(15)

x2 + a2

0

6.572

−μ

x



 12

x +a 2

2

 12

+a

μ

0



6.572

  Γ 1+ν−μ dx 2 J ν (bx) √ = W 12 μ, 12 ν (ab) M − 12 μ, 12 ν (ab) ab Γ(ν + 1) x2 + a2 [Re a > 0, b > 0, Re(ν − μ) > −1] ET II 26(40)

∞

2.

x−μ



x2 + a2

 12

+a

μ

K ν (bx) √

0

dx + a2

x2

 1+ν−μ   1−ν−μ  Γ 2 2 W 12 μ, 12 ν (iab) W 12 μ, 12 ν (−iab) = 2ab Re b > 0, Re μ + |Re ν| < 1] ET II 130(18), BU 87(6a) Γ

[Re a > 0, 

∞

3.

x−μ



x2 + a2

 12

−a

μ

0

6.573



∞

1.

xν−M+1 J ν (bx)

0

k +

dx + a2 %  

 Γ 1+ν+μ ν −μ 1 2 tan π M 12 μ, 12 ν (ab) = − W − 12 μ, 12 ν (ab) ab Γ(ν + 1) 2

 $ ν −μ + sec π W 12 μ, 12 ν (ab) 2  Re a > 0, b > 0, |Re ν| < 12 + 12 Re μ ET II 105(42)

Y ν (bx) √

x2

J μi (ai x) dx = 0

i=1 )

ai > 0,

k "

M=

k "

μi

i=1

ai < b < ∞,

*

−1 < Re ν < Re M +

1 2k

−

1 2

ET II 54(42)

i=1

 2. 0

∞

xν−M−1 J ν (bx) )

k +

J μi (ai x) dx = 2ν−M−1 b−ν Γ(ν)

i=1

ai > 0,

k " i=1

k + i=1

ai < b < ∞,

aμi i , Γ (1 + μi ) *

0 < Re ν < Re M + 12 k +

3 2

M=

k "

μi

i=1

WA 460(16)a, ET II 54(43)

6.576

6.574

Bessel functions and powers

 ν +μ−λ+1  ∞ α Γ 2 

J ν (αt) J μ (βt)t−λ dt = −ν + μ + λ+1 0 λ ν−λ+1 Γ(ν + 1) 2 β Γ 2 

α2 ν +μ−λ+1 ν −μ−λ+1 , ; ν + 1; 2 ×F 2 2 β [Re(ν + μ − λ + 1) > 0, Re λ > −1, 0 < α < β] ν

1.8

2.

691

WA 439(2)a, MO 49

If we reverse the positions of ν and μ and at the same time reverse the positions of α and β, the function on the right hand side of this equation will change. Thus, the right hand side represents α α a function of that is not analytic at = 1. β β For α = β, we have the following equation 

ν +μ−λ+1 λ−1  ∞ Γ(λ) Γ α 2   

J ν (αt) J μ (αt)t−λ dt = ν +μ+λ+1 ν−μ+λ+1 −ν + μ + λ + 1 0 Γ Γ 2λ Γ 2 2 2 [Re(ν + μ + 1) > Re λ > 0, α > 0] MO 49, WA 441(2)a

If μ − ν + λ + 1 (or ν − μ + λ + 1) is a negative integer, the right hand side of equation 6.574 1 (or 6.574 3) vanishes. 6.575  1.11

∞

J ν+1 (αt) J μ (βt)tμ−ν dt = 0

0

= 

∞

2. 0



 2 ν−μ

[α < β] μ

β α −β Γ(ν − μ + 1) 2

2ν−μ αν+1 √

[α ≥ β] [Re(ν + 1) > Re μ > −1]

J ν (x) J μ (x) π Γ(ν + μ)      dx = ν+μ  ν+μ x 2 Γ ν + μ + 12 Γ ν + 12 Γ μ + 12 [Re(ν + μ) > 0]

6.576  1.

∞

xμ−ν+1 J μ (x) K ν (x) dx =

0

1 2

Γ(μ − ν + 1)

 2.11 0

∞

MO 51

x−λ J ν (ax) J ν (bx) dx =

aν b ν Γ ν +

[Re μ > −1, 1−λ 2

KU 147(17), WA 434(1)

Re(μ − ν) > −1]





1+λ 2λ (a + b)2ν−λ+1 Γ(ν + 1) Γ 2 

1 4ab 1−λ , ν + ; 2ν + 1; ×F ν + 2 2 (a + b)2 [a > 0, b > 0, 2 Re ν + 1 > Re λ > −1]

ET II 370(47)

ET II 47(4)

692

Bessel Functions

  ν−λ−μ+1 ν −λ+μ+1  ∞ Γ b Γ 2 2 x−λ K μ (ax) J ν (bx) dx = λ+1 ν−λ+1 2 a Γ(1 + ν) 0 

b2 ν−λ+μ+1 ν −λ−μ+1 , ; ν + 1; − 2 ×F 2 2 a [Re (a ± ib) > 0, Re(ν − λ + 1) > |Re μ|] EH II 52(31), ET II 63(4), WA 449(1) ν

3.



∞

4.

−λ

x

0



∞

5.

−λ

x

0



∞

6.

   − 12 λ + 12 μ + 12 ν Γ 12 − 12 λ − 12 μ + 12 ν K μ (ax) I ν (bx) dx = 2λ+1 Γ(ν + 1)a−λ+ν+1 

1 1 1 1 1 1 1 1 b2 − λ + 2 μ + 2 ν, 2 − 2 λ − 2 μ + 2 ν; ν + 1; a2 ×F 2 2 [Re (ν + 1 − λ ± μ) > 0, a > b] EH II 93(35) bν Γ



∞

1 2

 π(ν − μ − λ) ∞ −λ 2 sin x K μ (ax) I ν (bx) dx π 2 0 Re λ > −1, Re (ν − λ + 1 ± μ) > 0] (see 6.576 5)

x−λ Y μ (ax) J ν (bx) dx = [a > b,

8



  1−λ+μ+ν 1−λ−μ+ν 2−2−λ a−ν+λ−1 bν Γ Γ K μ (ax) K ν (bx) dx = 2  2 

Γ(1 − λ) 1−λ−μ−ν 1−λ+μ−ν Γ ×Γ 2 2 

b2 1−λ+μ+ν 1−λ−μ+ν , ; 1 − λ; 1 − 2 ×F 2 2 a [Re(a + b) > 0, Re λ < 1 − |Re μ| − |Re ν|] ET II 145(49), EH II 93(36)

0

7.

6.577

xμ+ν+1 J μ (ax) K ν (bx) dx = 2μ+ν aμ bν

0

EH II 93(37)

Γ(μ + ν + 1) (a2 + b2 )μ+ν+1 [Re μ > |Re ν| − 1,

Re b > |Im a|]

ET 137(16), EH II 93(36)

6.577



∞

1.8 0

[a > 0,

 2.

∞

12 0

dx = (−1)n cν−μ+2n I μ (ac) K ν (bc) x2 + c2 Re c > 0, 2 + Re μ − 2n > Re ν > −1 − n, n ≥ 0 an integer]

ET II 49(13)

dx = (−1)n cμ−ν+2n I ν (bc) K μ (ac) + c2 Re ν − 2n + 2 > Re μ > −n − 1, n ≥ 0 an integer]

ET II 49(15)

xν−μ+1+2n J μ (ax) J ν (bx) b > a,

xμ−ν+1+2n J μ (ax) J ν (bx)

[b > 0,

a > b,

Re c > 0,

x2

6.578

6.578

Bessel functions and powers



∞

1. 0



∞

2.

2−1 aλ bμ c−λ−μ− Γ λ+μ+ν+ 2 

x−1 J λ (ax) J μ (bx) J ν (cx) dx = λ+μ−ν + Γ(λ + 1) Γ(μ + 1) Γ 1 − 2

 a2 b 2 λ+μ−ν + λ+μ+ν + , ; λ + 1, μ + 1; 2 , 2 × F4 2 2 c c   5 Re(λ + μ + ν + ) > 0, Re  < , a > 0, b > 0, c > 0, c > a + b ET II 351(9) 2 x−1 J λ (ax) J μ (bx) K ν (cx) dx

0

=



∞

3. 0



∞

4. 0



∞

5.

693

 

2−2 aλ bμ c−−λ−μ +λ+μ+ν +λ+μ−ν Γ Γ Γ(λ + 2 2

1) Γ(μ + 1)  a2 +λ+μ−ν +λ+μ+ν b2 , ; λ + 1, μ + 1; − 2 , − 2 × F4 2 2 c c [Re( + λ + μ) > |Re ν|, Re c > |Im a| + |Im b|] ET II 373(8)

xλ−μ−ν+1 J ν (ax) J μ (bx) J λ (cx) dx = 0  Re λ > −1, Re(λ − μ − ν) < 12 ,



c > b > 0,

0 b > 0,

0 −1, Re(μ + ν) > −1 WA 452(2), ET II 64(12)

∞

 − 1 μ− 1 μ+ 1 1 1 1 xμ+1 I ν (ax) K μ (bx) J ν (cx) dx = √ a−μ−1 bμ c−μ−1 e−(μ− 2 ν+ 4 )πi v 2 + 1 2 4 Q ν− 12 (iv) 2 2π  2acv = b2 − a2 + c2 , Re b > |Re a| + |Im c|, Re ν > −1, Re(μ + ν) > −1 ET II 66(22)

∞

x1−μ J μ (ax) J ν (bx) J ν (cx) dx  1 2 −μ μ− 12 μ−1 (μ− 12 )πi Q 2 −μ1 (cosh u) a (bc) (sinh u) sin[(μ − ν)π]e = ν− 2 π3 1 1 −μ μ− 12 μ−1 2 −μ = √ a (bc) (sin v) P ν− 1 (cos v) 2 2π

[a > b + c]

=0

[0 < a < |b − c|]

2bc cosh u = a − b − c , 2

2

2

2bc cos v = b + c − a , 2

2

2

b > 0,

c > 0;

[|b − c| < a < b + c]

Re ν > −1, Re μ > − 12



694

Bessel Functions



∞

9.

J ν (ax) J ν (bx) J ν (cx)x1−ν dx = 0

0

[0 < c ≤ |a − b| or c ≥ a + b] ν−1

=  Δ=

10.11

6.578

1 4

2ν−1

2 Δ     (abc)ν Γ ν + 12 Γ 12

 [c2 − (a − b)2 ] [(a + b)2 − c2 ],

[|a − b| < c < a + b]

a > 0,

b > 0,

c > 0,

Re ν > − 12

(Δ > 0 is equal to the area of a triangle whose sides are a, b, and c.) √ ν  ∞ πc Γ(ν + μ + 1) Γ(ν − μ + 1) −ν− 12 ν+1 x K μ (ax) K μ (bx) J ν (cx) dx = P μ− 1 (u) 1 2 ν+ 1 2 0 2 3 (ab)ν+1 (u2 − 1) 2 4  2abu = a2 + b2 + c2 , Re(a + b) > |Im c|, Re (ν ± μ) > −1, Re ν > −1 

11.11 0



∞

12.8

ET II 67(30)

ν+ 1

1

(ab)−ν−1 cν e−(ν+ 2 )πi Q μ− 21 (u) 2 x K μ (ax) I μ (bx) J ν (cx) dx = √ 1 1 2 2 ν+ 4 2π (u − 1)  2abu = a2 + b2 + c2 , Re a > |Re b| + |Im c|, Re ν > −1, Re(μ + ν) > −1

∞



ν+1

xν+1 [J ν (ax)] Y ν (bx) dx = 0 2

0

−ν− 12 23ν+1 a2ν b−ν−1  2 1  b − 4a2 = √ πΓ 2 −ν

 0 < b < 2a,  0 < 2a < b,

ET II 66(24)

|Re ν|
0,

=0 −ν− 12 23ν+1 a2ν b−ν−1  2 1  b − 4a2 =− √ πΓ 2 −ν

|Re ν| < 12 , 2a < b,

0 < b < 2a |Re ν| < 12



ET II 55(49)



∞

14.

xν+1 J μ (xa sin ψ) J ν (xa sin ϕ) K μ (xa cos ϕ cos ψ) dx

0

ν

=  tan 12 α = tan ψ cos ϕ,  15. 0

∞

a > 0,

2ν Γ(μ + ν + 1) (sin ϕ)

π > ϕ > 0, 2

xν+1 J ν (ax) K ν (bx) J ν (cx) dx =

cos α2

2ν+2

aν+2 (cos ψ) π 0 < ψ < , Re ν > −1, 2 ν



1 2

2ν+1 P −μ ν (cos α)  Re(μ + ν) > −1 ET II 64(11)



2 (abc) Γ ν + ν+ 12  √ 2 π (a2 + b2 + c2 ) − 4a2 c2  Re b > |Im a|, 3ν



c > 0,

Re ν > − 12



ET II 63(8)

6.581

Bessel functions and powers

 16.

∞

8

ν+1

x

I ν (ax) K ν (bx) J ν (cx) dx =

0

695

  23ν (abc)ν Γ ν + 12

ν+ 12 √  2 2 π (b − a2 + c2 ) + 4a2 c2  Re b > |Re a| + |Im c|;

Re ν > − 21



ET II 65(18)

6.579



∞

1.

x2ν+1 J ν (ax) Y ν (ax) J ν (bx) Y ν (bx) dx

0

=



∞

2. 0



∞

3.

x2ν+1 J ν (ax) K ν (ax) J ν (bx) K ν (bx) dx        Γ ν + 12 Γ 3ν+1 2ν−3 a2ν Γ ν+1 a4 2 2 1 3ν + 1 √ ; 2ν + 1; 1 − 4 F ν + 2, = 2 b πb4ν+2 Γ(ν + 1)  0 < a < b, Re ν > − 31 ET II 373(10) 4

x1−2ν [J ν (ax)] dx =

0



∞

4.



a2ν Γ(3ν + 1) 3 a2 1  1   ; F ν + , 3ν + 1; 2ν + 3 2 2 b2 2πb4ν+2  Γ 2 − ν Γ 1 2ν + 2 1 EH II 94(45), ET II 352(15) 0 < a < b, − 3 < Re ν < 2

x

1−2ν

0

Γ(ν) Γ(2ν)  2   2π Γ ν + 12 Γ(3ν)

[Re ν > 0]

a2ν−1 Γ(ν)   F [J ν (ax)] [J ν (bx)] dx = 2πb Γ ν + 12 Γ 2ν + 12 2



2

ET II 342(25)



1 a2 1 ν, − ν; 2ν + ; 2 2 2 b ET II 351(10)

6.581



a

1.

xλ−1 J μ (x) J ν (a − x) dx = 2λ

0



∞ " (−1)m Γ(λ + μ + m) Γ(λ + m) J λ+μ+ν+2m (a) m! Γ(λ) Γ(μ + m + 1) m=0 [Re(λ + μ) > 0, Re ν > −1]

ET II 354(25) a

2.8

xλ−1 (a − x)−1 J μ (x) J ν (a − x) dx

0

=

∞ 2λ " (−1)m Γ(λ + μ + m) Γ(λ + m) (λ + μ + ν + 2m) J λ+μ+ν+2m (a) aν m=0 m! Γ(λ) Γ(μ + m + 1)

[Re(λ + μ) > 0,



a

3. 0

 4. 0

a

Re ν > 0]

ET II 354(27)

    Γ μ + 12 Γ ν + 12 μ+ν+ 1 2 J a x (a − x) J μ (x) J ν (a − x) dx = √ μ+ν+ 12 (a) 2π Γ(μ + ν + 1)  Re μ > − 12 , Re ν > − 12 μ

ν



ET II 354(28), EH II 46(6)

   Γ μ + Γ ν + 32 μ+ν+ 3 μ ν+1 2 J a x (a − x) J μ (x) J ν (a − x) dx = √ μ+ν+ 12 (a) 2π Γ(μ + ν + 2)  Re ν > −1, Re μ > − 21 1 2

ET II 354(29)

696

Bessel Functions

 5.

a

μ

  2μ Γ μ + 12 Γ(ν − μ) μ √ a J ν (a) J μ (x) J ν (a − x) dx = π Γ(μ + ν + 1)  Re ν > Re μ > − 21

−μ−1

x (a − x)

0

 6.582

∞

−μ

xμ−1 |x − b|

0

 6.583

∞

μ−1

x

0

6.582

  1 K μ (|x − b|) K ν (x) dx = √ (2b)−μ Γ 12 − μ Γ(μ + ν) Γ(μ − ν) K ν (b) π  b > 0, Re μ < 12 , Re μ > |Re ν|

−μ

(x + b)

ET II 355(30)

√ π Γ(μ + ν) Γ(μ − ν)   K ν (b) K μ (x + b) K ν (x) dx = 2μ bμ Γ μ + 12 [|arg b| < π, Re μ > |Re ν|]

ET II 374(14)

ET II 374(15)

6.584  1.

8

∞ x−1

0

 2.

∞

8 0



∞

3. 0

 4. 0

∞

   (1) πi πi H (1) axe (ax) − e H ν ν

m   d πi (1) −2 r dx = H (ar) ν m+1 2m m! r dr (x2 − r2 ) m = 0, 1, 2, . . . , Im r > 0, a > 0, |Re ν| < Re  < 2m + 72

WA 465

  1 1 x−1 cos ( − ν)π J ν (ax) + sin ( − ν)π Y ν (ax) m+1 dx 2 2 (x2 + k 2 )

m  −2 d (−1)m+1 = m k K ν (ak) 2 · m! k dk  m = 0, 1, 2, . . . , Re k > 0, a > 0, |Re ν| < Re  < 2m + 72 WA 466(2) x1−ν dx am K ν+m (ak) {cos νπ J ν (ax) − sin νπ Y ν (ax)} = m m+1 2 · m!k ν+m (x2 + k 2 )  m = 0, 1, 2, . . . , Re k > 0, a > 0, −2m −

3 2

< Re ν < 1



WA 466(3)

'     ( x−1 cos 12  − 12 ν − μ π J ν (ax) + sin 12  − 12 ν − μ π Y ν (ax) μ+1 dx (x2 + k 2 ) ⎡  ν  1  

1 Γ 2  + 12 ν a2 k 2 +ν +ν πk −2μ−2 2 ak ⎣   1F 2 ; − μ, ν + 1; = 2 sin νπ · Γ(μ + 1) Γ(ν + 1) Γ 12  + 12 ν − μ 2 2 4 ⎤  1 −ν  1  

Γ  − 1ν a2 k 2 ⎦ −ν −ν 2 ak 1 2 1 2  1F 2 ; − μ, 1 − ν; − 2 2 4 Γ(1 − ν) Γ 2  − 2 ν − μ  WA 407(1) a > 0, Re k > 0, |Re ν| < Re  < 2 Re μ + 72

6.591

Powers and Bessel functions of complicated arguments



∞

5.12 0

697

⎤⎧ ⎡ ⎛ ⎞ ⎤ n ⎨ " 1 ⎣ J μj (bn x)⎦ cos ⎣ ⎝ + μj − ν ⎠ π ⎦ J ν (ax) ⎩ 2 j=1 j=1 ⎫ ⎞ ⎤ ⎡ ⎛ n ⎬ x−1 " 1 μj − ν ⎠ π ⎦ Y ν (ax) dx + sin ⎣ ⎝ + ⎭ x2 + k 2 2 j=1 ⎤ ⎡ n + I μj (bn k)⎦ K ν (ak)k −2 = −⎣ ⎡

n +

j=1

⎡ ⎣Re k > 0,

a>

n "

⎛ Re ⎝ +

|Re bj |,

j=1

n "

⎞

⎤

μj ⎠ > |Re ν|⎦

WA 472(9)

j=1

6.59 Combinations of powers and Bessel functions of more complicated arguments 6.591



∞

1. 0



∞

0

x

K ν (bx) dx =

√

1

2πb−ν−1 aν+ 2 J 1+2ν

√

0

∞

4. 0

[a > 0,

1

x2ν+ 2 Y ν+ 12

a

x

K ν (bx) dx =

∞

5.

1

x2ν+ 2 K ν+ 12

[a > 0,

a

1

x−2ν+ 2 J ν− 12

−2ν+ 12

x

0

 6.12 0

Re b > 0,

Re ν > −1]

√

√

√ 1 2πb−ν−1 aν+ 2 Y 2ν+1 2ab K 2ν+1 2ab Re b > 0,

Re ν > −1]

x

1 √

1 √

√ 1 K ν (bx) dx = 2πb−ν−1 aν+ 2 K 2ν+1 e 4 iπ 2ab K 2ν+1 e− 4 iπ 2ab

ET II 146(56) [Re a > 0, Re b > 0]

√ √ 1 K ν (bx) dx = 2πbν−1 a 2 −ν K 2ν−1 2ab x √

√   2ab + cos(νπ) Y 2ν−1 2ab × sin(νπ) J 2ν−1

a

[a > 0, 

√

2ab K 1+2ν 2ab

ET II 143(41) ∞

3. 

a

ET II 142(35)

2.



1

x2ν+ 2 J ν+ 12

∞

1

Y ν− 12

a

x

a

Re b > 0,

Re ν < 1]

ET II 142(34)



√

π ν−1 1 −ν b a 2 sec(νπ) K 2ν−1 2ab √

√   2 2ab − J 1−2ν 2ab × J 2ν−1

K ν (bx) dx = −

[a > 0,

Re ν < 1]

ET II 143(40)

J ν (bx) dx x−2ν+ 2 J 12 −ν  x  1 π i cosec(2νπ)bν−1 a 2 −ν e2νπi J 1−2ν (u) J 2ν−1 (v) − e−2νπi J 2ν−1 (u) J 1−2ν (v) =− 2   √ √ u = 2abeπi/4 , v = 2abe−πi/4 , a > 0, b > 0, − 21 < Re ν < 3 ET II 58(12)

698

Bessel Functions



∞

7. 0

1

x−2ν+ 2 K ν− 12

a

x

Y ν (bx) dx =

6.592

√

√

√ 1 2πbν−1 a 2 −ν Y 2ν−1 2ab K 2ν−1 2ab  b > 0, Re a > 0, Re ν > 16 ET II 113(30)

 

 ∞ aν− bν Γ 12 μ + 12  − 12 ν b −1   dx = 2ν−+1 x J μ (ax) J ν x 2 Γ(ν + 1) Γ 12 μ + 12 ν − 12  + 1 0 

ν +μ− a2 b 2 ν −μ− + 1, + 1; × 0 F 3 ν + 1, 16 2 2 aμ bμ+ Γ 12 ν − 12 μ − 12    + 2μ++1 2 Γ(μ + 1) Γ 12 μ + 12 ν + 12  + 1 

μ−ν + ν+μ+ a2 b 2 × 0 F 3 μ + 1, + 1, + 1; 2    2  16  3 a > 0, b > 0, − Re μ + 2 < Re  < Re ν + 32

 8.

6.592 1.



1

λ

μ−1

x (1 − x)

12 0

 2.12

1

WA 480(1)

   √  Γ(μ) Γ λ + 1 + 12 ν −ν ν   Y ν a x dx = 2 a cot(νπ) Γ(1 + ν) Γ λ + 1 + μ + 12 ν 

a2 1 1 × 1 F 2 λ + 1 + 2 ν; 1 + ν, λ + 1 + μ + 2 ν; −   4 Γ(μ) Γ λ + 1 − 12 ν ν −ν   −2 a cosec(νπ) Γ(1 − ν) Γ λ + 1 + μ − 12 ν 

a2 × 1 F 2 λ − 12 ν + 1; 1 − ν, λ + 1 + μ − 12 ν; − 4  Re λ > −1 + 12 |Re ν|, Re μ > 0 ET II 197(76)a

 √  xλ (1 − x)μ−1 K ν a x dx

0

=2

+2

ν−1 −ν

a

  

Γ(ν) Γ(μ) Γ λ + 1 − 12 ν a2 1 1   1 F 2 λ + 1 − 2 ν; 1 − ν, λ + 1 + μ − 2 ν; 4 Γ λ + 1 + μ − 12 ν

  

Γ(−ν) Γ λ + 1 + 12 ν Γ(μ) 1 a2 1   a 1 F 2 λ + 1 + 2 ν; 1 + ν, λ + 1 + μ + ν; 2 4 Γ λ + 1 + μ + 12 ν

−1−ν ν

2ν−1 = ν Γ(μ) G 21 13 a

a2 4

!ν  ! −λ !2 ! ν, 0, ν − λ − μ 2 OB 159 (3.16)



Re λ > −1 + 12 |Re ν|,

Re μ > 0



ET II 198(87)a

6.592

Powers and Bessel functions of complicated arguments

 3.

∞

11 1



∞

4. 1

5. 6. 7.12

8. 9.



1



∞

λ

μ−1

x (x − 1)

 √  J ν a x dx = 22λ a−2λ G 20 13

a2 4

!  !0 ! ! −μ, λ + 1 ν, λ − 1 ν Γ(μ) 2 2  a > 0, 0 < Re μ < 34 − Re λ

 √  xλ (x − 1)μ−1 K ν a x dx = Γ(μ)22λ−1 a−2λ G 30 13 

 √  1 1 x− 2 (1 − x)− 2 J ν a x dx = π J 1 ν

2

699

ET II 205(36)a

!  a !! 0 4 ! −μ, 12 ν + λ, − 12 ν + λ 2

[Re a > 0,

Re μ > 0]

ET II 209(60)a

1 a [Re ν > −1] ET II 194(59)a 2 2 0

2   1  √  1 − 12 − 12 a x (1 − x) I ν a x dx = π I 1 [Re ν > −1] ET II 197(79) ν 2 2 0  1 a  a

 a

 √  1 1 1√ + I − ν2 K ν2 x− 2 (1 − x)− 2 K ν a x dx = π sec (νπ) I ν2 2 2 2 2 0 [|Re ν| < 1] ET II 198(85)a  ∞

  2   √ 1 1 a x− 2 (x − 1)− 2 K ν a x dx = K ν2 [Re a > 0] ET II 208(56)a 2 1 #  1 a 2 a 2 $    √  − 12 − 12 x (1 − x) Y ν a x dx = π cot(νπ) J ν2 − cosec(νπ) J − ν2 2 2 0 [|Re ν| < 1]

10. 1



0 < Re μ
0,

ET II 195(68)a

 √  1 x− 2 ν (x − 1)μ−1 J −ν a x dx = Γ(μ)2μ a−μ [cos(νπ) J ν−μ (a) − sin(νπ) Y ν−μ (a)]  a > 0, 0 < Re μ < 12 Re ν + 34 ET II 205(35)a

∞

12.

 √  1 x− 2 ν (x − 1)μ−1 K ν a x dx = Γ(μ)2μ a−μ K ν−μ (a)

1



[Re a > 0, ∞

13. 1

 1

15. 1

0 < Re μ
0,

Re μ > 0]

 √  1 (1) a x dx = 2μ a−μ H ν−μ (a) Γ(μ) x− 2 ν (x − 1)μ−1 H (1) ν [Re μ > 0,

∞

Im a > 0]

ET II 206(45)a

Im a < 0]

ET II 207(48)a

 √  1 (2) a x dx = 2μ a−μ H ν−μ (a) Γ(μ) x− 2 ν (x − 1)μ−1 H (2) ν [Re μ > 0,

700

Bessel Functions



1

16. 0

 17. 0

1

6.593

 √  1 22−ν a−μ s μ+ν−1,μ−ν (a) x− 2 ν (1 − x)μ−1 J ν a x dx = Γ(ν) [Re μ > 0]

ET II 194(64)a

 √  1 22−ν a−μ cot(νπ) s μ+ν−1,μ−ν (a) x− 2 ν (1 − x)μ−1 Y ν a x dx = Γ(ν) −2μ a−μ cosec(νπ) J μ−ν (a) Γ(μ) [Re μ > 0,

6.593  1. 2.

Re ν < 1]

ET II 196(75)a

2   √  a 1 b > 0, Re ν > − 12 x J 2ν−1 a x J ν (bx) dx = ab−2 J ν−1 2 4b 0 

2

2   ∞  √  √ a a πa − Lν−1 x J 2ν−1 a x K ν (bx) dx = 2 I ν−1 4b 4b 4b 0  Re b > 0, Re ν > − 21

6.594  1.

∞√

∞

0



 √   √  √ 1 x I 2ν−1 a x J 2ν−1 a x K ν (bx) dx = π2−ν a2ν−1 b−2ν− 2 J ν− 12 ν

[Re b > 0, ∞

2.

a2 2b

ET II 58(15)

ET II 144(44)



Re ν > 0]

ET II 148(65)

 √   √  xν I 2ν−1 a x Y 2ν−1 a x K ν (bx) dx

0

√ −ν−1 2ν−1 −2ν− 1 2 cosec(νπ) π2 a b

2

2

2   a a a + cos(νπ) J ν− 12 + sin(νπ) Y ν− 12 × H 12 −ν 2b 2b 2b [Re b > 0, Re ν > 0] ET II 148(66)  ∞  √   √  xν J 2ν−1 a x K 2ν−1 a x K ν (bx) dx 0

2

2   a a 2 −ν−2 2ν−1 −2ν− 12 − Y 12 −ν =π 2 a b cosec(νπ) H 12 −ν 2b 2b [Re b > 0, Re ν > 0] ET II 148(67) =

3.

6.595  1. 0

∞

)

ν+1

x

J ν (cx)

n + i=1

 zi = x2 + b2i ,

zi−μi J μi (ai zi ) dx = 0 ai > 0,

Re bi > 0,

n "

 ai < c;

Re

i=1

n " 1 1 μi − n+ 2 2 i=1



* > Re ν > −1

EH II 52(33), ET II 60(26)

 2. 0

∞

n +

n +

 −μi bi J μi (ai bi ) xν−1 J ν (cx) zi−μi J μi (ai zi ) dx = 2ν−1 Γ(ν)c−ν i=1 i=1 )  *  n n  " " 1 3 zi = x2 + b2i , ai > 0, Re bi > 0, n+ > Re ν > 0 ai < c, Re μi + 2 2 i=1 i=1 EH II 52(34), ET II 60(27)

6.596

6.596

Powers and Bessel functions of complicated arguments



∞

1. 0

701



x2μ+1 2μ Γ(μ + 1) J ν a x2 + z 2  dx = μ+1 ν−μ−1 J ν−μ−1 (az) ν a z (x2 + z 2 )   

1 1 ν− > Re μ > −1 a > 0, Re 2 4

 √  ∞ a

a

J ν a t2 + 1 π √ Y ν2 [Re ν > −1, dt = − J ν2 2 2 2 t2 + 1 0  ∞ 

x2μ+1 2μ Γ(μ + 1) K ν a x2 + z 2  dx = K ν−μ−1 (az) ν aμ+1 z ν−μ−1 (x2 + z 2 ) 0

WA 457(5)

 2. 3.

 4.8 0

∞

J ν (bx)

' √ ( J μ−1 a x2 + z 2 (x2 + z 2 )

1 1 2 μ+ 2

[a > 0, xν+1 dx =

aμ−1 z ν K ν (bz) 2μ−1 Γ(μ) [a < b,

a > 0]

MO 46

Re μ > −1]

WA 457(6)

Re(μ + 2) > Re ν > −1]

' √ ( ∞ J μ a x2 + z 2 ν−1 2ν−1 Γ(ν) J μ (az) J ν (bx)  x dx = μ bν zμ (x2 + z 2 ) 0 [Re(μ + 2) > Re ν > 0,

ET II 59(19)

 5.8

b > a > 0] WA 459(12)

 √  ∞ J μ a x2 + z 2 ν+1 J ν (bx)  dx μ x (x2 + z 2 ) 0

 6.6

=0 =

7.8

8.8

bν aμ

√

a2 − b 2 z

μ−ν−1

[0 < a < b]    J μ−ν−1 z a2 − b2

[a > b > 0]

[Re μ > Re ν > −1] WA 415(1)    √  √ μ−ν−1  ∞ 

K μ a x2 + z 2 ν+1 a2 + b 2 bν J ν (bx)  x dx = μ K μ−ν−1 z a2 + b2 μ a z (x2 + z 2 ) 0  π a > 0, b > 0, Re ν > −1, |arg z| < KU 151(31), WA 416(2) 2 

 ν  ∞ 

  μ 1 u π 2 2 − 2 ν+1 2 2 · μ x −y J ν (ux) K μ v x − y x dx = exp −iπ μ − ν − 2 2 v 0 *μ−ν−1 )√ 

2 2 u +v (2) · H μ−ν−1 y u2 + v 2 y   1α  1α  2 1 Re μ < 1, Re ν > −1, u > 0, v > 0, y > 0; x − y 2 2 = e 2 απi y 2 − n2 2 if x < y

702

Bessel Functions

 9.

∞

8 0



 − μ v x2 + y 2 x2 + y 2 2 xν+1 dx J ν (ux) H (2) μ uν = μ v ⎡ ⎣ Re μ > Re ν > −1,

u > 0,

6.597

*μ−ν−1 )√ 

v 2 − u2 (2) H μ−ν−1 y v 2 − u2 y [u < v]

v > 0,

y > 0;

arg



v 2 − u2 = 0, for v > u;

⎤   2 μ − ν − 1 1 σ ⎦ arg v − u2 = −πσ for v < u, where σ = or σ = 2 2 MO 43



∞

10.8 0







xν−1 2ν−1 Γ(ν) J μ (az) J μ (γz) J ν (bx) J μ a x2 + z 2 J μ γ x2 + z 2 dx = μ μ bν  zμ (x2 + z 2 )   z a > 0; b > a + γ; γ > 0, Re 2μ + 52 > Re ν > 0 WA 459(14)

n 

 +  −μ x J μ (ak x) J μ ak t2 + x2 (t2 + x2 )−nμ dt = 2ν−1 b−ν Γ(ν) 0 k=1 k=1 ) * 

n + 1 1 x > 0, ai > 0, b > > Re ν > 0 ak ; Re nμ + n + MO 43 2 2 k=1     ∞ 2 √ 2  Jμ a + x2 2ν−2 Γ ν − 12 8 √ Hν (2a) 12. x dx = Re ν > 12 WA 457(8) 2 + x2 )ν ν+1 π (a 2a 0  ∞   1   − 1 μ  −1 6.597 tν+1 J μ b t2 + y 2 2 t2 + y 2 2 t2 + β 2 J ν (at) dt 0   1   − 1 μ = β ν J μ b y 2 − β 2 2 y 2 − β 2 2 K ν (aβ)

11.

∞

8

J ν (bt)tν−1

n +

[a ≥ b, 

1

6.598

Re β > 0,

−1 < Re ν < 2 + Re μ]

EH II 95(56)



 √   √  − 1 (ν+μ+1)  μ ν x 2 (1 − x) 2 J μ a x J ν b 1 − x dx = 2aμ bν a2 + b2 2 J ν+μ+1 a2 + b 2

0

[Re ν > −1,

Re μ > −1]

EH II 46a

6.61 Combinations of Bessel functions and exponentials 6.611  1.

∞

0

 2. 0

e

−αx

J ν (bx) dx =

b−ν

√ ν α2 + b2 − α √ α2 + b2

[Re ν > −1,

Re (α ± ib) > 0] EH II 49(18), WA 422(8)

∞

− 1  e−αx Y ν (bx) dx = α2 + b2 2 cosec(νπ) #   −ν ν $  2 1 1 ν 2 2 −ν 2 2 2 × b α +b α +b +α cos(νπ) − b +α [Re α > 0,

b > 0,

|Re ν| < 1]

MO 179, ET II 105(1)

6.612

Bessel functions and exponentials



∞

3.

e−αx K ν (bx) dx =

 π as b → ∞ 2   ET II 131(22)



  ν −ν π cosec(νπ) −ν ν 2 2 2 2 b α+ α −b = √ −b α −b +α 2 α2 − b2 [|Re ν| < 1, Re(α + b) > 0]

0



sin(νθ) π b sin(νπ) sin θ

703

∞

4.8 0



cos θ =

α ; b

with θ →

ET I 197(24), MO 180

√  ν b−ν α − α2 − b2 −αx √ e I ν (bx) dx = α2 − b2

[Re ν > −1,

Re α > |Re b|] MO 180, ET I 195(1)

) √  √ 2ν *& ν %  ∞ α + α2 + b2 α2 + b2 − α i (1,2) −αx √ cos(νπ) − 1± e H ν (bx) dx = sin(νπ) b2ν bν α2 + b2 0

5.

(2) [−1 < Re ν < 1; a plus sign corresponds to the function H (1) ν , a minus sign to the function H ν ].



∞

6. 0



∞

7. 0

9.12

 10.

1 (2) e−αx H 0 (bx) dx = √ α2 + b2

1−

α 2i ln + π b )

% 1+

α 2i ln + π b

 1+

10 a

∞

b

 α dα

∞

dk J 1 (kα)e

−k|β|

0

 = a

b



|β|

1−  α2 + β 2

α 2

*&

MO 180, ET I188(54, 55)

b

[Re α > |Im b|] *& α 2 1+ b

MO 180, ET I 188(53)

[Re α > |Im b|]

MO 180, ET I 188(53)

[Re α > |Im b|]

MO 47, ET I 187(44)



√ −2 α + α2 + b2 −αx e Y 0 (bx) dx = √ ln b π α2 + b2 0  ∞ α arccos b e−αx K 0 (bx) dx = √ b2 − α2  0   α2 1 α + = √ ln −1 b b2 α2 − b2

 8.

1 (1) e−αx H 0 (bx) dx = √ 2 α + b2

)

%

[0 < a < b] [0 ≤ b < a]

WA 424, ET II 131(22) MO 48

 dα (see 3.241 6)

6.612 1.



∞

e−2αx J 0 (x) Y 0 (x) dx =

  − 1  K α α2 + 1 2 1

π (α2 + 1) 2  ∞ 1  1  e−2αx I 0 (x) K 0 (x) dx = K 1 − α2 2 2 ) 0 1 * 1 2 1 K 1− 2 = 2α α

[Re α > 0]

ET II 347(58)

0

2.

[0 < α < 1] [1 < α < ∞] ET II 370(48)

704

Bessel Functions



∞

3.

e

−αx

0

5. 6.



2 1 α + b2 + γ 2 J ν (bx) J ν (γx) dx = √ Q ν− 12 2bγ π γb  Re (α ± ib ± iγ) > 0, γ > 0, Re ν > − 12

WA 426(2), ET II 50(17)



2 2b 2 e−αx [J 0 (bx)] dx = √ K √ π α2 + 4b2 α2 + 4b2 0



  2    ∞ 2α + b2 K √α2b+b2 − 2 α2 + b2 E √α2b+b2 √ e−2αx J12 (bx) dx = πb2 α2 + b2 0  ∞ r2 e−3x I l (x) I m (x) I n (x) dx = r1 g + 2 + r3 π g 0 where √

  3−1 2 1 11 Γ2 Γ g= 3 96π 24 24 

4.

6.612

∞

MO 178

WA 428(3)

and (lmn) 000 100 110 111 200 210 211 220 221 222 300 310 311 320 321 322 330 331 332 333 400 410 411 420 421 422 430 431

r1 1 1 5/12 − 1/8 10/3 3/8 − 2/3 73/36 − 15/16 5/8 35/2 − 79/36 − 11/4 319/48 − 125/36 35/16 50/3 − 35/3 35/9 − 35/16 994/9 − 515/16 − 9/2 12907/120 − 229/16 35/3 2641/48 − 1505/36

r2 0 0 − 1/2 3/4 2 − 9/4 2 − 29/6 21/8 − 27/20 21 − 85/6 21/2 − 119/8 269/30 − 213/40 − 1046/25 148/5 − 1012/105 1587/280 542/3 − 879/8 357/5 − 13903/10 1251/40 − 1024/35 − 28049/200 118051/1050

r3 0 − 1/3 0 0 −2 1/3 0 0 0 0 −13 4 − 2/3 − 1/3 0 0 0 0 0 0 −92 115/3 −12 −6 1 0 1/3 0

(lmn) 432 433 440 441 442 443 444 500 510 511 520 521 522 530 531 532 533 540 541 542 543 544 550 551 552 553 554 555

r1 525/32 − 595/72

6025/36 − 29175/224 2975/48 − 539/32 77/8 9287/12 − 189029/180 275/4 2897/16 − 937/12 509/8 3589/18 − 1329/8 2555/36 − 2233/48 18471/32 − 1390/3 7777/32 − 5621/72 1155/32 197045/108 − 12023/8 1683/2 − 5159/16 24563/312 − 9251/208

r2 − 4617/112 8809/420 − 620161/1470 131379/400 − 31231/200 119271/2800 − 186003/7700 3005/2 − 138331/50 5751/10 − 15123/20 27059/30 − 4209/28 − 1993883/3075 297981/700 − 187777/1050 164399/1400 − 28493109/19600 286274/245 − 1715589/2800 4550057/23100 − 560001/6160 − 101441689/22050 18569853/4900 − 5718309/2695 2504541/3080 − 1527851/77000 12099711/107800

r3 0 0 0 0 0 0 0 − 2077/3 348 −150 − 229/3 24 0 0 − 4/3 0 0 − 1/3 0 0 0 0 0 0 0 0 0 0

6.616

Bessel functions and exponentials

 6.613 6.614

∞

11

e

−xz

e

−αx

0



∞

1. 0



∞

2. 0

J ν+ 12

 dx =



 π  Γ(ν + 1) πi √ D −ν−1 ze 4 i D −ν−1 ze− 4 π

√

e e−αx Y 2ν 2 bx dx = √ αb

b − 12 α

#

 b Γ(ν + 1) M 12 ,ν − cosec(νπ) W cot(νπ) Γ(2ν + 1) α

MO 122

[ν = 0] MO 178

$ b 1 2 ,nu α

[Re α > 0, 

∞ √

Γ(ν + 1) b e M − 12 ,ν e−αx I 2ν 2 bx dx = √ Γ(2ν + 1) α αb 0

|Re ν| < 1]

ET I 188(50)a

[Re α > 0,

 1 b ∞ √

e2 α b −αx e K 2ν 2 bx dx = √ Γ(ν + 1) Γ(1 − ν) W − 12 ,ν α 2 αb 0

Re ν > −1]

ET I 197(20)a

1 b 2 α

 4.

[Re ν > −1]



2

2 

2   √  π b b b b I 12 (ν−1) − I 12 (ν+1) J ν b x dx = exp − 3 4 α 8α 8α 8α 1 −b2 /4α = e α

 3.

x2 2

705

ET I 199(37)a [Re α > 0, |Re ν| < 1]

2   √  b π b b b K1 − K0 5. e−αx K 1 b x dx = exp MO 181 3 8 α 8α 8α 8α 0 



 ∞  √   √  β2 + γ2 2βγ 1 exp − [Re ν > −1] 6.615 e−αx J ν 2β x J ν 2γ x dx = I ν α α α 0





∞

2



2



MO 178

6.616 1. 2. 3.







  1 e−αx J 0 b x2 + 2γx dx = √ exp γ α − α2 + b2 α2 + b2 0  ∞ 



 1 e−αx J 0 b x2 − 1 dx = √ exp − α2 + b2 α2 + b2 1 √  ∞ 2 2 

eiα r +x (1) itx 2 2 e H 0 r α − t dt = −2i √ 2 2 −∞  r + x 0 ≤ arg α2 − t2 < π, 0 ≤ arg α < π; 

∞

∞

4. −∞



1

5.3 −1

MO 179 MO 179

 r and x are real

MO 49

 r and x are real

MO 49

√

e

−itx

(2) H0

2 2 

e−iα r +x r α2 − t2 dt = 2i √ 2 2   r +x 2 2 −π < arg α − t ≤ 0,

−π < arg α ≤ 0,



  −1/2 e−ax I 0 b 1 − x2 dx = 2 a2 + b2 sinh a2 + b2 [a > 0,

b > 0]

706

Bessel Functions



∞

6.8 0

6.617



∞

1.

∞ 

" P n (x) e−xy J 0 y 1 − x2 /(α + y) dy = n! n+1 α n=0

K q−p (2z sinh x) e(p+q)x dx =

0



∞

2.

6.617

K 0 (2z sinh x) e−2px dx = −

0

π 4

π2 [J p (z) Y q (z) − J q (z) Y p (z)] 4 sin[(p − q)π] [Re z > 0, −1 < Re(p − q) < 1] # J p (z)

∂ Y p (z) ∂ J p (z) − Y p (z) ∂p ∂p

MO 44

$

[Re z > 0] 6.618



∞

1.

e

−αx2

0

2

2 √ π b b I 12 ν J ν (bx) dx = √ exp − 8α 8α 2 α

[Re α > 0,

MO 44

b > 0,

Re ν > −1] WA 432(5), ET II 29(8)

2 

2

2  √ ∞ νπ

π b νπ 1 b b −αx2 √ exp − tan I 12 ν + sec K 12 ν e Y ν (bx) dx = − 2 α 8α 2 8α π 2 8α 0 [Re α > 0, b > 0, |Re ν| < 1]

 2.



∞

3. 0



∞

4. 0

 5.

0

∞

2

2 νπ √π 2 b b 1 √ exp K 12 ν e−αx K ν (bx) dx = sec 4 2 8α 8α α [Re α > 0,

2

2 √ π b b −αx2 I 12 ν e I ν (bx) dx = √ exp 2 α 8α 8α

WA 432(6), ET II 106(3)

|Re ν| < 1] EH II 51(28), ET II 132(24)

[Re ν > −1,

Re α > 0]

EH II 92(27)

2

e−αx J μ (bx) J ν (bx) dx

  Γ μ+ν+1 2 α b =2 Γ(μ + 1) Γ(ν + 1) 

b2 ν +μ+1 ν +μ+2 ν +μ+1 , , ; μ + 1, ν + 1, ν + μ + 1; − × 3F 3 2 2 2 α [Re(ν + μ) > −1, Re α > 0] EH II 50(21)a −ν−μ−1 − ν+μ+1 ν+μ 2

6.62–6.63 Combinations of Bessel functions, exponentials, and powers 6.621

Notation:   1  1 = (a + b)2 + z 2 − (a − b)2 + z 2 , 2

2 =

  1  (a + b)2 + z 2 + (a − b)2 + z 2 2

6.621

Bessel functions, exponentials, and powers



∞

1.

 e

−αx

μ−1

J ν (bx)x

dx =

0

 =



b ν Γ(ν + μ) 2α μ α Γ(ν + 1)



b ν Γ(ν + μ) 2α αμ Γ(ν + 1)

 b ν

F

ν+μ ν +μ+1 b2 , ; ν + 1; − 2 2 2 α



WA 421(2)

 1 −μ  b2 2 b2 ν −μ+1 ν −μ 1+ 2 , + 1; ν + 1; − 2 F α 2 2 α

Γ(ν + μ)

WA 421(3)

b2 ν +μ 1−μ+ν , ; ν + 1; 2 2 2 α + b2

=  F ν+μ (α2 + b2 ) Γ(ν + 1) [Re(ν + μ) > 0, 2

707

Re (α + ib) > 0,

Re (α − ib) > 0] WA 421(3)

  − 1 μ  1  2 2 −2 α α = α2 + b2 2 Γ(ν + μ) P −ν + b μ−1 [α > 0,



b > 0,

Re(ν + μ) > 0] ET II 29(6)



∞

2.

e−αx Y ν (bx)xμ−1 dx

 b ν

0

= cot νπ 

ν+μ

(α2 + b2 )

− cosec νπ 

Γ(ν + μ)

2

 b −ν 2

F

Γ(ν + 1)

b2 ν +μ ν −μ+1 , ; ν + 1; 2 2 2 α + b2

Γ(μ − ν)

(α2 + b2 )μ−ν Γ(1 − ν)

F



b2 μ−ν 1−ν −μ , ; 1 − ν; 2 2 2 α + b2

[Re μ ≥ |Re ν|,    − 1 μ  1 2 2 2 −2 = − Γ(ν + μ) b2 + α2 2 Q −ν μ−1 α α + b π [α > 0, b > 0,



Re (α ± ib) > 0] WA 421(4)

Re μ > |Re ν|] ET II 105(2)



∞

3. 0



∞

4.

xμ−1 e−αx K ν (bx) dx =

√ π(2b)ν Γ(μ + ν) Γ(μ − ν)   F μ + ν, ν + (α + b)μ+ν Γ μ + 12 [Re μ > |Re ν|,

xm+1 e−αx J ν (bx) dx = (−1)m+1 b−ν

0

 5.10

dm+1 dαm+1

) √

ν * α2 + b2 − α √ α2 + b2 [b > 0,

∞

e−zx J 1 (ax) J 1/2 (bx) x−3/2 dx

0

1 = a



2 πb

#

1 2

 1 α−b 1 ;μ + ; 2 2 α+b Re(α + b) > 0]

ET II 131(23)a, EH II 50(26)

Re ν > −m − 2]

ET II 28(3)

  $  1 a2 2 2 2 2 arcsin +z a − 1 + b − 1 − b 2 2 [arg a > 0, arg b > 0, arg z > 0]

708

Bessel Functions

6.

7.10

8.10

9.10

10.10

11.12

 12.10 0

14.10

∞

e

−zx

−1/2

J 1 (ax) J 1/2 (bx) x

1 dx = a

[arg a > 0, ∞

arg b > 0,

arg z > 0]

e−zx J 1 (ax) J 5/2 (bx) x−5/2 dx ⎧     2  5/2 2 5/2

 2 b2 z2 1 ⎨ 2 b − b − 1 1 3a 1 2 + za − − arcsin =√ 15 a 8 2 2 2π b5/2 a ⎩ ⎫   2  3a2 z2 2 z 3 a2  1 ⎬ b − + − 1 +  + z1 a2 − 21 2 8 6 4 3 a2 − 21 ⎭

[arg a > 0, arg b > 0,  ∞ 22 − b2 2 2 3/2 a b e−zx J 2 (ax) J 3/2 (bx) x1/2 dx = 2 π (2 − 21 ) 42 0 [arg a > 0, arg b > 0, )    2 3/2 *  ∞ b − 21 2 b3/2 2 b2 − 21 −zx −1/2 − + e J 2 (ax) J 3/2 (bx) x dx = π a2 3 b 3b3 0 [arg a > 0, arg b > 0, 

13.10



   2 b − b2 − 21 πb 0 [arg a > 0, arg b > 0, arg z > 0]    ∞ 2 2  1 a2 −  1 1 e−zx J 1 (ax) J 1/2 (bx) x1/2 dx = a πb 22 − 21 0 [arg a > 0, arg b > 0, arg z > 0]    ∞ 2 21 b2 − 21 e−zx J 1 (ax) J 3/2 (bx) x1/2 dx = π b3/2 a (22 − 21 ) 0 [arg a > 0, arg b > 0, arg z > 0]  

  ∞  1 1 1 −zx −1/2 2 2 2 − 1 a − 1 a arcsin e J 1 (ax) J 3/2 (bx) x dx = √ a 2π b3/2 a 0 [arg a > 0, arg b > 0, arg z > 0] )

*  ∞  z 1 2a2 1 1 −zx −1/2 2 2 2 1 a − 1 +  e J 1 (ax) J 5/2 (bx) x dx = √ − 3a arcsin 5/2 2 2 a 2π b a a − 1 0 [arg a > 0, arg b > 0, arg z > 0]  ∞ e−zx J 1 (ax) J 5/2 (bx) x−3/2 dx 0 ⎡ 

4 1 ⎣ 1 5a2 21 1 7a 41 2 2  = √ − a − z − 8 4 8 2π b5/2 a a2 − 21 ⎤ 

  4  1 2 1 3a 3  1 ⎦  + 22 1 a2 − 21 + arcsin a2 z 2 + a2 b 2 − − 2 1 a 2 2 8 

10

6.621



arg z > 0]

arg z > 0]

arg z > 0]

6.623

Bessel functions, exponentials, and powers

 15.

∞

10 0



∞

16.10

18.10

19.10

e−zx J 3 (ax) J 1/2 (bx) x−1/2 dx  # $  2  ' ( 2 1 2 2 2 − 2 122 − 16b2 + 42 − 3a2 b 3a − − 4b + 12z b = 2 1 1 πb 3a3 [arg a > 0, arg b > 0, arg z > 0] e−zx J 3 (ax) J 3/2 (bx) x1/2 dx 

&  2  3/2 * b − 21 a 22 − a2 b2 − 21 + − 2 = b 3b2 (2 − 21 ) 32 [arg a > 0, arg b > 0, arg z > 0] )   *     ∞ 3/2  4b2 2b2 − 21 − 41 2b −zx −1/2 2 2 − 8z e J 3 (ax) J 3/2 (bx) x dx = 2 − b π 3a3 b4 0 [arg a > 0, arg b > 0, arg z > 0]  ∞ e−zx J 3 (ax) J 3/2 (bx) x−3/2 dx 0  #  2 $  2 b3/2 24b 821 a2 41 42 4 2 2 2 2 − 2 b − + − + a − + 4z − b = 1 π 3a3 5 b 5 5b b 5b3 [arg a > 0, arg b > 0, arg z > 0]  ∞ e−zx J 3 (ax) J 3/2 (bx) x−5/2 dx 0 ⎧ 

 2 b3/2 ⎨ 2 4 2 4z 3 a =− b z + − π 3a3 ⎩ 5 3    32 2 12 2 4 2 241 a4 21 a2 21 61 2 2 2 + 2 − b a + b − 1 − 2 + 2 + + + 15⎫ 5 3 5b 16b4 24b4 30b4

⎬ b a6 arcsin − 16b3 2 ⎭ 0

17.10

2 3/2 b π

%

4 a3

)

2 − 3



[arg a > 0, 6.622  1.

∞



J 0 (x) − e−αx

0

 2.

3.8

709

 dx = ln 2α x

arg b > 0,

arg z > 0]

[α > 0]

NT 66(13)

∞ i(u+x)

π e (1) J 0 (x) dx = i H 0 (u) u + x 2 0 μ− 1   ∞ Q ν− 12 (cosh α) 1 2 2 e−(μ− 2 )πi e−x cosh α I ν (x)xμ−1 dx = 1 π 0 sinhμ− 2 α [Re(μ + ν) > 0,

MO 44

Re (cosh α) > 1] WA 388(6)a

6.623  1. 0

∞

e

−αx

  (2b)ν Γ ν + 12 J ν (bx)x dx = √ ν+ 1 π (α2 + b2 ) 2 ν

 Re ν > − 12 ,

Re α > |Im b| WA 422(5)

710

Bessel Functions

  2α(2b)ν Γ ν + 32 e J ν (bx)x dx = √ ν+ 3 0 π (α2 + b2 ) 2 √ ν  ∞ α2 + b2 − α dx −αx = e J ν (bx) x νbν 0 [Re ν > 0; 

2.

3.

6.624 1. 2. 3.

∞

−αx

6.624

ν+1

[Re ν > −1,

Re α > |Im b|]

∞

e−tz(z

2

e−tz(z

2

−1/2

−1)

I −μ (t)tν dt =

0



∞

5.

−1 2

−1)

I μ (t)tν dt =

0



∞

6. 0



∞

7. 0

WA 422(7)

Γ(−ν − μ) (z 2

− 1)

1 2ν

P μν (z)

Γ(ν + μ + 1) (z 2

− 12 (ν+1)

− 1)

(2b)ν Γ ν + J ν (bx)xν √ dx = eπx − 1 π

 1 2



1

1.

EH II 57(7)

[Re(ν + μ) < 0]

EH II 57(8)

[Re(ν + μ) > −1]

EH II 57(9)

 Re(ν + μ) > −1,

0 ≤ θ < 12 π



EH II 57(10) ∞ "

1

n=1

(n2 π 2 + b2 )ν+ 2

1

xλ−ν−1 (1 − x)μ−1 e±iαx J ν (αx) dx =

0

MO 181

P −μ ν (z)

e−t cos θ J μ (t sin θ) tν dt = Γ(ν + μ + 1) P −μ ν (cos θ)



MO 181

[Re (ν ± μ) > −1]

[Re ν > 0,

|Im b| < π]

WA 423(9)



1 2−ν αν Γ(λ) Γ(μ) ; λ + μ, 2ν + 1; ±2iα λ, ν + 2F 2 Γ(λ + μ) Γ(ν + 1) 2 

[Re λ > 0, 

1

Re μ > 0]

ET II 194(58)a

 1 ; μ + 2ν + 1; ±2iα ν + 1F 1 2 0  ET II 194(57)a Re μ > 0, Re ν > − 12   

 1 1 (2α)ν Γ ν + 2 Γ(μ) 1 ; μ + 2ν + 1; ±2α ν + xν (1 − x)μ−1 e±αx I ν (αx) dx = √ 1F 1 π Γ(μ + 2ν + 1)  2 0 Re μ > 0, Re ν > − 12 

3.

(cf. 6.611 1)

∞

4.

2.

WA 422(6)

% ) * & 

2 α α 1 α √ + xe−αx K 0 (bx) dx = 2 ln −1 −1 α − b2 b b α2 − b2 0   ∞ √ −αx π 1 xe K ± 12 (bx) dx = 2b α + b 0  ∞ −1/2 2 Γ(ν − μ + 1) −iμπ μ e−tz(z −1) K μ (t)tν dt = e Q ν (z) − 1 (ν+1) 0 (z 2 − 1) 2





6.625

Re α > |Im b|]

1

xν (1 − x)μ−1 e±iαx J ν (αx) dx =

(2α)ν Γ(μ) Γ ν + 2 √ π Γ(μ + 2ν + 1)

BU 9(16a), ET II 197(77)a

6.626

Bessel functions, exponentials, and powers



1

λ−1

x

4.

μ−1 ±αx

(1 − x)

0



1

5.

e

 1 ν α Γ (λ + ν) Γ(μ) I ν (αx) dx = 2 Γ(ν + 1) Γ(λ + μ + ν)  1 × 2 F 2 ν + , λ + ν; 2ν + 1, μ + λ + ν; ±2α 2 [Re μ > 0, Re(λ + ν) > 0] ET II 197(78)a

xμ−κ (1 − x)2κ−1 I μ−κ

0

711

 1 1 1 Γ(2κ) xz e− 2 xz dx = √ ez/2 z −κ− 2 M κ,μ (z) 2 π Γ(1 + 2μ)   Re κ − 12 − μ < 0, Re κ > 0 BU 129(14a)



!1 ∞ ! − λ, 0 (2α)λ Γ(μ) 21 2 ! √ 2α x−λ (x − 1)μ−1 e−αx I ν (αx) dx = G 23 ! π 1  −μ, ν − λ,1−ν − λ 0 < Re μ < 2 + Re λ, Re α > 0

 6.



! 1 ∞ ! 0, − λ √ ! 2 2α x−λ (x − 1)μ−1 e−αx K ν (αx) dx = Γ(μ) π(2α)λ G 30 23 ! −μ, ν − λ, −ν − λ 1

 7.

[Re μ > 0, Re α > 0]  ∞ (2α)ν−μ Γ 2 − μ + ν Γ(μ) −ν μ−1 −αx √ x (x − 1) e I ν (αx) dx = π Γ(1 − μ + 2ν) 1 

1 − μ + ν; 1 − μ + 2ν; −2α × 1F 1 2 0 < Re μ < 12 + Re ν, Re α > 0 1

 8.



∞

9.

x−ν (x − 1)μ−1 e−αx K ν (αx) dx =

1

 10.12

1

ET II 208(55)a

ET II 207(49)a

√ 1 1 π Γ(μ)(2α)− 2 μ− 2 e−α W − 12 μ,ν− 12 μ (2α) [Re μ > 0,

∞

ET II 207(50)a

x−μ− 2 (x − 1)μ−1 e−αx K ν (αx) dx =

Re α > 0]

ET II 208(53)a

√ 1 π Γ(μ)(2α)− 2 e−α W −μ,ν (2α)

1



[Re μ > 0, 1

11.3



−1

1 − x2

−1/2



1.11 0

∞

ET II 207(51)a



   −1/2 2 sinh a − a a2 + b2 xe−ax I 1 b 1 − x2 dx = sinh a2 + b2 b [a > 0,

6.626

Re α > 0]

xλ−1 e−αx J μ (bx) J ν (cx) dx =

b > 0]

∞ " Γ(λ + μ + ν + 2m) b μ cν 2−ν−μ α−λ−μ−ν Γ(ν + 1) m! Γ(μ + m + 1) m=0

m 2 c b2 × F −m, −μ − m; ν + 1; 2 − 2 b 4α [Re(λ + μ + ν) > 0, Re (α ± ib ± ic) > 1]

EH II 48(15)

712

Bessel Functions



∞

2. 0

 3.

∞

e

−2αx

ν+μ

J ν (bx) J μ (bx)x

6.627

  Γ ν + μ + 12 bν+μ √ dx = π3  π2 cosν+μ ϕ cos(ν − μ)ϕ  × dϕ ν+μ 2 2 2 α2 + b2 cos2 ϕ 0 (α + b cos ϕ)  Re α > |Im b|, Re(ν + μ) > − 12

e−2αx J 0 (bx) J 1 (bx)x dx =

K



√ b α2 +b2



−E



√ b α2 +b2

WA 427(1)



√ 2πb α2 + b2 #

$

  ∞ 1 b α 1 b − K e−2αx I 0 (bx) I 1 (bx)x dx = E 2 2 2πb α − b α α α 0

WA 427(2)

0

4.

5.10

[Re α > Re b] WA 428(5)



1 a μ−ν−2n−1 ρ ν xν−μ+2n e−zx J μ (αx) J ν (ρx) dx = √ π 2 a  0   ∞ Γ ν +n+q+ 1 " ν − μ + n + 12 q 1 2    ×  Γ μ − ν − n + 12 q=0 q! Γ ν + q + 12

q  1 /ρ z2 dx −2q 2ν+2q 2 √ ρ + ×a x 2 1 − x2 0   1 − x  1 2 2 2 2 μ > ν + 2n, n = 0, 1, . . . , ν > − 21 where 1 = 2 (a + ρ) + z − (a − ρ) + z



∞



∞

6.627 6.628

0



∞

1. 0

2.

3.

πea K ν (a) x−1/2 −x e K ν (x) dx = √ x+a a cos(νπ)

 |arg a| < π,

|Re ν|
−1, 0 < β < WA 424(4) 2 

 1 u xu π B(2ν, 2μ − 2ν + 1) e 2 ixu dx = 22(ν−μ) e 2 (μ−ν)i e 2 (1 − x)2ν−1 xμ−ν J μ−ν 1 M ν,μ (u) 2 Γ(μ − ν + 1) uν+ 2 0

4.8 0

e−x cos β Y ν (x sin β) xμ dx = −

MO 118a

e−x cosh α I ν (x sinh α) xμ dx = Γ(ν + μ + 1) P −ν μ (cosh α)  Re(μ + ν) > −1,

|Im α| < 12 π



WA 423(1)

6.631

Bessel functions, exponentials, and powers



∞

5.

e−x cosh α K ν (x sinh α) xμ dx =

0

713

sin μπ Γ(μ − ν + 1) Q νμ (cosh α) sin(ν + μ)π [Re(μ + 1) > |Re ν|]



∞

6.

e−x cosh α I ν (x)xμ−1 dx =

0

WA 423(2)

μ− 12 ν− 12

(cosh α) Q cos νπ  sin(μ + ν)π π (sinh α)μ− 12 2 [Re(μ + ν) > 0,

Re (cosh α) > 1] WA 424(6)



∞

7.

 e

−x cosh α

μ−1

K ν (x)x

dx =

0

1 2 −μ ν− 12

P (cosh α) π Γ(μ − ν) Γ(μ + ν) 1 2 (sinh α)μ− 2 [Re μ > |Re ν|,

Re (cosh α) > −1] WA 424(7)

 6.6298

∞

x−1/2 e−xα cos ϕ cos ψ J μ (αx sin ϕ) J ν (αx sin ψ) dx

0

 α > 0, 6.631



∞

1.

2

xμ e−αx J ν (bx) dx =

0

bν Γ

  1 (cos ϕ) P −ν = Γ μ + ν + 12 α− 2 P −μ 1 (cos ψ) ν− 12  μ− 2 π 1 π 0 < ϕ < , 0 < ψ < , Re(μ + ν) > − ET II 50(19) 2 2 2 

1

1 1 2ν + 2μ + 2 1 2ν+1 α 2 (μ+ν+1) Γ(ν +

1

1)

1F 1

b2 ν +μ+1 ; ν + 1; − 2 4α

1 2ν + 2μ +

1 2

BU 8(15)



2

2 b b 1 1 M = exp − 1 μ, 2 ν μ 2 8α 4α bα 2 Γ(ν + 1) [Re α > 0, Γ



Re(μ + ν) > −1]

EH II 50(22), ET II 30(14), BU 14(13b)



∞

2.

2

xμ e−αx Y ν (bx) dx



2 1 b ν −μ = −α− 2 μ b−1 sec π exp − 2 8α %  



2 Γ 12 + 12 μ + 12 ν ν −μ b sin π M 12 μ, 12 ν +W × Γ(1 + ν) 2 4α

0

[Re α > 0,  3.

∞

12 0

 4.11 0

∞

μ −αx2

x e

1 1 K ν (bx) dx = α− 2 μ b−1 Γ 2

2

xν+1 e−αx J ν (bx) dx =

bν (2α)ν+1

1+ν +μ 2

2 b exp − 4α

Re μ > |Re ν| − 1,

1 1 2 μ, 2 ν

b > 0]

b2 4α

&

ET II 106(4)

 

2

2 1−ν+μ b b Γ exp W − 12 μ, 12 ν 2 8α 4α [Re α > 0, Re μ > |Re ν| − 1] ET II 132(25)

[Re α > 0,

Re ν > −1] WA 431(4), ET II 29(10)

714

Bessel Functions



b2 [Re α > 0, Re ν > 0] x e J ν (bx) dx = 2 b γ ν, 4α 0    ∞ 2 b2 bν ν +1 π − xν+1 e±iαx J ν (bx) dx = exp ±i (2α)ν+1 2 0  4α α > 0, −1 < Re ν < 12 , 

5.

12

6.

8.

∞

ν−1 −αx2

ν−1 −ν

2

2 

2  2 b b b 1 1 1 1 I − I xe−αx J ν (bx) dx = exp − 3 2 ν− 2 2 ν+ 2 8α 8α 8α 2 8α 0 [Re α > 0, Re ν > −2] ) *  1 n " 2 1 eα − e−α xn+1 e−αx I n (2αx) dx = I r (2α) 4α 0 r=−n 

7.

6.632



√ πb

∞

∞

9.

)

2

x1−n e−αx I n (2αx) dx =

1



∞

10. 0

ET II 30(11)

b>0



ET II 30(12)

ET II 29(9)

[n = 0, 1, . . .] *

ET II 365(8)a

[n = 1, 2, . . .]

ET II 367(20)a

n−1 " 1 eα − e−α I r (2α) 4α r=1−n

 √  2 1 n! e−x x2n+μ+1 J μ 2x z dx = e−z z 2 μ Lμn (z) 2

[n = 0, 1, . . . ;

n + Re μ > −1] BU 135(5)

 6.632

∞

  1   − 1 1 x− 2 exp − x2 + a2 − 2ax cos ϕ 2 x2 + a2 − 2ax cos ϕ 2 K ν (x) dx

0 1

= πa− 2 sec(νπ) P ν− 12 (− cos ϕ) K ν (a)  |arg a| + |Re ϕ| < π, |Re ν| < 12 ET II 368(32) 6.633



∞

1.

λ+1 −αx2

x

e

0

 

2 aβ 1 a + β2 I J p (ax) J p (βx)x dx = 2 exp − p 2 42 22 0   π Re p > −1, |arg | < , a > 0, β > 0 KU 146(16)a, WA 433(1) 4 



 ∞ 1 1 − 3 ν− 1 1 2ν+1 −αx2 2 2 W 12 ν, 12 ν x e J ν (x) Y ν (x) dx = − √ α exp − 2α α 2 π 0  Re α > 0, Re ν > − 21 ET II 347(59) 

2.

3.

  m μ+ν+λ+2 ∞ " 2 Γ m + 12 ν + 12 μ + 12 λ + 1 b2 bμ cν α− − J μ (bx) J ν (cx) dx = ν+μ+1 2 Γ(ν + 1) m=0 m! Γ(m + μ + 1) 4α 

2 c × F −m, −μ − m; ν + 1; 2 b [Re α > 0, Re(μ + ν + λ) > −2, b > 0, c > 0] EH II 49(20)a, ET II 51(24)a

∞

2

e−

x2

6.637

Bessel functions, exponentials, and powers



∞

4.

xe

−αx2

0



1 exp I ν (bx) J ν (cx) dx = 2α

b 2 − c2 4α



Jν

Re ν > −1]

ET II 63(1)

2

xλ−1 e−αx J μ (bx) J ν (bx) dx

 + 12 μ + 12 ν α b =2 Γ(μ + 1) Γ(ν + 1)   μ 1 ν μ ν +μ+λ b2 ν + + , + + 1, ; μ + 1, ν + 1, μ + ν + 1; − × 3F 3 2 2 2 2 2 2 α [Re(ν + λ + μ) > 0, Re α > 0] WA 434, EH II 50(21)

0

−ν−μ−1 − 12 (ν+λ+μ) ν+μ Γ

6.634



[Re α > 0, ∞

5.



bc 2α

715

∞

1

x2

xe− 2a [I ν (x) + I −ν (x)] K ν (x) dx = aea K ν (a)

2λ

[Re a > 0,

−1 < Re ν < 1]

0

ET II 371(49)

6.635



∞

1.

α

x−1 e− x J ν (bx) dx = 2 J ν

√ √



2αb K ν 2αb

0



∞

2.

α

x−1 e− x Y ν (bx) dx = 2 Y ν

√

0



∞

3.

α

x−1 e− x −βx J ν (γx) dx = 2 J ν

0

√



2αb K ν 2αb

[Re α > 0,

b > 0]

ET II 30(15)

[Re α > 0,

b > 0]

ET II 106(5)

# #  12 $  12 $ √  √  Kν 2α β2 + γ2 − β 2α β2 + γ2 + β [Re α > 0,

Re β > 0,

γ > 0] ET II 30(16)

√  ∞ 1 1

1

√  2  1 1 1 − 12 −α x x e J ν (bx) dx = √ Γ ν + 12 D −ν− 12 2− 2 αe 4 πi b− 2 D −ν− 12 2− 2 αe− 4 πi b− 2 πb 0  Re α > 0, b > 0, Re ν > − 12

6.636

ET II 30(17)

6.637



∞



− 12

∞



− 12

1.

β 2 + x2

0

 2. 0

β 2 + x2

 1   exp −α β 2 + x2 2 J ν (γx) dx #  #  $ $ 1 1 1  2 1  2 β α + γ 2 2 − α K 12 ν β α + γ2 2 + α = I 12 ν 2 2 [Re α > 0, Re β > 0, γ > 0, Re ν > −1] ET II 31(20)

 1   exp −α β 2 + x2 2 Y ν (γx) dx #  νπ

$ 1 1  2 K 12 ν β α + γ2 2 + α = − sec 2 # 2 # 

$ νπ

$ 1 1 1  2 1  2 1 2 2 2 2 1 1 K β α +γ I 2ν β α +γ + α + sin −α × π 2ν 2 2 2 [Re α > 0, Re β > 0, γ > 0, |Re ν| < 1] ET II 106(6)

716

Bessel Functions



∞

3.



x2 + β 2

− 12

0

6.641

 1   exp −α x2 + β 2 2 K ν (γx) dx

  νπ

 2  2 1  1  1  1  1 2 2 2 2 K 12 ν β α+ α −γ β α− α −γ = sec K 12 ν 2 2 2 2 [Re α > 0, Re β > 0, Re(γ + β) > 0, |Re ν| < 1] ET II 132(26)

6.64 Combinations of Bessel functions of more complicated arguments, exponentials, and powers 

6.641 6.642 1.

∞√

  xe−αx J ± 14 x2 dx =

0



∞

10

−1 −αx

x

e

0



∞

2.12 0

6.643



∞

1.

2.

3.

∞



∞

μ− 12 −αx

x

e

0



[Re α > 0]

MC MI 44, EH II 91(26)

 

2

2  √  Γ μ + ν + 12 b b exp − α−μ M μ,ν J 2ν 2b x dx = b Γ(2ν + 1) 2α   α Re μ + ν + 12 > 0 

 1

 √  Γ μ + ν + 2 −1 1 b exp xμ− 2 e−αx I 2ν 2b x dx = Γ(2ν + 1)

BU 14(13a), MI 42a



 2

∞

MI 47a

2

2  √  b 1 b α−n−ν−1 Lνn xn+ 2 ν e−αx J ν 2b x dx = n!bν exp − α α

− 12 −αx

 √  Y 2ν b x dx = −

 exp − b2  8α π

[n + ν > −1]

b2 8α



1 + Kν π 1

MO 178a

b2 8α



sin(νπ) I ν  |Re ν| < 2

 12 m− 12



  ∞  √  1 1 Γ(m + 1) 1 1 W − 12 (m+1),− 12 m x 2 m e−αx K m 2 x dx = exp 2α α 2α α 0 0

6.

MI 42

b b2 α−μ M −μ,ν 2α α 0    1 Re μ + ν + 2 > 0 MI 45    

 

 ∞  √  Γ μ + ν + 12 Γ μ − ν + 12 1 b2 b2 exp α−μ W −μ,ν xμ− 2 e−αx K 2ν 2b x dx = 2b 2α  α 0   Re μ + ν + 12 > 0 , (cf. 6.631 3)

4.

5.

  √   √  2 dx = 2 K ν 2 α Y ν 2 α Yν x

 √  √  2 (1,2) −1 −αx dx = 2 H (1,2) x e Hν α Kν α ν x

0





2

2  √ πα α α H∓ 14 − Y ∓ 14 4 4 4

x

e

α

cos(νπ)

MI 44 MI 48a

6.647

Bessel functions of complicated arguments, exponentials, and powers



∞

6.644

e

−βx

J 2ν

717





√  a2 β a2 b 1  Jν 2a x J ν (bx) dx = exp − 2 2 2 2 2 β +b β +b β + b2  Re β > 0, b > 0, Re ν > − 12



0

ET II 58(17)

6.645



∞

1.



x2 − 1

    



 1 1 e−αx J ν β x2 − 1 dx = I 12 ν α2 + β 2 − α K 12 ν α2 + β 2 + α 2 2

− 12

1



∞

2.



x2 − 1

 12 ν



e−αx J ν β x2 − 1 dx =



1

 3.3

MO 179a



− 1 ν− 1 2 ν 2 β α + β 2 2 4 K ν+ 12 α2 + β 2 π MO 179a



  −1/2 −ax  2 cosh a2 + b2 − cosh a 1 − x2 e I 1 b 1 − x2 dx = b −1 1

[a > 0, 6.646 1.

2.

3.7

b > 0]

 √  ν 

exp − α2 + b2 b 2 √ √ e J ν b x − 1 dx = α2 + b2 α + α2 + b2 1 [Re ν > −1] EF 89(52), MO 179 √   1  ν

ν  ∞

exp − α2 − b2 x − 1 2 −αx  2 b √ √ e I ν b x − 1 dx = x+1 α2 − b2 α + α2 − b2 1 [Re ν > −1, α > b] MO 180  12 ν

 ∞ 

x−b Γ(ν + 1)  ν −bs x e e−px K ν a x2 − b2 dx = Γ(−ν, bx) − y ν ebs Γ(−ν, by) ν x+b 2sa b  2 1/2 where x = p − s, y = p + s, s = p − a2 [Re(p + a) > 0, |Re(ν)| < 1] . 

∞

x−1 x+1

 12 ν

−αx

ME 39a

6.647  1.

∞

1

x−λ− 2 (b + x)

λ− 12

e−αx K 2μ

  x(b + x) dx

0

=

  2.

∞

1

1

(a + x)− 2 x− 2 e−x cosh t K ν

   1 21 αb  1 e Γ 2 − λ + μ Γ 12 − λ − μ W λ,μ (z1 ) W λ,μ (z2 ) b

 z1 = 12 b α + α2 − 1 ,

|arg b| < π,

  x(a + x) dx

0

=

Re α > −1,



 z2 = 12 b α − α2 − 1 ET II 377(37) Re λ + |Re μ| < 12



 νπ 1 1 t 1 −t 1 sec e 2 a cosh t K 12 ν ae K 12 ν ae 2 2 4 4 [−1 < Re ν < 1] ET II 377(36)

718

Bessel Functions

 3.

a

11

1

1

=e  6.648 6.649

∞

12 −∞



  x(a − x) dx 1  1  −(a/2) sinh t 2 Γ 2 + λ + μ Γ 2 − λ + μ

xλ− 2 (a − x)−λ− 2 e−x sinh t I 2μ

0

∞

1.

e

x

a + bex aex + b

 K 2ν

0

∞

3.

1  a2 + b2 + 2ab cosh x 2 dx = 2 K ν+ (a) K ν− (b) [Re a > 0,

K μ−ν (2z sinh x) e(ν+μ)x dx =

Re b > 0]

ET II 379(45)

π2 [J ν (z) Y μ (z) − J μ (z) Y ν (z)] 4 sin[(ν − μ)π] [Re z > 0, −1 < Re(ν − μ) < 1]

J ν+μ (2x sinh t) e(ν−μ)t dt = K ν (x) I μ (x)  Re(ν − μ) < 32 , Y ν−μ (2x sinh t) e−(ν+μ)t dt =

0





 1 t 1 −t ae ae M −λ,μ 2 2 a [Γ(2μ + 1)]2  ET II 377(32) Re μ > |Re λ| − 12

MO 44 ∞

2.



M λ,μ



0



6.648

∞

4.

K 0 (2z sinh x) e−2νx dx = −

0

π 4

Re(ν + μ) > −1,

x>0



EH II 97(68)

1 {I μ (x) K ν (x) − cos[(ν − μ)π] I ν (x) K μ (x)} sin[π(μ − ν)]  |Re(ν − μ)| < 1, Re(ν + μ) > − 12 , x > 0 EH II 97(73) # $ ∂ Y ν (z) ∂ J ν (z) J ν (z) − Y ν (z) ∂ν ∂ν

6.65 Combinations of Bessel and exponential functions of more complicated arguments and powers 6.651



∞

1. 0

1

1

2

xλ+ 2 e− 4 α

x2

Iμ

1

3 1 = √ 2λ+1 b−λ− 2 2π



J ν (bx) dx 

2 ! b !! 1 − μ, 1 + μ 21 G 23 2α2 ! h, 12 , k

2 2 4α x

h=

 π |arg α| < , 4  2. 0

∞

b > 0,

3 4

+ 12 λ + 12 ν,

k=

3 4

− 23 − Re(2μ + ν) < Re λ < 0

+ 1 λ − 12 ν  2 ET II 68(8)

  1 1 2 2 xλ+ 2 e− 4 α x K μ 14 α2 x2 J ν (bx) dx  ! 

π λ+1 −λ− 3 12 b2 !! 1 − μ, 1 + μ 2 2 b = G 23 2 2α2 ! h, 12 , k 

h=

|arg α|
− 23

3

4

+ 12 λ − 12 ν ET II 69(15)

6.652

Bessel and exponential functions and powers



∞

3.

2

1

x2μ−ν+1 e− 4 αx I μ

0



∞

4.

2

1

x2μ+ν+1 e− 4 α

1

x2

Kμ =



∞

2

1

x2μ+ν+1 e− 2 αx I μ

1

2 2 4α x



J ν (bx) dx

√ μ −2μ−2ν−2 ν Γ (1 + 2μ + ν)   π2 α b F 1 1 1 + 2μ + ν; μ + ν + Γ μ + ν + 32  |arg α| < 14 π, Re ν > −1, Re(2μ + ν) > −1, b > 0

1

2 2 αx



b2 3 ;− 2 2 2α



ET II 69(13)

K ν (bx) dx

2

2 1   b b 2μ− 2 −μ− 32 − 12 μ− 12 ν− 14 1 W k,m α Γ(2μ + ν + 1) Γ μ + 2 exp = √ b π 8α 4α 1 2k = −3μ − ν − , 2m = μ + ν + 12 2  1 Re α > 0, Re μ > − 2 , Re (2μ + ν) > −1 ET II 146(53)

0





J ν (bx) dx  

bν−2μ−1 1 b2 1 1 μ−ν+ 12 − 12   1F 1 +μ + μ; − μ + ν; − =2 (πα) Γ 2 2 2 2α Γ 12 − μ + ν  Re α > 0, b > 0, Re ν > 2 Re μ + 12 > − 21 ET II 68(6) 2 4 αx

0

5.

719

∞

6. 0

1

2

xe− 4 αx J 12 ν





  1  2 bc2 αc2 2 2 −2 1 J J bx (cx) dx = 2 α + b exp − ν 4 2ν α2 + b2 α2 + b2 [c > 0, Re α > |Im b|, Re ν > −1]

1

ET II 56(2)



∞

7. 0



∞

8.

1

2

xe− 4 αx I 12 ν

2

1

x1−ν e− 4 α

x2

1

 2 J ν (bx) dx = 4 αx

Iν

0



∞

9.

1

2

x−ν−1 e− 4 α

x2

1

2 2 4α x

I ν+1



− 12

2 1 b πα b−1 exp − 2 2α [Re α > 0, 

J ν (bx) dx =

1

0

2 2 4α x



b > 0,

Re ν > −1] ET II 67(3)



 2 bν−1 b2 b exp − 2 D −2ν π α 4α α  |arg α| < 14 π, b > 0, 

J ν (bx) dx =

Re ν > − 12



ET II 67(1)





2 ν b b2 b exp − 2 D −2ν−3 π 4α α  |arg α| < 14 π, Re ν > −1,

b>0



ET II 67(2)

 6.652 0

∞

2ν −

x e



x2 8



+αx

Iν

x2 8



2

α   Γ(4ν + 1) e 2 dx = 4ν W − 32 ν, 12 ν α2 ν+1 2 Γ(ν + 1) α    Re ν + 14 > 0

MI 45

720

Bessel Functions

6.653  1.

∞

0

6.653

    1  2 ab dx 1 a + b2 I ν = 2 I ν (a) K ν (b) exp − x − 2 2x x x = 2 K ν (a) I ν (b) [Re ν > −1]



∞

2. 0

6.655

[0 < b < a] WA 482(2)a, EH II 53(37), WA 482(3)a

  zw dx  1  2 1 z + w2 K ν = 2 K ν (z) K ν (w) exp − x − 2 2x x x  |arg z| < π, |arg w| < π, arg(z + w) < 14 π

WA 483(1), EH II 53(36)

 √  √  1 β2 dx = 4πα− 2 K 2ν β α x e Kν ME 39 8x 0  

∞  2 2 − 1 α x α β √ Jν J ν (γx) dx = γ −1 e−βγ J 2ν (2α γ) x β 2 + x2 2 exp − 2 2 2 2 β +x β +x 0  Re β > 0, γ > 0, Re ν > − 12

 6.654

[0 < a < b]

∞

2

− 12 − β 8x −αx

ET II 58(14)

6.656  1.

∞

  1 e−(ξ−z) cosh t J 2ν 2(zξ) 2 sinh t dt = I ν (z) K ν (ξ)

0



 Re ν > − 12 ,

Re(ξ − z) > 0

EH II 98(78)

∞

2. 0

  1 1 e−(ξ+z) cosh t K 2ν 2(zξ) 2 sinh t dt = K ν (z) K ν (ξ) sec(νπ) 2  |Re ν| < 12 ,



2 1 1 Re z 2 + ξ 2 ≥ 0 EH II 98(79)

6.66 Combinations of Bessel, hyperbolic, and exponential functions Bessel and hyperbolic functions 6.661



1. 0

∞

π cosec sinh(ax) K ν (bx) dx = 2

 νπ  2

   sin ν arcsin ab √ b 2 − a2 [Re b > |Re a|,

  ∞ π cos ν arcsin ab νπ

cosh(ax) K ν (bx) dx =  0 2 b2 − a2 cos 2

 2.

|Re ν| < 2] ET II 133(32)



[Re b > |Re a|,

|Re ν| < 1] ET II 134(33)

6.662

Notation:   1  1 = (b + c)2 + a2 − (b − c)2 + a2 , 2

2 =

  1  (b + c)2 + a2 + (b − c)2 + a2 2

6.663

Bessel, hyperbolic, and exponential functions

 1.

∞

10 0

K(k) cosh(βx) K 0 (αx) J 0 (γx) dx = √ u+v

721

# $ 1 2 2 2 2 2 2 2 2 2 u= (α + β + γ ) − 4α β + α − β − γ 2 # $ 1 2 2 2 2 2 2 2 2 2 (α + β + γ ) − 4α β − α + β + γ v= 2 k 2 = v(u + v)−1

[Re α > |Re β|,

γ > 0]

ET II 15(23)

alternatively, with a = γ, b = β, c = α,  ∞ K(k) cosh(bx) K 0 (cx) J 0 (ax) dx =  2 2 − 21 0 k2 = 

∞

2.10 0

22 − c2 , 22 − 21

[Re c > |Re b|,

a > 0]

  K(k) snu dn u sinh(βx) K 1 (αx) J 0 (γx) dx = a−1 u E(k) − K(k) E(u) + cn u # $−1  12   2 cn 2 u = 2γ 2 α2 + β 2 + γ 2 − 4α2 β 2 − α2 + β 2 + γ 2 # − 12 $    2 2 1 1 − α2 − β 2 − γ 2 α + β 2 + γ 2 − 4α2 β 2 k2 = 2 [Re α > |Re β|, γ > 0] ET II 15(24)

alternatively, with a = γ, b = β, c = α,    ∞ K(k) snu dn u −1 u E(k) − K(k) E(u) + sinh(bx) K 1 (cx) J 0 (ax) dx = c cn u 0 2 2 2 − c a  , k 2 = 22 [Re c > |Re b|, cn 2 u = 2  2 − c2 2 − 21 6.663  1.

∞

K ν±μ (2z cosh t) cosh [(μ ∓ ν) t] dt =

0

 2. 0

 0

4. 0

Y μ+ν (2z cosh t) cosh[(μ − ν)t] dt =

WA 484(1), EH II 54(39)

π [J μ (z) J ν (z) − Y μ (z) Y ν (z)] 4 [z > 0]

∞

3. 

1 K μ (z) K ν (z) 2 [Re z > 0]

∞

J μ+ν (2z cosh t) cosh[(μ − ν)t] dt = −

EH II 96(64)

π [J μ (z) Y ν (z) + J ν (z) Y μ (z)] 4 [z > 0]

∞

a > 0]

1 J μ+ν (2z sinh t) cosh[(μ − ν)t] dt = [I ν (z) K μ (z) + I μ (z) K ν (z)]  2 Re(ν + μ) > −1, |Re(μ − ν)| < 32 ,

EH II 97(65)

z>0



EH II 97(71)

722

Bessel Functions



∞

5. 0

6.664



∞

1.

1 J μ+ν (2z sinh t) sinh[(μ − ν)t] dt = [I ν (z) K μ (z) − I μ (z) K ν (z)]  2 Re(ν + μ) > −1, |Re(μ − ν)| < 32 ,

J 0 (2z sinh t) sinh(2νt) dt =

0



∞

6.664

sin(νπ) 2 [K ν (z)] π

 |Re ν| < 34 ,

z>0

z>0





EH II 97(72)

EH II 97(69)

 cos(νπ) 2 [K ν (z)] |Re ν| < 34 , z > 0 EH II 97(70) π 0    ∞ 1 1 ∂ K ν (z) ∂ I ν (z) 2 I ν (z) − K ν (z) − cos(νπ) [K ν (z)] Y 0 (2z sinh t) sinh(2νt) dt = π ∂ν ∂ν π 0  |Re ν| < 34 , z > 0 EH II 97(75)  ∞ 2 ' ( π Jν2 (z) + Nν2 (z) K 0 (2z sinh t) cosh 2νt dt = [Re z > 0] MO 44 8 0

   ∞ 1 1 1 Γ +μ−ν Γ − μ − ν W ν,μ (iz) W ν,μ (−iz) K 2μ (z sinh 2t) coth2ν t dt = 4z 2 2 0  π |arg z| ≤ , |Re μ| + Re ν < 12 2

2. 3.

4. 5.

Y 0 (2z sinh t) cosh(2νt) dt = −

MO 119



∞

6.

cosh(2μx) K 2ν (2a cosh x) dx =

0



∞

6.665

1 K μ+ν (a) K μ−ν (a) 2

sech x cosh(2λx) I 2μ (a sech x) dx =

0

[Re a > 0]     Γ 12 + λ + μ Γ 12 − λ + μ

ET II 378(42)

M λ,μ (a) M −λ,μ (a) 2 2a [Γ(2μ  + 1)] ET II 378(43) |Re λ| − Re μ < 12

Bessel, hyperbolic, and algebraic functions  ∞ ∞ 2" xν+1 sinh(αx) cosech(πx) J ν (βx) dx = (−1)n−1 nν+1 sin(nα) K ν (nβ) 6.666 π n=1 0 [|Re α| < π, Re ν > −1] ET II 41(3), WA 469(12)

6.667 1.

2.

3

 √

  a2 − x2 sinh t I 2ν (x) 1 t 1 −t π √ ae I ν ae dx = I ν 2 2 a2 − x2 0 2 Re ν > − 12 ET II 365(10) √   a          cosh a2 − x2 sinh t K 2ν (x) π2 √ cosec(νπ) I −ν aet I −ν ae−t − I ν aet I ν ae−t dx = 4 a2 − x2 0  |Re ν| < 12 ET II 367(25) 

a

cosh

6.669

Bessel, hyperbolic, and exponential functions

723

Exponential, hyperbolic, and Bessel functions 6.668

Notation:   1  1 = (b + c)2 + a2 − (b − c)2 + a2 , 2 

1.10

2.12

6.669

∞

1

2 =

  1  (b + c)2 + a2 + (b − c)2 + a2 2 1

−1

e−αx sinh(βx) J 0 (γx) dx = (αβ) 2 r1−1 r2−1 (r2 + r1 ) 2 (r2 − r1 ) 2 0   r1 = γ 2 + (β − α)2 , r2 = γ 2 + (β + α)2 , [Re α > |Re β|, γ > 0] alternatively, with a = γ, b = β, c = α,  ∞ 1 e−cx sinh(bx) J 0 (ax) dx = 2 2  0 2 − 1 [Re c > |Re b|, a > 0]  ∞ 1 1 −1 e−αx cosh(βx) J 0 (γx) dx = (αβ) 2 r1−1 r2−1 (r2 + r1 ) 2 (r2 − r1 ) 2 0   r1 = γ 2 + (β − α)2 , r2 = γ 2 + (β + α)2 , [Re α > |Re β|, γ > 0] alternatively, with a = γ, b = β, c = α,  ∞ 2 e−cx cosh(bx) J 0 (ax) dx = 2 2 − 21 0 [Re c > |Re b|, a > 0] 

∞

1. 0



∞

2. 0

 3.12 0

ET II 12(52)

ET II 12(54)

  

2λ   1 Γ 12 − λ + μ 1 −β cosh x x M −λ,μ α2 + β 2 2 − β coth e J 2μ (α sinh x) dx = 2 α Γ(2μ + 1)   1 × W λ,μ α2 + β 2 2 + β  Re β > |Re α|, Re(μ − λ) > − 12 BU 86(5b)a, ET II 363(34) 

2λ 1 x coth e−β cosh x Y 2μ (α sinh x) dx 2  



sec[(μ + λ)π] =− α2 + β 2 + β W −λ,μ α2 + β 2 − β W λ,μ α    



tan[(μ + λ)π] Γ 12 − λ + μ W λ,μ α2 + β 2 + β M −λ,μ α2 + β 2 − β − α Γ(2μ + 1)  Re β > |Re α|, Re λ < 12 − |Re μ| ET II 363(35)



2ν √ 1 1 x e− 2 (a1 +a2 )t cosh x coth K 2μ (t a1 a2 sinh x) dx 2     Γ 12 + μ − ν Γ 12 − μ − ν W ν,μ (a1 t) W ν,μ (a2 t) = √ 2t a1 a2    √ √ 2 1 ± 2μ Re ν < Re , Re t ( a1 + a2 ) > 0 BU 85(4a) 2

∞

724

Bessel Functions

 4.

∞

12

e

− 12 (a1 +a2 )t cosh x

0

6.

   x 2ν Γ 12 + μ − ν √ W ν,μ (a1 t) M ν,μ (a2 t) coth I 2μ (t a1 a2 sinh x)dx = √ 2  1  t a1 a2 Γ(1 + 2μ) Re 2 + μ − ν > 0, Re μ > 0, a1 > a2 BU 86(5c)

    

√ Γ 12 + μ + ν Γ 12 + μ − ν xy ds = M ν,μ (x) M −ν,μ (y) √ cosh s cosh s xy [Γ(1 + 2μ)]2 −∞    Re ±ν + 12 + μ > 0 BU 83(3a)a 1  1  

√  ∞ Γ 2 +μ+ν Γ 2 +μ−ν x+y xy ds = e2νs− 2 tanh s J 2μ M ν,μ (x) M ν,μ (y) √ 2 cosh s cosh s xy [Γ(1 + 2μ)] −∞    Re ∓ν + 12 + μ > 0 BU 84(3b)a 

5.

6.671

∞

e2νs−

x−y 2

tanh s

I 2μ

6.67–6.68 Combinations of Bessel and trigonometric functions 6.671



∞

1. 0



∞

2. 0

  sin ν arcsin ab √ J ν (ax) sin bx dx = a2 − b 2

[b < a]

= ∞ or 0

[b = a]

aν cos νπ = √ √2  ν b 2 − a2 b + b 2 − a2

[b > a]



J ν (ax) cos bx dx =

cos ν arcsin √ a2 − b 2

b a



[b = a] ν

−a = √  2 b − a2 b

3. 0

∞

WA 444(4)

[b < a]

= ∞ or 0



[Re ν > −2]

sin νπ √2 + b2

− a2

ν

[b > a] [Re ν > −1]

Y ν (ax) sin(bx) dx 

 νπ   1 b 2 2 −2 a −b = cot sin ν arcsin 2 a νπ   1 1 2 2 −2 b −a = cosec 2# 2 $     1 ν  1 −ν  × a−ν cos(νπ) b − b2 − a2 2 − aν b − b2 − a2 2

WA 444(5)

[0 < b < a, |Re ν| < 2]

[0 < a < b,

|Re ν| < 2] ET I 103(33)

6.671

Bessel and trigonometric functions



∞

4. 0



Y ν (ax) cos(bx) dx   

 tan νπ b 2 cos ν arcsin = 1 a (a2 − b2 ) 2 # νπ   1    1 ν − = − sin b2 − a2 2 a−ν b − b2 − a2 2 + cot(νπ) 2 $    1 −ν ν +a b − b2 − a2 2 cosec(νπ)

=

∞

6.

∞

7. 0

8.

K ν (ax) cos(bx) dx

J 0 (ax) sin(bx) dx = 0

[0 < b < a]

1 = √ 2 b − a2

[0 < a < b] ET I 99(1)

∞

J 0 (ax) cos(bx) dx = √

1 a2 − b 2

[0 < b < a]

=∞

[a = b]

=0

[0 < a < b]

 ∞ 1 b J 2n+1 (ax) sin(bx) dx = (−1)n √ T 2n+1 2 − b2 a a 0

ET I 43(1)



=0

 ∞ 1 b J 2n (ax) cos(bx) dx = (−1)n √ T 2n 2 − b2 a a 0

[0 < b < a] [0 < a < b] ET I 99(2)

 10.

|Re ν| < 1]

# νπ  ν  ν $ − 1  2 1 1 1 −ν πa cosec a2 + b 2 2 b + a2 2 + b − b 2 + a2 2 − b 4 2 [Re a > 0, b > 0, |Re ν| < 2, ν = 0] ET I 105(48)

0

9.

[0 < a < b,

$   νπ #  2  2  1  1 ν  1 −ν π 2 2 −2 −ν 2 2 ν 2 2 b +a a = b+ b +a sec +a b+ b +a 4 2 [Re a > 0, b > 0, |Re ν| < 1] ET I 49(40)

0



|Re ν| < 1]

K ν (ax) sin(bx) dx

0



[0 < b < a,

ET I 47(29) ∞

5.



725

=0

[0 < b < a] [0 < a < b] ET I 43(2)

726

Bessel Functions



∞

11. 0



  2 arcsin ab Y 0 (ax) sin(bx) dx = √ π a2 − b 2 ) *  b2 1 2 b − = √ ln −1 π b 2 − a2 a a2 Y 0 (ax) cos(bx) dx = 0

0

∞

0

[0 < a < b]

1 K 0 (βx) sin αx dx =  ln α2 + β 2



α + β



ET I 47(28)



α2 +1 β2

[α > 0, ∞

14.8 0

6.672  1.

[0 < a < b]

[0 < b < a]

1 = −√ 2 b − a2

13. 

[0 < b < a]

ET I 103(31) ∞

12.



6.672

∞

π K 0 (βx) cos αx dx =  2 α2 + β 2

β > 0]

WA 425(11)a, MO 48

[α > 0]

WA 425(10)a, MO 48

J ν (ax) J ν (bx) sin(cx) dx

0

=0



b +a −c 1 = √ P ν− 12 2 ab 

2ab2 b + a2 − c2 cos(νπ) =− √ Q ν− 12 − 2ab π ab 

∞

2.

2

2

J ν (x) J −ν (x) cos(bx) dx =

0

2

1 P 1 2 ν− 2

1 2 b −1 2

[Re ν > −1,

0 < c < b − a,

[Re ν > −1,

b − a < c < b + a,

[Re ν > −1,

b + a < c,

0

∞

4. 0

 5.

ET I 102(27)

[0 < b < 2] [2 < b]

π K ν (ax) K ν (bx) cos(cx) dx = √ sec(νπ) P ν− 12 4 ab 2

1 K ν (ax) I ν (bx) cos(cx) dx = √ Q ν− 12 2 ab

 2  a + b2 + c2 (2ab)−1  Re(a + b) > 0, c > 0,

|Re ν|
|Re b|,

c > 0,

Re ν > − 12



ET I 49(47) ∞

2

  1 P 1 1 − 2a2 2 ν− 2   1 = cos(νπ) Q ν− 12 2a2 − 1 π

sin(2ax) [J ν (x)] dx = 0

0 < a < b]

ET I 46(21) ∞

3.



0 < a < b]



=0 

0 < a < b]

[0 < a < 1, [a > 1,

Re ν > −1]

Re ν > −1] ET II 343(30)

6.673

Bessel and trigonometric functions



∞

6.

  1 Q 1 1 − 2a2 π ν− 2   1 = − sin(νπ) Q ν− 12 2a2 − 1 π

cos(2ax) [J ν (x)]2 dx =

0



∞

7.

sin(2ax) J 0 (x) Y 0 (x) dx = 0

0

K

=− 

∞

8. 0



1−a

ET II 344(32)

[0 < a < 1]

1  −2 2

[a > 1]

πa

1 K 0 (ax) I 0 (bx) cos(cx) dx =  K c2 + (a + b)2

 0 < a < 1, Re ν > − 12  a > 1, Re ν > − 12

%

& √ 2 ab  c2 + (a + b)2 [Re a > |Re b|,

∞

9.

1 K(a) π

 1 1 =− K πa a

cos(2ax) J 0 (x) Y 0 (x) dx = −

0





727

c > 0]

ET II 348(60)

ET I 49(46)

[0 < a < 1] [a > 1] ET II 348(61)

∞

10.

2



1 K 1 − a2 π   1 2 K 1− 2 = πa a

cos(2ax) [Y 0 (x)] dx = 0



[0 < a < 1] [a > 1] ET II 348(62)

6.673



∞

1. 0

 νπ

νπ  − Y ν (ax) sin sin(bx) dx J ν (ax) cos 2 2 =0 =

 2. 0

2aν

1 √ b 2 − a2

# $  2  1 ν   1 ν  2 2 2 2 2 b+ b −a + b− b −a

[0 < b < a,

|Re ν| < 2]

[0 < a < b,

|Re ν| < 2] ET I 104(39)

∞

 νπ

νπ  + J ν (ax) sin cos(bx) dx Y ν (ax) cos 2 2 =0 =−

2aν

1 √ b 2 − a2

# $  2  1 ν   1 ν  2 2 2 2 2 b+ b −a + b− b −a

[0 < b < a,

|Re ν| < 1]

[0 < a < b,

|Re ν| < 1] ET I 48(32)

 3. 0

π/2

[cos x I 0 (a cos x) + I 1 (a cos x)] dx =

ea − 1 a

728

6.674

Bessel Functions



a

1.

sin(a − x) J ν (x) dx = a J ν+1 (a) − 2ν

0



(−1)n J ν+2n+2 (a)

n=0

[Re ν > −1] a

2.

cos(a − x) J ν (x) dx = a J ν (a) − 2ν

0



∞ "

6.674

∞ "

(−1)n J ν+2n+1 (a)

n=0

a

3.

)

[Re ν > −1]

sin(a − x) J 2n (x) dx = a J 2n+1 (a) + (−1)n 2n cos a − J 0 (a) − 2

0



a

)

cos(a − x) J 2n (x) dx = a J 2n (a) − (−1)n 2n sin a − 2

n−1 "

(−1)m J 2m (a) ET II 334(10)

*

m=0

ET II 335(21) [n = 0, 1, 2, . . .] ) * n a " n m sin(a − x) J 2n+1 (x) dx = a J 2n+2 (a) + (−1) (2n + 1) sin a − 2 (−1) J 2m+1 (a) m=0

a

6.

[n = 0, 1, 2, . . .] ) cos(a − x) J 2n+1 (x) dx = a J 2n+1 (a) + (−1)n (2n + 1) cos a − J 0 (a) − 2

0



*

(−1)m J 2m+1 (a)

0



n "

[n = 0, 1, 2, . . .]

0

5.

ET II 336(23)

m=1

4.



ET II 334(12)

ET II 334(11) n "

*

(−1)m J 2m (a)

m=1

[n = 0, 1, 2, . . .] z

7.

ET II 336(22)

sin(z − x) J 0 (x) dx = z J 1 (z)

WA 415(2)

cos(z − x) J 0 (x) dx = z J 0 (z)

WA 415(1)

0

 8.

z

0

6.675



∞

1. 0





2 

2

2

2  √   √  νπ νπ a π a a a a 1 1 1 1 − J − sin − J cos J ν a x sin(bx) dx = 3 ν− 2 ν+ 2 2 2 8b 4 8b 8b 4 8b 4b 2 [a > 0, b > 0, Re ν > −4] ET I 110(23)

∞

2.

 √  J ν a x cos(bx) dx

0

=−

 3. 0

∞

√  a π 3

4b 2

sin

νπ a2 − 8b 4



 √  1 J 0 a x sin(bx) dx = cos b

a2 4b





2 

2  a νπ a2 a J 12 ν− 12 + cos J 12 ν+ 12 − 8b 8b 4 8b [a > 0, b > 0, Re ν > −2] ET I 53(22)a

[a > 0,

b > 0]

ET I 110(22)

6.677

Bessel and trigonometric functions



∞

4. 0

6.676



∞

1. 0



∞

2. 0

 3. 4.

a2 4b

 [a > 0,

 √   √  1 J ν a x J ν b x sin(cx) dx = J ν c

 √   √  1 J ν a x J ν b x cos(cx) dx = J ν c



ab 2c

ab 2c

∞



cos



sin

b > 0]

 a2 + b 2 νπ − 4c 2 [a > 0, b > 0,

∞

5. 0

2

ET I 53(21)

c > 0,

νπ a2 + b 2 − 4c 2 [a > 0, b > 0,

c > 0,

[Re a > 0,

b > 0]

ET I 111(31)

b > 0]

ET I 54(29)

 √ Re a > 0, b > 0  ∞ √

√

a

a  π2  1 1 H0 −Y0 K0 axe 4 πi K 0 axe− 4 πi cos(bx) dx = 8b 2b 2b 0 [Re a > 0, b > 0]

6.677



∞

1.



J 0 b x2 − a2 sin(cx) dx = 0

 √  cos a c2 − b2 √ = c2 − b 2

a

√  ∞ 

exp −a b 2 − c2 √ J 0 b x2 − a2 cos(cx) dx = 2 2 a b √− c  − sin a c2 − b2 √ = c2 − b 2

2.

 3.6 0

∞

 

cos z a2 − β 2 2 2 J 0 a x + z cos βx dx =  a2 − β 2 =0

ET I 54(30)

ET I 54(31)

[0 < c < b] [0 < b < c] ET I 113(47)





Re ν > −1] ET I 54(27)



a

√  √  1 K0 ax Y 0 ax cos(bx) dx = − K 0 2b 2b

Re ν > −2] ET I 111(29)a



 √   √  a 1 K0 [Re a > 0, J 0 a x K 0 a x sin(bx) dx = 2b 2b 0  ∞ a  √  √  π  a

I0 − L0 J0 ax K 0 ax cos(bx) dx = 4b 2b 2b 0 

6.

 √  1 J 0 a x cos(bx) dx = sin b

729

[0 < c < b] [0 < b < c] ET I 57(48)a

[0 < β < a,

z > 0]

[0 < a < β,

z > 0] MO 47a

730

Bessel Functions



∞

4. 0



6.678





 1 Y 0 a x2 + z 2 cos βx dx =  sin z a2 − β 2 a2 − β 2

 1 exp −z β 2 − a2 = − β 2 − a2

[0 < β < a,

z > 0]

[0 < a < β,

z > 0] MO 47a

∞

5. 0

  

 π K 0 a x2 + β 2 cos(γx) dx =  exp −β a2 + γ 2 2 a2 + γ 2 [Re a > 0, Re β > 0,

γ > 0] ET I 56(43)

 √  a 

sin a b2 + c2 √ J 0 b a2 − x2 cos(cx) dx = [b > 0] b 2 + c2 0  √   ∞ 

cosh a b2 − c2 √ J 0 b x2 − a2 cos(cx) dx = [0 < c < b, b 2 − c2 0 

6. 7.

=0

8.

a > 0] ET I 57(49)

γ>0



 

i exp −iβ α2 + γ 2 (2)  H 0 α β 2 − x2 cos(γx) dx = α2 + γ 2   −π < arg β 2 − x2 ≤ 0,

α > 0,

γ>0

∞

0

 6.678

∞

0



∞

1. 0



∞

0



   √  π  √  1 π sin K 0 2 x + Y 0 2 x sin(bx) dx = 2 2b b

 ET I 59(59)



[b > 0]

ET I 58(58)

ET I 111(34)

 x  sin(bx) dx = −i [I ν−ib (a) K ν+ib (a) − I ν+ib (a) K ν−ib (a)] J 2ν 2b sinh 2 [a > 0, b > 0, Re ν > −1] ET I 115(59)

2.

4.

[0 < b < c,

α > 0,

9.

3.

a > 0]



 ∞ 

exp iβ α2 + γ 2 (1)  H 0 α β 2 − x2 cos(γx) dx = −i α2 + γ 2 0   π > arg β 2 − x2 ≥ 0, 

6.679

MO 48a, ET I 57(47)

 x  cos(bx) dx = I ν−ib (a) K ν+ib (a) + I ν+ib (a) K ν−ib (a) J 2ν 2a sinh 2  a > 0, b > 0, Re ν > − 12 ET I 59(64)

 x  π cos(bx) dx = − [J ν+ib (a) Y ν−ib (a) + J ν−ib (a) Y ν+ib (a)] J 2ν 2a cosh 2 2 0  ∞  x  2 sin(bx) dx = sinh(πb) [K ib (a)]2 J 0 2a sinh 2 π 0 ∞

[a > 0,

b > 0]

ET I 59(63)

ET I 115(58)

6.681

Bessel and trigonometric functions



∞

5. 0



∞

6. 0



∞

7. 0

731

 x  cos(bx) dx = [I ib (a) + I −ib (a)] K ib (a) J 0 2a sinh 2 

Y 0 2a sinh

x  2

[a > 0, cos(bx) dx = −

b > 0]

ET I 59(62)

b > 0]

ET I 59(65)

2 2 cosh(πb) [K ib (a)] π

[a > 0,  x   2  π 2 2 cos(bx) dx = [J ib (a)] + [Y ib (a)] K 0 2a sinh 2 4

[Re a > 0, 6.681



1.

π 2

π J ν+μ (a) J ν−μ (a) 2

cos(2μx) J 2ν (2a cos x) dx =

0

 2.

π 2

cos(2μx) Y 2ν (2a cos x) dx =

0

 3.

π 2

cos(2μx) I 2ν (2a cos x) dx =

0

 4.

π 2

0



π

5. 0



π

6. 0

 7.

π 2

0



π

8.

cos(νx) K ν (2a cos x) dx =

b > 0]

 Re ν > − 12

ET II 361(23)

π [cot(2νπ) J ν+μ (a) J ν−μ (a) − cosec(2νπ) J μ−ν (a) J −μ−ν (a)] 2  ET II 361(24) |Re ν| < 12 π I ν−μ (a) I ν+μ (a) 2

π I 0 (a) K ν (a) 2

 Re ν > − 12

ET I 59(61)

[Re ν < 1]

WA 484(3)

J 0 (2z cos x) cos 2nx dx = (−1)n πJn2 (z).

MO 45

J 0 (2z sin x) cos 2nx dx = πJn2 (z). cos(2nx) Y 0 (2a sin x) dx =

ET I 59(66)

WA 43(3), MO 45

π J n (a) Y n (a) 2

[n = 0, 1, 2, . . .]

ET II 360(16)

sin(2μx) J 2ν (2a sin x) dx = π sin(μπ) J ν−μ (a) J ν+μ (a)

0



[Re ν > −1] π

9. 0

 10.

π 2

cos(2μx) J 2ν (2a sin x) dx = π cos(μπ) J ν−μ (a) J ν+μ (a)  Re ν > − 12 J ν+μ (2z cos x) cos[(ν − μ)x] dx =

0

 11. 0

ET II 360(13)

ET II 360(14)

π J ν (z) J μ (z) 2 [Re(ν + μ) > −1]

π 2

cos[(μ − ν)x] I μ+ν (2a cos x) dx =

π I μ (a) I ν (a) 2

MO 42

[Re(μ + ν) > −1] WA 484(2), ET II 378(39)

732

Bessel Functions

 12.12

π 2

cos[(μ − ν)x] K μ+ν (2a cos x) dx =

0

 13.8

6.682

π cosec[(μ + ν)π] [I −μ (a) I −ν (a) − I μ (a) I ν (a)] 2 [|Re(μ + ν)| < 1]

π 2

K ν−m (2a cos x) cos[(m + ν)x] dx = (−1)m

0

ET II 378(40)

π I m (a) K ν (a) 2 [|Re(ν − m)| < 1]

6.682 1.7



π 2

WA 485(4)



π J ν (x) 2x 0 [ν may be zero, a natural number, one half, or a natural number plus one half; x > 0]

 2.

π 2

0

6.683  1.

π 2

0

 2.

π 2

0



1

J ν− 12 (x sin t) sinν+ 2 t dt =



z

√ 1 z −ν Jν2 J ν (z sin x) sinν x cos2ν x dx = 2ν−1 π Γ ν + 2 2  Re ν > − 12

3. 0

4.

π 2

J μ (z sin θ) (sin θ)

1−μ

2ν+1

(cos θ)

dθ =

J μ (z sin θ) (sin θ)

0

Re μ > −1]

(see also 6.513 6)

π 2

WA 410(1)

dθ =

μ+1

(cos θ)

J μ (a sin θ) (sin θ)

2+1

WA 414(1)

s μ+ν,ν−μ+1 (z) 2μ−1 z ν+1 Γ(μ) WA 407(2)

Hμ− 12 (z)  2z π

1−μ

0



k=0

[Re ν > −1] π 2

5.

6.

Re μ > −1]

WA 407(4)

∞     1" J ν z cos2 x J μ z sin2 x sin x cos x dx = (−1)k J ν+μ+2k+1 (z) z

0



[Re ν > Re μ > −1] 

z1ν z2μ J ν+μ+1 z12 + z22  J ν (z1 sin x) J μ (z2 cos x) sinν+1 x cosμ+1 x dx = ν+μ+1 (z12 + z22 )

[Re ν > −1, 

MO 42a

 μ−ν Γ 2 2

 J μ (z) J ν (z sin x) I μ (z cos x) tanν+1 x dx = μ+ν Γ +1 2 z ν

[Re ν > −1, π 2

MO 42a

WA 407(3)

dθ = 2 Γ( + 1)a−−1 J +μ+1 (a) [Re  > −1,

Re μ > −1] WA 406(1),

EH II 46(5)

6.686

Bessel and trigonometric functions



π 2

ν

2ν

J ν (2z sin θ) (sin θ) (cos θ)

7. 0

733

dθ

    ∞ 1 " (−1)m z ν+2m Γ ν + m + 12 Γ ν + 12 = 2 m=0 m! Γ(ν + m + 1) Γ(2ν + m + 1)   √ 1 2 = z −ν π Γ ν + 12 [J ν (z)] 2



Re ν > − 21



EH II 47(10)

 8.

π 2

ν−1

z J ν (z sin θ) (sin θ)ν+1 (cos θ)−2ν dθ = 2−ν √ Γ π

1 − ν sin z 2  −1 < Re ν < 12 EH II 68(39)  1 Γ 2 + ν J 2ν+ 12 (z)     J ν z sin2 θ J ν z cos2 θ (sin θ)2ν+1 (cos θ)2ν+1 dθ = 2ν+ 3 √ 2 Γ(ν + 1) 2 z  1 WA 409(1) Re ν > − 2     Γ μ + 12 Γ ν + 12 J μ+ν+ 12 (z)     √ J μ z sin2 θ J ν z cos2 θ sin2μ+1 θ cos2ν+1 θ dθ = √ 2 π Γ(μ + ν + 1) 2z  WA 417(1) Re μ > − 12 , Re ν > − 12

0

 9.

π 2

0

 10.

π 2

0

6.684 

π

1.8

(sin x)

2ν

0



π

2.



(sin x)

0

2ν





Jν α2 + β 2 − 2αβ cos x √ 1 J ν (α) J ν (β) ν 

ν dx = 2 π Γ ν + 2 αν βν α2 + β 2 − 2αβ cos x  Re ν > − 12 



Yν α2 + β 2 − 2αβ cos x √ 1 J ν (α) Y ν (β) 

ν dx = 2ν π Γ ν + 2 αν βν α2 + β 2 − 2αβ cos x  |α| < |β|,

 6.685 6.686

π 2

0



∞

1. 0

 2. 0

∞

sec x cos(2λx) K 2μ (a sec x) dx =

π W λ,μ (a) W −λ,μ (a) 2a

Re ν > − 21

[Re a > 0]

2

2  √   π ν+1 b b sin ax2 J ν (bx) dx = − √ sin − π J 12 ν 8a 4 8a 2 a [a > 0, b > 0,

2 

2 √  2 π b ν+1 b − π J 12 ν cos ax J ν (bx) dx = √ cos 2 a 8a 4 8a [a > 0, b > 0,

ET II 362(27)



ET II 362(28)

ET II 378(41)

Re ν > −3] ET II 34(13)

Re ν > −1] ET II 38(38)

734

Bessel Functions



∞

3.

∞

4.

  cos ax2 Y ν (bx) dx

√ νπ

π = √ sec 4 a 2 

2 

2

2  3ν + 1 ν −1 b b b b2 − π J 12 ν + cos + π Y 12 ν × sin 8a 4 8a 8a 4 8a [a > 0, b > 0, −1 < Re ν < 1] ET II 107(8)

0



  sin ax2 Y ν (bx) dx

√ νπ

π = − √ sec 4 a 2 

2 

2

2  3ν + 1 b ν −1 b b b2 − π J 12 ν − sin + π Y 12 ν × cos 8a 4 8a 8a 4 8a [a > 0, b > 0, −3 < Re ν < 3] ET II 107(7)

0



∞

  b2 1 sin ax2 J 1 (bx) dx = sin b 4a 0

2  ∞  2 b 2 6. cos ax J 1 (bx) dx = sin2 b 8a 0

2  ∞   1 b 7. sin2 ax2 J 1 (bx) dx = cos 2b 8a 0

2  ∞  π  π x K 2ν xei 4 K 2ν xe−i 4 dx 6.687 cos 2a 0 5.

= 6.688



π 2

1. 0

 0

3.

π J ν (μz sin t) cos (μx cos t) dt = J ν2 2

Γ

1 4

[a > 0,

b > 0]

ET II 19(16)

[a > 0,

b > 0]

ET II 20(20)

[a > 0,

b > 0]

ET II 19(17)

  √  π + ν Γ 14 − ν π √ W 14 ,ν aei 2 W 8 a a > 0, |Re ν| < 14

1 4 ,ν



π

ae−i 2



ET II 372(1)

   √  √ x2 + z 2 + x x2 + z 2 − x J ν2 μ μ 2 2 [Re ν > −1,

π 2

2. 

6.687

MO 46 Re z > 0]     − 1 ν− 1 1 1√ ν+1 (sin x) cos (β cos x) J ν (α sin x) dx = 2− 2 παν α2 + β 2 2 4 J ν+ 12 α2 + β 2 2

[Re ν > −1] π 2

0

ET II 361(19)

   π cos [(z − ζ) cos θ] J 2ν 2 zζ sin θ dθ = J ν (z) J ν (ζ) 2  Re ν > − 12

EH II 47(8)

6.69–6.74 Combinations of Bessel and trigonometric functions and powers 

6.691 0

∞

x sin(bx) K 0 (ax) dx =

− 3 πb  2 a + b2 2 2

[Re a > 0,

b > 0]

ET I 105(47)

6.693

Bessel and trigonometric functions and powers

6.692  1.

∞

0

 0

∞

0



∞

2.8 0



   − 1 1 3 x K ν (ax) I ν (bx) sin(cx) dx = − (ab)− 2 c u2 − 1 2 Q 1ν− 1 (u), u = (2ab)−1 a2 + b2 + c2 2 2  Re a > |Re b|, c > 0, Re ν > − 32 ET I 106(54)

∞

2.

6.693  1.

735

 − 1     3 π x K ν (ax) K ν (bx) sin(cx) dx = (ab)− 2 c u2 − 1 2 Γ 32 + ν Γ 32 − ν P −1 (u) ν− 12 4    u = (2ab)−1 a2 + b2 + c2 Re(a + b) > 0, c > 0, |Re ν| < 32 ET I 107(61)

 1 b dx = sin ν arcsin J ν (ax) sin bx x ν a aν sin νπ 2 =  √ ν ν b + b 2 − a2

 b dx 1 = cos ν arcsin J ν (ax) cos bx x ν a aν cos νπ 2 =  √ ν ν b + b 2 − a2

[b ≤ a] [b ≥ a] [Re ν > −1]

WA 443(2)

[b ≤ a] [b ≥ a]

[Re ν > 0] WA 443(3)

∞

3.

Y ν (ax) sin(bx)

0

dx x



 νπ

b 1 sin ν arcsin = − tan ν 2 a

[0 < b < a, |Re ν| < 1] $ νπ #   1 ν  2   1 −ν  1 −ν 2 2 2 ν 2 2 sec a cos(νπ) b − b − a −a b− b −a = 2ν 2 [0 < a < b, |Re ν| < 1] 

ET I 103(35) ∞

4. 0

 5. 0

dx x√2       b cos ν arcsin ab a2 − b2 sin ν arcsin ab − = ν 2− ν (ν 2 − 1) 1 √  νπ ν −a cos 2 b + ν b2 − a2 = √ ν  ν (ν 2 − 1) b + b2 − a2

J ν (ax) sin(bx)

[0 < b < a,

Re ν > 0]

[0 < a < b,

Re ν > 0] ET I 99(6)

∞

dx J ν (ax) cos(bx) 2 x       a cos (ν + 1) arcsin ab a cos (ν − 1) arcsin ab + = 2ν(ν − 1)  2ν(ν + 1)   νπ aν sin 2 aν+2 sin νπ 2 = − √ √ ν−1 ν+1   2 2 2 2ν(ν − 1) b + b − a 2ν(ν + 1) b + b − a2

[0 < b < a,

Re ν > 1]

[0 < a < b,

Re ν > 1] ET I 44(6)

736

Bessel Functions

 6.

∞

12

J 0 (ax) sin x

0

 7.

8. 9.

dx π = x 2 = arccosec a

[0 < a < 1] [a > 1] WH

∞

dx π = x 2 0 = arcsin b π =− 2  ∞ dx [J 0 (x) − cos ax] = ln 2a x 0  z ∞ 2" dx = J ν (x) sin(z − x) (−1)k J ν+2k+1 (z) x ν 0 

10.

6.694

[b > 1]  2 b 0]

WA 416(4)

k=0

z

J ν (x) cos(z − x)

0

∞ 1 dx 2" = J ν (z) + (−1)k J ν+2k (z) x ν ν k=1

WA 416(5) [Re ν > 0]  *   √ √ 2  ∞ J 1 (ax) b 2a + b 2 2ab 2 2ab 2 2 2 − (2a − b) K (4a + b )E sin(bx) dx = − x 2 12πa2 2a + b 2a + b 0 [a > 0, b > 0]

)

6.69412

ET I 102(22)

6.695



∞

1. 0



∞

2. 0



[a > 0,

Re b > 0,

u > a]

π e−ab cos ax J (ux) dx = I 0 (bu) 0 b2 + x2 2 b

[a > 0,

Re b > 0,

−a < u < a]

MO 46

MO 46 ∞

3. 0



sin ax sinh ab K 0 (bu) J 0 (ux) dx = b2 + x2 b

x π sin(ax) J 0 (γx) dx = e−ab I 0 (γb) x2 + b2 2

[a > 0,

Re b > 0,

0 < γ < a] ET II 10(36)

∞

4.

x2

0

x cos(αx) J 0 (γx) dx = cosh(αβ) K 0 (βγ) + β2

[α > 0,

Re β > 0,

α < γ] ET II 11(45)

 6.696 0

∞

[1 − cos(ax)] J 0 (bx)

a

dx = arccosh x b =0

[0 < b < a] [0 < a < b] ET II 11(43)

6.698

6.697 1.

Bessel and trigonometric functions and powers



∞

sin[a(x + b)] J 0 (x) dx = 2 x+b −∞



a

0

cos bu √ du 1 − u2

[0 ≤ a ≤ 1]

= π J 0 (b) 

∞

2. 0



∞

3. 0

 4. 5.12 6.12

7.

737

WA 463(2)

[1 ≤ a < ∞]

WA 463(1), ET II 345(42)

π sin(x + t) J 0 (t) dt = J 0 (x) x+t 2

[x > 0]

WA 475(4)

cos(x + t) π J 0 (t) dt = − Y 0 (x) x+t 2

[x > 0]

WA 475(5)

∞

|x| sin[α(x + β)] J 0 (bx) dx = 0 [0 ≤ α < b] WA 464(5), ET II 345(43)a −∞ x + β  ∞ 2  2 sin[a(x + b)]  J n+ 12 (x) dx = π J n+ 12 (b) [2 ≤ a, n = 0, 1, . . .] ET II 346(45) x+b −∞  ∞ sin[a(x + b)] J n+ 12 (x) J −n− 12 (x) dx = π J n+ 12 (b) J −n− 12 (b) x+b −∞

[2 ≤ a, n = 0, 1, . . .]  √  ∞ Γ(μ + ν) π a2 J μ+ν− 12 [a(z − ζ)] J μ [a(z + x)] J ν [a(ζ + x)]     · dx = 1 μ (ζ + x)ν Γ μ + 12 Γ ν + 12 (z − ζ)μ+ν− 2 −∞ (z + x)

ET II 346(46)

[Re(μ + ν) > 0] 6.698



1. 0



∞√

x J ν+ 14 (ax) J −ν+ 14 (ax) sin(bx) dx =

  b  2 cos 2ν arccos 2a √ πb 4a2 − b2

2. 0



∞√

x J ν− 14 (ax) J −ν− 14 (ax) cos(bx) dx =

[0 < b < 2a] [0 < 2a < b]

=0 

WA 463(3)





2 cos 2ν arccos √ πb 4a2 − b2

b 2a

ET I 102(26)

 [0 < b < 2a] [0 < 2a < b]

=0

ET I 46(24)

 3. 0

∞√

x I 14 −ν

 

1 1 ax K 14 +ν ax sin(bx) dx = 2 2





√

2ν

π −2ν b + a2 + b2 √ a 2b  a2 + b 2 Re a > 0, b > 0,

Re ν
0, b > 0, Re ν < 34 ET I 50(49)

738

6.699

Bessel Functions



∞

1. 0

 0



 2 + λ + ν 2 + λ − ν 3 b2 , ; ; 2 F x J ν (ax) sin(bx) dx = 2 a b 2 2 2 a  0 < b < a, − Re ν − 1 < 1 + Re λ < 32  

ν Γ (ν + λ + 1) 1+λ+ν 1 a b−(ν+λ+1) sin π = 2 Γ(ν + 1) 2 

a2 2+λ+ν 1+λ+ν , ; ν + 1; 2 ×F 2 2  b 0 < a < b, − Re ν − 1 < 1 + Re λ < 32 λ

1+λ −(2+λ)



2+λ−μ Γ 2λ b Γ 2+μ+λ 2 2

∞

xλ K μ (ax) sin(bx)dx =

∞

4. 0



∞

5. 0

a2+λ

6. 0

∞

ET I 45(13)

 b2 2+μ+λ 2+λ−μ 3 , ; ;− 2 2 2 2 a [Re (−λ ± μ) < 2, Re a > 0, b > 0]

F

  1+λ−μ μ+λ+1 λ λ−1 −λ−1 Γ x K μ (ax) cos(bx) dx = 2 a Γ 2 2

b2 μ+λ+1 1+λ−μ 1 , ; ;− 2 ×F 2 2 2 a [Re (−λ ± μ) < 1, Re a > 0,

ET I 106(50)

−ν− 12 √ ν ν 2 π2 b a − b2   x sin(ax) J ν (bx) dx = Γ 12 − ν ν

=0 

 2  Γ ν−λ 2

xλ J ν (ax) cos(bx) dx    2λ a−(1+λ) Γ 1+λ+ν 1 + λ + ν 1 + λ − ν 1 b2 2  ν−λ+1  , ; ; 2 F = 2 2 2 a Γ 2  0 < b < a, − Re ν < 1 + Re λ < 32  a ν −(ν+1+λ)   b Γ (1 + λ + ν) cos π2 (1 + λ + ν) a2 1+λ+ν 2+λ+ν 2 = F , ; ν + 1; 2 Γ(ν + 1) 2 2 b  0 < a < b, − Re ν < 1 + Re λ < 32

0



Γ

ET I 100(11) ∞

2.

3.

 2+λ+ν 

6.699



 0 < b < a,

−1 < Re ν
0, b > 0, 

xν cos(ax) J ν (ax) dx =

∞

12. 0



Re ν > − 23



ET I 105(49)

b > 0,

Re μ > − 21



ET I 49(41) ∞

13.



xν Y ν−1 (ax) sin(bx) dx = 0

√ −ν− 12 2ν πaν−1 b  2 1  b − a2 = Γ 2 −ν

0





−μ− 12 1√ 1  2 μ μ b + a2 x K μ (ax) cos(bx) dx = π(2a) Γ μ + 2 2  Re a > 0,

ET II 335(20)



0 < b < a,

|Re ν|
0,

ET I 47(26)

  

Γ 32 − ν a 3 3 2   − ν, − 2ν; 2 − ν; a F sin(2ax) J ν (x) Y ν (x) dx = − 2 2 2 Γ 2ν − 12 Γ(2 − ν) 0 < Re ν < 32 , 0 < a < 1 ET II 348(63)



ρ2 a2 2z 2 Γ (ν) aμ ρ−ν − − μ−ν+3 2 Γ (μ + 1) ν − 1 μ + 1 3 0  2   ∞ μ −ν 2 a Γ (ν) a ρ ρ − − 2z 2 cos (zx)xν−μ−3 J μ (ax) J ν (ρx) dx = μ−ν+3 2 Γ (μ + 1) ν − 1 μ + 1 0 arg sin (zx)xν−μ−4 J μ (ax) J ν (ρx) dx = z

√ −ν− 12 π(2a)ν  2  b + 2ab x [J ν (ax) cos(ax) + Y ν (ax) sin(ax)] sin(bx) dx =  1 Γ 2 −ν 0  b > 0, −1 < Re ν < 12 √  ∞ −ν− 12 π(2a)ν  2 ν  b + 2ab x [Y ν (ax) cos(ax) − J ν (ax) sin(ax)] cos(bx) dx = −  1 Γ 2 −ν 0 



∞

ν

xν [J ν (ax) cos(ax) − Y ν (ax) sin(ax)] sin(bx) dx

0

=0 =

4.

ET I 104(40)

ET I 48(35) ∞

3.





√ −ν− 12 2ν πbν  2 1  b − 2ab Γ 2 −ν

 

0 < b < 2a, 2a < b,

−1 < Re ν < −1 < Re ν < 12

1 2



ET I 104(41) ∞

xν [J ν (ax) sin(ax) + Y ν (ax) cos(ax)] cos(bx) dx

0

=0

√ −ν− 12 π(2a)ν  2  b − 2ab = − 1 Γ 2 −ν

 0 < b < 2a,  0 < 2a < b,

|Re ν|
−1]



6.718

∞

sin[a(x + b)] J ν+2n (x) dx = πb−ν J ν+2n (b) ν (x + b) x −∞  1 ≤ a < ∞,

6.717



∞

1. 0



∞

2. 0



∞

3. 0



ET II 335(26)

n = 0, 1, 2, . . . ;

xν sin(ax) J ν (cx) dx = bν−1 sinh(ab) K ν (bc) x2 + b2  0 < a ≤ c,

Re b > 0,

xν+1 cos(ax) J ν (cx) dx = bν cosh(ab) K ν (bc) x2 + b2  0 < a ≤ c,

Re b > 0,



−1 < Re ν
0,

ET II 345(44)

3 2

1 2





ET II 33(8)

ET II 37(33)

Re ν > − 21



ET II 33(9) ∞

4.

 0 < c ≤ a,

−ν

x π cos(ax) J ν (cx) dx = b−ν−1 e−ab I ν (bc) 2 +b 2

x2

0

Re b > 0,

Re ν > − 23



ET II 37(34)

6.719 1.



a

∞ "     sin(bx) √ J ν (x) dx = π (−1)n J 2n+1 (ab) J 12 ν+n+ 12 12 a J 12 ν−n− 12 12 a a2 − x2 n=0

a

  2     cos(bx) π √ J ν (x) dx = J 0 (ab) J 12 ν 12 a +π (−1)n J 2n (ab) J 12 ν+n 12 a J 12 ν−n 12 a 2 a2 − x2 n=1

6 0

 2. 0

[Re ν > −2]

[Re ν > −1] 6.721



1. 0

ET II 335(17)

∞ "

√   x J 14 a2 x2 sin(bx) dx = 2−3/2 a−2 πb J 14

∞√

b2 4a2

ET II 336(27)

 [b > 0]

ET I 108(1)

6.722

Bessel and trigonometric functions and powers

 2. 0

 3. 0

5.

6.



x J − 14 a x

2 2



cos(bx) dx = 2

−3/2 −2

a

√ πb J − 14

√   x Y 14 a2 x2 sin(bx) dx = −2−3/2 πba−2 H 14

∞√



b2 4a2



[b > 0] 

b 4a2

√   x Y − 14 a2 x2 cos(bx) dx = −2−3/2 πba−2 H− 14

∞√

[b > 0] 6.722



1. 0

 2.12 0

 3.12 0

 4.12 0

∞√



x K 18 +ν a x

2 2





I 18 −ν a x

2 2



ET I 51(1)

2

 b2 4a2 0 



2   ∞ √  2 2 √ b2 b −5/2 −2 3 I 14 − L 14 x K 14 a x sin(bx) dx = 2 π ba 2 2 4a 4a 0   π |arg a| < , b > 0 4 

2 

2   ∞ √  2 2 √ b b 1 − L x K − 14 a x cos(bx) dx = 2−5/2 π 3 ba−2 I − 14 −4 2 4a 4a2 0 

4.

∞√

743

ET I 108(7) ET I 52(7)

ET I 109(11)

ET I 52(10)

5 

2 

2  √ b b −3/2 Γ 8−ν M −ν, 18 W ν, 18 sin(bx) dx = 2πb 5 2 8a 8a2   Γ 4 π 5 Re ν < , |arg a| < , b > 0 8 4 ET I 109(13)

∞√

    x J − 18 −ν a2 x2 J − 18 +ν a2 x2 cos(bx) dx  

2 −πi/2 

2 −πi/2  2 b e b e iπ/8 e W−nu,−1/8 Wnu,−1/8 = b3 π 8a2 8a2

2 πi/2 

2 πi/2  b e b e W−nu,−1/8 + e−iπ/8 Wnu,−1/8 2 8a 8a2  2 a > 0, Im b = 0 MC

∞√

    x J 18 −ν a2 x2 J 18 +ν a2 x2 sin(bx) dx 



2 πi/2 

2 πi/2  2 −3/2 πi/8 b e b e b e W −ν, 18 W ν, 18 = π 8a2 8a2  ⎤

2 −πi/2  πi b2 e− 2 ⎦ b e W −ν, 18 + e−iπ/8 W ν, 18 2 8a 8a2  2 ET I 108(6) a > 0, b > 0

∞√

    x K 18 −ν a2 x2 I − 18 −ν a2 x2 cos(bx) dx =

√

3 

2 

2  b b 8−ν M −ν,− 18 W ν,− 18 2πb 3 2 8a 8a2 Γ 4  3 π Re ν < 8 , |arg a| < 4 , b > 0 ET I 52(12) −3/2 Γ

744

Bessel Functions

 6.723

∞

0

6.724



∞

1.

          1 x J ν x2 sin(νπ) J ν x2 − cos(νπ) Y ν x2 J 4ν (4ax) dx = J ν a2 J −ν a2 4 ET II 375(20) [a > 0, Re ν > −1]

x2λ J 2ν

0



∞

2. 0

6.723

x2λ J 2ν

a

x

a

x

sin(bx) dx 

√ 2ν πa Γ(λ − ν + 1)b2ν−2λ−1 1 a2 b 2  0 F 3 2ν + 1, ν − λ, ν − λ + ;

= 1 2 16 42ν−λ Γ(2ν + 1) Γ ν − λ + 2 

a2 b 2 3 a2λ+2 Γ(ν − λ − 1)b , λ − ν + 2, λ + ν + 2; + 2λ+3 0F 3 2 Γ(ν + λ + 2) 2 16 − 45 < Re λ < Re ν, a > 0, b > 0 ET I 109(15) cos(bx) dx

  

√ 2ν 2ν−2λ−1 Γ λ − ν + 12 a2 b 2 1 =4 πa b 0 F 3 2ν + 1, ν − λ + , ν − λ; Γ(2ν + 2 16  1) Γ(ν − λ)   1 2 2 Γ ν − λ − b 3 3 a 1 2 ,λ − ν + ,ν + λ+ ; +4−λ−1 a2λ+1 0F 3 3 2 2 2 16 Γ ν +λ+ 2 3 − 4 < Re λ < Re ν − 12 , a > 0, b > 0 ET I 53(14) λ−2ν

6.725



∞

1. 0



∞

2. 0



∞

3.

 √  sin(bx) √ J ν a x dx = − x

 √  cos(bx) √ J ν a x dx = x





π sin b

π cos b



a2 νπ π − − 8b 4 4

a2 νπ π − − 8b 4 4

 √  1 x 2 ν J ν a x sin(bx) dx = 2−ν aν b−ν−1 cos

0

 4. 0

∞

 √  1 x 2 ν J ν a x cos(bx) dx = 2−ν b−ν−1 aν sin





 a2 J ν2 8b [Re ν > −3,



b > 0] ET I 110(27)

 2

a 8b [Re ν > −1,

J 12 ν

a > 0,

a > 0,

b > 0] ET I 54(25)



νπ a2 − 4b 2 −2 < Re ν < 12 ,

a > 0,



ET I 110(28)



νπ a2 − 4b 2 −1 < Re ν < 12 ,

b>0

a > 0,

b>0



ET I 54(26)

6.727

Bessel and trigonometric functions and powers

6.726  1.

∞

0



− 1 ν  x x2 + b2 2 J ν a x2 + b2 sin(cx) dx  

 1 ν− 34 π −ν −ν+ 3  2 2 c a − c2 2 a b J ν− 32 b a2 − c2 = 2 =0



∞

2.



x +b J ν a x2 + b2 cos(cx) dx  

 1 ν− 1 π −ν −ν+ 1  2 2 a b a − c2 2 4 J ν− 12 b a2 − c2 = 2



=0 

∞

3.





∞



∞



x2 + b2

0

 5. 0

6.727  1.9 0



a

a

1 2

0 < a < c,

Re ν >

1 2



ET I 111(37)

 0 < c < a,  0 < a < c,

b > 0,

Re ν > − 21

b > 0,

Re ν > − 21



ET I 55(37) 

2 2 K ±ν a x + b sin(cx) dx  

− 1 ν− 3 π ν ν+ 3  2 a b 2 c a + c2 2 4 K −ν− 32 b a2 + c2 = 2 [Re a > 0, Re b > 0, c > 0] ET I 113(45)

∓ 12 ν



K ν a x2 + b2 cos(cx) dx  

± 1 ν− 1 π ∓ν 1 ∓ν  2 a b2 a + c2 2 4 K ±ν− 12 b a2 + c2 = 2 [Re a > 0, Re b > 0, c is real] ET I 56(45)

2



− 1 ν x2 + a2 2 Y ν b x2 + a2 cos(cx) dx  

  1 ν− 1 aπ (ab)−ν b2 − c2 2 4 Y ν− 12 a b2 − c2 = 2 

  1 ν− 1 2a (ab)−ν c2 − b2 2 4 K ν− 12 a c2 − b2 =− π

 0 < c < b,

a > 0,

Re ν > − 21

 0 < b < c,

a > 0,

Re ν > − 21





 a   a 



 cos(cx) π √ J ν b a2 − x2 dx = J 12 ν b2 + c2 − c J 12 ν b 2 + c2 + c 2 2 2 a2 − x2 [Re ν > −1, c > 0, a > 0] ET I 113(48)

∞



a

 a

   sin(cx) π √ c − c2 − b2 J − 12 ν c + c2 − b 2 J ν b x2 − a2 dx = J 12 ν 2 2 2 x2 − a2 [0 < b < c, a > 0, Re ν > −1]

∞



a

 a

   cos(cx) π √ c − c2 − b2 Y − 12 ν c + c2 − b 2 J ν b x2 − a2 dx = − J 12 ν 2 2 2 x2 − a2 [0 < b < c, a > 0, Re ν > −1]

a

3.

Re ν >

ET I 56(41)

2.12





0 < c < a,

 12 ν

x x +b 2

0

4.11



 1 2 −2ν

2

0

745

ET I 113(49)

ET I 58(54)

746

Bessel Functions

 4.

8

a



a −x 2

2

 12 ν



cos x I ν a2 − x2 dx =

0

6.728



∞

1.

∞

  x cos ax2 J ν (bx) dx



2  

2

2

2  √ πb νπ b νπ b b b − J 12 ν+ 12 + sin − J 12 ν− 12 = 3/2 cos 8a 4 8a 8a 4 8a 8a [a > 0, b > 0, Re ν > −2] ET II 38(39)

0

∞

3. 0



∞

4. 0



∞

5.

  1 b2 J 0 (bx) sin ax2 x dx = cos 2a 4a

[a > 0,

b > 0]

MO 47

  b2 1 sin J 0 (bx) cos ax2 x dx = 2a 4a

[a > 0,

b > 0]

MO 47

 νπ b2 − 4a 2  a > 0,

b > 0,

  xν+1 sin ax2 J ν (bx) dx =

0



WA 409(2)



2  

2

2

2  √ πb νπ νπ b b b b − J 12 ν− 12 − sin − J 12 ν+ 12 = 3/2 cos 8a 4 8a 8a 4 8a 8a [a > 0, b > 0, Re ν > −4] ET II 34(14)

2.



√ 2ν+1 πa   ν+1 2 Γ ν + 32  Re ν > − 12

  x sin ax2 J ν (bx) dx

0



6.728

∞

6.

  xν+1 cos ax2 J ν (bx) dx =

0

bν cos ν+1 2 aν+1

bν 2ν+1 aν+1



sin

−2 < Re ν
0,

b > 0,

−1 < Re ν
0, b > 0, 

c > 0,

ET II 51(26)



b 2 + c2 νπ bc − Jν 4a 2 2a [a > 0, b > 0,

Re ν > −2]

c > 0,

Re ν > −1] ET II 51(27)

6.735

6.731

Bessel and trigonometric functions and powers



∞

1.11

747

    x sin ax2 J ν bx2 J 2ν (2cx) dx





1 bc2 ac2 = √ Jν sin 2 2 2 2 2 b 2 − a2

b −2a 

b −2a  1 bc ac Jν cos = √ a2 − b 2 a2 − b 2 2 a2 − b 2

0

[0 < a < b,

Re ν > −1]

[0 < b < a,

Re ν > −1] ET II 356(41)a



∞

2.10

    x cos ax2 J ν bx2 J 2ν (2cx) dx

 

1 bc2 ac2 = √ Jν cos 2 2 2 2 2 b 2 − a2

b −2a  b −2 a  bc ac 1 Jν sin = √ a2 − b 2 a2 − b 2 2 a2 − b 2

0

 0 < a < b,

Re ν > − 21

 0 < b < a,

Re ν > − 21



ET II 356(42)a

 6.73212 6.733

∞

x3 cos

0



∞

sin

1. 0



∞

2. 0

 x2 Y 1 (x) K 1 (x) dx = −a3 K 0 (a) 2a

[a > 0]

ET II 371(52)

a

√  √  dx [sin x J 0 (x) + cos x Y 0 (x)] = π J0 a Y0 a 2x x

[a > 0] a

√ √   dx [sin x Y 0 (x) − cos x J 0 (x)] = π J0 cos a Y0 a 2x x

ET II 346(51)

[a > 0] ET II 347(52) a

√  √  πa K 0 (x) dx = J1 3. x sin a K1 a [a > 0] ET II 368(34) 2x 2 0  ∞ a

√  √  πa K 0 (x) dx = − Y1 4. x cos a K1 a [a > 0] ET II 369(35) 2x 2 0  ∞  √  dx 6.734 cos a x K ν (bx) √ x 0



    a a a a π D −ν− 12 − √ + D ν− 12 − √ D −ν− 12 √ = √ sec(νπ) D ν− 12 √ 2 b 2b 2b 2b 2b Re b > 0, |Re ν| < 12 ET II 132(27) 6.735



∞



∞

1. 0



∞

2. 0

 3.

0

∞

   √  √ x1/4 sin 2a x J − 14 (x) dx = πa3/2 J 34 a2

[a > 0]

ET II 341(10)

   √  √ x1/4 cos 2a x J 14 (x) dx = πa3/2 J − 34 a2

[a > 0]

ET II 341(12)

   √  √ x1/4 sin 2a x J 34 (x) dx = πa3/2 J − 14 a2

[a > 0]

ET II 341(11)

748

Bessel Functions



∞

4. 0

6.736 1.



∞

11 0

   √  √ x1/4 cos 2a x J − 34 (x) dx = πa3/2 J 14 a2

6.736

[a > 0]

ET II 341(13)

   √  √  π  2 π

J 0 a − sin a2 − Y 0 a2 x−1/2 sin x cos 4a x J 0 (x) dx = −2−3/2 π cos a2 − 4 4

ET II 341(18) [a > 0] ∞    √  √  π  2 π

J 0 a + cos a2 − Y 0 a2 x−1/2 cos x cos 4a x J 0 (x) dx = −2−3/2 π sin a2 − 4 4 0

 2.



∞

3.

 √  x−1/2 sin x sin 4a x J 0 (x) dx =



[a > 0]

  π π cos a2 + J 0 a2 2 4

ET II 342(22)



[a > 0]

  π π J 0 a2 cos a2 − 2 4

ET II 341(16)

0



∞

4.

 √  x−1/2 cos x sin 4a x J 0 (x) dx =

0



∞

5. 0



∞

1. 0



∞

2. 0

4.

 √  x−1/2 cos x cos 4a x Y 0 (x) dx

  √  



   sin a x2 + b2 b b π 2 2 2 2 √ a− a −c a+ a −c J − 12 ν J ν (cx) dx = J 12 ν 2 2 2 x2 + b2 [a > 0, Re b > 0, c > 0, a > c, Re ν > −1] ET II 35(19)   √  



   cos a x2 + b2 b b π √ a − a2 − c2 Y − 12 ν a + a 2 − c2 J ν (cx) dx = − J 12 ν 2 2 2 x2 + b2 [a > 0, Re b > 0, c > 0, a > c, Re ν > −1] ET II 39(44)

  √  a   a 



 cos b a2 − x2 π √ J ν (cx) dx = J 12 ν b2 + c2 − b J 12 ν b 2 + c2 + b 2 2 2 a2 − x2 0 ET II 39(47) [c > 0, Re ν > −1]  √ √  a cos a2 − x2 πa2ν+1 

[Re ν > −1] xν+1 √ I ν (x) dx = ET II 365(9) 3 a2 − x2 0 2ν+1 Γ ν + 2 

3.

ET II 347(55)

  √  π  2 π

J 0 a + sin a2 − Y 0 a2 = −2−3/2 π 3 cos a2 − 4 4 [a > 0] ET II 347(56)

0



[a > 0] ET II 342(20)

       √ √ π π

J 0 a2 − cos a2 − Y 0 a2 x−1/2 sin x cos 4a x Y 0 (x) dx = 2−3/2 π 3 sin a2 − 4 4 [a > 0]

∞

6.

6.737



a

6.741

Bessel and trigonometric functions and powers

  √ a b2 + x2 √ x J ν (cx) dx b2 + x2 0  

π 1 +ν ν  2 2 − 14 − 12 ν b 2 c a −c J −ν− 12 b a2 −c2 = 2

 5.

∞

749

ν+1 sin

 0 < c < a,  0 < a < c,

=0

Re b > 0,

−1 < Re ν
0,

−1 < Re ν
0, −1 < Re ν < 2  

1 1  2 1 +ν ν  2 − − ν = b 2 c c − a2 4 2 K ν+ 12 b c2 − a2 π   1 0 < a < c, Re b > 0, −1 < Re ν < 2 ET II 39(45)

6.738   a



 − 1 ν− 3 π ν+ 3  a 2 b 1 + b2 2 4 J ν+ 32 a 1 + b2 xν+1 sin b a2 − x2 J ν (x) dx = 1. 2 0 [Re ν > −1] ET II 335(19)  ∞

 2. xν+1 cos a x2 + b2 J ν (cx) dx 0  



 − 1 ν− 3  π ν+ 3 ν  2 ab 2 c a − c2 2 4 cos(πν) J ν+ 32 b a2 − c2 − sin (πν) Y ν+ 32 b a2 − c2 = 2  0 < c < a, Re b > 0, −1 < Re ν < − 21 =0



0 < a < c,

Re b > 0,

−1 < Re ν < − 21



ET II 39(43)

 √   √  t

  √t 

  √  t −1/2 cos b t − x 2 2 2 2 √ J 2ν a x dx = π J ν x a + b + b Jν a +b −b 2 2 t−x 0  Re ν > − 12 EH II 47(7)

 6.739 6.741  1.

2.

a

a

cos (μ arccos x) π √ J 12 (ν−μ) J ν (ax) dx = J 12 (μ+ν) 2 2 2 1 − x2 0 [Re(μ + ν) > −1,   1

a

cos [(ν + 1) arccos x] π a √ J ν+ 12 J ν (ax) dx = cos a 2 2 1 − x2 0 

3. 0

1

1

cos [(ν − 1) arccos x] √ J ν (ax) dx = 1 − x2



a

π sin J ν− 12 a 2

[Re ν > −1, a

a > 0]

a > 0]

ET II 41(54)

ET II 40(53)

2 [Re ν > 0,

a > 0]

ET II 40(52)a

750

Bessel Functions

6.751

6.75 Combinations of Bessel, trigonometric, and exponential functions and powers 6.751 1.

2.

3.10

    1  1  (b + c)2 + a2 − (b − c)2 + a2 , 2 = (b + c)2 + a2 + (b − c)2 + a2 2 2

   ∞  1 1 1 1 e− 2 ax sin(bx) I 0 b + b 2 + a2 ax dx = √ √ 2 2 2b b + a2 0 [Re a > 0, b > 0] ET I 105(44) 

 ∞ 1 1 a 1  ax dx = √ √ e− 2 ax cos(bx) I 0 √ 2 2 2 2b a + b b + a2 + b2 0 [Re a > 0, b > 0] ET I 48(38) 1/2  2  ∞ (b2 + c2 − a2 ) + 4a2 b2 + b2 + c2 − a2 e−bx cos(ax) J 0 (cx) dx = √  2 0 2 (b2 + c2 − a2 ) + 4a2 b2 Notation: 1 =

[c > 0] alternatively, with a and b interchanged,   ∞ 2 − b2 −ax e cos(bx) J 0 (cx) dx = 2 2 2 2 − 1 0 6.752



∞

10

1.

0

 2.10 0

∞

e

−ax

dx = arcsin J 0 (bx) sin(cx) x



ET II 11(46)

[c > 0]

2c



  a2 + (c + b)2 + a2 + (c − b)2

= arcsin

c 2



ET I 101(17) [Re a > |Im b|, c > 0]  b b − b2 − 21 dx = (1 − r) = , e−ax J 1 (cx) sin(bx) x c c   c2 a2 b2 = − , c > 0 1 − r2 r2 ET II 19(15)

Notation: For integrals 6.752 3–6.752 5 we define the auxiliary functions   1  1 (a) ≡ 1 (a, ρ, z) = (a + ρ)2 + z 2 − (a − ρ)2 + z 2 2   1  2 (a) ≡ 1 (a, ρ, z) = (a + ρ)2 + z 2 + (a − ρ)2 + z 2 2 when a ≥ 0, ρ ≥ 0, and z ≥ 0.   ∞ √ π 10 e−zx J ν+1/2 (ax) J ν+1 (ρx) x dx 3. 2 0

  a ρ2 − 21 2ν+2 1  ρ =a 2  (2 − 2 ) 2 1  ρ − 1 1 2 ν+1 2 − a2 ρ  = aν+1/2 2ν+2 2 2 2 2 − 1 2 −ν−3/2 −ν−1

[Re z > |Im a| + |Im ρ|]

6.753

4.

10

5.10

Combinations of Bessel, trigonometric, and exponential functions and powers

751

  ∞ π dx e−zx J ν+1/2 (ax) J ν (ρx) √ 2 0 x

  1/ 2 1 1 1 ν+1/2 ν  =a ρ d 2ν 2 2   1 − a /2 2 0 2  a/ 2  dx ν > − 12 , Re z > |Im a| + |Im ρ| x2ν √ = a−ν−1/2 ρν 2 1−x 0

2  

 2 − a2 a −  ∞ 2 − 2  a 2 1 ρ a dx −zx + arcsin e sin(ax) J 1 (ρx) 2 = x 2aρ 2  2 0 [Re z > |Im a| + |Im ρ|]

6.753  1.8

sin (xa sin ψ) −xa cos ϕ cos ψ ϕ ν e J ν (xa sin ϕ) dx = ν −1 tan sin(νψ) x 2 0  π π Re ν > −1, a > 0, 0 < ϕ < , 0 < ψ < ET II 33(10) 2 2  ∞ cos (xa sin ψ) −xa cos ϕ cos ψ ϕ ν e J ν (xa sin ϕ) dx = ν −1 tan cos(νψ) x 0  2 π Re ν > 0, a > 0, 0 < ϕ, ψ < 2

2.



∞

ET II 38(35) ∞

3.8 0

ν

2(2a) Γ(ν + 32 )R xν+1 e−sx sin(bx) J ν (ax) dx = − √ π

 −2ν−3 

b cos(ν + 32 )ϕ + s sin(ν + 32 )ϕ

Re ν > − 23 , 2  R4 = s2 + a2 − b2 + 4b2 s2 , 

∞

8

4.

0

xν+1 e−sx cos(bx) J ν (ax) dx =

Re s > |Im a| + |Im b|,    ϕ = arg s2 + a2 − b2 − 2ibs

 2(2a)ν √ Γ(ν + 32 )R−2ν−3 s cos(ν + 32 )ϕ − b sin(ν + 32 )ϕ , π  Re ν > −1,



∞

5.10 0

 6. 0

∞

2  R4 = s2 + a2 − b2 + 4b2 s2 ,

Re s > |Im a| + |Im b|,   2  2 2 ϕ = arg s + a − b − 2ibs

xν e−ax cos ϕ cos ψ sin (ax sin ψ) J ν (ax sin ϕ) dx   1  −ν− 12  ν νΓ ν + 2 √ a−ν−1 (sin ϕ) cos2 ψ + sin2 ψ cos2 ϕ sin ν + 12 β =2 π   π π β a > 0, 0 < ϕ < , 0 < ψ < , Re ν > −1 ET II 34(12) tan = tan ψ cos ϕ 2 2 2 xν e−ax cos ϕ cos ψ cos (ax sin ψ) J ν (ax sin ϕ) dx   1  −ν− 12  ν νΓ ν+ 2 √ a−ν−1 (sin ϕ) cos2 ψ + sin2 ψ cos2 ϕ cos ν + 12 β =2 π   π 1 β a > 0, 0 < ϕ, ψ < , Re ν > − tan = tan ψ cos ϕ ET II 38(37) 2 2 2

752

6.754 1. 2.

Bessel Functions

6.754

2 √ π − b2 b 8 I [b > 0] e ET I 108(9) 0 3/2 8 2 0   2

2 

2 

2  ∞  2  2 a π a π a a 1 π −ax J0 cos − −Y0 cos + e cos x J 0 x dx = 4 2 16 16 4 16 16 4 0





∞

∞

3. 0

  2 e−x sin(bx) I 0 x2 dx =

    1 e−ax sin x2 J 0 x2 dx = 4





π J0 2

 2

a 16

sin

MI 42 [a > 0]  



a2 π a2 π a2 − −Y0 sin + 16 4 16 16 4

[a > 0] 6.755



∞

1. 0

 2.

3.

4.

6.

7.

8. 9.

1 3 1 1 2 − 2 ν, 2 − 2 ν

[a > 0,



2a2



Re ν > 0]

ET II 366(14)

∞

   √  2 1 3 x−ν− 2 e−x cos 4a x I ν (x) dx = 2 2 ν−1 aν−1 e−a W − 32 ν, 12 ν 2a2 0  ET II 366(16) a > 0, Re ν > − 12    ∞

ν−1 Γ 3 − 2ν    √  2  ea W 3 ν− 1 , 1 − 1 ν 2a2 x−ν ex sin 4a x K ν (x) dx = 23/2 a π 21 2 2 2 2 Γ 2 +ν 0  a > 0, 0 < Re ν < 34 ET II 369(38) 1   ∞    √  Γ − 2ν a2 1 3  e W 3 ν,− 1 ν 2a2 x−ν− 2 ex cos 4a x K ν (x) dx = 2 2 ν−1 πaν−1 21 2 2 Γ 2+ 0 ν a > 0, − 12 < Re ν < 14 ET II 369(42)

 √  √  πa Γ( + ν) Γ( − ν) 1 3 − 32 −x 2   x e sin 4a x K ν (x) dx = 2 F 2  + ν,  − ν; ,  + ; −2a 2 2 2−2 Γ  + 12 0 ET II 369(39) [Re  > |Re ν|]

 √  ∞  √  π Γ( + ν) Γ( − ν) 1 1 2   ,  + ; −2a  + ν,  − ν; x−1 e−x cos 4a x K ν (x) dx = F 2 2 2 2 2 Γ  + 12 0 ET II 370(43) [Re  > |Re ν|]  ∞    √  2 1 x−1/2 e−x cos 4a x I 0 (x) dx = √ e−a K 0 a2 2π 0 ET II 366(15) [a > 0]   ∞    √  π a2 e K 0 a2 [a > 0] x−1/2 ex cos 4a x K 0 (x) dx = ET II 369(40) 2 0  ∞    √  2 1 x−1/2 e−x cos 4a x K 0 (x) dx = √ π 3/2 e−a I 0 a2 ET II 369(41) 2 0 

5.



ν−1  √  2 x−ν e−x sin 4a x I ν (x) dx = 23/2 a e−a W

MI 42

∞

6.761

Bessel, trigonometric, and hyperbolic functions

6.756  1.

∞

1

√ x

 √  sin a x J ν (bx) dx 



   ia i 1 a ia D −ν− 12 √ =√ D −ν− 12 √ − D −ν− 12 − √ Γ ν+ 2 2πb b b b [a > 0, b > 0, Re ν > −1] ET II 34(17)

1

√ x

 √  cos a x J ν (bx) dx 



   a ia ia 1 1 D −ν− 12 √ =√ Γ ν+ D −ν− 12 √ + D −ν− 12 − √ 2 2πb b b  b a > 0, b > 0, Re ν > − 21 ET II 39(42)

x− 2 e−a

0



∞

2.

x− 2 e−a

0



∞

3.

x−1/2 e−a

√ x

0



∞

4. 0

6.757  1.

∞

x−1/2 e−a

√ x

 √  1 sin a x J 0 (bx) dx = a I 14 2b

 √  a I 1 cos a x J 0 (bx) dx = 2b − 4

2 a K 14 4b  π |arg a| < , 4

2

2 a a K 14 4b 4b  π |arg a| < , 4

a2 4b



b>0

b>0

 ET II 11(40)



=2

∞ " (−1)n Γ(ν − b + 2n + 1) Γ (ν + b) (ν + 2n − 1) J ν+2n+1 (a) Γ(ν − b + 1) Γ(ν + b + 2n + 2) n=0

[Re b > − Re ν]

∞

2.

∞

=

Γ(ν − b + 2n) Γ(ν + b) J ν (a) " + (ν + 2n) J ν+2n (a) 2(−1)n ν + b n=0 Γ(ν − b + 1) Γ(ν + b + 2n + 1) [Re b > − Re ν]



ET I 193(26)

     e−bx cos a 1 − e−x J ν ae−x dx

0

6.758

ET II 12(49)

     e−bx sin a 1 − e−x J ν ae−x dx

0



753

π 2

π −2

ei(μ−ν)θ (cos θ)

ν+μ

(λz)−ν−μ J ν+μ (λz) dθ = π(2az)−μ (2bz)−ν J μ (az) J ν (bz)  [Re(ν + μ) > −1] λ = 2 cos θ (a2 eiθ + b2 e−iθ )

ET I 193(27)

EH II 48(12)

6.76 Combinations of Bessel, trigonometric, and hyperbolic functions  6.761 0

∞

x



cosh x cos (2a sinh x) J ν (be ) J ν be

−x



 √  J 2ν 2 b2 − a2 √ dx = 2 b 2 − a2 =0

[0 < a < b,

Re ν > −1]

[0 < b < a,

Re ν > −1] ET II 359(10)

754

Bessel Functions

 6.762

∞

     cosh x sin (2a sinh x) J ν (bex ) Y ν be−x − Y ν (bex ) J ν be−x dx

0

=0 =−  6.763

∞

6.762

    −1/2 1/2  2 cos(νπ) a2 − b2 K 2ν 2 a2 − b2 π x



cosh x cos (2a sinh x) Y ν (be ) Y ν be

−x

|Re ν|
0, b > 0]

  ∞ √ 2 x + a2 + x ab dx [Re a > 0, b > 0] 6.774 ln √ = K 20 J 0 (bx) √ 2 + a2 − x 2 + a2 2 x x 0 ∞ 

   1  6.77512 x ln a + a2 + x2 − ln x J 0 (bx) dx = 2 1 − e−ab b 0 [Re a > 0, b > 0]  

  ∞ a2 2 1 − a K 1 (ab) 6.776 x ln 1 + 2 J 0 (bx) dx = [Re a > 0, b > 0] x b b 0  ∞ 2 6.777 J 1 (tx) arctan t2 dt = − kei x x 0 1.

ET II 10(27) ET II 19(11) ET II 19(12) MO 46

ET II 10(28) ET II 10(29)

ET II 12(55) ET II 10(30) MO 46

6.784

Combinations of Bessel and other special functions

755

6.78 Combinations of Bessel and other special functions 

6.781

∞

0

1 si(ax) J 0 (bx) dx = − arcsin b

 b a

[0 < b < a] [0 < a < b]

=0

ET II 13(6)

6.782 1. 2. 3. 4. 5. 6.12 7.12 6.783 1. 2. 3.



∞

 √  e−z − 1 Ei(−x) J 0 2 zx dx = z 0  ∞  √  sin z si(x) J 0 2 zx dx = − z 0  ∞  √  cos z − 1 ci(x) J 0 2 zx dx = z 0  ∞  √  dx Ei(−z) − C − ln z √ Ei(−x) J 1 2 zx √ = x z 0  ∞ π  √  dx − si(z) si(x) J 1 2 zx √ = − 2 √ x z 0  ∞  √  dx ci(x) − C − ln z √ ci(x) J 1 2 zx √ = x z 0  ∞  √  C + ln z − ez Ei(−z) Ei(−x) Y 0 2 zx dx = πz 0

NT 60(6) NT 60(5) NT 60(7) NT 60(9) NT 60(8) NT 63(5)

2  2 b [a > 0] x si a x J 0 (bx) dx = − 2 sin 2 b 4a 0 

2   ∞   b 2 [a > 0] x ci a2 x2 J 0 (bx) dx = 2 1 − cos 2 b 4a 0  2 

2    ∞  2 2 b b 1 ci + ln + 2C ci a x J 0 (bx) dx = b 4a2 4a2 0





∞

∞

4. 0

6.784

NT 60(4)



1. 0

∞



2 2





  1 − si si a2 x2 J 1 (bx) dx = b

ν+1

x

[1 − Φ(ax)] J ν (bx) dx = a

b 4a2 2

−ν

 −

π 2



ET II 13(7)a ET II 13(8)a

[a > 0]

ET II 13(8)a

[a > 0]

ET II 20(25)a

 



2  Γ ν + 32 b2 b 1 1 1 1 exp − M 2 2 2 ν+ 2 , 2 ν+ 2 b2 Γ(ν + 2) 8a 4a  π |arg a| < 4 , b > 0, Re ν > −1 ET II 92(22)

756

Bessel Functions



∞

2.

 ν

x [1 − Φ(ax)] J ν (bx) dx =

0

6.785

  

2 

1 b2 2 a 2 −ν Γ ν + 12 b   1 1 1 1 M exp − 2 ν− 4 , 2 ν+ 4 π b3/2 Γ ν + 32 8a2 4a2  |arg a| < π4 , Re ν > − 12 , b > 0 ET II 92(23)

 6.785

∞ exp



∞

0

 6.787

−x 

x

0

6.786

a2 2x

1−Φ

a √ 2x

 K ν (x) dx =

  π 5/2 2 2 sec(νπ) [J ν (a)] + [Y ν (a)] 4  Re a > 0, |Re ν| < 12 ET II 370(46)

  xν−2μ+2n+2 ex2 Γ μ, x2 Y ν (bx) dx 3  3 

2

2 b b nΓ 2 − μ + ν + n Γ 2 − μ + n exp W μ− 12 ν−n−1, 12 ν = (−1) b Γ(1 − μ) 8 4  n is an integer, b > 0, Re(ν − μ + n) > − 32 , Re(−μ + n) > − 32 , Re ν < 12 − 2n

ET II 108(2) ∞

0

1

xν+2n− 2 J ν (bx) dx = 0 B(a + x, a − x)



π ≤ b < ∞,

−1 < Re ν < 2a − 2n −

7 2



ET II 92(21)

6.79 Integration of Bessel functions with respect to the order 6.791



∞

1. −∞  ∞

2. −∞  ∞

3. −∞

K ix+iy (a) K ix+iz (b) dx = π K iy−iz (a + b)

[|arg a| + |arg b| < π]

J ν−x (a) J μ+x (a) dx = J μ+ν (2a)

[Re(μ + ν) > 1]

ET II 382(21) ET II 379(1)

J κ+x (a) J λ−x (a) J μ+x (a) J ν−x (a) dx =

Γ(κ + λ + μ + ν + 1) Γ (κ + ⎛ λ + 1) Γ(λ + μ + 1) Γ(μ + ν + 1) Γ (ν + κ + 1) × 4F 5 ⎝

κ+λ+μ+ν +1 κ+λ+μ+ν +1 κ+λ+μ+ν κ+λ+μ+ν , , + 1, + 1; 2 2 2 2 ⎞

κ + λ + μ + ν + 1, κ + λ + 1, λ + μ + 1, μ + ν + 1, ν + κ + 1; −4a2⎠ [Re(κ + λ + μ + ν) > −1] 6.792



∞

1. −∞

ET II 379(3)

eπx K ix+iy (a) K ix+iz (b) dx = πe−πz K i(y−z) (a − b) [a > b > 0]

ET II 382(22)

6.794

Integration of Bessel functions

 2.

∞

12 −∞



e

iρx

K ν+ix (α) K ν−ix (β) dx = π

α + βeρ αeρ + β

ν K 2ν



757

α2 + β 2 + 2αβ cosh ρ



[|arg α| + |arg β| + |Im ρ| < π] ET II 382(23)

∞

3. −∞

e(π−γ)x K ix+iy (a) K ix+iz (b) dx = πe−βy−αz K iy−iz (c)

[0 < γ < π,

a > 0,

b > 0,

c > 0,

α, β, γ—the angles of the triangle with sides a, b, c] ET II 382(24), EH II 55(44)a

2ν ∞ h (2) (2) (2) e−cxi H ν−ix (a) H ν+ix (b) dx = 2i H 2ν (hk) k −∞  

 4.11

h= 

∞

5. −∞

1

1

ae 2 c + be− 2 c ,

1

1

ae− 2 c + be 2 c

k=

[a, b > 0,

c is real]

ET II 380(11)

a−μ−x b−ν+x ecxi J μ+x (a) J ν−x (b) dx * 12 μ+ 12 ν )   #  c

1/2 $ c 2 cos 2c 2 − 12 ci 2 12 ci a e exp (ν − μ)i J μ+ν 2 cos +b e = 1 1 2 2 a2 e− 2 ci + b2 e 2 ci [a > 0, b > 0, |c| < π, Re(μ + ν) > 1] =0 [a > 0,

b > 0,

|c| ≥ π,

Re(μ + ν) > 1]

EH II 54(41), ET II 379(2)

6.793 1.

2ν h e−cxi [J ν−ix (a) Y ν+ix (b) + Y ν−ix (a) J ν+ix (b)] dx = −2 J 2ν (hk) k −∞   1 1 1 1 k = ae− 2 c + be 2 c [a, b > 0, Im c = 0] h = ae 2 c + be− 2 c ,





∞

∞

2. −∞

e

−cxi

2ν h [J ν−ix (a) J ν+ix (b) − Y ν−ix (a) Y ν+ix (b)] dx = 2 Y 2ν (hk) k   h=



∞

3.10 −∞

6.794



∞

1. 2.

1

1

1

ae− 2 c + be 2 c

k=



K ix (a) K ix (b) cosh[(π − ϕ)x] dx =

∞

cosh 0

1

ae 2 c + be− 2 c ,

[a, b > 0,

Im c = 0]

ET II 380(10)



eiγx sech(πx) [J −ix (α) J ix (β) − J ix (α) J −ix (β)] dx = 2i H(σ) sign(β − α) J 0 σ 1/2

0



ET II 380(9)

π

π x K ix (a) dx = 2 2

α, β, γ ∈ R,

α, β > 0,

σ = α2 + β 2 − 2αβ cosh γ



π K0 a2 + b2 − 2ab cos ϕ 2 [a > 0]



EH II 55(42) ET II 382(19)

758

Bessel Functions



∞

3.

cosh(x) K ix+ν (a) K −ix+ν (a) dx =

0

5. 6.

   π K 2ν 2a cos 2 2

[2|arg a| + |Re | < π] π

sech [a > 0] x J ix (a) dx = 2 sin a 2 −∞  ∞ π

x J ix (a) dx = −2i cos a cosech [a > 0] 2 −∞  ∞   sech(πx) [J ix (a)]2 + [Y ix (a)]2 dx = − Y 0 (2a) − E0 (2a) 

4.

6.794

ET II 383(28)

∞

ET II 380(6) ET II 380(7)

0



∞

7. 0



∞

8. 0

9.

10.

π

πa x K ix (a) dx = x sinh 2 2 x tanh(πx) K ix (b) K ix (a) dx =

[a > 0]

ET II 380(12)

[a > 0]

ET II 382(20)

π √ exp(−b − a) ab 2 a+b

ET II 175(4) [|arg b| < π, |arg a| < π] 

2 π a a √ exp −b − x sinh(πx) K 2ix (a) K ix (b) dx = 5/2 8b 2 b 0  π b > 0, |arg a| < ET II 175(5) 4  ∞ x sinh(πx) π2 I n (b) K n (a) K ix (a) K ix (b) dx = [0 < b < a; n = 0, 1, 2, . . .] 2 2 x +n 2 0 2 π I n (a) K n (b) [0 < a < b; n = 0, 1, 2, . . .] = 2



∞



∞

11. 0

3/2



 c a b ab π2 exp − + + 2 x sinh(πx) K ix (a) K ix (b) K ix (c) dx = 4 2 b a c

|arg a| + |arg b|
0

* √ ∞ π

π2 c (a + b) c2 + 4ab √ x K 12 ix (a) K 12 ix (b) K ix (c) dx = √ x sinh exp − 2 2 c2 + 4ab 2 ab 0 [|arg a| + |arg b| < π, c > 0] )

 12.

ET II 176(8)

 13. 0



ET II 176(9)

ET II 176(10) ∞

x sinh(πx) K 12 ix+λ (α) K 12 ix−λ (α) K ix (γ) dx = 0 π γ 22λ+1  α2λ z z = γ 2 − 4α2 2

=

[0 < γ < 2α]   2λ (γ + z) + (γ − z)2λ [0 < 2α < γ]

ET II 176(11)

6.797

6.795

Integration of Bessel functions



∞

cos(bx) K ix (a) dx =

1. 0



J x (ax) J −x (ax) cos(πx) dx =

0 ∞

3.

x sin(ax) K ix (b) dx =

0



−∞

4. −∞



∞

5.

1. 2. 3. 4.



∞

∞

5.

πb sinh a exp (−b cosh a) 2



∞

1. 0

 0



ET II 380(4)

 |Im a|
0



ET II 175(1)

[0 < a < b;

n = 0, 1, . . .]

[0 < b < a;

n = 0, 1, . . .] ET II 382(25)

π 2,

b>0



ET II 175(6)

1

2

π J0 4 2



b > 0]

ET II 380(8) EH II 55(47) EH II 55(48)

[a > 0,

 b 2a sinh 2

b > 0]

ET II 383(27)

[a > 0,

b > 0]

ET II 382(26)

xeπx sinh(πx) Γ(ν + ix) Γ(ν − ix) H ix (a) H ix (b) dx (2)

(2)

  √ = i2ν π Γ 12 + ν (ab)ν (a + b)−ν K ν (a + b) [a > 0,

2.

a>0

[|a| < 1]

 

b2 1 π 3/2 b πx K 12 ix (a) K ix (b) dx = √ exp −a − 2 8a 2a  |arg a|
0, −∞ sinh(πx) 

 ∞ 1 π πx K ix (a) dx = cos (a sinh b) cos(bx) cosh 2 2 0 

 ∞ 1 π πx K ix (a) dx = sin (a sinh b) sin(bx) sinh 2 2 0 

  ∞ b π2 2 cos(bx) cosh(πx) [K ix (a)] dx = − Y 0 2a sinh 4 2 0 

π 2,

sin[(ν + ix)π] K ν+ix (a) K ν−ix (b) dx = π 2 I n (a) K n+2ν (b) n + ν + ix = π 2 K n+2ν (a) I n (b)

0

6.796

 |Im b|
0,

Re ν > 0]

ET II 381(14)

iπ 3/2 2ν (2) (2)  (b−a)−ν H (2) xeπx sinh(πx) cosh(πx) Γ(ν +ix) Γ(ν −ix) H ix (a) H ix (b) dx =  1 ν (b−a) Γ − ν 2  0 < a < b, 0 < Re ν < 12 ET II 381(15)

760

Functions Generated by Bessel Functions



∞

3.

xe

πx

sinh(πx) Γ

0

ν + ix 2

6.811

  ν − ix (2) (2) Γ H ix (a) H ix (b) dx 2



 − 1 ν a2 + b 2 = iπ22−ν (ab)ν a2 + b2 2 H (2) ν [a > 0,

 4.

∞

11 0

b > 0,

Re ν > 0]

ET II 381(16)

  x sinh(πx) Γ(λ + ix) Γ(λ − ix) K ix (a) K ix (b) dx = 2λ−1 π 3/2 (ab)λ (a + b)−λ Γ λ + 12 K λ (a + b) [|arg a| < π,

Re λ > 0,

b > 0] ET II 176(12)



∞

5.

x sinh(2πx) Γ(λ + ix) Γ(λ − ix) K ix (a) K ix (b) dx =

0



∞

6. 0



∞

7. 0

λ

5 2

ab 2λ π 1  K λ (|b − a|) |b − a| Γ 2 − λ a > 0, 0 < Re λ < 12 , b > 0



ET II 176(13)



    ab √ K 2λ x sinh(πx) Γ λ + 12 ix Γ λ − 12 ix K ix (a) K ix (b) dx = 2π 2 a2 + b 2 2 a2 + b 2  |arg a| < π2 , Re λ > 0, b > 0 x tanh(πx) K ix (a) K ix (b) 1  3 1   3 1  dx = 2 Γ 4 + 2 ix Γ 4 − 2 ix



ET II 177(14)

 πab 2 + b2 exp − a a2 + b 2  |arg a| < π2 ,

b>0 ,

ET II 177(15)

6.8 Functions Generated by Bessel Functions 6.81 Struve functions 6.811



∞

1.

Hν (bx) dx = −

0



∞

2.

Hν

0



∞

3. 0

a2 x

Hν−1

cot

 νπ  b

2

 Hν (bx) dx = − a x

2



√

J 2ν 2a b b

[−2 < Re ν < 0,

b > 0]

 a > 0,

Re ν > − 32

b > 0,

ET II 158(1)



ET II 170(37)

√

1 dx Hν (bx) = − √ J 2ν−1 2a b x a b

 a > 0,

b > 0,

Re ν > − 12



ET II 170(38)

6.812



1. 0

∞

H1 (bx) dx π [I 1 (ab) − L1 (ab)] = x2 + a2 2a

[Re a > 0,

b > 0]

ET II 158(6)

6.821

Combinations of Struve functions, exponentials, and powers



∞

2. 0

761

  

b cot νπ Hν (bx) 3 − ν 3 + ν a2 b 2 π 2

L ; ; 1; dx = − (ab) + F ν 2 1 νπ x2 + a2 1 − ν2 2 2 2 2a sin 2 [Re a > 0, b > 0, |Re ν| < 2] ET II 159(7)

6.813



∞

1. 0



∞

2.

s−1

x

  

2s−1 Γ s+ν s+ν 2  π tan Hν (ax) dx = s  1 1 2   a Γ 2ν − 2s + 1

3 , 1 − Re ν a > 0, −1 − Re ν < Re s < min 2

x−ν−1 Hν (x) dx =

0



∞

3.

x−μ−ν Hμ (x) Hν (x) dx =

0



 Re ν > − 32

1

xν+1 Hν (ax) dx =

1 Hν+1 (a) a

x1−ν Hν (ax) dx =

1 aν−1   √ 1 − a Hν−1 (a) ν−1 2 πΓ ν + 2

0 1

5. 0

ET II 383(2)

√ 2−μ−ν π Γ(μ + ν)       Γ μ + 12 Γ ν + 12 Γ μ + ν + 12 [Re(μ + ν) > 0]

4. 

2−ν−1 π Γ(ν + 1)

WA 429(2), ET I 335(52)

 a > 0,

WA 435(2), ET II 384(8)

Re ν > − 32



[a > 0] 6.814  1.

∞

(x2 +

0

6.815



1.

xν+1 Hν (bx)

1

ET II 158(3)a

2μ−1 πaμ+ν b−μ [I −μ−ν (ab) − Lμ+ν (ab)] Γ(1 − μ) cos[(μ + ν)π]  Re a > 0, b > 0, Re ν > − 32 , Re(μ + ν) < 12 , Re(2μ + ν) < 32

a2 )1−μ

ET II 158(2)a

dx =

ET II 159(8)

 √  1 x 2 ν (1 − x)μ−1 Hν a x dx = 2μ a−μ Γ(μ) Hμ+ν (a)

0

2.

 Re ν > − 32 , Re μ > 0 ET II 199(88)a 

 1  √  1 3 3 a2 3 B(λ, μ)aν+1   , ν + , λ + μ; − 1, λ; xλ− 2 ν− 2 (1 − x)μ−1 Hν a x dx = ν √ 2F 3 2 2 4 2 π Γ ν + 32 0 [Re λ > 0,

Re μ > 0]

ET II 199(89)a

6.82 Combinations of Struve functions, exponentials, and powers 6.821  1.6 0

∞



−n− 12  1 1  e−αx H−n− 12 (βx) dx = (−1)n β n+ 2 α + α2 + β 2 2 α + β2 [Re α > |Im β|]

ET I 206(6)

762

Functions Generated by Bessel Functions

 2.

∞

6 0





−n− 12  1 1  e−αx L−n− 12 (βx) dx = β n+ 2 α + α2 − β 2 2 α − β2

√

∞

6.822

e−αx H0 (βx) dx =

2 π

α2 +β 2 +β α

ln

[Re α > |Re β|]



ET I 208(26)

 [Re α > |Im β|] ET II 205(1) α2 + β 2

β  ∞ arcsin α 2 −αx 4. e L0 (βx) dx =  [Re α > |Re β|] ET II 207(18) π α2 + β 2 0   ∞  a

a

a

a  π (ν+1)x cosec(νπ) sinh I ν+ 12 − cosh I −ν− 12 6.822 e Hν (a sinh x) dx = a 2 2 2 2 0 [Re a > 0, −2 < Re ν < 0] 3.

0

ET II 385(11)

6.823



∞

1.

λ −αx

x e

0



∞

2.

 λ+ν +3 3 3 b2 λ+ν bν+1 Γ(λ + ν + 2)  3 F 2 1,

+ 1, ; ,ν + ;− 2 Hν (bx) dx = √ 3 2 2 2 2 a 2ν aλ+ν+2 π Γ ν + 2 [Re a > 0, b > 0, Re(λ + ν) > −2] ET II 161(19)

ν

xν e−αx Lν (βx) dx =

0

 

 Γ(2ν + 1) αβ (2β)ν Γ ν + 12 β −ν− 12  − P

2ν+1 −ν− 12 √  2 1 1   α π ν+ π α − β2 α β 2 − α2 2 4 2  Re α > |Re β|, Re ν > − 12 ET I 209(35)a

6.824  1.

∞

√

tν e−at L2ν 2 t dt =

0

6.825

a

1

ea Φ 2ν+1

1 √ a

 MI 51



1 1 1 1 a  − 2ν, t e e γ MI 51 2 a Γ 2 − 2ν a2ν+1 0  

  ∞ β ν+1 Γ 12 + 2s + ν2 ν+s+1 3 3 β2 s−1 −α2 x2  1, x e Hν (βx) dx = ν+1 √ ν+s+1  ; , ν + ; − F 2 2 2 2 2 4α2 2 πα Γ ν + 32 0 Re s > − Re ν − 1, |arg α| < π4

 2.

1

∞

ν −at

√

L−2ν t dt =

1

ET I 335(51)a, ET II 162(20)

6.83 Combinations of Struve and trigonometric functions 

6.831

∞



x−ν sin(ax) Hν (bx) dx = 0

0

=

√

 π2

−ν −ν

b

ν− 12 b 2 − a2   Γ ν + 12



0 < b < a,

Re ν > − 12

0 < a < b,

Re ν > − 12

ET II 162(21)

6.847

Combinations of Struve and Bessel functions



∞√

6.832



x sin(ax) H 14 b x

0

2 2



dx = −2

−3/2 √

π

√

a

b2

Y

1 4

763

 a2 4b2 [a > 0]

ET I 109(14)

6.84–6.85 Combinations of Struve and Bessel functions 

∞

6.841

 0 < b < a,  0 < a < b,

Hν−1 (ax) Y ν (bx) dx = −aν−1 b−ν

0

=0 

∞

6.842 0

6.843





 √  1 J 2ν a x Hν (bx) dx = − Y ν b

∞

 √  2 Γ(ν + 1) S −ν−1,ν K 2ν 2a x Hν (bx) dx = πb

0

2. 0

a2 4b

ν

1 2

|Re ν|
0]

 a > 0,

b > 0,

a b

2

ET II 114(36)

 |a − b| a+b [a > 0,



∞

1. 

4 K [H0 (ax) − Y 0 (ax)] J 0 (bx) dx = π(a + b)

|Re ν|
0,

Re ν > −1] ET II 168(27)

 

 ∞  √   √   √  μ−ν μ−ν π J μ a x − sin π Y μ a x K μ a x Hν (bx) dx cos 2 2 0

2

2 a a 1 W − 12 ν, 12 μ = 2 W 12 ν, 12 μ a 2b 2b  |arg a| < π4 , b > 0, Re ν > |Re μ| − 2 ET II 169(35)

 6.844

6.845  1.

∞

0



 a

a  √

4 − Y −ν J ν (bx) dx = cos(νπ) K 2ν 2 ab H−ν x x πb  |arg a| < π,

b > 0,

|Re ν|
0, b > 0, − 32 < Re ν < 0 ∞

0

 6.847 0

∞





ET II 170(39)



 √   √  2 a 1 K 2ν 2a x + Y 2ν 2a x Hν (bx) dx = J ν π b  b a > 0, 2

b > 0,

|Re ν|
0, Re k > 0,

1 2

ET II 169(30)

− 12 < Re ν < 2



ET II 384(5)a, WA 467(8)

764

6.848

Functions Generated by Bessel Functions



∞

1.

x [I ν (ax) − L−ν (ax)] J ν (bx) dx =

0



6.848

2 a ν−1 1 cos(νπ) 2 π b a + b2  Re a > 0,

b > 0,

−1 < Re ν < − 21



ET II 74(12) ∞

2.

x [H−ν (ax) − Y −ν (ax)] J ν (bx) dx = 2

0

cos(νπ) ν−1 1 b aν π a + b |arg a| < π,

− 21 < Re ν,

b>0



ET II 73(5)

6.849



∞

1.

x K ν (ax) Hν (bx) dx = a−ν−1 bν+1

0



∞

2. 0

6.851



∞

x [K μ (ax)]2 H0 (bx) dx = −2−μ−1 πa−2μ  z = 4a2 + b2

0

#

2  J 12 ν (ax) − Y

 Re a > 0,

1 + b2 

x

1.

a2

(z + b)

Re ν > − 23

2μ

2 $ 1 (ax) Hν (bx) dx = 0 2ν

 0 < b < 2a,  0 < 2a < b,

4 √ 2 πb b − 4a2



ET II 164(12)



+ (z − b) sec(μπ), bz  Re a > 0, b > 0, |Re μ| < 32 2μ

= 

b > 0,

ET II 166(18)

− 32 < Re ν < 0 − 32 < Re ν < 0



ET II 164(7) ∞

2.





xν+1 [J ν (ax)]2 − [Y ν (ax)]2 Hν (bx) dx

0



=0 −ν− 12 23ν+2 a2ν b−ν−1  2 1  b − 4a2 = √ πΓ 2 −ν



0 < b < 2a,

− 43 < Re ν < 0

0 < 2a < b,

− 43 < Re ν < 0



ET II 163(6)

6.852



∞

1.

x1−μ−ν J ν (x) Hμ (x) dx =

0

 2.

(2ν − 1)2−μ−ν     (μ + ν − 1) Γ μ + 12 Γ ν + 12 Re ν > 12 ,

Re(μ + ν) > 1

ET II 383(4)

∞

xμ−ν+1 Y μ (ax) Hν (bx) dx

0

=0 =

ν−μ−1 21+μ−ν aμ b−ν  2 b − a2 Γ(ν − μ)

 

0 < b < a,

Re(ν − μ) > 0,

− 32 < Re μ
0,

− 32 < Re μ
0, b > 0, Re ν > − 23 , Re(μ + ν) > − 32 ET II 165(13)

μ+ν+1

x

0

6.853  1.

∞

x1−μ [sin (μπ) J μ+ν (ax) + cos(μπ) Y μ+ν (ax)] Hν (bx) dx

0



=0

 μ−1 b ν b 2 − a2 = μ−1 μ+ν 2 a Γ(μ)





0 < b < a,

1 < Re μ < 32 ,

Re ν > − 23 ,

Re(ν − μ)
− 23 ,

Re(ν − μ)
− 32 , − Re ν − 52 < Re(λ − μ) < 1 ET II 76(21)

0

 

765

∞

3.

Re a > 0,

b > 0,

⎛

1

xλ+ 2 [Hμ (ax) − Y μ (ax)] J ν (bx) dx

0

! ⎞ !1−μ μ μ ! , 1 − , 1 + 2 2 1 cos(μπ) 3 ⎜b ! 2 ⎟ b−λ− 2 G 23 = 2λ+ 2 ⎠ 33 ⎝ 2 ! 3 π2 a ! + λ+ν , 1−μ , 3 + λ−ν ! 2 2 4 2 4 |arg a| < π, Re(λ + μ) < 1, Re(λ + ν) + 32 > |Re μ| ET II 73(6) ⎛

2

  4.

   2 ν− 1 −ν 1 a 2b √ x I ν− 12 (ax) − Lν− 12 (ax) J ν (bx) dx = 2 π a + b2  Re a > 0, b > 0,

∞√

0

 0

∞

6.



1. 0

1 2





a b 1 b2 1 2 μ−ν+1   F 1, ; ν − μ + ; − 2 x [I μ (ax) − Lμ (ax)] J ν (bx) dx = √ 2 2 a π Γ ν − μ + 12  1 −1 < 2 Re μ + 1 < Re ν + 2 , Re a > 0, b > 0 ET II 74(13) μ−ν+1 μ−1 ν−2μ−1



1 b2 1 2μ−ν+1 a−μ−1 bν−1    F 1, + μ; + ν; − 2 [I μ (ax) − L−μ (ax)] J ν (bx) dx =  1 2 2 a Γ 2 − μ Γ 12 + ν  1 Re a > 0, Re ν > − 2 , Re μ > −1, b > 0 ET II 75(18)

μ−ν+1

x

0

6.854

|Re ν|
0,

∞

xH

1 2ν



ax

2



K ν (bx) dx =

1

2ν + 1 1 1− 2 2 ν aπ

Γ



S

− 12 ν−1, 12 ν

 b2 4a [a > 0,

Re b > 0,

Re ν > −2] ET II 150(75)

766

Functions Generated by Bessel Functions



∞

2. 0



x H 12 ν ax

2



1 J ν (bx) dx = − Y 2a

1 2ν

b2 4a



 a > 0,

6.855

b > 0,

−2 < Re ν
0, |Re ν| < 12

∞

2. 0



1  a

a  √

√

3 aν+ 2 2 √ − Lν+ 12 J ν (bx) dx = 2 I ν+ 12 J 2ab K 2ab 2ν+1 2ν+1 x x πbν+1 Re a > 0, b > 0, −1 < Re ν < 12

ET II 74(8) ∞

3.

1

x2ν+ 2

0



∞

6.856 0

 a

a  − Y ν+ 12 J ν (bx) dx Hν+ 12 x x

√

√

1 1 1 2abe 4 πi K 2ν+1 2abe− 4 πi = −25/2 π −3/2 aν+ 2 b−ν−1 sin(νπ) K 2ν+1  |arg a| < π, b > 0, −1 < Re ν < − 61 ET II 74(9)

2  √   √  1 a x Y ν a x K ν a x Hν (bx) dx = 2 exp − 2b 2b b > 0,

|arg a|
− 32



ET II 169(32)

6.857



∞

1.

x exp

0

a2 x2 8



K 12 ν

k = 14 ν,  2. 0

∞

σ−2

x

a2 x2 8

m=

1 2

 Hν (bx) dx

2 

2 νπ 1  b b 2 − ν −1 ν −1 2 2 Γ − ν exp W k,m b cos =√ a 2 π  2 2 2a a2 1 3 3 + 4ν |arg a| < 4 π, b > 0, − 2 < Re ν < 0 ET II 167(24)





1 2 2 1 2 2 a x Hν (bx) dx exp − a x K μ 2 2     √ + μ Γ ν+σ −μ π −ν−σ ν+1 Γ ν+σ 2 2       b = ν+2 a 2 Γ 32 Γ ν + 32 Γ ν+σ 2

 ν+σ 3 3 ν+σ b2 ν +σ + μ, − μ; , ν + , ;− 2 × 3 F 3 1, 2 2 2 2 2 4a  b > 0, |arg a| < π4 , Re(σ + ν) > 2|Re μ| ET II 167(23)

6.866

Lommel functions

767

6.86 Lommel functions 6.861



∞

1.

λ−1

x

Γ

S μ,ν (x) dx =

1

2 (1

0

6.862 1.



u

12

   + λ + μ) Γ 12 (1 − λ − μ) Γ 12 (1 + μ + ν) Γ 12 (1 + μ − ν)   22−λ−μ Γ 12 (ν − λ) + 1 Γ 1 − 12 (λ + ν)  − Re μ < Re λ + 1 < 52 ET II 385(17)

 √  1 1 xλ− 2 μ− 2 (u − x)σ−1 s μ,ν a x dx

0

aμ uλ+σ Γ(λ + 1) (μ − ν + 1) (μ + ν + 1) Γ(λ + σ + 1)  a2 u μ−ν+3 μ+ν +3 , , λ + σ + 1; − × 2 F 3 1, 1 + λ; 2 2 4 [Re λ > −1, Re σ > 0] ET II 199(92)  1 1  ∞  √   √  B μ, 12 (1 − λ − ν) − μ u 2 μ+ 2 ν 1 ν μ−1 2 x (x − u) Sλ,ν a x dx = S λ+μ,μ+ν a u μ a u !  √ ! !arg a u ! < π, 0 < 2 Re μ < 1 − Re(λ + ν) ET II 211(71) = Γ(σ)

2.

 6.863

∞√

0

 6.864 0

6.865

xe−αx s μ, 14

x2 2



∞



2

√ α 3 dx = 2−2μ−1 α Γ 2μ + S −μ−1, 14 2  2 Re α > 0, Re μ > − 34

ET I 209(38)

exp[(μ + 1)x] s μ,ν (a sinh x) dx = 2μ−2 π cosec(μπ) Γ() Γ(σ) a

a   a a

Iσ − I − I −σ × I 2 2 2 2 2 = μ + ν + 1, 2σ = μ − ν + 1 [a > 0, −2 < Re μ < 0] ET II 386(22)    ∞√ μ−ν 1 B 14 − μ+ν 2 , 4 − 2 sinh x cosh(νx) S μ, 12 (a cosh x) dx = S μ+ 12 ,ν (a) √ μ+ 3 a2 2 0 |arg a| < π, Re μ + |Re ν| < 12 ET II 388(31)

6.866 1.



∞

12

x−μ−1 cos(ax) s μ,ν (x) dx

0

=0 =2  2. 0

μ− 12

√

πΓ

μ+ν +1 2

   1 μ+ 1 −μ− 1 μ−ν+1  Γ 1 − a2 2 4 P ν− 1 2 (a) 2 2

[a > 1] [0 < a < 1] ET II 386(18)

∞

 

 1 μ− 1 μ− 1 √ μ−ν  2 μ+ν −μ −μ− 12 Γ 1− a − 1 2 4 P ν− 12 (a) x sin(ax) S μ,ν (x) dx = 2 πΓ 1− 2 2 2 [a > 1, Re μ < 1 − |Re ν|]

ET II 387(23)

768

6.867

Functions Generated by Bessel Functions



π/2

6.867

cos(2μx) S 2μ−1,2ν (a cos x) dx

1. 0

a

a

a

a  π22μ−3 a2μ cosec(2νπ)  J μ+ν Y μ−ν − J μ−ν Y μ+ν Γ(1 − μ − ν) Γ (1 − μ + ν) 2 2 2 2 [Re μ > −2, |Re ν| < 1] ET II 388(29)  π/2 a

a

Jσ cos [(μ + 1) x] s μ,ν (a cos x) dx = 2μ−2 π Γ() Γ(σ) J  2 2 0 2 = μ + ν + 1, 2σ = μ − ν + 1 [Re μ > −2] ET II 386(21) =

2.

 6.868

π/2

0

6.869



∞

1. 0



x

π

π

cos(2μx) π22μ−1 S 2μ,2ν (a sec x) dx = W μ,ν aei 2 W μ,ν ae−i 2 cos x a [|arg a| < π, Re μ < 1]

1−μ−ν

√ ν−1  12 (μ+ν−1) μ+ν−1 πa Γ(1 − μ − ν)  2   a J ν (ax) S μ,−μ−2ν (x) dx = − 1 P μ+ν (a) 1 2μ+2ν Γ ν + 2  a > 1, Re ν > − 12 , Re(μ + ν) < 1 ET II 388(28)

∞

2. 0

x−μ J ν (ax) s ν+μ,−ν+μ+1 (x) dx  μ = 2ν−1 Γ(ν)a−ν 1 − a2 =0



ET II 388(30)

 

0 < a < 1, 1 < a,

Re μ > −1,

Re μ > −1,

−1 < Re ν < −1 < Re ν < 32

3 2



ET II 388(28)

∞

3. 0

 

2  2 1 1 b 1 Γ μ + ν + 1 Γ μ − ν + 1 S −μ−1, 12 ν x K ν (bx) s μ, 12 ν ax dx = 4a 2 2 4a  Re μ > 12 |Re ν| − 2, a > 0, Re b > 0 ET II 151(78)

6.87 Thomson functions 6.871

1/2 β4 + 1 + β2  e−βx ber x dx = 2 (β 4 + 1) 0

1/2   ∞ β4 + 1 − β2  e−βx bei x dx = 2 (β 4 + 1) 0

 1.

2.

∞



ME 40

ME 40

6.874

6.872

Thomson functions



∞

1. 0



∞

2.

⎡



 π⎣ 1 3νπ 1 cos + J 12 (ν−1) β 2β 2β 4 ⎤



 3ν + 6 ⎦ 1 1 cos + π − J 12 (ν+1) 2β 2β 4

 √  1 e−βx berν 2 x dx = 2β

e

−βx

0

 3. 4. 5. 6. 7.

2.

3. 4.

⎡

MI 49

 1 3ν sin + π 2β 4 ⎤



 3ν + 6 ⎦ 1 1 sin + π − J 12 (ν+1) 2β 2β 4

 √  1 beiν 2 x dx = 2β



π⎣ J 12 (ν−1) β

1 2β



MI 49

 √  1 1 e−βx ber 2 x dx = cos β β 0  ∞   √ 1 1 e−βx bei 2 x dx = sin β β 0    ∞   √ 1 1 1 1 1 cos ci + sin si e−βx ker 2 x dx = − 2β β β β β 0    ∞  √  1 1 1 1 1 −βx sin ci − cos si e kei 2 x dx = − 2β β β β β 0



  ∞     √ √ 3νπ 2 2 1 Jν sin + e−βx berν 2 x beiν 2 x dx = 2β β β 2 0

 ∞  √   √  1 2 ber2ν 2 x + bei2ν 2 x e−βx dx = I ν β β 0

6.873

1.



∞



6.874

769

ME 40 MI 50 MI 50

[Re ν > −1]

MI 49

[Re ν > −1]

ME 40



  √

e π 1 3π 3νπ 1 √ ber2ν 2 2x dx = Jν cos − + x β β β 4 2 0  Re ν > − 12



  ∞ −βx  √

e π 1 3π 3νπ 1 √ bei2ν 2 2x dx = Jν sin − + β β β 4 2 x 0  Re ν > − 12 

 ∞ √  −βx ν 3νπ 2−ν 1 + [Re ν > −1] x 2 berν x e dx = 1+ν cos β 4β 4 0 

 ∞ √  −βx ν 3νπ 2−ν 1 2 + [Re ν > −1] x beiν x e dx = 1+ν sin β 4β 4 0



ME 40

∞ −βx

MI 49

MI 49 ME 40 ME 40

770

6.875 1. 2. 6.876 1. 2.

Mathieu Functions

6.875

     √   √  1 1 π 1 1 ln β cos + sin ker 2 x − ln x ber 2 x dx = e 2 β β 4 β 0      ∞  √   √  1 1 π 1 1 ln β sin − cos e−βx kei 2 x − ln x bei 2 x dx = 2 β β 4 β 0



∞



∞

−βx

1 arctan a2 2a 0  ∞  1 ln (1 + a4 ) x ker x J 1 (ax) dx = 2a 0 x kei x J 1 (ax) dx = −

MI 50 MI 50

[a > 0]

ET II 21(32)

[a > 0]

ET II 21(33)

6.9 Mathieu Functions (m)

Notation: k 2 = q. For definition of the coefficients Ap

(m)

and Bp

see section 8.6.

6.91 Mathieu functions 6.911



2π

cem (z, q) cep (z, q) dz = 0

1.

[m = p]

MA

0



2π

2.

∞  2 2  " (2n) (2n) 2 A2r [ce2n (z, q)] dz = 2π A0 +π =π

0



2π

2

[ce2n+1 (z, q)] dz = π

3. 0



MA

r=1 ∞  2 " (2n+1) A2r+1 =π

MA

r=0 2π

sem (z, q) sep (z, q) dz = 0

4.

[m = p]

MA

0



2π

2

[se2n+1 (z, q)] dz = π

5. 0



MA

r=0 2π

2

[se2n+2 (z, q)] dz = π

6. 0



∞  2 " (2n+1) B2r+1 =π ∞  2 " (2n+2) B2r+2 =π

MA

r=0 2π

sem (z, q) cep (z, q) dz = 0

7.

[m = 1, 2, . . . ;

p = 1, 2, . . .]

MA

0

6.92 Combinations of Mathieu, hyperbolic, and trigonometric functions 6.921



1. 0

π

(2n)

cosh (2k cos u sinh z) ce2n (u, q) du =

πA0   (−1)n Ce2n (z, −q) ce2n π2 , q [q > 0]

MA

6.922

Mathieu, hyperbolic, and trigonometric functions



π

2.

(2n)

cosh (2k sin u cosh z) ce2n (u, q) du =

0



πA0 (−1)n Ce2n (z, −q) ce2n (0, q) [q > 0]

π

3.

πkB1 (−1)n Ce2n+1 (z, −q) se2n+1 (0, q) [q > 0]

π

4.

MA

(2n+1)

sinh (2k cos u sinh z) ce2n+1 (u, q) du =

0



MA

(2n+1)

sinh (2k sin u cosh z) se2n+1 (u, q) du =

0



771

πkA1   (−1)n+1 Se2n+1 (z, −q) ce2n+1 π2 , q [q > 0]

π

5. 0

MA

(2n+1)

sinh (2k sin u sin z) se2n+1 (u, q) du =

πkB1 se2n+1 (z, q) se2n+1 (0, q) [q > 0]

6.922



π

1.

(2n+1)

πA1 Ce2n+1 (z, q) 2 ce2n+1 (0, q)

cos u cosh z cos (2k sin u sinh z) ce2n+1 (u, q) du =

0



[q > 0] π

2.

sin u sinh z cos (2k cos u cosh z) se2n+1 (u, q) du =

0



π

3.

sin u sinh z sin (2k cos u cosh z) se2n+2 (u, q) du =

0



π

0

πB1 (2n+1) π Se2n+1 (z, q) ,q 2 se2n+1 2 [q > 0] (2n+2) πkB2   − 2 se2n+2 π2 , q

πkB2 Se2n+2 (z, q) 2 se2n+2 (0, q)

sin u cosh z cosh (2k cos u sinh z) se2n+1 (u, q) du =

cos u sinh z cosh (2k sin u cosh z) ce2n+1 (u, q) du =

0

7. 0

πB1

MA

(2n+1)

2 se2n+1

π 2

,q

(−1)n Ce2n+1 (z, −q)

[q > 0]

6. 

MA

(2n+2)

cos u cosh z sin (2k sin u sinh z) se2n+2 (u, q) du =

0

π

MA

(2n+1) πA1

2 ce2n+1 (0, q)

(−1)n Se2n+1 (z, −q)

[q > 0] π

MA

Se2n+2 (z, q)

[q > 0] π

5.



MA

[q > 0]

4. 

MA

sin u cosh z sinh (2k cos u sinh z) se2n+2 (u, q) du =

MA

(2n+2) πkB2 π (−1)n+1  ,q 2 se2n+2

Se2n+2 (z, −q)

2

[q > 0]

MA

772

Mathieu Functions



π

8.

6.923

(2n+2)

cos u sinh z sinh (2k sin u cosh z) se2n+2 (u, q) du =

0

πkB2 (−1)n Se2n+2 (z, −q) 2 se2n+2 (0, q) [q > 0]

6.923



∞

1.

πB1 (2n+1)   Se2n+1 (z, q) 4 se2n+1 12 π, q

sin (2k cosh z cosh u) sinh z sinh u Se2n+1 (u, q) du = −

0



[q > 0] ∞

2.

cos (2k cosh z cosh u) sinh z sinh u Se2n+1 (u, q) du = −

0

 3.

[q > 0] ∞

4.

cos (2k cosh z cosh u) sinh z sinh u Se2n+2 (u, q) du = −

0



∞

πA0   Ce2n (z, q) 2 ce2n 12 π, q

cos (2k cosh z cosh u) Ce2n (u, q) du =

(2n) πA0 1  − 2 ce2n 2 π, q

[q > 0] ∞

6. 0



∞

kπA1   Fey2n+1 (z, q) 2 ce2n+1 12 π, q

cos (2k cosh z cosh u) Ce2n+1 (u, q) du =

(2n+1) kπA1   2 ce2n+1 12 π, q

[q > 0] ∞

8. 0



π

1. 0

 2. 0

MA

(2n)

cos (2k cos u cos z) ce2n (u, q) du =

πA0 1

ce2n

2 π, q

 ce2n (z, q) [q > 0]

π

MA

Ce2n+1 (z, q)

[q > 0] 6.924

MA

(2n+1)

sin (2k cosh z cosh u) Ce2n+1 (u, q) du =

0



MA

Fey2n (z, q)

[q > 0]

7.

MA

(2n)

sin (2k cosh z cosh u) Ce2n (u, q) du =

0



MA

kπB2 (2n+2)   Se2n+2 (z, q) 4 se2n+2 12 π, q

[q > 0]

5.

MA

kπB2 (2n+2)   Gey2n+2 (z, q) 4 se2n+2 12 π, q

sin (2k cosh z cosh u) sinh z sinh u Se2n+2 (u, q) du = −

0



MA

πB1 (2n+1)   Gey2n+1 (z, q) 4 se2n+1 12 π, q

[q > 0] ∞

MA

MA

(2n+1)

sin (2k cos u cos z) ce2n+1 (u, q) du = −

πkA1   ce2n+1 (z, q) ce2n+1 12 π, q [q > 0]

MA

6.926

Mathieu, hyperbolic, and trigonometric functions



π

3.

773

(2n)

cos (2k cos u cosh z) ce2n (u, q) du =

0

πA0 1

ce2n

2 π, q

 Ce2n (z, q) [q > 0]



π

4.

MA

(2n)

cos (2k sin u sinh z) ce2n (u, q) du =

0

πA0 Ce2n (z, q) ce2n (0, q) [q > 0]



π

5.

(2n+1) πkA1   −  ce2n+1 12 π, q

sin (2k cos u cosh z) ce2n+1 (u, q) du =

0

MA

Ce2n+1 (z, q)

[q > 0] 

π

6.

MA

(2n+1)

sin (2k sin u sinh z) se2n+1 (u, q) du =

0

πkB1 Se2n+1 (z, q) se2n+1 (0, q) [q > 0]

6.925 1.

MA

 Notation: z1 = 2k cosh2 ξ − sin2 η, and tan α = tanh ξ tan η  2π sin [z1 cos(θ − α)] ce2n (θ, q) dθ = 0.

MA

0



(2n)

2π

cos [z1 cos(θ − α)] ce2n (θ, q) dθ =

2. 0



2πA0   Ce2n (ξ, q) ce2n (η, q) ce2n (0, q) ce2n 12 π, q (2n+1)

2π

sin [z1 cos(θ − α)] ce2n+1 (θ, q) dθ = −

3.

MA

0

2πkA1   Ce2n+1 (ξ, q) ce2n+1 (η, q) ce2n+1 (0, q) ce2n+1 12 π, q MA



2π

cos [z1 cos(θ − α)] ce2n+1 (θ, q) dθ = 0

4.

MA

0



(2n+1)

2π

sin [z1 cos(θ − α)] se2n+1 (θ, q) dθ =

5. 0

2πkB1   Se2n+1 (ξ, q) se2n+1 (η, q) se2n+1 (0, q) se2n+1 12 π, q MA



2π

6.

cos [z1 cos(θ − α)] se2n+1 (θ, q) dθ = 0

MA

sin [z1 cos(θ − α)] se2n+2 (θ, q) dθ = 0

MA

0



2π

7. 0



2π

cos [z1 cos(θ − α)] se2n+2 (θ, q) dθ =

8. 0

2πk 2 B2 (2n+2)    se2n+2 (0, q) se2n+2 12 π, q

Se2n+2 (ξ, q) se2n+2 (η, q) MA

 6.926 0

π

(2n+2)

sin u sin z sin (2k cos u cos z) se2n+2 (u, q) du = −

πkB2   se2n+2 (z, q) 2 se2n+2 π2 , q [q > 0]

MA

774

Mathieu Functions

6.931

6.93 Combinations of Mathieu and Bessel functions 6.931

2  (2n) π A0 π ce2n (z, q) 0 ,q ce2n (0, q) ce2n 2 2  (2n)  2π   2π A0 1/2 π Fey2n (z, q) ce2n (u, q) du = Y 0 k [2 (cos 2u + cosh 2z)] 0 ce2n (0, q) ce2n ,q 2

 1.

2.

π

  J 0 k [2 (cos 2u + cos 2z)]1/2 ce2n (u, q) du =

MA

MA

6.94 Relationships between eigenfunctions of the Helmholtz equation in different coordinate systems Notation: Particular solutions of the Helmholtz equation in three-dimensional infinite space ∇2 Ψ + k 2 Ψ = 0 in Cartesian (x, y, z), spherical (r, θ, φ), and cylindrical (ρ, z, φ) coordinates are Ψkx ky kz (x, y, z) ∝ ei(kx x+ky y+kz z) with k 2 = kx2 + ky2 + kz2  k imφ Z l+1/2 (kr) P m Ψlm (r, θ, φ) ∝ e l (cos θ) r 

Ψmkz (ρ, z, φ) ∝ ei(mφ+kz z) Z l+1/2 ρ k 2 − kz2 with P m l (cos θ) the associated Legendre function, Z is any Bessel function, m = 0, 1, . . . , l; l ∈ N, r2 = ρ2 + z 2 , ρ = r sin θ, z = r cos θ, φ = arccot(x/y), and kt2 = k 2 − kz2 . 6.941   k 



z

2πk m p iρz l−m 2 2 dp = i J l+1/2 (kr) P m e J m ρ k − ρ Pl 1. l k r r −k 

∞

2. −∞

 3. 0

∞

e−iρz J l+1/2 (kr) P m l

z

r

 dz = im−l

[ρ > 0, l ≥ m ≥ 0]

ρ

 2πr J m ρ k 2 − ρ2 P m l k k

[ρ > 0, l ≥ m ≥ 0]

 x dx J m (ρkt ) cos kx x + m arcsin ρ  

 (−1)m kx 2 2 = cos y kt − kx + m arccos 2 2 kt kt − kx 

=0

 2 kx < kt2  2 kx > kt2

6.941

Eigenfunctions of Helmholtz equation

 x dx Y m (ρkt ) cos kx x + m arcsin 0 ρ 

 m (−1) kx 2 2 =  sin y kt − kx + m arccos 2 2 kt kt − kx   

(−1)m |kx | 2 2 =  exp −y kx − kt − m sign (kx ) arccosh kt kx2 − kt2 

  ∞ z

2πr (j)  2 kz (j) −ikz z m−l 2 Pm e H ρ H l+1/2 (kr) P m dz = i k − k z l m l r k k −∞ 

4.

5.

6.

7.

8. 9.

10.

∞

775



 2 kx < kt2  2 kx > kt2

[ρ > 0]   The result is true for j = 1 if π > arg k 2 − kz2 ≥ 0, for j = 2 if −π < arg k 2 − kz2 ≤ 0. 

  ∞ 

z

2πk (j) kz (j) m ikz z l−m 2 2 e H l+1/2 (kr) P m H m ρ k − kz P l dkz = i l k r r −∞   The result is true for j = 1 if π > arg k 2 − kz2 ≥ 0, for j = 2 if −π < arg k 2 − kz2 ≤ 0. 

  ∞



 2 2πr kz m z −ikz z m−l e J m ρ k 2 − kz2 P m kz < k 2 J l+1/2 (kr) P l dz = i l r k k −∞  2 =0 kz > k 2 

  k 

z

2πk kz m ikz z l−m 2 2 e J l+1/2 (kr) P m J m ρ k − kz P l dkz = i l k r r −k  

 ∞



 2 2πr kz m z m −ikz z m−l 2 2 e Y m ρ k − kz P l kz < k 2 Y l+1/2 (kr) P l dz = i r k k −∞ 

 

 2 2r kz K m ρ kz2 − k 2 P m kz > k 2 = −2im−l l kπ k 

 k 

kz eikz z dkz il−m Y m ρ k 2 − kz2 P m l k −k

  

 m kz 4 ∞ 1 K m ρ kz2 − k 2 eikz z dkz − cos kz z + 2 π(m − l) P l π k k  z

2πk Y l+1/2 (kr) P m = l r r

776

Associated Legendre Functions

7.111

7.1–7.2 Associated Legendre Functions 7.11 Associated Legendre functions 

7.111 7.112

1

cos ϕ



1

1. −1

P ν (x) dx = sin ϕ P −1 ν (cos ϕ)

m Pm n (x) P k (x) dx = 0

= 

[n = k]

2 (n + m)! 2n + 1 (n − m)!

[n = k] SM III 185, WH

1

2. −1



MO 90

1

3. −1

m m Qm n (x) P k (x) dx = (−1)

1 − (−1)n+k (n + m)! (k − n)(k + n + 1)(n − m)!

P ν (x) P σ (x) dx 2π sin π(σ − ν) + 4 sin(πν) sin(πσ) [ψ(ν + 1) − ψ(σ + 1)] π 2 (σ − ν) (σ + ν + 1) 2  2 π − 2 (sin πν) ψ (ν + 1)   = 1 2 π ν+ 2

=



1

4. −1

EH I 171(18)

Q ν (x) Q σ (x) dx =

=

[σ + ν + 1 = 0] [σ = ν]

EH I 170(7)

EH I 170(9)a

[ψ(ν + 1) − ψ(σ + 1)] [1 + cos(πσ) cos(νπ)] − π2 sin π(ν − σ) (σ − ν) (σ + ν + 1) [σ + ν + 1 = 0; ν, σ = −1, −2, −3, . . .] 1 2 2π

EH I 170(11)

  2 − ψ (ν + 1) 1 + (cos νπ) 

2ν + 1

[ν = σ,

ν = −1, −2, −3, . . .] EH I 170(12)



1

5. −1

P ν (x) Q σ (x) dx =

1 − cos π(σ − ν) − 2π −1 sin(πν) cos(πσ) [ψ(ν + 1) − ψ(σ + 1)] (ν − σ) (ν + σ + 1) [Re ν > 0, Re σ > 0,

=−

sin(2νπ) ψ  (ν + 1) π(2ν + 1)

σ = ν]

EH I 170(13)

[Re ν > 0,

σ = ν]

EH I 171(14)

7.113

Notation: 

1. 0

1

    Γ 12 + ν2 Γ 1 + σ2  A = 1 σ  Γ 2 + 2 Γ 1 + ν2

P ν (x) P σ (x) dx =

πν πσ −1 A sin πσ sin πν 2 cos 2 − A 2 cos 2 1 2 π(σ − ν)(σ + ν + 1)

EH I 171(15)

7.124

2.12

Associated Legendre functions and powers



1

Q ν (x) Q σ (x) dx =

0



1

P ν (x) Q σ (x) dx =

3. 0

7.114



ψ(ν + 1) − ψ(σ + 1) −

∞

1. 1

π 2



777

    π(σ−ν) −1 A − A−1 sin π(σ+ν) sin − A + A 2 2

(σ − ν)(σ + ν + 1) [Re ν > 0, Re σ > 0] A−1 cos π(ν−σ) −1 2 (σ − ν)(σ + ν + 1)

P ν (x) Q σ (x) dx =

1 (σ − ν)(σ + ν + 1)

[Re ν > 0,

Re σ > 0]

[Re(σ − ν) > 0,

EH I 171(16) EH I 171(17)

Re(σ + ν) > −1] ET II 324(19)



∞

2. 1



Q ν (x) Q σ (x) dx =

∞

3. 

7.115

ψ  (ν + 1) 2ν + 1

2

[Q ν (x)] dx =

1

∞

1

Q ν (x) dx =

ψ(σ + 1) − ψ(ν + 1) (σ − ν)(σ + ν + 1) [Re(ν + σ) > −1;

1 ν(ν + 1)

σ, ν = −1, −2, −3, . . .]



Re ν > − 12

EH I 170(5)

EH I 170(6)

[Re ν > 0]

ET II 324(18)

7.12–7.13 Combinations of associated Legendre functions and powers 7.12112 7.122



1

cos ϕ





sin ϕ sin ϕPν (cos ϕ) + cos ϕ P 1ν (cos ϕ) (ν − 1)(ν + 2)

2

1

[P m 1 (n + m)! n (x)] dx = 1 − x2 2m (n − m)!

1. 0



x P ν (x) dx =

1

[P μν (x)]2

2. 0

dx Γ(1 + μ + ν) =− 2 1−x 2μ Γ(1 − μ + ν)

MO 90

[0 < m ≤ n] [Re μ < 0,

MO 74

ν + μ is a positive integer] EH I 172(26)



1



3. 0

2 P n−ν (x) ν

dx n! =− 2 1−x 2(n − ν) Γ(1 − n + 2ν)

[n = 0, 1, 2, . . . ;

Re ν > n] ET II 315(9)

 7.123

7.12412

1

−1



k Pm n (x) P n (x)

dx =0 1 − x2

[0 ≤ m ≤ n,

0 ≤ k ≤ n;

m = k] MO 74

1

−1

 1m  2  12 m m xk (z − x)−1 1 − x2 2 P m Q n (z)z k n (x) dx = 2 z − 1

[m ≤ n; k = 0, 1, . . . , n − m;

z is in the complex plane with a cut along the interval (−1, 1) on the real axis] ET II 279(26)

778

Associated Legendre Functions

 7.125

1



1 − x2

−1

 12 m

7.125

m m m −3/2 (k + m)!(l + m)! (n + m)!(s − m)! Pm k (x) P l (x) P n (x) dx = (−1) π   (k − m)!(l − m)!(n  − m)!(s  −  k)!  Γ m + 12 Γ t − k + 12 Γ t − l + 12 Γ t − n + 12   × (s − l)!(s − n)! Γ s + 32 [2s = k + l + n + m and 2t = k + l − n − m are both even l ≥ m, m ≤ k − l − m ≤ n ≤ k + l + m]

ET II 280(32)

7.126  1.

1

0



1

2. 0

√

P ν (x)xσ dx =



xσ P m ν (x) dx =

> −1]

EH I 171(23)

ET II 313(2)

    π 1/2 22μ−1 Γ 1+σ μ 3+σ−μ ν −μ+1 μ+ν 2   3+σ−μ  3 F 2  ,− , 1 − ; 1 − μ, ;1 x 2 2 2 2 Γ Γ 1−μ 0 2 2 [Re σ > −1, Re μ < 2] ET II 313(3)  ∞ 1   μ xμ−1 Q ν (ax) dx = eμπi Γ(μ)a−μ a2 − 1 2 Q −μ ν (a) 

4.



 3+σ+m m+ν+1 m−ν m , , + 1; m + 1, ;1 2 2 2 2 [Re σ > −1; m = 0, 1, 2, . . .]

× 3F 2

3.

π2−σ−1 Γ(1 + σ)  

[Re σ Γ 1 + 12 σ − 12 ν Γ 12 σ + 12 ν + 32   (−1)m π 1/2 2−2m−1 Γ 1+σ Γ(1 + m + ν) 2 1 1  3 σ m Γ 2 +2 m Γ 2 + 2 + 2 Γ(1 − m + ν)

1

σ

P μν (x) dx =

1

[|arg(a − 1)| < π,

 7.127 7.128  1.

1

−1 1

(1 + x)σ P ν (x) dx = 1

1

2

2σ+1 [Γ(σ + 1)] Γ(σ + ν + 2) Γ(1 + σ − ν)

μ− 1

Re μ > 0,

Re(ν − μ) > −1]

[Re σ > −1]

ET II 325(26)

ET II 316(15)

μ− 3

(1 − x)− 2 μ (1 + x) 2 2 (z + x) 2 P μν (x) dx −1   1 −1/2 Γ μ − 12 (z − 1)μ− 2 (z + 1) = − 1/2 2μπi π e Γ (μ + ν) Γ(μ − ν − 1)    1/2 1/2 1/2 1/2    1 + z 1+z 1+z 1+z μ−1 μ μ μ−1 Q −ν−1 + Qν Q −ν−1 × Qν 2 2 2 2 [− 12 < Re μ < 1,

z is in the complex plane with a cut along the interval (−1, 1) of the real axis] 

ET II 317(20) 1

2. −1

1

1

μ− 12

μ− 1

(z + x) 2 P μν (x) dx     1/2 1/2 2e−2μπi Γ 12 + μ 1+z 1+z μ μ μ Q −ν−1 (z − 1) Q ν = 1/2 2 2 π Γ(μ − ν) Γ (μ + ν + 1) [− 12 < Re μ < 1,

(1 − x)− 2 μ (1 + x) 2

z is in the complex plane with a cut along the interval (−1, 1) of the real axis] ET II 316(18)

7.132

Associated Legendre functions and powers

 7.129

1

−1

7.131  1.

∞

1

P ν (x) P λ (x)(1 + x)λ+ν dx =

1

1

(x − 1)− 2 μ (x + 1) 2

μ− 12

779

4

2λ+ν+1 [Γ(λ + ν + 1)]

[Γ(λ + 1) Γ(ν + 1)]2 Γ(2λ + 2ν + 2) [Re(ν + λ + 1) > 0] μ− 12

(z + x)

EH I 172(30)

P μν (x) dx

  1/2 2 − ν) Γ (1 − μ + ν) 1+z μ μ   Pν (z − 1) =π 2 Γ 12 − μ [Re(μ + ν) < 0, Re(μ − ν) < 1, |arg(z + 1)| < π] ET II 321(6) 1/2 Γ(−μ

 2.

∞

7.132 12

2.

3.

μ− 32

(z + x)

P μν (x) dx

9

    π2μ Γ λ + 12 μ Γ λ − 12 μ         1−x Γ λ + 12 ν + 1 Γ λ − 12 ν Γ − 12 μ + 12 ν + 1 Γ − 12 μ − 12 ν + 12 −1 ET II 316(16) [2 Re λ > |Re μ|]        ∞  2 λ−1 μ 2μ−1 Γ λ − 12 μ Γ 1 − λ + 12 ν Γ 12 − λ − 12 ν      x −1 P ν (x) dx =  Γ 1 − 12 μ + 12 ν Γ 12 − 12 μ − 12 ν Γ 1 − λ − 12 μ 1 [Re λ > Re μ, Re(1 − 2λ − ν) > 0, Re(2 − 2λ + ν) > 0] ET II 320(2) 

1



∞





1

4. 0



1

5. 0

12

 0

1

 2 λ−1



2

x −1

1

6.

μ− 12

  1/2 1/2 1 −1/2 1+z 1+z π 1/2 Γ(1 − μ − ν) Γ (2 − μ + ν) (z − 1)μ− 2 (z + 1) μ μ−1   Pν = Pν 2 2 Γ 32 − μ [Re μ < 1, Re(μ + ν) < 1, Re(μ − ν) < 2, |arg(1 + z)| < π] ET II 321(7) 1

1.

1

1

(x − 1)− 2 μ (x + 1) 2

P μν (x) dx =

λ−1

Q μν (x) dx

=e

μπi Γ

 − 1 μ xσ 1 − x2 2 P μν (x) dx =



1 2

Γ 1+

       + 12 ν + 12 μ Γ 1 − λ + 12 ν Γ λ + 12 μ Γ λ − 12 μ     22−μ Γ 1 + 12 ν − 12 μ Γ 12 + λ + 12 ν [|Re μ| < 2 Re λ < Re ν + 2] 2 1 2σ

μ−1

−

Γ

1 2ν

1

 

ET II 324(23)



1 1 2 + 2 σ Γ 1 + 2 σ − 12 μ Γ 12 σ + 12 ν − 12 μ

+

3 2



[Re μ < 1, Re σ > −1] EH I 172(24)    1 1  1m (−1)m 2−m−1 Γ 2 + 2 σ Γ 1 + 2 σ Γ (1 + m + ν)     xσ 1 − x2 2 P m ν (x) dx = Γ(1 − m + ν) Γ 1 + 12 σ + 12 m − 12 ν Γ 32 + 12 σ + 12 m + 12 ν [Re σ > −1, m is a positive integer] EH I 172(25), ET II 313(4) 1

σ

x



 2 η

1−x

P μν (x) dx =

    2μ−1 Γ 1 + η − 12 μ Γ 12 + 12 σ   Γ(1 −μ) Γ 32 + η + 12 σ − 12 μ  μ 3+σ−μ ν −μ+1 μ+ν ,− , 1 + η − ; 1 − μ, + η; 1 × 3F 2 2 2

2  2 Re η − 12 μ > −1, Re σ > −1 ET II 314(6)

780

Associated Legendre Functions



∞

7. 1

7.133



x

∞

u

∞

∞

1

∞

2. 1

7.135



1

1. −1

2.

P μν (x) dx

=

 ρ+μ+ν   ρ+μ−ν−1  Γ 2 √2 π Γ(ρ) Re(ρ + μ + ν) > 0, Re(ρ + μ − ν) > 1]

2ρ+μ−2 Γ



x2 − 1

 12 λ

ET II 320(3)

0 < Re μ < 1 + Re ν]

MO 90a

  1 λ+ 1 μ μ−1 Q −λ dx = Γ(μ)eμπi u2 − 1 2 2 Q −λ−μ (u) ν (x)(x − u) ν [|arg(u − 1)| < π,

1.



x −1

− 12 μ

 1μ Q ν (x)(x − u)μ−1 dx = Γ(μ)eμπi u2 − 1 2 Q −μ ν (u)

u



2

[|arg(u − 1)| < π,

2.

7.134



[Re μ < 1,

1.



−ρ

7.133

0 < Re μ < 1 + Re(ν − λ)]

 1μ 2λ+μ Γ(λ) Γ(−λ − μ − ν) Γ(1 − λ − μ + ν) (x − 1)λ−1 x2 − 1 2 P μν (x) dx = Γ(1 − μ + ν) Γ(−μ − ν) Γ(1 − λ − μ) [Re λ > 0, Re(λ + μ + ν) < 0, Re(λ + μ − ν) < 1]

ET II 204(30)

ET II 321(4)

 − 1 μ 2λ−μ sin πν Γ(λ − μ) Γ(−λ + μ − ν) Γ(1 − λ + μ + ν) (x − 1)λ−1 x2 − 1 2 P μν (x) dx = − π Γ(1 − λ) [Re(λ − μ) > 0, Re(μ − λ − ν) > 0, Re(μ − λ + ν) > −1] ET II 321(5)



1 − x2

− 12 μ

 − 1 μ (z − x)−1 P μμ+n (x) dx = 2e−iμπ z 2 − 1 2 Q μμ+n (z)

[n = 0, 1, 2, . . . , Re μ + n > −1, z is in the complex plane with a cut along the interval (−1, 1) of the real axis.] ET II 316(17)  ∞  μ/2 −ρ (x − 1)λ−1 x2 − 1 (x + z) P μν (x) dx 1

=

2λ+μ−ρ Γ(λ − ρ) Γ(ρ − λ − μ − ν) Γ (ρ − λ − μ + ν + 1) Γ(1 − μ + ν) Γ(−μ − ν) Γ (1 + ρ − λ − μ)

 1+z × 3 F 2 ρ, ρ − λ − μ − ν, ρ − λ − μ + ν + 1; ρ − λ + 1, ρ − λ − μ + 1; 2   1+z Γ(ρ − λ) Γ(λ) μ λ−ρ 2 (z + 1) λ, −μ − ν, 1 − μ + ν; 1 − μ, 1 − ρ + λ; + 3F 2 Γ(ρ) Γ(1 − μ) 2 [Re λ > 0, Re(ρ − λ − μ − ν) > 0, Re(ρ − λ − μ + ν + 1) > 0, |arg(z + 1)| < π] ET II 322(9)

7.137

Associated Legendre functions and powers



∞

3. 1

781

 −μ/2 (x − 1)λ−1 x2 − 1 (x + z)−ρ P μν (x) dx =−

sin(νπ) Γ(λ − μ − ρ) Γ (ρ − λ + μ − ν) Γ(ρ − λ + μ + ν + 1) 2ρ−λ+μ π Γ (1 + ρ − λ) 

 1+z × 3 F 2 ρ, ρ − λ + μ − ν, ρ − λ + μ + ν + 1; 1 + ρ − λ, 1 + ρ − λ + μ; 2 Γ(λ − μ) Γ (ρ − λ + μ) λ−ρ−μ (z + 1) + Γ(ρ)   Γ(1 − μ) 1+z × 3 F 2 λ − μ, −ν, ν + 1; 1 + λ − μ − ρ, 1 − μ; 2 [Re(λ − μ) > 0, Re (ρ − λ + μ − ν) > 0, Re(ρ − λ + μ + ν + 1) > 0, |arg(z + 1)| < π] ET II 322(10)

7.136  1.

1



1 − x2

λ−1 

1 − a2 x2

P ν (ax) dx

  μ+ν 1−μ+ν 1 π2μ Γ(λ) 2  1 1    , ; + λ; a − 2F 1 2 2 2 Γ 2 + λ Γ 2 − 2 μ − 12 ν Γ 1 − 12 μ + 12 ν ET II 318(31) [Re λ > 0, −1 < a < 1]  ∞  2 λ−1  2 2 μ/2 μ x −1 a x −1 P ν (ax) dx 1     Γ(λ) Γ 1 − λ − 12 μ + 12 ν Γ 12 − λ − 12 μ − 12 ν     = Γ 1 − 12 μ + 12 ν Γ 12 − 12 ν − 12 μ Γ(1 − λ − μ)  μ−ν 1 1−μ+ν ,1 − λ − ; 1 − λ − μ; 1 − 2 ×2μ−1 aμ−ν−1 2 F 1 2 2 a [Re a > 0, Re λ > 0, Re(ν − μ − 2λ) > −2, Re(2λ + μ + ν) < 1] ET II 325(25) −1

=

2.

μ/2



∞

3. 1

7.137  1.

∞

1



2

x −1

1

1

1

λ−1 

2 2

a x −1

1

Q μν (ax) dx =

 1

∞

Γ

2

1

x− 2 μ− 2 (x − 1)−μ− 2 (1 + ax) 2 μ Q μν (1 + 2ax) dx

2.

 μ+ν+1 

  μ−2 μπi −μ−ν−1 Γ(λ) Γ 1 − λ + μ+ν 2 e a 2   3 Γ ν+2   μ+ν 3 −2 μ+ν +1 ,1 − λ + ;ν + ;a × 2F 1 2 2 2 [|arg(a − 1)| < π, Re λ > 0, Re(2λ − μ − ν) < 2] ET II 325(27)

− 12 μ

1

1

3

1

2  1μ  μ  1/2 2 Q (1 + a) − μ a ν 2

Re μ < 12 , Re(μ + ν) > −1 ET II 325(28)

= π −1/2 e−μπi Γ |arg a| < π,

1

μ

x− 2 μ− 2 (x − 1)−μ− 2 (1 + ax) 2 Q μν (1 + 2ax) dx

    1 1 −1/2 μ+1   1/2 (1 + a) Q μν (1 + a)1/2 Qν = −π −1/2 e−μπi Γ −μ − 12 a 2 μ+ 2 1 + a2

|arg a| < π, Re μ < − 21 , Re(μ + ν + 2) > 0 ET II 326(29)

782

Associated Legendre Functions



1

3. 0



1

4. 0



2   1   1 1 1 1 x− 2 μ− 2 (1 − x)−μ− 2 (1 + ax) 2 μ P μν (1 + 2ax) dx = π 1/2 Γ 12 − μ a 2 μ P μν (1 + a)1/2

Re μ < 12 , |arg a| < π ET II 319(32) 1

1

1

3

μ

x− 2 μ− 2 (1 − x)−μ− 2 (1 + ax) 2 P μν (1 + 2ax) dx

      1 1 1/2 μ 1/2 (1 + a) P (1 + a) = π 1/2 Γ − 21 − μ a 2 μ+ 2 P μ+1 ν ν

Re μ < − 21 , |arg a| < π ET II 319(33)

1

5. 0

1

1

1

1

x 2 μ− 2 (1 − x)μ− 2 (1 + ax)− 2 μ P μν (1 + 2ax) dx = π 1/2 Γ



1

6. 0

1

7. 0

1 2

1

1

1

3

μ

1

1

      1 1 1 1/2  −1/2 1/2 π Γ μ − 12 a 2 − 2 μ (1 + a) P 1−μ (1 + a) P μν (1 + a)1/2 ν 2     (1 + a)1/2 P μν (1 + a)1/2 + (μ + ν)(1 − μ + ν) P −μ ν

Re μ > 12 , |arg a| < π ET II 319(35) 1

x− 2 − 2 (1 − x)−μ− 2 (1 + ax) 2 μ Q μν (1 + 2ax) dx = π 1/2 Γ



1

8. 0

μ



y

0

1 2

     1 − μ a 2 μ P μν (1 + a)1/2 Q μν (1 + a)1/2

Re μ < 12 , |arg a| < π ET II 320(38)

1

x− 2 − 2 (1 − x)−μ− 2 (1 + ax) 2 μ Q μν (1 + 2ax) dx 1

3

=

9.

     1 1/2 (1 + a) + μ a− 2 μ P μν (1 + a)1/2 P −μ ν

Re μ > − 21 , |arg a| < π ET II 319(34)

x 2 μ− 2 (1 − x)μ− 2 (1 + ax)− 2 μ P μν (1 + 2ax) dx =



7.137

 1 1/2  −1/2 12 μ+ 12 π Γ −μ − 12 (1 + a) a 2         (1 + a)1/2 Q μν (1 + a)1/2 + P μν (1 + a)1/2 Q μ+1 (1 + a)1/2 × P μ+1 ν ν

Re μ < − 21 , |arg a| < π ET II 320(39)

  − 12 λ λ (y − x)μ−1 x 1 + 12 γx P ν (1 + γx) dx

 12 μ− 12 λ   12 μ   1 2 y 1 + γy P λ−μ (1 + γy) ν γ 2 [Re λ < 1, Re μ > 0, |arg γy| < π] ET II 193(52)

= Γ(μ)  10. 0

y

 − 1 λ 1 (y − x)μ−1 xσ+ 2 λ−1 1 + 12 γx 2 P λν (1 + γx) dx  γ − 12 λ   Γ(σ) Γ(μ)y σ+μ−1 1 γy −ν, 1 + ν, σ; 1 − λ, σ + μ; − = 2 3F 2 Γ(1 − λ) Γ(σ + μ) 2 [Re σ > 0, Re μ > 0, |γy| < 1] ET II 193(53)

7.142

Associated Legendre functions, exponentials, and powers



y

11. 0

 12. 0

1

1

1

(y − x)μ−1 [x(1 − x)]− 2 λ P λν (1 − 2x) dx = Γ(μ)[y(1 − y)] 2 μ− 2 λ P λ−μ (1 − 2y) ν [Re λ < 1,

Re μ > 0,

0 < y < 1] ET II 193(54)

y

1

1

(y − x)μ−1 xσ+ 2 λ−1 (1 − x)− 2 λ P λν (1 − 2x) dx Γ(μ) Γ(σ)y σ+μ−1 3 F 2 (−ν, 1 + ν, σ; 1 − λ, σ + μ; y) Γ (σ + μ) Γ(1 − λ) [Re σ > 0, Re μ > 0, 0 < y < 1] ET II 193(155)

=

 7.138

783

∞

0

−μ−ν−2

(a + x)

 Pμ

a−x a+x



 Pν

a−x a+x

 dx =

a−μ−ν−1 [Γ(μ + ν + 1)]4 2

[Γ(μ + 1) Γ(ν + 1)] Γ(2μ + 2ν + 2) [|arg a| < π, Re(μ + ν) > −1] ET II 326(3)

7.14 Combinations of associated Legendre functions, exponentials, and powers 7.141



∞

1. 1



 1μ e−ax (x − 1)λ−1 x2 − 1 2 P μν (x) dx =

[Re a > 0, ∞

2. 1

∞

3. 1

Re λ > 0]

e

−ax

λ−1

(x − 1)



2

x −1

− 12 μ

    1 + μ, 1 Γ(ν + μ + 1)eμπi −λ−μ −a 22 a e G 23 2a  λ + μ, ν + 1, −ν 2 Γ(ν − μ + 1) [Re a > 0, Re λ > 0, Re(λ + μ) > 0] ET II 325(24)

P μν (x) dx

= −π

−1

sin(νπ)a

μ−λ −a

e

[Re a > 0,

    1, 1 − μ  2a  λ − μ, 1 + ν, −ν Re(λ − μ) > 0]

31 G 23

ET II 323(15)

   ∞  1 − μ, 1  − 1 μ 1  2a e−ax (x − 1)λ−1 x2 − 1 2 Q μν (x) dx = eμπi aμ−λ e−a G 22 23  λ − μ, ν + 1, −ν 2 1 [Re a > 0, Re λ > 0, Re(λ − μ) > 0]

 4.

ET II 323(13)

 1μ e−ax (x − 1)λ−1 x2 − 1 2 Q μν (x) dx =



    1 + μ, 1 a−λ−μ e−a 31  2a G  λ + μ, −ν, 1 + ν Γ(1 − μ + ν) Γ(−μ − ν) 23



ET II 323(14) ∞

5. 1

  e−ax x2 − 1

− 12 μ

1

P μν (x) dx = 21/2 π −1/2 aμ− 2 K ν+ 12 (a) [Re a > 0,

Re μ < 1] ET II 323(11), MO 90

 7.142

1

∞

1

e− 2 ax



x+1 x−1

 12 μ

P μν− 1 (x) dx = 2

2 W μ,ν (a) a

Re μ < 1,

ν−

1 2

= 0, ±1, ±2, . . . BU 79(34), MO 118

784

7.143

Associated Legendre Functions



∞

1. 0



∞

2. 0

7.144  1.

∞

0

[x(1 + x)]

− 12 μ −βx

e

1

P μν (1

β μ− 2 1 + 2x) dx = √ e 2 β K ν+ 12 π

1μ  1 1 2 −βx μ e2β W μ,ν+ 12 (β) 1+ e P ν (1 + 2x) dx = x β

1



∞

0

7.145



∞

1. 0



∞

2. 0



  β 2

[Re μ < 1,

Re β > 0]

ET I 179(1)

[Re μ < 1,

Re β > 0]

ET I 179(2)

1

e−βx xλ+ 2 μ−1 (x + 2) 2 μ Q μν (1 + x) dx =

2.

7.143

1

 sin(νπ) Γ(ν + μ + 1) E (−ν, ν + 1, λ + μ; μ + 1 : 2β) λ+μ Γ(ν − μ + 1) 2β sin(μπ)  sin [(μ + ν) π] − 1−μ λ E (ν − μ + 1, −ν − μ, λ : 1 − μ : 2β) 2 β sin(μπ) [Re β > 0, Re λ > 0, Re (λ + μ) > 0] ET I 181(16) sin(νπ) E (−ν, ν + 1, λ − μ : 1 − μ : 2β) sin(μπ) sin[(μ − ν)π] − 1+μ λ E (μ + ν + 1, μ − ν, λ : 1 + μ : 2β) 2 β sin(μπ) [Re β > 0, Re λ > 0, Re(λ − μ)] > 0 ET I 181(17)

1

e−βx xλ− 2 μ−1 (x + 2) 2 μ Q μν (1 + x) dx = −

2β λ−μ

  e−βx 1 eβ Pν W ν+ 12 ,0 (β) W −ν− 12 ,0 (β) − 1 dx = 1+x (1 + x)2 β   π2 x−1 e−βx Q − 12 1 + 2x−2 dx = 8



 J0

1 β 2

2

[Re β > 0]  2   1 β + Y0 2 [Re β > 0]

∞

3. 0



∞

1. 0

 2. 0

1

x− 2 μ e−βx P μν

√  β 1 5 1 + x dx = 2μ β 2 μ− 4 e 2 W

1 1 1 1 2 μ+ 4 , 2 ν+ 4

√  β 1 e−βx 1 3 P μν x− 2 μ √ 1 + x dx = 2μ β 2 μ− 4 e 2 W 1+x

Re ν > −1]

ET II 327(6)

(β)

[Re μ < 1, ∞

ET II 327(5)

  1 x−1 e−ax Q ν 1 + 2x−2 dx = [Γ(ν + 1)]2 a−1 W −ν− 12 ,0 (ai) W −ν− 12 ,0 (−ai) 2 [Re a > 0,

7.146

ET I 180(6)

1 1 1 1 2 μ+ 4 , 2 ν+ 4

Re β > 0]

ET I 180(7)

(β)

[Re μ < 1, Re β > 0]

ET I 180(8)a

7.153

Associated Legendre and hyperbolic functions



∞√

3.

xe

0

P 1/4 ν

−βx



1+

x2



P −1/4 ν

785

       1 1 1 π (1) (2) 2 H 1 β H ν+ 1 β 1 + x dx = 2 2 2β ν+ 2 2 2 [Re β > 0]

 7.147

∞

0

 1ν xλ−1 x2 + a2 2 e−βx P μν



x (x2

ET I 180(9)

1/2

+ a2 )

dx

2−ν−2 aλ+ν 32 = G π Γ(−μ − ν) 24

   1 − λ , 1−λ  2 2  1 λ+μ+ν  0, 2 , − 2 , − λ−μ+ν 2 Re β > 0, Re λ > 0] ET II 327(7) 

a2 β 2 4

[a > 0,   1 1−x 1 1 1 1 1 7.148 (1 − x)− 2 μ (1 + x) 2 μ+ν−1 exp − y P μν (x) dx = 2ν y 2 μ+ν− 2 e 2 y W 1+x −1 [Re y > 0]  ∞    2 −1/2 1/2  2 2 2 P ν (x) dx α + β + 2αβx 7.149 exp − α + β + 2αβx 

1 1 1 2 μ−ν− 2 , 2 μ

(y)

ET II 317(21)

1

= 2π −1 (αβ)−1/2 K ν+ 12 (α) K ν+ 12 (β) [Re α > 0,

Re β > 0]

ET II 323(16)

7.15 Combinations of associated Legendre and hyperbolic functions 7.151



∞

1. 0



∞

2. 0

α−1

P −μ ν

α−1

Q μν

(sinh x)

(sinh x)

      2−1−μ Γ 12 α + 12 μ Γ 12 ν − 12 α + 1 Γ 12 − 12 α − 12 ν       (cosh x) dx = Γ 12 μ + 12 ν + 1 Γ 12 + 12 μ − 12 ν Γ 1 + 12 μ − 12 α [Re(α + μ) > 0, Re(ν − α + 2) > 0, Re(1 − α − ν) > 0] EH I 172(28)

    eiμπ 2μ−α Γ 12 + 12 ν + 12 μ Γ 1 + 12 ν − 12 α     (cosh x) dx Γ 1 + 12 ν − 12 μ Γ 12 + 12 ν + 12 α     × Γ 12 α + 12 μ Γ 12 α − 12 μ [Re (α ± μ) > 0,

 7.152

7.153∗

∞

0

e

−αx

sinh

2μ

1

EH I 172(29)

     1  −2μ  1 

Γ 2μ + 12 Γ(α − n − μ) Γ α + n − μ + 12   cosh 2 x dx = μ √ 2 x P 2n 4 π Γ(α + n + μ + 1) Γ α − n + μ + 12

Re α > n + Re μ, Re μ > − 14

√ 2 2 −(n+1/2)|a| −3/2 e (cosh a − x) Pn (x) dx = sinh |a| −1



Re(ν − α + 2) > 0]

ET I 181(15)

786

Associated Legendre Functions

7.161

7.16 Combinations of associated Legendre functions, powers, and trigonometric functions 7.161



1

1. 0

 − 1 μ xλ−1 1 − x2 2 sin(ax) P μν (x) dx =

 2. 0

1

π 1/2 2μ−λ−1 Γ (λ + 1) a    Γ 3+λ−μ+ν Γ 1 + λ−μ−ν 2 2   λ 3 λ−μ−ν 3+λ−μ+ν a2 1+λ ,1+ ; ,1 + , ;− × 2F 3 2 2 2 2 2 4 [Re λ > −1, Re μ < 1] ET II 314(7) 

 − 1 μ xλ−1 1 − x2 2 cos(ax) P μν (x) dx =



∞

3.



x2 − 1

0

 12 μ

π 1/2 2μ−λ Γ(λ)     λ−μ+ν 1+λ−μ−ν Γ Γ 1+ 2 2   λ−μ+ν a2 λ λ+1 1 1+λ−μ−ν , ; , ,1 + ;− × 2F 3 2 2 2 2 2 4 [Re λ > 0, Re μ < 1] ET II 314(8) 1

sin(ax) P μν (x) dx =

Γ

1 2

2μ π 1/2 a−μ− 2   S 1 1 (a) 1 − 2 μ − 12 ν Γ 1 − 12 μ + 12 ν μ+ 2 ,ν+ 2

a > 0, Re μ < 32 , Re(μ + ν) < 1

ET II 320(1)

7.162



∞

1. a



0

πa − 1 sin(bx) dx = − 4 cos(νπ)



2   2  ab ab − J −ν− 12 2 2 [a > 0, b > 0, −1 < Re ν < 0]

 J ν+ 12

  P ν 2x2 a−2 − 1 cos(bx) dx

∞



x2 + 2

−1/2

  2  2  x + 1 dx = 2−1/2 π −1 a sin(νπ) K ν+ 12 2−1/2 a sin(ax) P −1 ν

[a > 0, −2 < Re ν < 1] ET I 98(22) ∞      −1/2  x2 + 2 sin(ax) Q 1ν x2 + 1 dx = −2−3/2 πa K ν+ 12 2−1/2 a I ν+ 12 2−1/2 a 0

ET 98(23) a > 0, Re ν > − 32

 4.

P ν 2x a



         ab ab ab ab π J −ν− 12 − Y ν+ 12 Y −ν− 12 = − a J ν+ 12 4 2 2 2 2 [a > 0, b > 0, −1 < Re ν < 0] ET II 326(2)

a

3.

2 −2

ET II 326(1) ∞

2.





7.164

Associated Legendre functions, powers, and trigonometric functions

√   2 2 a sin(νπ) K ν+ 12 √ cos(ax) P ν 1 + x dx = − π 2 0 [a > 0,      ∞   a a π 2 cos(ax) Q ν 1 + x dx = √ K ν+ 12 √ I ν+ 12 √ 2 2 2 0 [a > 0,  1       a a J −ν− 12 cos(ax) P ν 2x2 − 1 dx = π2 J ν+ 12 2 2 0 

5.

6.

7.

∞



2



−1 < Re ν < 0]

ET I 42(23)

Re ν > −1]

ET I 42(24)

[a > 0] 7.163



∞

1. a

 2. 0

1



x2 − a2

 12 ν− 14

1

sin(bx) P 02

−ν

  1 π νπ + ax−1 dx = b−ν− 2 cos ab − 2 4

a > 0, |Re ν| < 12

∞

1. 0

2.12

∞

3. 0

 4. 0

 2 2 x2 x1/2 sin(bx) P −1/4 1 + a dx = ν





Γ

−1 < Re ν < 0]

ET II 327(4)

 2  b  K ν+ 1 2 2a ν

Re a > 0, b > 0, − 54 < Re ν < 14

2 −1 −1/2 b πa  1 Γ 4− 4 +ν

5

ET II 327(8)

∞

0



ET I 98(24)

  1 x−1 cos(ax) P ν 2x−2 − 1 dx = − π cosec(νπ) 1 F 1 (ν + 1; 1; ai) 1 F 1 (ν + 1; 1; −ai) 2

7.164



ET I 42(25)



[a > 0, 

787

∞

    2 x2 Q −1/4 2 x2 dx x1/2 sin(bx) P −1/4 1 + a 1 + a ν ν   π − 1 πi      4 Γ ν + 54 b b 2e K ν+ 12 = I ν+ 12   1 3 2a 2a 2 ab Γ ν + 4

Re a > 0, b > 0, Re ν > − 45 ET II 328(9) x1/2 sin(bx) P −1/4 ν

x1/2 sin(bx) P 1/4 ν

    dx −1/4 1 + a2 x2 P ν−1 1 + a2 x2 √ 1 + a2 x2     b b a−2 b1/2 5  5  K ν− 1 K ν+ 12 = √ 2 2a 2a 2π Γ 4 + ν Γ 4 − ν 5

Re a > 0, b > 0, − 4 < Re ν < 54 ET II 328(10)

    dx 2 x2 √ 1 + a2 x2 P −3/4 1 + a ν 1 + a2 x2  2  b a−2 b1/2 7  3  K ν+ 1 = √ 2 2a 2π Γ 4 + ν Γ 4 − ν

Re a > 0, b > 0, − 47 < Re ν < 34 ET II 328(11)

788

Associated Legendre Functions



∞

5. 0

 0

∞

7. 0



∞

8. 0

 7.165

x

 cos(bx)

P 1/4 ν



1+

a2 x2

2

 −1/2  2  a−1 πb b 2   1  K ν+ 1 dx =  3 2 2a Γ 4 + ν Γ − 4 − ν

Re a > 0, b > 0, − 34 < Re ν < − 14

ET II 328(12)

∞

6.



1/2

7.165

∞

x1/2 cos(bx) P 1/4 ν



x1/2 cos(bx) P −1/4 ν

x1/2 cos(bx) P 1/4 ν



1 + a2 x2 Q 1/4 ν





1 + a2 x2 dx   π 1 πi      4 Γ ν + 34 b b 2e   K ν+ 12 = I ν+ 12 1/2 Γ ν + 5 2a 2a ab 4

Re a > 0, b > 0, Re ν > − 34 ET II 328(13)

    dx 1 + a2 x2 P 3/4 1 + a2 x2 √ ν 1 + a2 x2  2  b a−2 b1/2 5  1  K ν+ 12 = √ 2a 2π Γ + ν Γ − ν 4 4

Re a > 0, b > 0, − 45 < Re ν < 14 ET II 328(14)



   dx 1/4 1 + a2 x2 P ν−1 1 + a2 x2 √ 1 + a2 x2     b b a−2 b1/2 1 3  3  K ν− 1 K = √ ν+ 2 2 2a 2a 2π Γ 4 +ν Γ 4 −ν

Re a > 0, b > 0, |Re ν| < 34 ET II 329(15)

cos(ax) P ν (cosh x) dx

       ν + iα ν − iα 1 + ν − iα Γ Γ − Γ − 2 2 2 [a > 0, −1 < Re ν < 0] ET II 329(18)      π 2−μ π Γ 12 α + 12 μ Γ 12 α − 12 μ −μ α−1        7.166 P ν (cos ϕ) sin ϕ dϕ =  1 1 Γ 2 + 2 α + 12 ν Γ 12 α − 12 ν Γ 12 μ + 12 ν + 1 Γ 12 μ − 12 ν + 12 0 MO 90, EH I 172(27) [Re (α ± μ) > 0]   η   a η 1 η 2 Γ(μ − η) Γ η + dx sin(a − x) −η 2 (sin a) √ = P −μ 7.167 P −μ (cos x) P [cos(a − x)] ν ν ν (cos a) sin x sin x π Γ(η + μ + 1) 0 Re μ > Re η > − 21 ET II 329(16) 0

=−

sin(νπ) Γ 4π 2



1 + ν + iα 2

7.17 A combination of an associated Legendre function and the probability integral 

7.171

1

∞



x2 − 1

− 12 μ

  exp a2 x2 [1 − Φ(ax)] P μν (x) dx     3 a2 μ−ν 1+μ+ν −1 μ−1 Γ aμ− 2 e 2 W =π 2 Γ 2 2 [Re a > 0,

Re μ < 1,

Re (μ + ν) > −1,

1 1 1 1 4 − 2 μ, 4 + 2 ν

 2 a

Re(μ − ν) > 0] ET II 324(17)

7.182

Associated Legendre and Bessel functions

789

7.18 Combinations of associated Legendre and Bessel functions 7.181  1.

∞

1



∞

2. 1

7.182



∞

1. 1



∞

2.

∞

3.

 0



1

∞

6.

∞

7.

1

P λ−1 (x) J ν (ax) dx λ

  2λ+ν a−λ Γ 12 + ν S λ−ν,λ+ν (a) = π 1/2 Γ(1 − λ) a > 0, Re ν < 52 ,

ET II 344(36)a

Re(2λ + ν)
0, |Re ν| < 2 + 2 Re μ ET II 344(37)a

 − 1 μ 1 x 2 −μ x2 − 1 2 P μν− 1 (x) Y ν (ax) dx

 a a a  a  1 J μ− 12 −Yν Y μ− 12 = 2−3/2 π 1/2 aμ− 2 J ν 2 2 2

2 1 − 4 < Re μ < 1, a > 0, Re(2μ − ν) > − 12 ET II 349(67)a 

    π μ− 1 a 2 J 12 −μ 12 a J ν+ 12 12 a 2 [Re μ < 1, Re(μ − ν) < 2]

 1 −μ

x2

x2 − 1

− 12 μ

1

P μν− 1 (x) K ν (ax) dx = (2π)−1/2 aμ− 2 2

− 1 μ 1  xμ+ 2 x2 − 1 2 P μν− 1 (x) K ν (ax) dx =



− 1 μ 3  xμ− 2 x2 − 1 2 P μν− 1 (x) K ν (ax) dx =



2

1

8.

x −1

 12 λ− 12

2

1



2

ET II 108(3)a

ET II 337(33)a ∞

1



x



 − 1 μ 1 x 2 −μ 1 − x2 2 P μν (x) J ν+ 12 (ax) dx =

5. 

ν

2

1

4.

     

1 1  cos 2 a Y ν 12 a + sin 12 a J ν 12 a P ν− 12 (x)x1/2 J ν (ax) dx = − √ 2a

|Re ν| < 12

2

1



     

  P ν− 12 (x)x1/2 Y ν (ax) dx = 2−1/2 a−1 cos 12 a J ν 12 a − sin 12 a Y ν 12 a

a > 0, Re ν < 12

∞

− 1 μ 1  xμ− 2 x2 − 1 2 P μν− 3 (x) K ν (ax) dx = 2



    K ν 12 a K μ− 12 12 a

[Re μ < 1,

Re a > 0]

ET II 135(5)a

π −3/2 − 1 a a e 2 W μ,ν (a) 2 [Re μ < 1,

Re a > 0]

ET II 135(3)a

π −1/2 − 1 a a e 2 W μ−1,ν (a) 2 [Re μ < 1,

Re a > 0]

ET II 135(4)a

π −1 − 1 a a e 2 W μ− 12 ,ν− 12 (a) 2 [Re μ < 1]

ET II 135(6)a

790

Associated Legendre Functions



∞

  a 2   1 ν− 1 1 −ν  2  2x − 1 K ν (ax) dx = π −1/2 a−ν 2ν−1 K μ+ 12 x1/2 x2 − 1 2 4 P μ2 2

1 Re ν > − 2 , Re a > 0 ET II 136(11)a

∞

  1 ν− 1 1 −ν  2  2x − 1 Y ν (ax) dx x1/2 x2 − 1 2 4 P μ2  a a a  a  J −μ− 12 − Y μ+ 12 Y −μ− 12 = π 1/2 2ν−2 a−ν J μ+ 12 2 2 2

2 Re ν > − 12 , a > 0, Re ν + |2 Re μ + 1| < 32 ET II 108(5)a

∞

  1 ν− 1 1 −ν  2  2x − 1 J ν (ax) dx x1/2 x2 − 1 2 4 P μ2

9. 1

 10. 1

 11. 1



∞

12. 1



  a 2   a 2  J μ+ 12 − J −μ− 12 2 2

< 2 Re μ < 12 − Re ν ET II 345(39)a

= −2ν−2 a−ν π 1/2 sec(μπ) Re ν > − 12 ,

a > 0,

Re ν −

3 2

  − 1 ν  x x2 − 1 2 P νμ 2x2 − 1 K ν (ax) dx = 2−ν aν−1 K μ+1 (a) [Re a > 0,

∞

13. 0



7.182



x x2 + a

1 2 2ν

 ν

P μ 1 + 2x2 a

 −2

Re ν < 1]

ET II 136(10)a

K ν (xy) dx = 2−ν ay −ν−1 S 2ν,2μ+1 (ay) [Re a > 0,

Re y > 0,

Re ν < 1] ET II 135(7)

∞

14. 0

1ν    

 x x2 + a2 2 (μ − ν) P νμ 1 + 2x2 a−2 + (μ + ν) P ν−μ 1 + 2x2 a−2 K ν (xy) dx = 21−ν μy −ν−2 S 2ν+1,2μ (ay) [Re a > 0,



∞

15. 0

Re y > 0,

Re ν < 1]

ET II 136(8)

 1 ν−1 ν    

 P μ 1 + 2x2 a−2 + P ν−μ 1 + 2x2 a−2 K ν (xy) dx = 21−ν y −ν S 2ν−1,2μ (ay) x x2 + a2 2 [Re a > 0,

Re y > 0,

Re ν < 1] ET II 136(9)





∞

16. 0

 17. 0

 −1/2 2 −1/2 12 −ν −1/2 1 K y 2 2 π y 1 1 1     μ+ − ν− −ν−   2   x1/2 x2 + 2 2 4 P μ 2 x2 + 1 J ν (xy) dx = 3 1

Γ3 ν + μ + 2 Γ ν − μ + 2 1 − 2 − Re ν < Re μ < Re ν + 2 , y > 0

ET II 44(1)

∞

 − 1 ν− 1 ν+ 1   x1/2 x2 + 2 2 4 Q μ 2 x2 + 1 J ν (xy) dx

    1 1 = 2−ν− 2 π 1/2 e(ν+ 2 )πi y ν K μ+ 12 2−1/2 y I μ+ 12 2−1/2 y

Re ν > −1, Re(2μ + ν) > − 25 , y > 0 ET II 46(12)

7.184

Associated Legendre and Bessel functions

 7.183

∞

 − 1 μ− 1 μ+ 1 x1−μ 1 + a2 x2 2 4 Q ν− 12 (±iax) J ν (xy) dx 2

0

1

7.184 1.12





∞

1

0

ν− 12

Pμ



 2x−2 − 1 K ν (ax) dx

  2 1 + 2 J ν (ax) dx x 3  2  1  iπν − 12 ν −ν−2 Γ 2 +μ+ν Γ 2 +ν −μ = −ie π 2 a

sin(νπ) cos(μπ)  W μ+ 1 ,ν+ 1 (a)  1 (a) + × W −μ− 12 ,ν+ 12 (a) M 1 2 2 Γ(2 + 2ν) μ+ 2 ,ν+ 2 Γ ν + μ + 32

3 1 ET II 46(14) a > 0, Re(μ + ν) > − 2 , Re(μ − ν) < 2

  1 + ν ν+ 1 xν 1 + x2 4 2 Q μ 2

  1 ν+ 1 −ν− 1   xν 1 − x2 2 4 P μ 2 2x−2 − 1 J ν (xy) dx 3  1  ν+ 12 ν Γ 2 + μ + ν Γ 2 + ν − μ y =2  

1/2 Γ 3 + ν 2 (2π) 2     3 3 × 1 F 1 ν + μ + ; 2ν + 2; iy 1 F 1 ν + μ + ; 2ν + 2; −iy 2 2

y > 0, − 32 − Re ν < Re μ < Re ν + 12 ET II 45(3)

∞

0

6.

1 1 4−2ν

= π 1/2 2−ν a−2+ν W μ+ 12 ,ν− 12 (a) W −μ− 12 ,ν− 12 (a)

Re ν < 32 , a > 0 ET II 370(45)a

5.



y > 0,

  x−ν x2 − 1

0

0

Re ν < Re μ < 1 + Re ν,

−1

ET II 44(2)a

3.



1 2

   y K μ 12 a−1 y

Re a > 0 ET II 46(11) 2a

∞

1

4.

− 43 −

1

  1 μ− 1 12 −μ  −1 

x J ν (xa) dx = 21/2 a−1−μ π −1/2 cos a + 12 (ν − μ)π x1/2 x2 − 1 2 4 P − 1 +ν 2

|Re μ| < 12 , Re ν > −1, a > 0

2.



1

= i(2π)1/2 eiπ(μ∓ 2 ν∓ 4 ) a−1 y μ−1 I ν

∞

1



791

  1 − 1 ν 1 −ν   1 + 2a2 x−2 K ν (xy) dx x−ν x2 + a2 4 2 Q μ2   2 1 W −μ− 12 ,ν− 12 (iay) W −μ− 12 ,ν− 12 (−iay) = ie−iπν π 1/2 2−ν−1 a−ν− 2 y ν−2 Γ 32 + μ − ν

Re a > 0, Re y > 0, Re μ > − 23 , Re(μ − ν) > − 32 ET II 137(13)

∞

   1 − 1 ν 1 −ν  1 + 2x−2 J ν (ax) dx x−ν x2 + 1 4 2 Q μ2 =2

−ν −ν−2 ie

a

  Γ 32 + μ − ν M μ+ 12 ,ν− 12 (a) W −μ− 12 ,ν− 12 (a) Γ(2ν)

a > 0, 0 < Re ν < Re μ + 32 ET II 47(15)a

−iνπ 1/2

π

792

Associated Legendre Functions



∞

7.

  1 − 1 ν 12 −ν   1 + 2a2 x−2 K ν (xy) dx x−ν x2 + a2 4 2 Q − 1 2

0

= ie 

7.185

−iπν 3/2 −ν−3

π

2

a

1 2 −ν

y

ν−1

  ay 2   ay 2  [Γ(1 − ν)] × J ν− 12 + Y ν− 12 2 2 [Re a > 0, Re y > 0, Re ν < 1] ET II 136(12) 2

   2 

1/2    a + x2 x−1 J ν (xy) dx = 2−1/2 πy −1 exp − a2 − 14 y J ν 12 y 0

ET II 46(10) Re ν > − 12 , y > 0    ∞ 2   1 − x −ν−1 −2 J 0 (xy) dx = y 2ν [2ν Γ(ν + 1)] K 0 (y) x 1 + x2 Pν 1 + x2 0 [Re ν > 0] ET II 13(10)

7.185

7.186 7.187  1.

∞

∞

0



∞

2. 0



∞

3. 0



∞

4. 0

 5. 0

x1/2 Q ν− 12

x P νμ



 1 + x2 K ν (xy) dx = y −3/2 S ν+ 12 ,μ+ 12 (y)

[Re ν < 1, Re y > 0]  2  y 2   x P λ− 12 1 + a2 x2 J 0 (xy) dx = 2π −2 y −1 a−1 cos(λπ) K λ 2a Re a > 0, |Re λ| < 14 , −1/2 ν  x 1 + x2 Pμ

− 12 ν

x Pμ

∞

x P μσ− 1 2







∞

6.

x P μσ− 1 2

0

− 12 ν

    1 + a2 x2 Q μσ− 1 1 + a2 x2 J 0 (xy) dx 2

=y Re a > 0,

 y y  +σ−μ W μ,σ M −μ,σ Γ(1 + 2σ) a

a Re σ > − 41 , Re μ < 1 ET II 14(15)

−2 μπi Γ

e

y > 0,

1 2

   2 x2 J (xy) dx 1 + a2 x2 P −μ 1 + a 1 0 σ− 2

Re a > 0,  0

∞

ET II 13(11)

[Re ν < 1, Re y > 0] ET II 137(15)   1 + a2 x2 J ν (xy) dx   1 y  y  y −1 e− 2 νπi Γ 1 + μ + 12 ν   1 1 K I = μ+ 2 μ+ 2 2a 2a a Γ 1 + μ − 12 ν

3 Re a > 0, y > 0, Re μ > − 4 , Re ν > −1 ET II 47(16)

= 2π −1 y −2 cos(σπ) W μ,σ

7.



1 + x2 K ν (xy) dx = y −1/2 S ν− 12 ,μ+ 12 (y)

1 + a2 x2 Q μ



y>0





ET II 137(14)

y > 0,

y

|Re σ|
0, y > 0, |Re σ| < 14 , Re μ < 1 ET II 14(13)

7.189

Associated Legendre and Bessel functions



∞

8. 0



∞

9. 0



∞

10. 0

 1 1   −1/2 − 12 − 12 ν   − ν x 1 + a2 x2 Pμ 1 + a2 x2 P μ2 2 1 + a2 x2 J ν (xy) dx   y 2 K μ+ 12 2a     = πa2 Γ ν2 + μ + 32 Γ ν2 − μ + 12 ET II 46(9) Re a > 0, y > 0, − 45 < Re μ < 14   y 2 −1    1 2 K y  2 μ+ 2 2a −1ν     x Pμ 2 1 + a2 x2 J ν (xy) dx = 1 1 πa Γ 1 + μ + ν Γ ν − μ 2 2

Re a > 0, y > 0, − 34 < Re μ < − 14 , Re ν > −1



∞

1. 0



∞

2. 0



∞

3. 0

ET II 45(7)

   −1/2 − 12 ν   − 12 ν x 1 + a2 x2 Pμ 1 + a2 x2 P μ+1 1 + a2 x2 J ν (xy) dx

Re a > 0, 7.188

793

y y K μ+ 32 2a K μ+ 12 2a     = πa2 Γ 2 + 12 ν + μ Γ 12 ν − μ ET II 45(8) y > 0, − 74 < Re μ < − 41

  − 1 μ  y μ−2 e−ay a √ J ν (xy) dx = x a2 + x2 2 P −ν μ−1 Γ(μ + ν) a2 + x2 Re a > 0, y > 0, Re ν > −1,  1ν xν+1 x2 + a2 2 P ν



x2 + 2a2 √ 2a x2 + a2

 − 1 ν x1−ν x2 + a2 2 P ν−1



2

 J ν (xy) dx =

2

x + 2a √ 2a x2 + a2

Re μ >

1 2



ET II 45(4)

 ya 2 (2a)ν+1 y −ν−1  K ν+ 12 π Γ(−ν) 2 [Re a > 0, −1 < Re ν < 0, y > 0] ET II 45(5)



 ay  y (2a) I ν− 12 K ν− 12 Γ(ν) 2 2 [Re a > 0, y > 0, 0 < Re ν < 1]

J ν (xy) dx =

1−ν ν−1

 ay 

ET II 45(6)

7.189



∞

1. 0

 2. 0

∞

μ −x

(a + x) e

P −2μ ν

−μ −x

(x + a)

e

  2x I μ (x) dx = 0 1+ a − 12 < Re μ < 0,

P −2μ ν



− 12 + Re μ < Re ν < − 21 − Re μ

ET II 366(18)

  2x I μ (x) dx 1+ a     2μ−1 Γ μ + ν + 12 Γ μ − ν − 12 ea W 12 −μ, 12 +ν (2a) = π 1/2 Γ (2μ + ν + 1) Γ(2μ− ν) 1 

|arg a| < π, Re μ > Re ν + 2  ET II 367(19)

794

Associated Legendre Functions



∞

3. 0



x

e

P 2μ ν

  2x K μ (x + a) dx 1+ a

    = π −1/2 2μ−1 cos(μπ) Γ μ + ν + 12 Γ μ − ν + 12 W 12 −μ, 12 +ν (2a)  

|arg a| < π, Re μ > Re ν + 12  ET II 373(11)

∞

4. 0



−μ x

7.191

− 12 μ

x

−1/2 −x

(x + a)

e

P μν− 1 2



a−x a+x



 K ν (a + x) dx =

π −1μ a 2 Γ(μ, 2a) 2

[a > 0, ∞

5. 0

Re μ < 1]

ET II 374(12)

3

(sinh x)μ+1 (cosh x)−2μ− 2 P −μ ν [cosh(2x)] I μ− 12 (a sech x) dx 1

=

2μ− 2 Γ(μ − ν) Γ (μ + ν + 1)

M ν+ 12 ,μ (a) M −ν− 12 ,μ (a) 3 2 π 1/2 aμ+ 2 [Γ (μ + 1)] [Re μ > Re ν, Re μ > − Re ν − 1] ET II 378(44)

7.19 Combinations of associated Legendre functions and functions generated by Bessel functions 7.191



∞

1. a



   2  1  2  − Jν 2a = 2−ν−2 π 1/2 a cosec(μπ) cos(νπ) Y ν 12 a

−1 < Re μ < 0, Re ν < 12 ET II 384(6)

∞

2. 0

7.192



1. 0

 − 1 − 1 ν ν+ 1   x1/2 x2 − a2 4 2 P μ 2 2x2 a−2 − 1 [Hν (x) − Y ν (x)] dx

1

 −1/4−ν/2 ν+1/2  2 −2  2x a − 1 [I −ν (x) − Lν (x)] dx x1/2 x2 − a2 Pμ   

  2  2 − I −ν 12 a = 2−ν−1 π 1/2 a cosec(2μπ) cos(νπ) I ν 12 a

−1 < Re μ < 0, Re ν < 12 ET II 385(15)

 (ν−μ−2)/4 (μ−ν+2)/2 x(ν−μ−1)/2 1 − x2 P ν−1/2 (x) S μ,ν (ax) dx       μ − 3ν + 3 μ−ν μ+ν +3 μ−3/2 1/2 −(ν−μ−1)/2 Γ cos π =2 π a Γ 4 4 2 1  1  1   1 

× J ν 2 a Y −(μ−ν+1)/2 2 a − Y ν 2 a J −(μ−ν+1)/2 2 a [Re(μ − ν) < 0,

a > 0,

|Re(μ + ν)| < 1,

Re(μ − 3ν) < 1]

ET II 387(24)a

7.215

Integration of associated Legendre functions



∞

2. 1

 −β/2 β x1/2 x2 − 1 P ν (x) S μ,1/2 (ax) dx = Re β < 1,

7.193 1.



∞



∞

795

   1 Γ β−μ−ν − 2 4  S μ−β+1,ν+1/2 (a) −μ

ET II 387(25)a Re(μ + ν − β) < − 12 , Re(μ − ν − β) < 12 

β−μ+ν 2  π 1/2 Γ 12

2−3/2+β−μ aβ−1 Γ

a > 0,

+

1 4

 1/4−ν/2 ν−1/2  −2  x−ν x2 − 1 P μ/2−ν/2 2x − 1 S μ,ν (ax) dx 1       2μ−ν aν−2 π 1/2 Γ 3ν−μ−1 2  1+ν−μ  W ρ,σ aeiπ/2 W ρ,σ ae−iπ/2 = Γ 2

Re(μ − ν) < 0, a > 0, Re ν < 32 , Re(3ν − μ) > 1 ρ = 12 (μ + 1 − ν), σ = ν − 12 , ET II 387(27)a

2. 1

 −ν/2 ν  2  x x2 − 1 P λ 2x − 1 S μ,ν (ax) dx

   + λ Γ ν−μ−1 −λ 2 2   1−μ+ν   1−μ−ν S μ−ν+1,2λ+1 (a) Γ 2Γ 2 2 Re(μ − ν + λ) < −1, Re(μ − ν + λ) < 0] ET II 387(26)a =

[Re ν < 1,

a > 0,

aν−1 Γ

 ν−μ+1

7.21 Integration of associated Legendre functions with respect to the order 7.211

  1 1 cosec θ [0 < θ < π] 2 2 0    ∞ 1 θ [0 < θ < π] 2. P x (cos θ) dx = cosec 2 −∞  ∞   1 7.212 x−1 tanh(πx) P − 12 +ix (cosh a) dx = 2e− 2 a K e−a 

∞

1.

P −x− 12 (cos θ) dx =

ET II 329(19) ET II 329(20)

0

[a > 0] x tanh(πx) P − 12 +ix (cosh b) dx = Q a− 12 (cosh b) [Re a > 0] 2 2 0 ∞ a + x 1 sinh(πx) cos(ax) P − 12 +ix (b) dx =  2 (b + cosh a) 0 [a > 0, |b| < 1] 

7.213 7.214  7.215

0

∞

∞

cos(bx) P μ− 1 +ix (cosh a) dx = 0

π

2

= Γ

1 2



2

ET II 330(22) ET II 387(23)

ET I 42(27)

[0 < a < b] μ

(sinh a)

μ+ 12

− μ (cosh a − cosh b)

[0 < b < a] ET II 330(21)

796

Associated Legendre Functions

 7.216

∞

0

cos(bx) Γ(μ + ix) Γ(μ − ix) P

1 2 −μ − 12 +ix

7.216

π

μ− 1

Γ(μ) (sinh a) 2 (cosh a) dx = (cosh a + cosh b)μ [a > 0, b > 0, Re μ > 0] 2

ET II 330(24)

7.217 1.

      1 1 1 1 2 −ν − ix Γ 2ν − + ix P ν+ix−1 ν − + ix Γ (cos θ) I ν− 12 +ix (a) K ν− 12 +ix (b) dx 2 2 2 −∞  ν √ ab ν− 1 K ν (ω) = 2π (sin θ) 2 ω    1/2 ω = a2 + b2 + 2ab cos θ ET II 383(29)



∞



∞

2. 0



∞

3.

2(ab)1/2 −ikR (2) (2) e xeπx tanh(πx) P − 12 +ix (− cos θ) H ix (ka) H ix (kb) dx = − ; πR  2  1/2 R = a + b2 − 2ab cos θ [a > 0, b > 0, 0 < θ < π, Im k ≤ 0] 1

−ν

(2)

(2)

2 xeπx sinh(πx) Γ (ν + ix) Γ(ν − ix) P − (− cos θ) H ix (a) H ix (b) dx 1 +ix 2

0

1/2

= i(2π)  1/2 R = a2 + b2 − 2ab cos θ 

∞

4. 0

7.218∗



∞

1. 0



∞

2. 0



∞

3. 0

[a > 0,

b > 0,

ET II 381(17)

ν− 12



(sin θ)

0 < θ < π,

Re ν > 0]

ab R

ν

H (2) ν (R)

ET II 381 (18)

 λ  2  1 λ− 1 π 1/2 ab β − 1 2 4 K λ (z) (β) dx = √ z 2

|arg(β − 1)| < π, Re λ > 0 ET II 177(16)

1 2 −λ − 12 +ix

x sinh(πx) Γ(λ+ix) Γ(λ−ix) K ix (a) K ix (b) P  |arg a| < π2 , z = a2 + b2 + 2abβ

√ cos(ax) 2 1 √ P− 12 +ix (z)dx = cosh(πx) 2 z + cosh a   √ cos(ax) 2 z − cosh a 1 √ arctan P− 12 +ix (z)dx = π z − cosh a 1 + cosh a cosh2 (πx)   √ cosh(bx) 2 1 + cos b 1 √ P 1 arccot (z)dx = cosh(πx) − 2 +ix π z − cos b z − cos b

[|b| < 2π]

7.22 Combinations of Legendre polynomials, rational functions, and algebraic functions 7.221



1

1. −1

P n (x) P m (x) dx = 0 =

2 2n + 1

[m = n] [m = n]

WH, EH I 170(8, 10)

7.224

2.6

Legendre polynomials, rational functions, and algebraic functions



1

1 2n + 1 =0

[m = n]

P n (x) P m (x) dx =

0

=

797

[n − m is even, m!n!  2    ! − n) (n + m + 1) n2 ! m−1 2 (−1)

2m+n−1 (m

1 2 (m+n−1)

m = n]

[n is even, m is odd] WH



2π

3. 0

7.222



1

1. 

−1 1

2. −1



1

3. −1



1

−1

2  2n −2n 2 . n

MO 70, EH II 183(50)

xm P n (x) dx = 0

[m < n]

(1 + x)m+n P m (x) P n (x) dx =

2m+n+1 [(m + n)!]

4

(1 + x)m−n−1 P m (x) P n (x) dx = 0  2 n

1−x

ET II 277(15)

2

(m!n!) (2m + 2n + 1)! [m > n]

2n2 P 2m (x) dx = (n − m)(2m + 2n + 1)



1



1 − x2

−1

n−1

[m < n] 1

5. 0



P 2n (cos ϕ) dϕ = 2π



4. 



x2 P n+1 (x) P n−1 (x) dx =

n(n + 1) (2n − 1)(2n + 1)(2n + 3)

ET II 278(16)

P 2m (x) dx WH WH

1

1 2 {P n (x) P n−1 (z) − P n−1 (x) P n (z)} dx = − WH n −1 z − x 7.224 [z belongs to the complex plane with a discontinuity along the interval from −1 to +1.]  1 1. (z − x)−1 P n (x) dx = 2 Q n (z) ET II 277(7) 7.223

−1



1

2. −1



1

3. −1



1

4. −1



1

5. −1



1

6. −1



1

7. −1

x(z − x)−1 P 0 (x) dx = 2 Q 1 (z) xn+1 (z − x)−1 P n (x) dx = 2z n+1 Q n (z) −

ET II 277(8) 2

2n+1 (n!) (2n + 1)!

ET II 277(9)

xm (z − x)−1 P n (x) dx = 2z m Q n (z)

[m ≤ n]

ET II 277(10)a

(z − x)−1 P m (x) P n (x) dx = 2 P m (z) Q n (z)

[m ≤ n]

ET II 278(18)a

(z − x)−1 P n (x) P n+1 (x) dx = 2 P n+1 (z) Q n (z) − x(z − x)−1 P m (x) P n (x) dx = 2z P m (z) Q n (z)

2 n+1 [m < n]

ET II 278(19)

ET II 278(21)

798

Associated Legendre Functions



1

2

8. −1

7.225 1. 2. 3. 4.

5.10

7.226 1.

2. 3.

x(z − x)−1 [P n (x)] dx = 2z P n (z) Q n (z) −

2 2n + 1

ET II 278(20)

−1  1 (x − t) P n (t) dt = n + (1 + x)−1/2 [T n (x) + T n+1 (x)] 2 −1 −1   1 1 (t − x)−1/2 P n (t) dt = n + (1 − x)−1/2 [T n (x) − T n+1 (x)] 2 x  1 23/2 (1 − x)−1/2 P n (x) dx = 2n + 1 −1 √  1 2 2 −1/2 exp[−(2n + 1)p] (cosh 2p − x) P n (x) dx = 2n + 1 −1



1 2

x

−1/2

EH II 187(43)

EH II 187(44)

EH II 183(49)

[p > 0]



1

P (z) dz  = P (x) Q (y) 2 (xy − z) − (x2 − 1) (y 2 − 1) −1 = P (y) Q (x)

1



WH

(1 < x ≤ y) (1 < y ≤ x)

  2 Γ 12 + m 1−x P 2m (x) dx = m! −1      1   Γ 12 + m Γ 32 + m 2 −1/2 x 1−x P 2m+1 (x) dx = m!(m + 1)! −1  1  −m−3/2 2 (−p)m (1 + p)−m−1/2 1 + px2 P 2m (x) dx = 2m + 1 −1



7.225

 2 −1/2

ET II 276(4)

ET II 276(5)

[|p| < 1]  7.227

1

0

−1/2    x a2 + x2 P n 1 − 2x2 dx =

2n + 1

[Re a > 0]  1   1 −μ/2 −iπμ Γ(1 + μ) P l (x)(z − x)−μ−1 dx = z 2 − 1 e Q μl (z) 2 −1

7.2286

MO 71

  1/2 −2n−1 a + a2 + 1

[l = 0, 1, 2, . . . ,

ET II 278(23)

|arg(z − 1)| < π]

7.23 Combinations of Legendre polynomials and powers 7.231



1. 0

1

    (−1)m Γ m − 12 λ Γ 12 + 12 λ     x P 2m (x) dx = 2 Γ − 12 λ Γ m + 32 + 12 λ λ

[Re λ > −1]

EH II 183(51)

7.243

2.

6

Legendre polynomials and elementary functions



1

0

3.∗



1

0

    (−1)m Γ m + 12 − 12 λ Γ 1 + 12 λ     x P 2m+1 (x) dx = 2 Γ 12 − 12 λ Γ m + 2 + 12 λ λ

xλ P n (x) dx =

[Re λ > −2]

 λ−n+3 



1

1. −1

1

2. −1

or

2a Γ(a) Γ(n − a + 1) 4 F 3 (−m, m + 1, a, a; 1, a + n + 1, a − n; 1) Γ(1 − a) Γ(n + a + 1) ET II 278(17) [Re a > 0]

(1 − x)a−1 (1 + x)b−1 P n (x) dx =

2a+b−1 Γ(a) Γ(b) 3 F 2 (−n, 1 + n, a; 1, a + b; 1) Γ(a + b) [Re a > 0,



1

3. 0

[n is odd, Re(ν) > −2]

(1 − x)a−1 P m (x) P n (x) dx =



EH II 183(52)

1 Γ (λ + 1) Γ  2  2n Γ (λ − n + 2) Γ λ+n+3 2 [n is even, Re(ν) > −1]

7.232

799

(1 − x)μ−1 P n (1 − γx) dx =

Re b > 0]

Γ(μ)n! P (μ,−μ) (1 − γ) Γ(μ + n + 1) n

[Re μ > 0]   1 1 Γ(μ) Γ(ν) γ −n, n + 1, ν; 1, μ + ν; (1 − x)μ−1 xν−1 P n (1 − γx) dx = 3F 2 Γ(μ + ν) 2 0

 4.

[Re μ > 0, 

7.23312

1

0

  x2μ−1 P n 1 − 2x2 dx =

ET II 276(6)

2

(−1)n [Γ(μ)] 2 Γ(μ + n) Γ(μ − n)

Re ν > 0]

[Re μ > 0]

ET II 190(37)a

ET II 190(38) ET II 278(22)

7.24 Combinations of Legendre polynomials and other elementary functions 

7.241

∞

0

 7.242 7.243

∞

0



1. 0

P n (1 − x)e

∞



Pn e

 −x

−ax

n  a  1 d e dx = e a a da   a   n 1 d 1 n =a 1+ 2 da an+1 −a n



[Re a > 0] (a − 1)(a − 2) · · · (a − n + 1) e−ax dx = (a + n)(a + n − 2) · · · (a − n + 2) [n ≥ 2, Re a > 0]

P 2n (cosh x) e

−ax

ET I 171(2)

ET I 171(3)

 2  

a − 12 a2 − 32 · · · a2 − (2n − 1)2 dx = a (a2 − 22 ) (a2 − 42 ) · · · [a2 − (2n)2 ]

[Re a > 2n]    

∞ a a2 − 22 a2 − 42 · · · a2 − (2n)2 P 2n+1 (cosh x) e−ax dx = 2 (a − 1) (a2 − 32 ) · · · [a2 − (2n + 1)2 ] 0

ET I 171(6)

 2.

[Re a > 2n + 1]

ET I 171(7)

800

Associated Legendre Functions



∞

3. 0



∞

4. 0

5.11



1

−1

P 2n (cos x) e

−ax

7.244

 2  

a + 12 a2 + 32 · · · a2 + (2n − 1)2 dx = a (a2 + 22 ) (a2 + 42 ) · · · [a2 + (2n)2 ]

[Re a > 0]  2  2

2 a a +2 a + 4 · · · a + (2n)2 −ax P 2n+1 (cos x) e dx = 2 (a + 12 ) (a2 + 32 ) · · · [a2 + (2n + 1)2 ] 

 eixα P n (x) dx = in

2

ET I 171(4)

2

[Re a > 0] 2π J 1 (α) α n+ 2

ET I 171(5)

[n = 0, 1, 2, . . . ,

a > 0] GH2 24 (171.10)

7.244 1. 2.



 a 2   π J n+ 12 P n 1 − 2x2 sin ax dx = [a > 0] 2 2 0  1 a a   π J −n− 12 P n 1 − 2x2 cos ax dx = (−1)n J n+ 12 2 2 2 0 1

ET I 94(2)

[a > 0] 7.245



2π

1. 0



π

2. 0

P 2m+1 (cos θ) cos θ dθ =

P m (cos θ) sin nθ dθ =

π 24m+1



2m m



2m + 2 m+1

 MO 70, EH II 183(5)

2(n − m + 1)(n − m + 3) · · · (n + m − 1) (n − m)(n − m + 2) · · · (n + m)

[n > m and n + m is odd] [n ≤ m or n + m is even]

=0

3.10

ET I 38(1)

  √ 3 2π n+1 2 π Γ n + 2 P1 P 2n+1 (sin α sin φ) sin φ dφ = (−1) (cos α) (2n + 1) Γ (n + 2) 2n+1 0

α = 12 (2n + 1)π, n an integer

MO 71





1

4. −1

cos(αx) P n (x) dx = 0 = (−1)v



[n is odd] 2π 1 (α) J α 2v+ 2

[n = 2v is even] GH2 24 (171.10a)

 7.246 7.247

π

  2 sin(2n + 1)θ P n 1 − 2 sin2 x sin2 θ sin x dx = 0 (2n + 1) sin θ 1 π dx J 2n+ 32 (a) P 2n+1 (x) sin ax √ = (−1)n+1 x 2a 0

MO 71

[a > 0]

ET I 94(1)

7.251

7.248

Legendre polynomials and Bessel functions



1

  −1/2 1/2  P n (x) dx = π(ab)−1/2 J n+ 12 (aλ) J n+ 12 (bλ) a2 + b2 − 2abx sin λ a2 + b2 − 2abx



1. −1



[a > 0, 1

−1

3.∗

1

−1

7.249  1.

b > 0]

ET II 277(11)

  −1/2 1/2  P n (x) dx = −π(ab)−1/2 J n+ 12 (aλ) Y n+ 12 (bλ) a2 + b2 − 2abx cos λ a2 + b2 − 2abx



2. 

801

1

−1

[0 ≤ a ≤ b]

√ Pn (x) 2 2 −(n+ 1 )a 2 e dx = sinh a (cosh a − x)3/2

P n (x) arcsin x dx = 0 =π

[n = 0, 1, . . . ,

⎫2 ⎪ ⎪ ⎬

⎧ ⎪ ⎪ ⎨

ET II 277(12)

(n − 2)!!   ⎪ 1 n+1 ⎪ (n+1) ⎪ 2 ⎭ ⎩2 !⎪ 2

a > 0]

[n is even]

[n is odd] WH

2.

P n (x) =

1 t

t−1  $

x+

t=0

 2πr x2 − 1 cos t

n [t > n]

7.25 Combinations of Legendre polynomials and Bessel functions 7.251  1   x P n 1 − 2x2 Y ν (xy) dx = π −1 y −1 [S 2n+1 (y) + π Y 2n+1 (y)] 1. 0



[n = 0, 1, . . . ; 1

2. 0



1

3. 0



1

4. 0



1

5. 0

 6. 0

  x P n 1 − 2x2 K 0 (xy) dx = y −1



ν > 0] ET II 108(1)

 i n+1 (−1) K 2n+1 (y) + S 2n+1 (iy) 2

  x P n 1 − 2x2 J 0 (xy) dx = y −1 J 2n+1 (y)   2 x P n 1 − 2x2 [J 0 (ax)] dx =

y > 0,

[y > 0]

ET II 134(1)

[y > 0]

ET II 13(1)

  1 2 2 [J n (a)] + [J n+1 (a)] 2(2n + 1)

  x P n 1 − 2x2 J 0 (ax) Y 0 (ax) dx =

ET II 338(39)a

1 [J n (a) Y n (a) + J n+1 (a) Y n+1 (a)] 2(2n + 1) ET II 339(48)a

1

  x2 P n 1 − 2x2 J 1 (xy) dx = y −1 (2n + 1)−1 [(n + 1) J 2n+2 (y) − n J 2n (y)] [y > 0]

ET II 20(23)

802

Orthogonal Polynomials



1

7. 0

x

 7.252

1

0

 7.253

μ−1

7.254

0

1

  2 2−ν−1 aν Γ 12 μ + 12 ν     2x − 1 J ν (ax) dx = Γ(ν + 1) Γ 12 μ + 12 ν + n + 1 Γ 12 + 12 ν − n   μ+ν μ+ν a2 μ+ν μ+ν , ; ν + 1, + n + 1, − n; − × 2F 3 2 2 2 2 4 [a > 0, Re(μ + ν) > 0] ET II 337(32)a



2



e−ax P n (1 − 2x) I 0 (ax) dx =

π/2

0

Pn

7.252

e−a [I n (a) + I n+1 (a)] 2n + 1 [a > 0]

ET II 366(11)a

sin(2x) P n (cos 2x) J 0 (a sin x) dx = a−1 J 2n+1 (a)

ET II 361(20)

  x P n 1 − 2x2 [I 0 (ax) − L0 (ax)] dx = (−1)n [I 2n+1 (a) − L2n+1 (a)] [a > 0]

ET II 385(14)a

7.3–7.4 Orthogonal Polynomials 7.31 Combinations of Gegenbauer polynomials Cnν (x) and powers 7.311



1

1. −1

3.

4.

1 − x2

ν− 12

C νn (x) dx = 0

n > 0,

Re ν > − 12



     ν− 12 ν Γ(2ν + n) Γ(2ρ + n + 1) Γ ν + 12 Γ ρ + 12 xn+2ρ 1 − x2 C n (x) dx = 2n+1 Γ(2ν) Γ(2ρ + 1)n! Γ(n + ν + ρ + 1) 0

Re ρ > − 21 , Re ν > − 12    1  1 2β+ν+ 2 Γ(β + 1) Γ ν + 12 Γ(2ν + n) Γ β − ν + ν ν− 12 β     (1 − x) (1 + x) C n (x) dx = n! Γ(2ν) Γ β − ν − n + 32 Γ β + ν + n + 32 −1

Re β > −1, Re ν > − 12  1 2α+β+1 Γ(α + 1) Γ(β + 1) Γ (n + 2ν) β (1 − x)α (1 + x) C νn (x) dx = −1  n! Γ(2ν) Γ(α + β + 2) 1 × 3 F 2 −n, n + 2ν, α + 1; ν + , α + β + 2; 1 2 [Re α > −1, Re β > −1] 

2.



ET II 280(1)

1

ET II 280(2) 3 2



ET II 280(3)

ET II 281(4)

7.312 In the following integrals, z belongs to the complex plane with a cut along the interval of the real axis from −1 to 1.  1 3  ν− 12 ν  1 ν− 1 ν− 12 π 1/2 2 2 −ν −(ν− 12 )πi m  2 e 1. xm (z − x)−1 1 − x2 C n (x) dx = z z − 1 2 4 Q n+ν− 1 (z) 2 Γ(ν) −1

1 ET II 281(5) m ≤ n, Re ν > − 2

Gegenbauer polynomials Cnν (x) and powers

7.314

2.12

3.6

7.313



1



1

803

1  ν− 12 ν  1 ν− 1 ν− 12 π 1/2 2 2 −ν −(ν− 12 )πi n+1  2 e z − 1 2 4 Q n+ν− xn+1 (z − x)−1 1 − x2 C n (x) dx = z 1 (z) 2 Γ(ν) −1 π21−2ν−n n! − Γ(ν) Γ(ν + n + 1)

Re ν > − 12 ET II 281(6)  1 3  ν− 12 ν  1 ν− 1 π 1/2 2 2 −ν −(ν− 12 )πi  2 ν− 12 e z − 1 2 4 C νm (z) Q n+ν− (z−x)−1 1 − x2 C m (x) C νn (x) dx = 1 (z) 2 Γ(ν) −1

1 ET II 283(17) m ≤ n, Re ν > − 2

1.



1 − x2

−1

ν− 12

m = n,

C νm (x) C νn (x) dx = 0

Re ν > − 21



ET II 282(12), MO 98a, EH I 177(16)



1

2. −1



1 − x2

ν− 12

2

[C νn (x)] dx =

π21−2ν Γ(2ν + n) n!(n + ν) [Γ(ν)]

2



Re ν > − 12 ET II 281(8), MO 98a, EH I 177(17)

7.314 1.

2.

  π 1/2 Γ ν − 12 Γ(2ν + n) (1 − x) (1 + x) dx = n! Γ(ν) Γ(2ν) −1

Re ν > 12  1  1 2  23ν− 2 [Γ(2ν + n)] Γ 2n + ν + 12 2 ν ν− 12 2ν−1 (1 − x) (1 + x) [C n (x)] dx =   (n!)2 Γ(2ν) Γ 3ν + 2n + 12 −1



 3.

4.

5.

1

ν− 32

ν− 12

[C νn (x)]2

[Re ν > 0] 1

ET II 281(9)

ET II 282(10)

(1 − x)3ν+2n− 2 (1 + x)ν− 2 [C νn (x)]2 dx −1   2     π 1/2 Γ ν + 12 Γ ν + 2n + 12 Γ (2ν + 2n) Γ 3ν + 2n − 12 =  

2   1 22ν+2n n! Γ ν + Γ 2ν + 2n + 12

n + 2 1Γ(2ν) ET II 282(11) Re ν > 6  1 1 3 (1 − x)ν− 2 (1 + x)ν+m−n− 2 C νm (x) C νn (x) dx −1     Γ ν − 12 + m − n Γ 12 − ν + m − n 22−2ν−m+n π 3/2 Γ(2ν + n) m     = (−1)  2  Γ 12 − ν − n Γ 12 + m − n m!(n − m)! [Γ(ν)] Γ 12 + ν + m

Re ν > − 12 ; n ≥ m ET II 282(13)a  1 ν− 1 (1 − x)2ν−1 (1 + x) 2 C νm (x) C νn (x) dx −1       1 23ν− 2 Γ ν + 12 Γ (2ν + m) Γ(2ν + n) Γ ν + 12 + m + n Γ 12 − ν + n − m       = m!n! Γ(2ν) Γ 12 − ν Γ ν + 12 + n − m Γ 3ν + 12 + m + n ET II 282(14) [Re ν > 0] 3

1

804

Orthogonal Polynomials

 6.

7.

1

1

7.315

3

(1 − x)ν− 2 (1 + x)3ν+m+n− 2 C νm (x) C νn (x) dx −1  

2     24ν+m+n−1 Γ ν + 12 Γ(2ν + m + n) Γ ν + m + n + 12 Γ 3ν + m + n − 12    =  Γ(2ν + n) Γ(4ν + 2m + 2n) Γ ν + m + 12 Γ ν + n + 12 Γ (2ν + m)

Re ν > 16 ET II 282(15)  1 ν− 1 (1 − x)α (1 + x) 2 C μm (x) C νn (x) dx −1     1 2α+ν+ 2 Γ(α + 1) Γ ν + 12 Γ ν − α + n − 12 Γ(2μ + m) Γ (2ν + n)     = 1 3 Γ(2μ) Γ(2ν) m!n!   Γ ν −α− 2 Γ ν−α+n+ 2 1 3 3 3 × 4 F 3 −m, m + 2μ, α + 1, α − ν + ; μ + , ν + α + n + , α − ν − n + ; 1 2 2 2 2

Re α > −1, Re ν > − 12 ET II 283(16)   1  π 1/2 Γ 12 ν ν  2 2a2 − 1 1−x = 1 1 Cn Γ 2ν + 2 −1 ET II 283(19) [Re ν > 0]  1 2 2ν−1   n! [Γ(ν)] 2 ν−1 C νn (cos α) C νn (cos β) 1 − x2 C νn (cos α cos β + x sin α sin β) dx = Γ(2ν + n) −1 [Re ν > 0] ET II 283(20) 

7.315

7.316

1



2

 12 ν−1

C ν2n (ax) dx

7.317    1 Γ (2λ + n) Γ λ + 12 Γ(μ) (α,β) λ μ−1 λ− 12  P n (1 − γ)  1. (1 − x) x C n (1 − γx) dx = Γ(2λ) Γ λ + μ + n + 12 0

Re λ > −1, λ = 0, − 12 , Re μ > 0 β = λ − μ − 12 α = λ + μ − 12 , 

1

2. 0

(1 − x)μ−1 xν−1 C λn (1 − γx) dx =

 7.318

1

0

2ν

x



1−x

Γ(2λ + n) Γ(μ) Γ(ν) n! Γ(2λ)  Γ(μ + ν)

 γ 1 × 3 F 2 −n, n + 2λ, ν; λ + , μ + ν; 2 2 [2λ = 0, −1, −2, . . . , Re μ > 0, Re ν > 0]

  Γ(2ν + n) Γ ν + 12 Γ(σ) (α,β)  P n (1 − y),  1 − x y dx = 2 Γ(2ν) Γ n + ν + σ + 12

Re ν > − 21 , Re σ > 0 β = ν − σ − 12 α = ν + σ − 12 ,

 2 σ−1

C νn



2

ET II 190(39)a

ET II 191(40)a



ET II 283(21)

7.319    1   1 Γ(λ + n) Γ(μ) Γ(ν) 2 (1 − x)μ−1 xν−1 C λ2n γx1/2 dx = (−1)n 1. 3 F 2 −n, n + λ, ν; , μ + ν; γ n! Γ(λ) Γ(μ + ν) 2 0  2. 0

[Re μ > 0,

1



(1 − x)μ−1 xν−1 C λ2n+1 γx1/2



Re ν > 0]   (−1) 2γ Γ(μ) Γ(λ + n + 1) Γ ν + 12   dx = 1 n! Γ(λ) Γ μ + ν + 2

ET II 191(41)a

n

 1 × 3 F 2 −n, n + λ + 1, ν + 12 ; 32 , μ + ν + ; γ 2

2 Re μ > 0, Re ν > − 12 ET II 191(42)

7.324

Gegenbauer functions and Bessel functions

805

7.32 Combinations of Gegenbauer polynomials Cnν (x) and elementary functions 

7.321

1

−1

 7.322 7.323

2a

0



π

1. 0

2.11





1 − x2

ν− 12

π21−ν in Γ(2ν + n) −ν a J ν+n (a) n! Γ(ν)

Re ν > − 12 ET II 281(7), MO 99a   a ν x π Γ(2ν + n) − 1 e−bx dx = (−1)n C νn e−ab I ν+n (ab) a n! Γ(ν) 2b

Re ν > − 12 ET I 171(9)

eiax C νn (x) dx = 1

[x(2a − x)]ν− 2

C νn (cos ϕ) (sin ϕ)2ν dϕ = 0

[n = 1, 2, 3, . . .]

= 2−2ν π Γ(2ν + 1) [Γ(1 + ν)]

−2

[n = 0] EH I 177(18)

π

0

2ν−1

C νn (cos ψ cos ψ  + sin ψ sin ψ  cos ϕ) (sin ϕ)

dϕ 2

−1

= 22ν−1 n! [Γ(ν)] C νn (cos ψ) C νn (cos ψ  ) [Γ(2ν + n)] [Re ν > 0] 7.324



1

1.



1 − x2

0

 2. 0

1



1 − x2

ν− 12

ν− 12

C ν2n+1 (x) sin ax dx = (−1)n π

C ν2n (x) cos ax dx =

EH I 177(20)

Γ(2n + 2ν + 1) J 2n+ν+1 (a) (2n + 1)! Γ(ν)(2a)ν

Re ν > − 12 , a > 0

(−1)n π Γ(2n + 2ν) J ν+2n (a) (2n)! Γ(ν)(2a)ν Re ν > − 12 ,

a>0



ET I 94(4)

ET I 38(3)a

7.325∗ Complete System of Orthogonal Step Functions Let sj (x) = (−1)2jx for j ∈ N and cj (x) = (−1)2jx+1/2 for j ∈ 0 + N where z denotes the integer part of z. Thus, cj (z) and sj (z) have minimal period j −1 and manifest even and odd symmetry about x = 1/2, respectively, and so are the discrete analogues of cos 2πjx and sin 2πjx. Furthermore, for j ∈ N let j denotes its odd part: the quotient of j by its highest power-of-two factor. Then for all j and k ∈ N, if (j, k) denotes their highest common factor and [j, k] denotes their lowest common multiple:  (j,k)  1 if jj = kk 1. sj (x)sk (x) dx = [j,k] 0 otherwise 0   1 (−1)(j+k)/2+1 (j,k) if jj = kk [j,k] 2. cj (x)ck (x) dx = 0 otherwise 0

806

Complete System of Orthogonal Step Functions

7.331

7.33 Combinations of the polynomials C νn (x) and Bessel functions. Integration of Gegenbauer functions with respect to the index. 7.331  1.

∞

1

2n+1−ν

x



2

x −1

ν−2n− 12

C ν−2n 2n

  1 J ν (xy) dx x −1

−1

= (−1)n 22n−ν+1 y −ν+2n−1 [(2n)!] Γ(2ν − 2n) [Γ(ν − 2n)] cos y

y > 0, 2n − 12 < Re ν < 2n + 12 ET II 44(10)a 7.332  1.

∞

0

  − 1 ν− 3 ν+ 12  2 −1/2  1/2  x + β2 xν+1 x2 + β 2 2 4 C 2n+1 β J ν+ 32 +2n x2 + β 2 a J ν (xy) dx      2 1/2 1    1 y −1/2 1/2 ν+ 2 = (−1)n 21/2 π −1/2 a 2 −ν y ν a2 − y 2 1− 2 C 2n+1 sin β a2 − y 2 a [0 < y < a] =0 [a < y < ∞]



[a > 0,

Re β > 0,

Re ν > −1] ET II 59(23)

∞

2. 0

     −1/2  1/2  J ν+ 12 +2n x2 + β 2 xν+1 x2 + β 2 C 2n β x2 + β 2 a J ν (xy) dx  1/2    2    ν+ 12 y2 n 1/2 −1/2 12 −ν ν 2 −1/2 2 2 1/2 = (−1) 2 π 1− 2 C 2n a y a −y cos β a − y a [0 < y < a] − 12 ν− 34

ν+ 12

=0 [a < y < ∞]

[a > 0,

Re β > 0,

Re ν > −1] ET II 59(24)

7.333  1.

π

0

(sin x)

ν+1

ν+ 12

cos (a cos θ cos x) C n

(cos x) J ν (a sin θ sin x) dx  1/2 n 2π ν+ 1 ν = (−1) 2 (sin θ) C n 2 (cos θ) J ν+ 12 +n (a) a

[n = 1, 3, 5, . . .]

=0  2. 0

[Re ν > −1] π

(sin x)

ν+1

ν+ 12

sin (a cos θ cos x) C n =0 = (−1)

[n = 0, 2, 4, . . .]

n−1 2



WA 414(2)a

(cos x) J ν (a sin θ sin x) dx

2π a

1/2

[n = 0, 2, 4, . . .] ν

ν+ 12

(sin θ) C n

(cos θ) J ν+ 12 +n (a)

[Re ν > −1]

[n = 1, 3, 5, . . .] WA 414(3)a

7.343

Chebyshev polynomials and powers

7.334  1.

π

(sin x)

0

2.

12



π

2ν

J ν (ω) π Γ(2ν + n) J ν+n (α) J ν+n (β) C νn (cos x) dx = ν−1 , ων 2 n! Γ(ν) αν βν  1/2 ω = α2 + β 2 − 2αβ cos x n = 0, 1, 2, . . . ;

(sin x)2ν C νn (cos x)

0

807

Re ν > − 12



Y ν (ω) π Γ(2ν + n) J ν+n (α) Y ν+n (β) dx = ν−1 , ων 2 n! Γ(ν) αν βν  2

 1/2 ω = α + β 2 − 2αβ cos x |α| < |β|, Re ν > − 12

ET II 362(29)

ET II 362(30)

Integration of Gegenbauer functions with respect to the index  c+i∞ −ν  −1 7.335 [sin(απ)] tα C να (z) dα = −2i 1 + 2tz + t2 c−i∞

|arg (z ± 1)| < π] EH I 178(25)

 1 sech(πx) ν − + ix K ν− 12 +ix (a) I ν− 12 +ix (b) C ν− 1 +ix (− cos ϕ) dx 2 2 −∞ 2−ν+1 (ab)ν −ν ω K ν (ω) = Γ(ν)  EH II 55(45) ω = a2 + b2 − 2ab cos ϕ

 7.336

[−2 < Re ν < c < 0, 

∞

7.34 Combinations of Chebyshev polynomials and powers 7.341  1.

1

−1

2.∗



1

−1

 −1 2 [T n (x)] dx = 1 − 4n2 − 1 Tm (x)Tn (x) dx =

1 1 + 2 1 − (m − n) 1 − (m + n)2



1



 1 − x2 Um (x)Un (x) dx =

−1

[m + n is even] [m + n is odd]

=0 3.∗

ET II 271(6)

1 1 − 2 1 − (m − n) 1 − (m + n)2

[m + n is odd]

=0  7.342

1

−1

7.343  1.

1

−1

   1/2  1/2 1 − z2 U n x 1 − y2 + yz dx =

T n (x) T m (x) √

dx =0 1 − x2 π = 2 =π

[m + n is even]

2 U n (y) U n (z) n+1 [|y| < 1, |z| < 1]

ET II 275(34)

[m = n] [m = n = 0] [m = n = 0] MO 104

808

Complete System of Orthogonal Step Functions

 2.

7.344 1.12

 1 − x2 U n (x) U m (x) dx = 0 −1 π = 2



1

1

−1

7.344

[m = n]

ET II 274(28)

[m = n]

ET II 274(27), MO 105a

 −1/2 (y − x)−1 1 − y 2 T n+1 (y) dy = πU n (x)

[|x| < 1]

sign(x) = πU n (x) − πTn+1 (x) √ x2 − 1

[|x| ≥ 1] EH II 187(47)

2.12



1

−1

 1/2 (y − x)−1 1 − y 2 Un (y) dy = −πTn+1 (x)

 = −πTn+1 (x) + πUn (x) sign(x) x2 − 1

[|x| < 1] [|x| ≥ 1] EH II 187(48)

7.345



1

1. −1



1

2. −1



1

3. −1



1

4. −1

5.12



1

−1



1

6. −1



1

7. −1

3

(1 − x)−1/2 (1 + x)m−n− 2 T m (x) T n (x) dx = 0 3

(1 − x)−1/2 (1 + x)m+n− 2 T m (x) T n (x) dx =

3

(1 − x)1/2 (1 + x)m+n+ 2 U m (x) U n (x) dx = 1

ET II 272(10)

π(2m + 2n − 2)! 2m+n (2m − 1)!(2n − 1)! [m + n = 0]

ET II 272(11)

π(2m + 2n + 2)! 2m+n+2 (2m + 1)!(2n + 1)!

ET II 274(31)

(1 − x)1/2 (1 + x)m−n− 2 U m (x) U n (x) dx = 0 (1 − x)(1 + x)1/2 U m (x) U n (x) dx =

[m > n]

[m > n]

√ √ 2 2 − 1 − 4(m − n)2 1 − 4(m + n + 2)2

ET II 274(30)

ET II 274(29)

−1/2

(1 − x)α−1 T m (x) T n (x) dx    1 π 1/2 2α− 2 Γ(α) Γ n − α + 12 1 1 1  1   = F 4 3 −m, m, α, α + ; , α + n + , α − n + 2 2 2 Γ 2 − α Γ α + n + 12 [Re α > 0] ET II

(1 + x)

1/2

(1 + x)

1 ;1 2



272(12)

(1 − x)α−1 U m (x) U n (x) dx

  1 π 1/2 2α− 2 (m + 1) (n + 1) Γ(α) Γ n − α + 32     = 3 3  Γ 2 −α Γ 2 +α+n 3 1 3 × 4 F 3 −m, m + 2, α, α − ; , α + n + , α − n − 2 2 2 [Re α > 0] ET II

1 ;1 2



275(32)

7.353

8.

∗

Chebyshev polynomials and elementary functions



1

−1



7.346 7.347 1.

12

(1 + x) 1

0



1

−1

2.12

−1/2



1

−1

809

√ √ 2 2 Tm (x)Tn (x) dx = + 2 1 − 4(m − n) 1 − 4(m + n)2

xs−1 T n (x) √

dx π  = s 1 1 2 s2 B 2 + 2 s + 12 n, 12 + 12 s − 12 n 1−x [Re s > 0] β

(1 − x)α (1 + x) T n (x) dx =

β

(1 − x)α (1 + x) U n (x) dx =

2α+β+1 Γ(α + 1) Γ (β + 1) Γ(α + β + 2)   1 × 3 F 2 −n, n, α + 1; , α + β + 2; 1 2 [Re α > −1, Re β > −1]

ET II 324(2)

ET II 271(2)

α+β+1

(n + 1) Γ(α + 1) Γ (β + 1)   Γ(α + β + 2) 3 × 3 F 2 −n, n + 2, α + 1; , α + β + 2; 1 2 2

ET II 273(22)

 7.348



1 − x2

−1 1 

 7.349

1

1 − x2

−1

−1/2

−1/2

  U 2n (xz) dx = π P n 2z 2 − 1

[|z| < 1]

  1 T n 1 − x2 y dx = π [P n (1 − y) + P n−1 (1 − y)] 2

ET II 275(33) ET II 222(14)

7.35 Combinations of Chebyshev polynomials and elementary functions 

7.351

1

0

7.352 

∞

1. 0



∞

2.

     − 1 2a x−1/2 1 − x2 2 e− x T n (x) dx = π 1/2 D n− 12 2a1/2 D −n− 12 2a1/2 [Re a > 0]   −1/2  x U n a a2 + x2

  a−n a+1 −n−1 −2 dx = ζ n + 1, 1 n+1 2n 2 (a2 + x2 ) 2 (eπx + 1)   −1/2  x U n a a2 + x2 (a2 + x2 )

0

1 2 n+1

(e2πx − 1)

[Re a > 0] dx =



1. 0

∞



a2 + x2

− 12 n

 sech

ET II 275(39)

a−n−1 a−n 1 ζ(n + 1, a) − − 2 4 2n [Re a > 0]

7.353

ET II 272(13)

ET II 276(40)

        −1/2  a+3 a+1 1 πx T n a a2 + x2 − ζ n, dx = 21−2n ζ n, 2 4 4   a+1 1−n Φ −1, n, =2 2 [Re a > 0] ET II 273(19)

810

Complete System of Orthogonal Step Functions



∞

2.



a2 + x2

0



− 12 n

 cosh

−2     −1/2  1 a+1 πx dx = π −1 n21−n ζ n + 1, T n a a2 + x2 2 2 [Re a > 0]

7.354 1.12



7.354

ET II 273(20)

 1/2  1/2  sin(xyz) √ 1 − y2 cos 1 − x2 z T 2n+1 (x) dx = (−1)n π T 2n+1 (y) J 2n+1 (x) 2 1 − x −1 1

ET II 271(4)



1

2.

sin(xyz) sin



1 − x2

−1

1/2 

1 − y2

1/2  1/2  z U 2n+1 (x) dx = (−1)n π 1 − y 2 U 2n+1 (y) J 2n+2 (z) ET II 274(25)

3.12 4.



 1/2  1/2  cos(xyz) √ 1 − y2 cos 1 − x2 z T 2n (x) dx = (−1)n π T 2n (y) J 2n (z) ET II 271(5) 1 − x2 −1  1  1/2  1/2  1/2  1 − y2 cos(xyz) sin 1 − x2 z U 2n (x) dx = (−1)n π 1 − y 2 U 2n (y) J 2n+1 (z) 1

−1

ET II 274(24)

7.355



1

1. 0



1

2. 0

π dx T 2n+1 (x) sin ax √ = (−1)n J 2n+1 (a) 2 2 1−x T 2n (x) cos ax √

π dx = (−1)n J 2n (a) 2 2 1−x

[a > 0]

ET I 94(3)a

[a > 0]

ET I 38(2)a

7.36 Combinations of Chebyshev polynomials and Bessel functions 

7.361

1



1−x

0

 7.362

 2 −1/2

∞



x2 − 1

1

− 12

1 T n (x) J ν (xy) dx = π J 12 (ν+n) 2

  1 π K 2μ (ax) dx = W Tn x 2a



1 2 n,μ

   1 1 y J 12 (ν−n) y 2 2 [y > 0, Re ν > −n − 1]

ET II 42(1)

(a) W − 12 n,μ (a) [Re a > 0]

ET II 366(17)a

7.37–7.38 Hermite polynomials 

7.371

x

0



7.372 7.373

H n (y) dy = [2(n + 1)]−1 [H n+1 (x) − H n+1 (0)] 1

−1



1. 0

x



1 − t2

2

α− 12

H 2n



 1

EH II 194(27)

Lα n (x)

√  (−1)n π 1/2 (2n)! Γ α + 2 xt dt = Γ(n + α + 1)

Re a > − 21 2

e−y H n (y) dy = H n−1 (0) − e−x H n−1 (x)

[see 8.956]

EH II 195(34)

EH II 194(26)

7.374

Hermite polynomials



∞

2. −∞

7.374



∞

1. −∞

2.11



∞

−∞

2

e−x H 2m (xy) dx =

811

m √ (2m)!  2 y −1 π m!

2

e−x H n (x) H m (x) dx = 0

EH II 195(28)

[m = n]

√ = 2n · n! π

SM II 567

[m = n]

2 m+n−1 m n e−2x H m (x) H n (x) dx = (−1) 2 + 2  2 2 Γ



m+n+1 2

ET II 289(10)a ∞

3. 4.12 5.12

[m + n is even] [m + n is odd]

=0 

SM II 568



−∞  ∞

2

e−x H m (ax) H n (x) dx = 0

[m < n]

ET II 290(20)a

m √ −m+ 1 (2m + n)!  2 2 a − 1 an π2 m! −∞    ∞   m+n m+n−1 m+n+1 −2α2 x2 −m−n−1 2 2 2 1 − 2α e H m (x) H n (x) dx = 2 α Γ −∞  2  α2 1−m−n ; 2 × 2 F 1 −m, −n; 2 2α − 1

Re α2 > 0, α2 = 12 , m + n is even 

∞

6. −∞  ∞

7. −∞



∞

8. −∞



∞

9. −∞

2

e−x H 2m+n (ax) H n (x) dx =

2

e−(x−y) H n (x) dx = π 1/2 y n 2n

ET II 289(12)a

ET II 288(2)a, EH II 195(31)

  2 −2y 2 e−(x−y) H m (x) H n (x) dx = 2n π 1/2 m!y n−m Ln−m n

 n 2 e−(x−y) H n (αx) dx = π 1/2 1 − α2 2 H n



αy

[m ≤ n]

BU 148(15), ET II 289(13)a ET II 290(17)a

(1 − α2 )1/2

2

e−(x−y) H m (αx) H n (αx) dx min(m,n)

=π

1/2

$

k=0



ET II 291(21)a



m 2 k! k k



   m+n αy n  −k 2 2 1−α H m+n−2k 1/2 k (1 − α2 ) ET II 291(26)a

∞

10. −∞

e−

(x−y)2 2u

n



H n (x) dx = (2πu)1/2 (1 − 2u) 2 H n y(1 − 2u)−1/2



0 ≤ u < 12

EH II 195(30)

812

7.375

Complete System of Orthogonal Step Functions



∞

2

1. −∞



∞

1

e−2x H k (x) H m (x) H n (x) dx = π −1 2 2 (m+n+k−1) Γ(s − k) Γ(s − m) Γ(s − n) 2s = k + m + n + 1

−∞

[k + m + n is even]

ET II 290(14)a

m+n+k

2

2.

7.375

e−x H k (x) H m (x) H n (x) dx =

2 2 π 1/2 k!m!n! , (s − k)!(s − m)!(s − n)! 2s = m + n + k

[k + m + n is even] ET II 290(15)a

7.376



∞

1. −∞



∞

2. 0

3.12

4.∗

eixy e−

x2 2

H n (x) dx = (2π)1/2 e−

y2 2

2

H n (y)in 3

1

e−2αx xν H 2n (x) dx = (−1)n 22n− 2 − 2 ν

[Re α > 0, Re ν > −1] BU 150(18a)       ν+1 ∞ Γ n + 12 2 1 Γ 3 1 ν 2 e−2αx xν H 2n+1 (x) dx = (−1)n 22n− 2 ν F −n, + 1; ; √ 1 ν+1 2 2 2α πα 2 0 [Re α > 0,



∞



7.3778

7.378

∞

2

∞

e−x H m (x + y) H n (x + z) dx = 2n π 1/2 m!y n−m Ln−m (−2yz) m [m ≤ n]

xα−1 e−βx H n (x) dx = 2n

n $ 2 n! Γ(α + n − 2m)

m=0

m!(n − 2m)!

[Re α > 0, if n is even;

∞

2

1. −∞  ∞

2. −∞

 7.381

xe−x H 2m+1 (xy) dx = π 1/2

0

(−1)m 2−2m β 2m−α−n

Re α > −1, if n is odd;

Re β > 0]

m (2m + 1)!  2 y y −1 m!

(x ± ic) e

−1

x

ν



2

x +a

−x2

H n (x) dx = 2

 2 −1 −x2 e

ET I 172(11)a

EH II 195(28)

2

∞

∞

ET II 292(30)a

xn e−x H n (xy) dx = π 1/2 n!Pn (y)

−∞



BU 150(18b)

2

0



Re ν > −2]

e−x xn−1 Hn (x) dx = Γ(n) = (n − 1)!

−∞



7.382

   ν+1    Γ n + 12 ν +1 1 1 2 ; ; F −n, √ 1 (ν+1) 2 2 2α πα 2



0

7.379

Γ

MO 165a

EH II 195(29)

n−1−ν 1/2 Γ

π

 n−ν  2

Γ(−ν)



exp ± 21 π(ν + n)i

n 1/2 −2

H 2n+1 (x) dx = (−2) π

a

[c > 0] ET II 288(3)a   √  1 2 n a 2 n! − (2n + 1)!e 2 D −2n−2 a 2 ET II 288(4)a

7.387

7.383

Hermite polynomials



∞

1. 0



∞

2. 0



∞

3. 0

4.12

e−xp H 2n+1

813

√  3 x dx = (−1)n 2n (2n + 1)!!π 1/2 (p − 1)n p−n− 2

e−(b−βx) H 2n+1



1 √ e−(b−β)x H 2n x

[Re p > 0]



EF 151(261)a, ET I 172(12)a

√  (2n + 1)! (b − α)n (α − β)x dx = (−1)n π α − β 3 n! (b − β)n+ 2



[Re(b − β) > 0]



√ (2n)! (b − α) (α − β)x dx = (−1)n π n! (b − β)n+ 12 [Re(b − β) > 0]



∞

0

1

xa− 2 n−1 e−bx H n

%n& 2

∞

5. 0

 7.384

terms are kept in the series for

x−1/2 e−px H 2n

∞

0

√  1 x dx = (−1)n 2n (2n − 1)!!π 1/2 (p − 1)n p−n− 2

    √ (2n)! Γ b + 12 b e−bx n 2n −x √ L (s) H 2n s (1 − e ) dx = (−1) 2 π Γ(n + b + 1) n ex − 1 0

Re b > − 21  ∞ √ √  √ (2n + 1)! Γ(b) b  Ln (s) e−bx H 2n+1 s 1 − e−x dx = (−1)n 22n πs  Γ n + b + 32 0



 7.387

⎦

∞

∞

1. 0

 2.

0

ET I 173(17)a

∞

0



MO 177a

b

[Re b > 0] 7.386

2F 1

⎛ ⎞     √  √  n  α 1 −bx 2π  α+ x a− x ⎠ √ e + Hn dx = 1 − λ−2 b−1 2 H n ⎝  Hn x λ λ b λ2 − 1 [Re b > 0]

2.

Re b > 0, ⎤

ET I 172(14)a



1.

ET I 172(16)a

√    x dx = 2n/2 Γ(a)b−a 2 F 1 − 12 n, 12 − 12 n; 1 − a; b  Re a > 12 n, if n is even, Re a > 12 n − 12 , if n is odd, If a is even, only the first 1 +

7.385

ET I 172(15)a

n

∞

x−

n+1 2

q2

e− 4x H n



q √ 2 x



e−px dx = 2n π 1/2 p

n−1 2

e−q

√

p

ET I 174(23)a

ET I 174(24)a EF 129(117)

2

√  1 1 2 2βx H 2n+1 (x) dx = 2n− 2 π 1/2 β 2n+1 e 2 β

ET II 289(7)a

2

√  1 2 2βx H 2n (x) dx = 2n−1 π 1/2 β 2n e 2 β

ET II 289(8)a

e−x sinh e−x cosh

814

7.388



∞

1. 0



∞

2. 0

 0

∞

4. 0

5.12



6.



∞

∞

0



√  1 1 2 2βx H 2n+1 (x) dx = (−1)n 2n− 2 π 1/2 β 2n+1 e− 2 β

ET II 288(5)a



√   n+ 12 − 1 β 2 sin 2βx H 2n+1 (ax) dx = (−1)n 2−1 π 1/2 a2 − 1 e 2 H 2n+1

aβ

∞

7. 0

√ 1/2 2 (a2 − 1)

√  2 1 2 e−x cos 2βx H 2n (x) dx = (−1)n 2n−1 π 1/2 β 2n e− 2 β e

−x2

ET II 289(6)a

√    −1 1/2 2 n − 12 β 2 1−a cos 2βx H 2n (ax) dx = 2 π e H 2n √

aβ 1/2

2 (a2 − 1)

2

e−y [H n (y)]2 cos e

−x2

√    2βy dy = π 1/2 2n−1 n! Ln β 2

sin(bx) H n (x) H n+2m+1 (x) dx = 2

n−1

2

1

e−x cos(bx) H n (x) H n+2m (x) dx = 2n− 2

m√

(−1)



EH II 195(33)

πn!b

2

2m+1 − b4

e

L2m+1 n

[b > 0] b2 π n!(−1)m b2m e− 4 L2m n 2



2

b 2



b2 2





[b > 0]  7.389



ET II 290(19)a

0

11

e

−x2

7.388

ET II 290(18)a ∞

3. 

2

e−x sin

Complete System of Orthogonal Step Functions

π

0

ET I 39(11)a

ET I 39(11)a

  (2n)! n 1/2 2 dx = 2−n (−1)n π (cos x) H 2n a (1 − sec x) 2 [H n (a)] (n!)

ET II 292(31)

7.39 Jacobi polynomials 7.391



1

1. −1

β

(1 − x)α (1 + x) P (α,β) (x) P (α,β) (x) dx n m =0 α+β+1

=

Γ(α + n + 1) Γ (β + n + 1) 2 n! (α + β + 1 + 2n) Γ(α + β + n + 1)

[m = n,

Re α > −1,

Re β > −1]

[m = n,

Re α > −1,

Re β > −1] ET II 285(5, 9)



1

2. −1

σ

(1 − x)ρ (1 + x) P (α,β) (x) dx = n

2

ρ+σ+1

Γ(ρ + 1) Γ(σ + 1) Γ(n + 1 + α) n! Γ(ρ + σ + 2) Γ(1 + α)

× 3 F 2 (−n, α + β + n + 1, ρ + 1; α + 1, ρ + σ + 2; 1)

3.12



[Re ρ > −1, 1

−1

(1 − x)α (1 + x)σ P (α,β) (x) dx = n

Re σ > −1]

ET II 284(3)

2α+σ+1 Γ(σ + 1) Γ(n + α + 1) Γ(σ − β + 1) n! Γ(σ − β − n + 1) Γ(α + σ + n + 2) [Re α > −1,

Re σ > −1]

ET II 284(1)

7.392

Jacobi polynomials



1

4. −1

2β+ρ+1 Γ(ρ + 1) Γ(β + n + 1) Γ(α − ρ + n) n! Γ(α − ρ) Γ(β + ρ + n + 2)

[Re ρ > −1, Re β > −1]  2 2α+β Γ(α + n + 1) Γ(β + n + 1) (1 − x)α−1 (1 + x)β P (α,β) (x) dx = n n!α Γ(α + β + n + 1) −1

 5.

(1 − x)ρ (1 + x)β P (α,β) (x) dx = n

7.

[Re α > 0, Re β > −1] ET II 285(6)  2 1  2 24α+β+1 Γ α + 12 [Γ(α + n + 1)] Γ(β + 2n + 1) (1 − x)2α (1 + x)β P (α,β) (x) dx = √ n π (n!)2 Γ(α + 1) Γ(2α + β + 2n + 2) −1

ET II 285(7) Re α > − 12 , Re β > −1  1 β (1 − x)ρ (1 + x) P (α,β) (x) P (ρ,β) (x) dx n n 

−1

= 

1

8. −1

9.7



(1 − x)ρ−1 (1 + x)β P (α,β) (x) P (ρ,β) (x) dx = n n

1

2ρ+β Γ(α + n + 1) Γ(β + n + 1) Γ(ρ) n! Γ(α + 1) Γ(ρ + β + n + 1)

1

−1

Re ρ > 0]

ET II 286(11)

(α,σ) (1 − x)α (1 + x)σ P (α,β) (x) P m (x) dx n

= 

2ρ+β+1 Γ(ρ + n + 1) Γ(β + n + 1) Γ (α + β + 2n + 1) n! Γ(β + ρ + 2n + 2) Γ(α + β + n + 1) [Re ρ > −1, Re β > −1] ET II285(10)

[Re β > −1,

−1

10.12

ET II 284(2)

1

 6.

815

2α+σ+1 Γ(α + n + 1) Γ (α + β + m + n + 1) Γ(σ + m + 1) Γ(σ − β + 1) m!(n − m)! Γ (α + β + n + 1) Γ(α + σ + m + n + 2) Γ (α − β + m − n + 1) [Re α > −1, Re σ > −1] ET II 286(12)

(1 − x)ρ (1 + x)β P (α,β) (x) P (ρ,β) (x) dx n m

2β+ρ+1 Γ(α + β + m + n + 1) Γ (β + n + 1) Γ(ρ + m + 1) Γ (ρ − α − m + n) n!(n − m)! Γ(α + β + n + 1) Γ (β + ρ + m + n + 2) Γ(ρ − α) [Re β > −1, Re ρ > −1] ET II 287(16)  x  1  (α+1,β+1) α+1 β+1 (α+1,β+1) P (1 − y)α (1 + y)β P (α,β) (y) dy = (0) − (1 − x) (1 + x) P (x) n n−1 n−1 2n 0 =

11.

EH II 173(38)

7.392



1. 0

1

xλ−1 (1 − x)μ−1 P (α,β) (1 − γx) dx n

  1 Γ(α + n + 1) Γ(λ) Γ(μ) = 3 F 2 −n, n + α + β + 1, λ; α + 1, λ + μ; γ n! Γ (α + 1) Γ(λ + μ) 2 [Re λ > 0, Re μ > 0] ET II 192(46)a

816

Complete System of Orthogonal Step Functions



1

2. 0

xλ−1 (1 − x)μ−1 P (α,β) (γx − 1) dx n = (−1)n



1

3. 0



7.393

  Γ(β + n + 1) Γ(λ) Γ(μ) 1 γ a −n, n + α + β + 1, λ; β + 1, λ + μ; F 3 2 n! Γ (β + 1) Γ(λ + μ) 2 [Re λ > 0, Re μ > 0] ET II 192(47)a

xα (1 − x)μ−1 P (α,β) (1 − γx) dx = n

Γ(α + n + 1) Γ(μ) (α+μ,β−μ) P (1 − γ) Γ(α + μ + n + 1) n [Re a > −1,

1

4. 0

xβ (1 − x)μ−1 P (α,β) (γx − 1) dx = n



1

1.



 2 ν

1−x

0



1

2. 0



1 − x2

ν

(ν,ν) sin bx P 2n+1 (x) dx

=

(ν,ν)

cos bx P 2n (x) dx =

ET II 191(43)a

Γ(β + n + 1) Γ(μ) (α−μ,β+μ) P (γ − 1) Γ(β + μ + n + 1) n [Re β > −1,

7.393

Re μ > 0]

Re μ > 0]

ET II 191(44)a

√ (−1)n π Γ(2n + ν + 2) J 2n+ν+ 32 (b) 1

1

2 2 −ν (2n + 1)!bν+ 2 (−1)n 2

ν− 1 √ 2

[b > 0,

Re ν > −1]

ET I 94(5)

π Γ(2n + ν + 1) J 2n+ν+ 12 (b) 1

(2n)!bν+ 2 [b > 0,

Re ν > −1]

ET I 38(4)

7.41–7.42 Laguerre polynomials 7.411



t

1. 0



t

2. 0



t

3. 0



t

4. 0

5.

Ln (x) dx = Ln (t) − Ln+1 (t)/(n + 1) Lα n (x) dx

=

Lα+1 n−1 (x) dx

Lα n (t) =

−

Lα n+1 (t)

− Lα n (t)

 +

 −

n+α n

n+α n 

MO 110



 +

n+1+α n+1

 EH II 189(16)a

EH II 189(15)a

Lm (x) Ln (t − x) dx = Lm+n (t) − Lm+n+1 (t)

2 ∞  t $ Lk (x) dx = et − 1 k! 0

EH II 191(31)

[t ≥ 0]

MO 110

k=0

7.412



1. 0

1

(1 − x)μ−1 xα Lα n (ax) dx =

Γ(α + n + 1) Γ(μ) α+μ L (a) Γ(α + μ + n + 1) n [Re α > −1,

Re μ > 0] EH II 191(30)a, BU 129(14c)

7.414

Laguerre polynomials



1

2. 0

(1 − x)μ−1 xλ−1 Lα n (βx) dx =

817

Γ(α + n + 1) Γ(λ) Γ(μ) 2 F 2 (−n, λ; α + 1, λ + μ : β) n! Γ(α + 1) Γ(λ + μ) [Re λ > 0,

 7.413 7.414 1.11

1

0



∞

y



∞

2. 0

3.8



α

−y Ln (y) − Lα e−x Lα n (x) dx = e n−1 (y) e−bx Ln (λx) Ln (μx) dx =

EH II 191(29)

  2 b − (λ + μ)b + 2λμ (b − λ − μ)n P n bn+1 b(b − λ − μ)

α e−x xα Lα n (x) Lm (x) dx = 0

= ∞

Γ(α + n + 1) n!

ET I 175(34)

[m = n,

Re α > −1]

BU 115(8), ET II 293(3)

[m = n,

Re α > 0]

BU 115(8), ET II 292(2)

Γ(m + n + α + 1) (b − λ)n (b − μ)m bm+n+α+1   m!n! 0 b(b − λ − μ) × F −m, −n; −m − n − α; (b − λ)(b − μ) [Re α > −1, Re b > 0]      ∞ Γ(α + n + 1)2 Γ(α + m + 1) Γ α + 32 Γ m − 12 α −x α+1/2 α 9   4(1) . e x Ln (x) Lm (x) dx = n!m! Γ(α + 1) Γ − 12 0   × 3 F 2 −n, α + 32 , 32 ; α + 1, 32 − m; 1 4.



∞

5. 0



∞

0



∞

7. 0

 0

9. 0

e

−bx

Lan (x) dx

 n  $ a + m − 1 (b − 1)n−m

=

m

bn−m+1

e−bx Ln (x) dx = (b − 1)n b−n−1 e−st tβ Lα n (t) dt =

Γ(β + 1) Γ(α + n + 1) −β−1 s F n! Γ(α + 1)

ET I 175(35)

[Re b > 0]

ET I 174(27)

[Re b > 0]

ET I 174(25)

  1 −n, β + 1; α + 1; s [Re β > −1, Re s > 0] BU 119(4b), EH II 191(133)

∞

8. 

α e−bx xα Lα n (λx) L m (μx) dx =

m=0

6.

ET II 192(50)a

(m + n)! Γ(α + m + 1) Γ(β + n + 1) α+β+1 Lm+n (y) m!n! Γ(α + β + m + n + 2) [Re α > −1, Re β > −1] ET II 293(7)

β xα (1 − x)β Lα m (xy) Ln [(1 − x)y] dx =

[Re b > 0] ∞

0



Re μ > 0]

∞

e−st tα Lα n (t) dt =

n

Γ(α + n + 1)(s − 1) n!sα+n+1

[Re α > −1,

Re s > 0]

   α+m β+n β m+n e−x xα+β Lα (x) L (x) dx = (−1) (α + β)! m n n m [Re(α + β) > −1]

EH II 191(32), MO 176a

ET II 293(4)

818

10.

Complete System of Orthogonal Step Functions

6



∞

0



∞

11. 0



∞

12. 0

e

−bx 2a

x

2 [Lan (x)]

dx =

e−x xγ−1 Lμn (x) dx = e−x(s+

a1 +a2 2

    22a Γ a + 12 Γ n + 12 2

π (n!) b2a+1  2   Γ(a + n + 1) × F −n, a + 12 ; 12 − n; 1 − 2b

Re a > − 21 , Re b > 0

Γ(γ) Γ(1 + μ + n − γ) n! Γ(1 + μ − γ)

[Re γ > 0]

ET I 174(30) BU 120(4b)

) xμ+β Lμ (a x) Lμ (a x) dx 1 2 k k

 ⎤⎫ ⎧ ⎡  1+μ+β μ+β A2 ⎬ ⎨ k , 1 + ; 1 + μ; F 2 2 2 B d ⎣ Γ(1 + μ + β) Γ(1 + μ + k) ⎦ = ⎭ ⎩ dhk (k!)2 Γ (1 + μ) (1 − h)1+μ B 1+μ+β 

13.12

7.415

A2 =

4a1 a2 h ; (1 − h)2

B =s+

h=0

a1 + a2 1 + h 1−h 2

  a1 + a 2 Re s + > 0, a1 > 0, a2 > 0, Re(μ + β) > −1 BU 142(19) 2   2     ∞ b1 a1 + a2 Γ(1 + μ + k) bk2 (μ,0) μ μ μ x Lk (a1 x) Lk (a2 x) dx = Pk exp −x s + 1+μ+k 2 k! b 0 b2 b0 0 a1 + a2 a1 + a2 , b21 = b0 b2 + 2a1 a2 , b2 = s − b0 = s + 2 2     a 1 + a2 >0 Re μ > −1, Re s + 2 BU 144(22)



1

Γ(α + n + 1) B(λ, μ) 2 F 2 (α + n + 1, λ; α + 1, λ + μ; −β) n! Γ(α + 1) 0 [Re λ > 0, Re μ > 0] ET II 193(51)a        ∞ 1/2   n+m iy iy (2π) 1 2 n−m − 2 √ √ i H x dx = 7.416 xm−n exp − (x − y)2 Lm−n 2 H n m n 2 n! 2 2 −∞ 7.415

(1 − x)μ−1 xλ−1 e−βx Lα n (βx) dx =

BU 149(15b), ET II 293(8)a

7.417



∞

1. 0

 0

3.12

0

b2n [(a − ib)−ν − (a + ib)−ν ] 2(2n)! [b > 0, Re a > 0, Re ν > 2n] ET I 95(12)

∞

2.



xν−2n−1 e−ax sin(bx) Lν−2n−1 (ax) dx = (−1)n i Γ(ν) 2n

b

2n+1

−ν

−ν

[(a + ib) + (a − ib) ] 2(2n + 1)! [b > 0, Re a > 0, Re ν > 2n + 1]

xν−2n−2 e−ax sin(bx) Lν−2n−2 (ax) dx = (−1)n+1 Γ(ν) 2n+1

ET I 95(13) ∞

n+1 xν−2n e−ax cos(bx) Lν−2n Γ(ν) 2n−1 (ax) dx = i(−1)

b2n−1 [(a − ib)−ν − (a + ib)−ν ] 2(2n − 1)! [b > 0, Re a > 0, Re ν > 2n − 1] ET I 39(12)

7.421

Laguerre polynomials



∞

4. 0

xν−2n−1 e−ax cos(bx) Lν−2n−1 (ax) dx = (−1)n Γ(ν) 2n

819

b2n [(a + ib)−ν + (a − ib)−ν ] 2(2n)! [b > 0, Re ν > 2n, Re a > 0] ET I 39(13)

7.418 1.

2.

3.



   1 2 i 1  2 2 [D −n−1 (ib)] − [D −n−1 (−ib)] e− 2 x sin(bx) Ln x2 dx = (−1)n n! √ 2 2π 0 [b > 0]    2  ∞   1 2 π 1 2 b −1 (n!) e− 2 b 2−n H n √ e− 2 x cos(bx) Ln x2 dx = 2 2 0 [b > 0]       ∞ 1 2 π 2n+1 − 1 b2 n+ 12 b2 1 2 n+ 1 x b dx = x2n+1 e− 2 x sin(bx) Ln 2 e 2 Ln 2 2 2 0 

∞

∞

4. 0



∞

5. 0



∞

6. 0

n− 12

2

1

x2n e− 2 x cos(bx) Ln

1

2

xe− 2 x Lα n

e

− 12 x2

Lα n









1 1 2 −α x Ln2 2



1 2 − 1 −α x Ln 2 2

1 2 x 2







 dx =

1 2 x 2



ET I 95(14)

ET I 39(14)

[b > 0]   π 2n − 12 b2 n+ 12 1 2 b e b Ln 2 2

ET I 95(15)

ET I 39(16) [b > 0]      π 1/2 1 1 2 1 2 1 2 2 −α y y sin(xy) dx = L ye− 2 y Lα n n 2 2 2



1 2 x cos(xy) dx = 2

 π 1/2 2

e

− 12 y 2

Lα n





1 2 −α− 1 y Ln 2 2



ET II 294(11)

1 2 y 2



ET II 294(12)

 7.419

7.421

∞

0



∞

1. 0

 0



0

xe

− 12 αx2

 Ln

   (α − β)n − 1 y2 1 2 βy 2 2α e Ln βx J 0 (xy) dx = 2 αn+1 2α(β − α) [y > 0,

∞

2. 3.

1

xn+2ν− 2 exp[−(1 + a)x] L2ν n (ax) K ν (x) dx       1/2 π Γ n + ν + 12 Γ n + 3ν + 12 1 1 1 , n + 3ν + ; 2ν + 1; − a F n + ν + = 1 2 2 2 2n+2ν+

2 n! Γ (2ν + 1) Re a > −2, Re(n + ν) > − 12 , Re(n + 3ν) > − 12 ET II 370(44)

∞

  2 2 xe−x Ln x2 J 0 (xy) dx = 1

2

x2n+ν+1 e− 2 x Lν+n n



−2n−1

n!

1

y 2n e− 4 y

Re α > 0]

2

ET II 13(4)a ET II 13(5)

   1 2 1 2 1 2 x J ν (xy) dx = y 2n+ν e− 2 y Lν+n y n 2 2 [y > 0,

Re ν > −1]

MO 183

820

Hypergeometric Functions



∞

4. 0



∞

5. 0



∞

6. 0

7.422



∞

1. 0

ν+1 −βx2

x

e

Lνn

2

1



e− 2q x xν+1 Lνn

2

αx





J ν (xy) dx = 2

−ν−1 −ν−n−1

β



2

2

(β − α) y e

n+ν+1

x q J ν (xy) dx = e− 2q(1 − q) (q − 1)n

qy2 2

y ν Lνn

2.

∞

0

[y > 0,



∞

0



∞

2. 0

1

2

e− 2 x L n

e

− 12 x2



 Ln

 ET II 43(5)



MO 183

Re β > 0,

 2 σ  2 2 αx Ln αx J ν (xy) dx xν+1 e−αx Lν−σ m = (−1)

1.

y 2

αy 2 4β(α − β)

 y2 yν  Γ n + 1 + 12 ν (2β)−ν−1 e− 4β πn!     2l   n $ (−1)l Γ n − l + 12 Γ l + 12 αy 2 2α − β ν   × L2l β 2β(2α − β) Γ l + 1 + 12 ν (n − l)! l=0

m+n

7.423

2



[ν > 0]

[y > 0,





Lνn

 1  2 ν Ln2 αx2 J ν (xy) dx =

12

2

n ν −y 4β

  2 1  y 2n+ν − 1 y2 xν+1 e−x Lνn x2 J ν (xy) dx = e 4 2n! 2

xν+1 e−βx

7.422

2

y −ν−1 ν − 4α

y e

(2α)

Re α > 0,

Re ν > −1]

ET II 43(7)

  2 y2 y ν−σ+m−n Lm 4α 4α n = 0, σ =  0] ET II 43(8)

Lσ−m+n n

Re ν > −1,



        π 1/2 1 2 1 2 x 1 2 y √ √ x H 2n+1 y H 2n+1 sin(xy) dx = e− 2 y Ln 2 2 2 2 2 2 2 

1 2 x H 2n 2



x √ 2 2

 cos(xy) dx =

 π 1/2 2

e

− 12 y 2

 Ln



1 2 y H 2n 2



ET II 294(13)a

y √ 2 2



ET II 294(14)a

7.5 Hypergeometric Functions 7.51 Combinations of hypergeometric functions and powers 

7.511

0

∞

Γ(a + s) Γ(b + s) Γ(c) Γ(−s) Γ(a) Γ(b) Γ(c + s) [c = 0, −1, −2, . . . , Re s < 0, Re(a + s) > 0,

F (a, b; c; −x)x−s−1 dx =

Re(b + s) > 0]

EH I 79(4)

7.512

7.512

Hypergeometric functions and powers



1

1. 0



1

2. 0



1

3. 0



1

4. 0



1

5. 0

6.12



1

0

7.11



   α α Γ(γ) Γ (α − γ + 1) Γ γ − − β Γ 1+ 2 2    xα−γ (1 − x)γ−β−1 F (α, β; γ; x) dx = α α Γ (1 + α) Γ 1 + − β Γ γ − 2   2  α Re α + 1 > Re γ > Re β, Re γ − − β > 0 ET II 398(1) 2 Γ(γ) Γ(ρ) Γ(β − γ + 1) Γ(γ − ρ + n) Γ(γ + n) Γ(γ − ρ) Γ(β − γ + ρ + 1) [n = 0, 1, 2 . . . ; Re ρ > 0, Re(β − γ) > n − 1]

ET II 398(2)

Γ(γ) Γ(ρ) Γ(β − ρ) Γ(γ − α − ρ) Γ(β) Γ(γ − α) Γ(γ − ρ) [Re ρ > 0, Re(β − ρ) > 0, Re(γ − α − ρ) > 0]

ET II 399(3)

Γ(γ) Γ(ρ) Γ(γ + ρ − α − β) Γ(γ + ρ − α) Γ(γ + ρ − β) [Re γ > 0, Re ρ > 0, Re(γ + ρ − α − β) > 0]

ET II 399(4)

Γ(ρ) Γ(σ) 3 F 2 (α, β, ρ; γ, ρ + σ; 1) Γ(ρ + σ) [Re ρ > 0, Re σ > 0, Re(γ + σ − α − β) > 0]

ET II 399(5)

xρ−1 (1 − x)β−γ−n F (−n, β; γ; x) dx =

xρ−1 (1 − x)β−ρ−1 F (α, β; γ; x) dx =

xγ−1 (1 − x)ρ−1 F (α, β; γ; x) dx =

xρ−1 (1 − x)σ−1 F (α, β; γ; x) dx =

xλ−1 (1 − x)β−λ−1 F (α, β; λ; zx) dx = B(λ, β − λ) F (α, λ; γ; z) [Re β > Re λ,

1

0

[0 < Re γ < Re δ, 8.



1

0

9.12

 0

1

|arg(1 − z)| < π]

BU 9

xγ−1 (1 − x)δ−γ−1 F (α, β; γ; xz) F (δ − α, δ − β; δ − γ; (1 − x)ζ) dx =

12

821

γ−1

x

−1

(1 − x)

−δ

(1 − xz)

Γ(γ) Γ(δ − γ) (1 − ζ)α+β−δ F (α, β; δ; z + ζ − zζ) Γ(δ) |arg(1 − z)| < π, |arg(1 − ζ)| < π] ET II 400(11)

 (1 − x)z dx F (α, β; γ; xz) F δ, β − γ; ; (1 − xz) Γ(γ) Γ() = F (α + δ, β; γ + ; z) Γ(γ + ) [Re γ > 0, Re  > 0, |arg(1 − z)| < π] ET II 400(12), Eh I 78(3) 

xγ−1 (1 − x)ρ−1 (1 − zx)−σ F (α, β; γ; x) dx =

Γ(γ) Γ(ρ) Γ(γ + ρ − α − β) (1 − z)σ Γ(γ + ρ − α) Γ(γ + ρ − β) × 3 F 2 ρ, σ, γ + ρ − α − β; γ + ρ − α, γ + ρ − β;

[Re γ > 0,

Re ρ > 0,

Re (γ + ρ − α − β) > 0,

|arg(1 − z)| < π]

z z−1



ET II 399(6)

822

Hypergeometric Functions



∞

10. 0

xγ−1 (x + z)−σ F (α, β; γ; −x) dx =

7.513

Γ(γ) Γ(α − γ + σ) Γ (β − γ + σ) Γ(σ) Γ(α + β − γ + σ) × F (α − γ + σ, β − γ + σ; α + β − γ + σ; 1 − z)

[Re γ > 0, 

1

11. 0

Re(α − γ + σ) > 0,

Re (β − γ + σ) > 0,

|arg z| < π]

ET II 400(10)

(1 − x)μ−1 xν−1 p F q (a1 , . . . , ap ; ν, b2 , . . . , bq ; ax) dx Γ(μ) Γ(ν) p F q (a1 , . . . , ap ; μ + ν, b2 , . . . , bq ; a) Γ(μ + ν) p ≤ q + 1; if p = q + 1, then |a| < 1] ET II 200(94) =

[Re μ > 0, 

1

12. 0

Re ν > 0,

(1 − x)μ−1 xν−1 p F q (a1 , . . . , ap ; b1 , . . . , bq ; ax) dx Γ(μ) Γ(ν) p+1 F q+1 (ν, a1 , . . . , ap ; μ + ν, b1 , . . . , bq ; a) Γ(μ + ν) Re ν > 0, p ≤ q + 1, if p = q + 1, then |a| < 1] ET II 200(95) =

[Re μ > 0, 13.∗



1

0

xγ−1 (1 − x)σ−1 F (α, β; γ; zx)F (−n, n + γ + σ − 1; γ; x) dx =

 7.513

1

0

Γ(α + n)Γ(β + n)Γ2 (γ)Γ(σ + n) × (−z)n F (α + n, β + n; γ + σ + 2n; z) Γ(α)Γ(β)Γ(γ + n)Γ(γ + σ + 2n) [Re γ > 0, Re σ > 0, n = 0, 1, 2, . . . , | arg(1 − z)| < π]

  ν   s s  s 1  xs−1 1 − x2 F −n, a; b; x2 dx = B ν + 1, 3 F 2 −n, a, ; b, ν + 1 + ; 1 2 2 2 2 [Re s > 0, Re ν > −1] ET I 336(4)

7.52 Combinations of hypergeometric functions and exponentials 

7.521 7.522 1.

11

∞

0



∞

0

2.6

 0

∞

e−st p F q (a1 , . . . , ap ; b1 , . . . , bq ; t) dt =

e−λx xγ−1 2 F 1 (α, β; δ; −x) dx =

e−bx xa−1 F



1 s

p+1 F q



1, a1 , . . . , ap ; b1 , . . . , bq ; s−1 [p ≤ q]

 EH I 192

Γ(δ)λ−γ E (α; β : γ; δ : λ) Γ(α) Γ(β)

1 x 1 + ν, − ν; a; − 2 2 2



[Re λ > 0,

Re γ > 0]

EH I 205(10)

1 1 dx = 2a eb √ Γ(a)(2b) 2 −a K ν (b) π

[Re a > 0,

Re b > 0]

ET I 212(1)

7.524

Hypergeometric functions and exponentials



∞

3. 0

4.6

e

−bx γ−1

x

F (2α, 2β; γ; −λx) dx = Γ(γ)b

−γ

823

 α+β− 12   b b b 2λ e W 12 −α−β,α−β λ λ [Re b > 0, Re γ > 0, |arg λ| < π] BU 78(30), ET I 212(4)



∞

0

e−xt tb−1 F (a, a − c + 1; b; −t) dt = xa−b Γ(b)Ψ(a, c; x) [Re b > 0,



∞

5. 0



∞

6. 0



∞

7. 0

e−x xs−1 p F q (a1 , . . . , ap ; b1 , . . . , bq ; ax) dx = Γ(s)

xβ−1 e−μx

xβ−1 e−μx

p+1 F q

∞

8. 0

[p < q, Re s > 0]   2 −β Pn 1 − 2 F 2 (−n, n + 1; 1, β; x) dx = Γ(β)μ μ [Re μ > 0, Re β > 0]      1 1 −β ; x dx = Γ(β)μ −n, n; β, cos 2n arcsin √ F 2 2 2 μ

xρn −1 e−μx

mF n

EH I 273(11)

(s, a1 , . . . , ap ; b1 , . . . , bq ; a)

[Re μ > 0,



Re x > 0]

Re β > 0]

ET I 337(11)

ET I 218(6)

ET I 218(7)

(a1 , . . . , am ; ρ1 , . . . , ρn ; λx) dx

  λ = Γ (ρn ) μ m F n−1 a1 , . . . , am ; ρ1 , . . . , ρn−1 ; μ Re μ > 0, if m < n; Re μ > Re λ, if m = n] ET I 219(16)a −ρn

[m ≤ n; 

∞

9. 0

xσ−1 e−μx

mF n

Re ρn > 0,

(a1 , . . . , am ; ρ1 , . . . , ρn ; λx) dx

  λ = Γ(σ)μ m+1 F n a1 , . . . , am , σ; ρ1 , . . . , ρn ; μ Re μ > 0, if m < n; Re μ > Re λ, if m = n] ET I 219(17) −σ

[m ≤ n,  7.523

∞

1

1

Re σ > 0, 1

1

(x − 1)μ−1 x−μ− 2 e− 2 ax W 2μ+ 12 ,λ (ax) dx = Γ(μ)e− 2 a W μ+ 12 ,λ (a) [Re μ > 0,

7.524  1.

∞ 0



∞

2. 0

 3. 0

∞

e

−λx

Re a > 0]

  1 2 dx = λα+β−1 S 1−α−β,α−β (λ) F α, β; ; −x 2 

e−st p F q a1 , . . . , ap ; b1 , . . . , bq ; t

e−st 0 F q



q

q−1 t 1 2 , ,..., , 1; q q q q q

 2



dx = s−1

ET II 401(13) [Re λ > 0]   1 4 ; b a , . . . , a , 1, , . . . , b ; 1 p 1 q p+2 F q 2 s2

[p < q]   dt = s−1 exp s−q

MO 176 MO 176

824

Hypergeometric Functions

7.525  1.12

∞

0



xσ−1 e−μx

7.525

 a1 , . . . , am ; ρ1 , . . . , ρn ; (λx)k dx   k  σ+k−1 σ σ+1 kλ −σ ,..., ; ρ1 , . . . , ρn ; = Γ(σ)μ m+k F n a1 , . . . , am , , k k k μ  m + k ≤ n + 1, Re σ > 0; Re μ > 0, if m + k ≤ n;    Re μ + kλe2πir/k > 0; r = 0, 1, . . . , k − 1 for m + k = n + 1

mF n



ET I 220(19) ∞

2. 0

  xe−λx F α, β; 32 ; −x2 dx = λα+β−2 S 1−α−β,α−β (λ) [Re λ > 0]

7.526



γ+i∞

1. γ−i∞



∞

2. 0



∞

3. 0

ext x−b F

ET II 401(14)

  Γ(a + b − c + 1) b−1 1 dx = 2πi t a, b; a + b − c + 1; 1 − Ψ(a; c; t) x Γ(b) Γ(b − c + 1) Re b > 0, Re(b − c) > −1, 

γ>

1 2



EH I 273(12)



t(x + y + t) dt = Γ(γ)Ψ(a, c; x)Ψ (β, c; y) , (x + t)(y + t) γ =a+β−c+1 [Re γ > 0, xy = 0] EH I 287(21)

e−t tγ−1 (x + t)−α (y + t)−β F α, β; γ;

  x(x + y + z) dx xγ−1 (x + y)−α (x + z)−β e−x F α, β; γ; (x + y)(x + z) 1

y+z

= Γ(γ)(zy)− 2 −μ e 2 W ν,μ (y) W λ,μ (z) 2ν = 1 − α + β − γ; 2λ = 1 + α − β − γ; 2μ = α + β − γ [Re γ > 0,

|arg y| < π,

|arg z| < π] ET II 401(15)

7.527  1.

∞



1 − e−x

0



∞



1 − e−x

0

3. 0

  e−μx F α, β; γ; δe−x dx = B(μ, λ) 3 F 2 (α, β, μ; γ, μ + λ; δ) [Re λ > 0,

2. 

λ−1

μ

Re μ > 0,



1 − e−x

ET I 213(9)

  B(α, μ + n + 1) B(α, β + n − α) e−αx F −n, μ + β + n; β; e−x dx = B(α, β − α) [Re α > 0,

∞

|arg(1 − δ)| < π]

γ−1

Re μ > −1]

  Γ(μ) Γ(γ − α − β + μ) Γ(γ) e−μx F α, β; γ; 1 − e−x dx = Γ(γ − α + μ) Γ(γ − β + μ) [Re μ > 0, Re μ > Re(α + β − γ), Re γ > 0]

ET I 213(10)

ET I 213(11)

7.542

Hypergeometric and Bessel functions



∞

4. 0



1 − e−x

γ−1

825



 e−μx F α, β; γ; δ 1 − e−x dx = B(μ, γ) F (α, β; μ + γ; δ) [Re μ > 0,

Re γ > 0,

|arg(1 − δ)| < π]

ET I 213(12)

7.53 Hypergeometric and trigonometric functions 7.531



∞

1. 0

μ   3 2 2 −α−β+1 −α−β α+β−2 K α−β c dx = 2 x sin μx F α, β; ; −c x πc μ 2 Γ(α) Γ(β) μ > 0, Re α > 12 ,

μ   ∞ 1 2 2 −α−β+1 −α−β α+β−1 K α−β c dx = 2 cos μx F α, β; ; −c x πc μ 2 Γ(α) Γ(β) 0 [μ > 0, Re α > 0,

Re β >

1 2



ET I 115(6)

 2.

Re β > 0,

c > 0] ET I 61(9)

7.54 Combinations of hypergeometric and Bessel functions 

7.541

∞

0

xα+β−2ν−1 (x + 1)−ν exz K ν [(x + 1)z] F (α, β; α + β − 2ν; −x) dx 1

γ = α + β − 2ν



= π − 2 cos(νπ) Γ Re(α + β − 2ν) > 0,

1

   1 1 − α + ν Γ 12 − β + ν Γ(γ)(2z)− 2 − 2 γ W 12 γ, 12 (β−α) (2z)    

Re 12 − α + ν > 0, Re 12 − β + ν > 0, |arg z| < 32 π 2

ET II 401(16)

7.542 1.12



∞

  xσ−1 p F p−1 a1 , . . . , ap ; b1 , . . . , bp−1 ; −λx2 Y ν (xy) dx

  y 2  b∗0 , . . . , b∗p−1 , l 1 4λ  h, k, a∗1 , . . . , a∗p , l 2λ 2 σ Γ (a1 ) . . . Γ (ap ) σ σ σ a∗j = aj − , j = 1, . . . , p; b∗0 = 1 − ; b∗j = bj − , 2 2 2 ν ν 1+ν j = 1, . . . , p − 1; h = , k = − , l = − 2 2 2

|arg λ| < π, Re σ > |Re ν|, Re aj > 12 Re σ − 34 , y > 0

0

=

 2. 0

Γ (b1 ) . . . Γ (bp−1 )

p+2,1 G p+2,p+3



ET II 118(53) ∞

  xσ−1 p F p a1 , . . . , ap ; b1 , . . . , bp ; −λx2 Y ν (xy) dx

  y 2  b∗0 , . . . , b∗p , l 1 4λ  h, k, a∗1 , . . . , a∗p , l 2λ 2 σ Γ (a1 ) . . . Γ (ap ) ν ν 1+ν σ bj ∗ = bj − ; j = 1, . . . , p; h = , k = − , l = − 2 2 2 2

Re λ > 0, Re σ > |Re ν|, Re aj > 12 Re σ − 34 , y > 0 =

b∗0 = 1 −

σ ; 2

a∗j = aj −

σ , 2

Γ (b1 ) . . . Γ (bp )

p+2,1 G p+2,p+3



ET II 119(54)

826

Hypergeometric Functions



∞

3.

7.542

  xσ−1 p F q a1 , . . . , ap ; b1 , . . . , bq ; −λx2 Y ν (xy) dx

 σ + ν  σ − ν  Γ = −π 2 y cos (σ − ν) Γ 2 2 2   σ+ν σ−ν 4λ , ; b1 , . . . , bq ; − 2 × p+2 F q a1 , . . . , ap , 2 2 y [y > 0, p ≤ q − 1, Re σ > |Re ν|] ET II 119(55)

0

π

−1 σ−1 −σ



∞

  xσ−1 p F q a1 , . . . , ap ; b1 , . . . , bq ; −λx2 K ν (xy) dx       σ−ν σ+ν σ+ν σ−ν 4λ σ−2 −σ Γ , ; b1 , . . . , bq ; 2 =2 y Γ p+2 F q a1 , . . . , ap , 2 2 2 2 y [Re y > 0, p ≤ q − 1, Re σ > |Re ν|] ET II 153(88)

∞

  x2ρ p F p a1 , . . . , ap ; b1 , . . . , bp ; −λx2 J ν (xy) dx

4. 0

 5. 0

y > 0, 

∞

6. 0

x2ρ



∞

7. 0

m+1 F m

y > 0,

∞

8. 0

 9. 0

∞

 a1 , . . . , am+1 ; b1 , . . . , bm ; −λ2 x2 J ν (xy) dx

  2  22ρ Γ (b1 ) . . . Γ (bm ) y −2ρ−1 m+2,1 y  1, b1 , . . . , bm = G m+1,m+3 Γ (a1 ) . . . Γ (am+1 ) 4λ2  h, a1 , . . . , am+1 , k 1 h = 2 + ρ + 12 ν, k = 12 + ρ − 12 ν, 1 Re λ > 0, Re(2ρ + ν) > −1, Re (ρ − ar ) < 4 ; r = 1, . . . , m + 1 ET II 91(19)

  xδ F α, β; γ; −λ2 x2 J ν (xy) dx





  2 22ρ Γ (b1 ) . . . Γ (bp ) y  1, b1 , . . . , bp p+1,1 = 2ρ+1 G y Γ (a1 ) . . . Γ (ap ) p+1,p+2 4λ  h, a1 , . . . , ap , k h = 12 + ρ + 12 ν, k = 12 + ρ − 12 ν 1 ET II 91(18) Re λ > 0, −1 − Re ν < 2 Re ρ < 2 + 2 Re ar , r = 1, . . . , p

 ⎞   1 − α, 1 − β  1+δ−ν ⎠ 1+δ+ν , 0, 1 − γ,  2

2 −1 − Re ν − 2 min (Re α, Re β) < Re δ < − 21 ET II 82(9)

⎛ 2δ Γ(γ) −δ−1 22 ⎝ y 2 = y G 24 Γ(α) Γ(β) 4λ2

y > 0,

Re λ > 0,

 ⎞  1, γ    2 y Γ(γ) y 31 xδ F α, β; γ; −λ2 x2 J ν (xy) dx = G 24 ⎝ 2  1 + δ + ν 1+δ−ν ⎠ Γ(α) Γ(β) 4λ  , α, β, 2

2 y > 0, Re λ > 0, − Re ν − 1 < Re δ < 2 max (Re α, Re β) − 12 ET II 81(6) δ −δ−1

⎛

2

     2ν+1 Γ(γ) −ν−2 30 y 2  γ y xν+1 F α, β; γ; −λ2 x2 J ν (xy) dx = G 13 Γ(α) Γ(β) 4λ2  ν + 1, α, β

y > 0, Re λ > 0, −1 < Re ν < 2 max (Re α, Re β) − 32 ET II 81(5)

7.542

Hypergeometric and Bessel functions



∞

y   2ν−α−β+2 Γ(ν + 1) α+β−ν−2 y xν+1 F α, β; ν + 1; −λ2 x2 J ν (xy) dx = K α−β λα+β Γ(α) Γ(β) λ

y > 0, Re λ > 0, −1 < Re ν < 2 max (Re α, Re β) − 32 ET II 81(3)

∞

y   xν+1 F α, β; ν + 1; −λ2 x2 K ν (xy) dx = 2ν+1 λ−α−β y α+β−ν−2 Γ(ν + 1) S 1−α−β,α−β λ [Re y > 0, Re λ > 0, Re ν > −1]

10. 0

 11. 0



∞

0

∞

13. 0



∞

14. 0

15.12







12.



827

∞

0

ν+1

x

ET II 152(86)

   y 2 y β−1 λ−ν−β−1  Γ β+ν+2 2 β+ν 2 2 + 1; −λ x Jν (xy) dx = K 12 (ν−β+1) F α, β; 1 β−1 2 2λ π 2 Γ(α)  Γ(β)2 

y > 0, −1 < Re ν < 2 max (Re α, Re β) − 32 ET II 81(4)

   1   1 λ−σ−1 y − 2 Γ(γ) 41 y 2  1 − p, γ − p, l xσ+ 2 F α, β; γ; −λ2 x2 Y ν (xy) dx = √ G 35 4λ2  h, k, α − p, β − p, l 2 Γ(α) Γ(β) 1 1 1 1 h = 4 + 2 ν, k = 4 − 2 ν, l = − 14 − 12 ν, p = 12 + 12 σ ET II 118(52) y > 0, Re λ > 0, Re σ > |Re ν| − 32 , Re σ < 2 Re α, Re σ < 2 Re β ν+2

x

 F

xν+2 F



 y  y  3 1 1 2ν y −ν−1 2 2 , − ν; ; −λ x Y ν (xy) dx = 1 K ν+1 1  Kν 2 2 2 2λ 2λ π 2 λ2 Γ 2 − ν

y > 0, Re λ > 0, − 23 < Re ν < − 21 3 1, 2ν + ; ν + 2; −λ2 x2 2



ET II 117(49)

 y 2 1 Γ(ν + 2)   Y ν (xy) dx = π − 2 2−ν λ−2ν−3 y ν  K ν 2λ Γ 2ν + 32

y > 0, Re λ > 0, − 21 < Re ν < 12

ET II 117(50)



∞

16. 0

17.

xν+2 F





y  3 3 π 2−μ−ν−1 λ−μ−2ν−3 y μ+ν   K 1, μ + ν + ; ; −λ2 x2 Y ν (xy) dx = μ 2 2 λ Γ μ + ν + 32

3 1 3 y > 0, Re λ > 0, − 2 < Re ν < 2 , Re(2μ + ν) > − 2 ET II 118(51) 1 2

  1 2 2 x F α − ν − , α; 2α; −λ x J ν (xy) dx 2  0 1   y   y  i Γ 2 + α Γ 12 + α + ν y W 12 −α,− 12 −ν e−iπ − W 12 −α,− 12 −ν eiπ W 12 −α,− 12 −ν = 1−ν−2α 2α−1 ν+2 π2 λ y λ λ λ

y > 0, Re λ > 0, Re ν < − 21 , Re(α + ν) > − 21 ET II 80(1)



∞



∞

18. 0

2α+ν

x2α−ν F

  1 ν + α − , α; 2α; −λ2 x2 J ν (xy) dx 2   y 22α−ν Γ 12 + α y ν−2 1 1 M W = α− ,ν− 2 2 λ2α−1 Γ(2ν) λ

1 1 2 −α,ν− 2

y λ

ET II 80(2)

828

7.543

Confluent Hypergeometric Functions



∞

1. 0



∞

2. 0

 7.544

∞

0

−2α−1

x

 F

xν+1−4α F

7.543

 4λ2 1 + α, 1 + α; 1 + 2α; − 2 J ν (xy) dx = λ−2α I 12 ν+α (λy) K 12 ν−α (λy) 2 x

y > 0, Re λ > 0, Re ν > −1, Re α > − 21 ET II 81(7)

  λ2 1 α, α + ; ν + 1; − 2 J ν (xy) dx 2 x     Γ(ν) ν 1−2α 2α−ν−1 1 1 2 λ λy K 2α−ν−1 λy = y Iν Γ(2α) 2 2

3 y > 0, Re λ > 0, Re α − 1 < Re ν < 4 Re α − 2 ET II 81(8)

  4x 1 J ν (xy) dx xν+1 (1 + x)−2α F α, ν + ; 2ν + 1; 2 (1 + x)2 Γ(ν + 1) Γ(ν − α + 1) 2ν−2α+1 2(α−ν−1) 2 = y J ν (y) Γ(α)

y > 0, −1 < Re ν < 2 Re α − 32 ET II 82(10)

7.6 Confluent Hypergeometric Functions 7.61 Combinations of confluent hypergeometric functions and powers 7.611



∞

1. 0



∞

2. 0

3.12



5.

Γ

3 4

π 2 2k sec(μπ)    − 12 k + 12 μ Γ 34 − 12 k − 12 μ

|Re μ| < 12

x−1 M k,μ (x) W λ,μ (x) dx =

x−1 W k,μ (x) W λ,μ (x) dx

Γ(2μ + 1)   (k − λ) Γ 12 + μ − λ

Re μ > − 12 ,

ET II 406(22)

Re(k − λ) > 0



BU 116(11), ET II 409(39)



1 π 1     − 1    = (k − λ) sin(2μπ) Γ 12 − k + μ Γ 12 − λ − μ Γ 2 − k − μ Γ 12 − λ + μ

BU 116(12), ET II 409(40) |Re μ| < 12  1  1  ∞ ψ 2 +μ−κ −ψ 2 −μ−κ dz π     = {W κ,μ (z)}2 z sin 2πμ Γ 12 + μ − κ Γ 12 − μ − κ 0

|Re μ| < 12 BU 117(12a)    ∞  1 ψ −κ 1 2 [W κ,0 (z)] dz =  2 BU 117(12b)  2 1 0 z Γ 2 −κ 0

4.

∞

3

x−1 W k,μ (x) dx =

7.613

Confluent hypergeometric functions and powers



∞

6. 0

7.11



ρ−1

x

829

    Γ(ρ + 1) Γ 12 ρ + 12 + μ Γ 12 ρ + 12 − μ     W k,μ (x) W −k,μ (x) dx = 2 Γ 1 + 12 ρ + k Γ 1 + 12 ρ − k [Re ρ > 2|Re μ| − 1]

∞

0

xρ−1 W k,μ (x) W λ,ν (x) dx =

Γ(1 − μ + ν + ρ) Γ(1 + μ + ν + ρ) Γ(−2ν)     Γ 12 − λ − ν Γ 32 − k + ν + ρ   × 3 F 2 1 − μ + ν + ρ, 1 + μ + ν + ρ, 12 − λ + ν; 1 + 2ν, 32 − k + ν + ρ; 1 Γ(1 + μ − ν + ρ) Γ(1 − μ − ν + ρ) Γ(2ν)     + Γ 12 − λ + ν Γ 32 − k − ν + ρ   × 3 F 2 1 + μ − ν + ρ, 1 − μ − ν + ρ, 12 − λ − ν; 1 − 2ν, 32 − k − ν + ρ; 1 [|Re μ| + |Re ν| < Re ρ + 1]

7.612



∞

1. 0



∞

2. 0

ET II 409(41)

tb−1 1 F 1 (a; c; −t) dt = tb−1 Ψ(a, c; t) dt =

Γ(b) Γ(c) Γ(a − b) Γ(a) Γ(c − b)

[0 < Re b < Re a]

Γ(b) Γ(a − b) Γ(b − c + 1) Γ(a) Γ(a − c + 1)

ET II 410(42)

EH I 285(10)

[0 < Re b < Re a

Re c < Re b + 1] EH I 285(11)

7.613



t

1. 0

xγ−1 (t − x)c−γ−1 1 F 1 (a; γ; x) dx = tc−1

Γ(γ) Γ(c − γ) 1 F 1 (a; c; t) Γ(c) [Re c > Re γ > 0] BU 9(16)a, EH I 271(16)



t

2. 0



xβ−1 (t − x)γ−1 1 F 1 (t; β; x) dx =

1

3. 0

xλ−1 (1 − x)2μ−λ 1 F 1



Γ(β) Γ(γ) β+γ−1 t 1 F 1 (t; β + γ; t) Γ(β + γ)

1 + μ − ν; λ; xz 2



[Re β > 0,

Re γ > 0] 1

ET II 401(1)

1

dx = B(λ, 1 + 2μ − λ)e 2 z z − 2 −μ M ν,μ (z) [Re λ > 0,

Re(2μ − λ) > −1] BU 14(14)



t

4. 0

xβ−1 (t − x)δ−1 1 F 1 (t; β; x) 1 F 1 (γ; δ; t − x) dx =

Γ(β) Γ(δ) β+δ−1 t 1 F 1 (t + γ; β + δ; t) Γ(β + δ) [Re β > 0, Re δ > 0] ET II 402(2), EH I 271(15)

 5. 0

t

1

1

xμ− 2 (t − x)ν− 2 M k,μ (x) M λ,ν (t − x) dx =

Γ(2μ + 1) Γ(2ν + 1) μ+ν t M k+λ,μ+ν+ 12 (t) Γ(2μ + 2ν + 2)

Re μ > − 12 , Re ν > − 12

BU 128(14), ET II 402(7)

830

Confluent Hypergeometric Functions

 6. 0

1

7.621

xβ−1 (1 − x)σ−β−1 1 F 1 (α; β; λx) 1 F 1 [σ − α; σ − β; μ(1 − x)] dx Γ(β) Γ(σ − β) λ e 1 F 1 (α; σ; μ − λ) Γ(σ) [0 < Re β < Re σ] ET II 402(3) =

7.62–7.63 Combinations of confluent hypergeometric functions and exponentials 7.621



∞

1. 0



e

    Γ α + ν + 32 1 2 3 t M μ,ν (t) dt =  α+ν+ 32 F α + ν + 2 , −μ + ν + 2 ; 2ν + 1; 2s + 1 1 2 +s  

Re α + μ + 32 > 0, Re s > 12

−st α

BU 118(1), MO 176a, EH I 270(12)a

∞

2. 0

1

1



e−st tμ− 2 M λ,μ (qt) dt = q μ+ 2 Γ(2μ + 1) s − 12 q

λ−μ− 12  

s + 12 q

−λ−μ− 12

1 Re μ > − , 2

|Re q| Re s > 2



BU 119(4c), MO 176a, EH I 271(13)a



3.

4.

   −α−μ− 32 1   ∞ Γ α + μ + 32 Γ α − μ + 32 q μ+ 2 1 −st α s+ q e t W λ,μ (qt) dt = Γ(α − λ + 2) 2 0   1 2s − q 3 × F α + μ + , μ − λ + ; α − λ + 2; 2 2     2s + q 3 q Re α ± μ + > 0, Re s > − , q > 0 EH I 271(14)a, BU 121(6), MO 176 2 2  ∞   [|s| > |k|] e−st tb−1 1 F 1 (a; c; kt) dt = Γ(b)s−b F a, b; c; ks−1   0 k = Γ(b)(s − k)−b F c − a, b; c; [|s − k > |k||] k−s [Re b > 0, Re s > max (0, Re k)] EH I 269(5) 

∞

5. 0



∞

6. 0

 7. 0

∞

 −a tc−1 1 F 1 (a; c; t)e−st dt = Γ(c)s−c 1 − s−1

[Re c > 0,

Re s > 1]

EH I 270(6)

Γ(b) Γ(b − c + 1) F (b, b − c + 1; a + b − c + 1; 1 − s) Γ(a + b − c + 1) [Re b > 0, Re c < Re b + 1,  Γ(b) Γ(b − c + 1) −b  s F a, b; a + b − c + 1; 1 − s−1 = Γ(a + b − c + 1)

tb−1 Ψ (a, c; t) e−st dt =

b

e− 2 x xν−1 M κ,μ (bx) dx =

1



|1 − s| < 1]

Re s >

1 2



EH I 270(7)

Γ(1 + 2μ) Γ (κ − ν) Γ 2 + μ + ν ν     b Γ 12 + μ + κ Γ 12 + μ − ν   Re ν + 12 + μ > 0,

Re (κ − ν) > 0



BU 119(3)a, ET I 215(11)a

7.622

Confluent hypergeometric functions and exponentials



∞

8. 0

9.12



∞

0

10.12



∞

0



∞

11. 0

12.

7.622

e

−sx

2 Γ(1 + 2μ)e−iπκ dx   = M κ,μ (x) x Γ 12 + μ + κ

e−sx W κ,μ (x)

π dx = x cos (πμ)



s− s+



s− s+

κ 1 2

2 1 2

1 2 1 2

 κ2

Q κμ− 1 (2s) 2  1 Re 2 + μ > 0,

P κμ− 1 (2s) 2

831

1  Re 2 ± μ > 0,

 

Γ k + μ + 12 Γ 14 (2k + 6μ + 5) k+2μ−1 − 32 x 

x e W k,μ (x) dx =  Γ k + 3μ + 12 Γ 14 (2μ − 2k + 3) Re(k + μ) > − 12 , 1

e− 2 x xν−1 W κ,μ (x) dx =

 Γ ν+

  1 2 −μ Γ ν + Γ (ν − κ + 1)

1 2

Re s >



∞

0



BU 121(7)

Re(k + 3μ) > − 21



BU 122(8b)

Re (κ + ν) < 0



BU 122(8c)a

e−st tc−1 1 F 1 (a; c; t) 1 F 1 (α; c; λt) dt

= Γ(c)(s − 1)−a (s − λ)−α sa+α−c F a, α; c; λ(s − 1)−1 (s − λ)−1

∞

0

BU 119(4a)

BU 122(8a), ET II 406(23)

 +μ

[Re c > 0, 2.



Re s > − 12



 Re ν + 12 ± μ > 0      ∞ Γ 1 + μ + ν Γ 12 − μ + ν 1  1  e 2 x xν−1 W κ,μ (x) dx = Γ (−κ − μ)  12 Γ 2 − μ − κ Γ 2 +μ − κ 0  Re ν + 12 ± μ > 0,

1.

1 2

Re s > Re λ + 1]

EH I 287(22)

e−t tρ 1 F 1 (a; c; t)Ψ (a ; c ; λt) dt

=C

 Γ(c) Γ(β) σ  λ F c − a, β; γ; 1 − λ−1 , Γ(γ) ρ = c − 1,

ρ = c + c − 2,

σ = −c,

β = c − c + 1,

σ = 1 − c − c ,

γ = c − a + a − c + 1,

β = c + c − 1,

γ = a − a + c,

C=

C=

Γ (a − a) , or Γ (a )

Γ (a − a − c + 1) Γ (a − c + 1) EH I 287(24)

832

3.

12

Confluent Hypergeometric Functions



∞

0

7.623

xν−1 e−bx M λ1 ,μ1 − 12 (a1 x) . . . M λn ,μn − 12 (an x) dx = aμ1 1 . . . aμnn (b + A)−ν−M Γ(ν + M )  × F A ν + M : μ1 − λ1 , . . . , μn − λn ; 2μ1 , . . . , 2μn :

 an a1 ,..., , b+A b+A M = μ1 + · · · + μn , A = 12 (a1 + · · · + an ) 1 1 Re(ν + M ) > 0, Re b ± 2 a1 ± · · · ± 2 an > 0 ET I 216(14)

7.623



∞

1. 0

3.

[− Re c < n < 1 − Re a, t

−1

k−1

|arg y| < π]

EH I 285(16)

1 2 (t−x)

ET II 402(6)

1

   t Γ(λ) Γ 2 − k − λ + μ Γ 12 − k − λ − μ 1 1  1  W k+λ,μ (t) x−k−λ−1 (t − x)λ−1 e 2 x W k,μ (x) dx = tk+1 Γ 2 − k + μ Γ 2 − k − μ 0

Re λ > 0, Re(k + λ) < 12 − |Re μ|

 4.

n = 0, 1, 2, . . . ,

  1 Γ(k) Γ(2μ + 1) 1 k− 1 2 2   t x (t − x) e M k,μ (x) dx = Iμ 1 π t 2 Γ k + μ + 0 2

Re k > 0, Re μ > − 21 ET II 402(5)   k+λ−1  t 1 Γ(λ) Γ k + μ + 2 t 1   M k,μ (t) xk−1 (t − x)λ−1 e 2 (t−x) M k+λ,μ (x) dx = 1 Γ k+λ+ 0

μ+ 2 Re(k + μ) > − 12 , Re λ > 0 

2.

e−x xc+n−1 (x + y)−1 1 F 1 (a; c; x) dx = (−1)n Γ(c) Γ(1 − a)y c+n−1 Ψ(c − a, c; y)

ET II 405(21)



∞

5. 1

6.11



∞

1

 7. 1

  Γ(μ) Γ 12 − k − λ − μ − 1 μ 1 a μ−1 λ− 12 12 ax 1  a 2 e 2 W k+ 12 μ,λ+ 12 μ (a) (x − 1) x e W k,λ (ax) dx = Γ − k − λ 2

ET II 211(72)a |arg(a)| < 32 π, 0 < Re μ < 12 − Re(k + λ) 1

1

1

1

(x − 1)μ−1 xλ− 2 e− 2 ax W k,λ (ax) dx = a− 2 μ Γ(μ)e− 2 a W k− 12 μ,λ− 12 μ (a) [Re μ > 0,

∞

1

Re a > 0]

ET II 211(74)a

Re a > 0]

ET II 211(73)a

1

(x − 1)μ−1 xk−μ−1 e− 2 ax W k,λ (ax) dx = Γ(μ)e− 2 a W k−μ,λ (a) [Re μ > 0,

7.624

Confluent hypergeometric functions and exponentials

 8. 0

1

833

1

(1 − x)μ−1 xk−μ−1 e− 2 ax W k,λ (ax) dx 1

= Γ(μ)e− 2 a sec[(k − μ − λ)π]     Γ k − μ + λ + 12 M k−μ,λ (a) + cos[(k − λ)π] W k−μ,λ (a) × sin(μπ) Γ(2λ + 1)

ET II 200(93)a 0 < Re μ < Re k − |Re λ| + 12 7.624



∞

1.

    1 , 1, 1 − k + ρ −σ Γ(2μ + 1)aσ 2 23 = 1 1  G 34 a  1  2 + μ + ρ, −σ, σ, 12 − μ + ρ π2 Γ 2 + k + μ

1 |arg a| < π, Re(μ + ρ) > − 2 , Re(k − ρ − σ) > 0 ET II 403(8)

0



∞

2. 0



∞

3.

ρ−1

x

    1 , 1, 1 − k + ρ  1  1 2σ 1 1  − x − σ 32 2 x 2 + (a + x) 2 e 2 W k,μ (x) dx = −π 2 σa G 34 a  1  2 + μ + ρ, 12 − μ + ρ, −σ, σ

ET II 406(24) |arg a| < π, Re ρ > |Re μ| − 12

 1  1 2σ 1 xρ−1 x 2 + (a + x) 2 e 2 x W k,μ (x) dx

   1  1 , 1, 1 + k + ρ σπ − 2 aσ  2    G 33 = − 1 34 a  1  2 + μ + ρ, 12 − μ + ρ, −σ, σ Γ 2 − k + μ Γ 12 − k − μ

|arg a| < π, Re ρ > |Re μ| − 12 , Re(k + ρ + σ) < 0 ET II 406(25)

0



 1  1 2σ 1 xρ−1 x 2 + (a + x) 2 e− 2 x M k,μ (x) dx

∞

4. 0

 1  1 1 2σ 1 xρ−1 (a + x)− 2 x 2 + (a + x) 2 e− 2 x M k,μ (x) dx

   1 1  0, , − k − ρ Γ(2μ + 1)aσ 23 2 2  a   G 34 1  −σ, ρ + μ, ρ − μ, σ π 2 Γ 12 + k + μ

Re(ρ + μ) > − 21 , Re(k − ρ − σ) > − 21 ET II 403(9) =

|arg a| < π, 

∞

5.

 1  1 1 2σ 1 xρ−1 (a + x)− 2 x 2 + (a + x) 2 e 2 x W k,μ (x) dx

   1 1 1  0, , + k + ρ π − 2 aσ 33 2 2     a = G 34  −σ, ρ + μ, ρ − μ, σ Γ 2 − k + μ Γ 12 − k − μ

|arg a| < π, Re ρ > |Re μ| − 12 , Re(k + ρ + σ) < 12 ET II 406(26)

0

1

 6. 0

∞

ρ−1

x

− 12

(a + x)

   1 1  1   0, , − k + ρ 1 2σ − 12 x − 12 σ 32 2 2  2 2 x + (a + x) e W k,μ (x) dx = π a G 34 a  −σ, ρ + μ, ρ − μ, σ |arg a| < π, Re ρ > |Re μ| − 12 ET II 406(27)

834

7.625

Confluent Hypergeometric Functions



∞

1. 0



xρ−1 exp − 21 (α + β)x M k,μ (αx) W λ,ν (βx) dx =

2.

12



∞

0

3.

12



∞

0



∞

4. 0

7.626 1.12



1

ρ−1

x

ρ−1

x

ρ−1

x



0

Γ(1 + μ + ν + ρ) Γ(1 + μ − ν + ρ) μ+ 1 −μ−ρ− 1 2 3  α 2β Γ − λ + μ + ρ 2   3 α 1 + k + μ, 1 + μ + ν + ρ, 1 + μ − ν + ρ; 2μ + 1, − λ + μ + ρ; − × 3F 2 2 2 β [Re α > 0, Re β > 0, Re (ρ + μ) > |Re ν| − 1] ET II 410(43)

 1 exp (α + β)x W k,μ (αx) W λ,ν (βx) dx 2         −1 = β −ρ Γ 12 − k + μ Γ 12 − k − μ Γ 12 − λ + ν Γ 12 − λ − ν    β  12 + μ, 12 − μ, 1 + λ + ρ 33 × G 33  α  12 + ν + ρ, 12 − ν + ρ, −k [|Re μ| + |Re ν| < Re ρ + 1, Re(k + λ + ρ) < 0] ET II 410(44)a 

    1 β  12 + μ, 12 − ν, 1 − λ + ρ −ρ 22 exp − (α + β)x W k,μ (αx) W λ,ν (βx) dx = β G 33  2 α  12 + ν + ρ, 12 − ν + ρ, k [Re(α + β) > 0, |Re μ| + |Re ν| < Re ρ + 1] ET II 411(46) 

 1 exp − (α − β)x W k,μ (αx) W λ,ν (βx) dx 2       

β  12 + μ, 12 − μ, 1 + λ + ρ −1 −ρ 23 1 1 Γ 2 −λ+ν Γ 2 −λ−ν =β  G 33 α  12 + ν + ρ, 12 − ν + ρ, k [Re α > 0, |Re μ| + |Re ν| < Re ρ + 1] ET II 411(45) 

   1 1 k − (ξ + η) exp − (ξ + η)x xc 1 F 1 (a; c; ξx) 1 F 1 (a; c; ηx) dx x 4 2 =0 a 2 = e−ξ [ 1 F 1 (a + 1; c; ξ)] ξ [where ξ and η are any two zeros of the function

 2. 1

∞

7.625



[ξ = η,

Re c > 0]

[ξ = η,

Re c > 0]

1F 1

(a; c; x)]

 1 1 k − (ξ + η) e− 2 (ξ+η)x xc Ψ (a, c; ξx) Ψ(a, c; ηx) dx = 0 x 4 = −ξ −1 e−ξ [Ψ(a − 1, c; ξ)]2 [where ξ and η are any two zeros of the function Ψ(a, c; x)]

EH I 285

[ξ = η] ; [ξ = η] EH I 286

7.627

7.627

Confluent hypergeometric functions and exponentials



∞

1. 0



∞

2. 0

2λ−1

x

−μ− 12

(a + x)

1

e

1 2x

1

  Γ(2λ) Γ 12 − k + μ − 2λ λ−μ− 1 2 W 1  a W k,μ (a + x) dx = k+λ,μ−λ (a) Γ − k + μ 2

ET II 411(50) |arg a| < π, 0 < 2 Re λ < 12 − Re(k + μ) −1x

x2λ−1 (a + x)−μ− 2 e− 2 x M k,μ2 (a + x) dx =



∞

3. 0

 4. 0



∞

5. 0



∞

7. 0

8. 0

1

Re λ > 0]

ET II 411(47)

1

xλ−1 (a + x)k−λ−1 e− 2 x W k,μ (a + x) dx = Γ(λ)ak−1 W k−λ,μ (a) [|arg a| < π, Re λ > 0] ET II 411(48)     0, 1 − k − σ ρ−1 −σ − 12 x ρ 21 a 30  x (a + x) e W k,μ (a + x) dx = Γ(ρ)a e G 23 a  −ρ, 12 + μ − σ, 12 − μ − σ Re ρ > 0]

ET II 411(49)

1

xρ−1 (a + x)−σ e 2 x W k,μ (a + x) dx

   1  k − σ + 1, 0 Γ(ρ)aρ e− 2 a 31     G 23 a  = 1 −ρ, 12 + μ − σ, 12 − μ − σ Γ 2 − k + μ Γ 12 − k − μ [|arg a| < π, 0 < Re ρ < Re(σ − k)] ET II 412(51)

0



1

x2λ−1 (a + x)−μ− 2 e− 2 x W k,μ (a + x) dx = Γ(2λ)aλ−μ− 2 W k−λ,μ−λ (a)

[|arg a| < π, ∞

6.



1

  Γ(2λ) Γ(2μ + 1) Γ k + μ − 2λ + 12 λ−μ− 1 2 M   a k−λ,μ−λ (a) 1 Γ k+

μ + 2 Γ(1 − 2λ + 2μ) ET II 405(20) Re λ > 0, Re(k + μ − 2λ) > − 12

[|arg a| < π, ∞

835

∞

1

e− 2 (a+x)

1

Γ (a + x)2κ−1 dx = W κ,μ (x) (ax)κ x

1 2

   − μ − κ Γ 12 + μ − κ W κ,μ (a) a Γ (1 − 2κ) 

1 Re 2 ± μ − κ > 0

BU 126(7a)

dx (x + a)α     Γ(1 + 2μ) Γ 12 + μ + γ Γ (κ − γ) 1 1     = 2 F 2 α, κ − γ; 2 + μ − γ, 2 − μ − γ; a 1 1 Γ 2 + μ − γ Γ  2 +  μ+κ  Γ α + γ + 12 + μ Γ −γ − 12 − μ γ+ 1 +μ a 2 + Γ(α)   × 2 F 2 α + γ + μ + 12 , κ + μ + 12 ; 1 + 2μ, 32 + μ + γ; a  

Re γ + α + 12 + μ > 0, Re (γ − κ) < 0 BU 126(8)a

e− 2 x xγ+α−1 M κ,μ (x)

836

Confluent Hypergeometric Functions

 9. 0

7.628  1.

2.

7.628

  dx 1 n+1 n+μ+ 12 12 a = (−1) − μ + κ W −κ,μ (a) e x M κ,μ (x) a e Γ(1 + 2μ) Γ x+a 2       1 n > 0, Re κ − μ − < n, |arg a| < π n = 0, 1, 2, . . . , Re μ + 1 + BU 127(10a)a 2 2

∞

− 12 x n+μ+ 12

    2 1 1 1 e−st e−t t2c−2 1 F 1 a; c; t2 dt = 21−2c Γ(2c − 1)Ψ c − , a + ; s2 2 2 4 0

1 EH I 270(11) Re c > 2 , Re s > 0  2  2  ∞ 2 1 2 1 t as dt = √ Γ(4ν + 1)a−ν s−4ν eas /8 K 2ν t2ν−1 e− 2a t e−st M −3ν,ν a 8 2 π 0

Re a > 0, Re ν > − 14 , Re s > 0 

∞

∞

3. 0

1

2

t2μ−1 e− 2a t e−st M λ,μ



2

t a

dt

=2

7.629  1.12

∞

0



∞

2. 0

tk exp

a 2t

e−st W k,μ

−3μ−λ

a t

ET I 215(12)

 1 2 (λ+μ−1)

λ−μ−1

∞

0

 2. 0

∞

as2 4



Γ(4μ + 1)a s e W − 12 (λ+3μ), 12 (λ−μ)

Re a > 0, Re μ > − 41 , Re s > 0 ET I 215(13)

 √  √ 1 dt = 21−2k as−k− 2 S 2k,2μ 2 as |arg a| < π, Re (k ± μ) > − 21 ,

Re s > 0



ET I 217(21)

a  a  √  √ 1 e−st W k,μ dt = 2 ask− 2 K 2μ 2 as t−k exp − 2t t [Re a > 0,

7.631  1.



as2 8

xρ−1 exp

xρ−1 exp



1 

α−1 x − βx−1

2

 1 2

−ν −ρ

ET II 412(55)

    W k,μ α−1 x W λ,ν βx−1 dx         −1 = β ρ Γ 12 − k + μ Γ 12 − k − μ Γ 12 − λ + ν Γ 12 − λ − ν    β  1 + k, 1 + λ − ρ × G 42 24 α  12 + μ, 12 − μ, 12 + ν − ρ, 12 − ν − ρ

3 |arg β| < 2 π, Re(λ − ρ) < 12 − |Re μ|, Re(k + ρ) < 12 − |Re ν|

α−1 x − βx−1

|arg α| < 32 π,

ET I 217(22)

    W k,μ α−1 x W λ,ν βx−1 dx

    −1 = β ρ Γ 12 − k + μ Γ 12 − k − μ   β  1 + k, 1 − λ − ρ 41 × G 24 α  12 + μ, 12 − μ, 12 + ν − ρ,

3 |arg α| < 2 π, Re β > 0, Re(k + ρ) < −|Re ν| − 12

1  2



Re s > 0]



ET II 412(57)

7.644

Confluent hypergeometric and trigonometric functions



∞

3. 0

 7.632

∞

0

xρ−1 exp

1  2

    W k,μ α−1 x W λ,ν βx−1 dx    β  1 − k, 1 − λ − ρ ρ 40 = β G 24 α  12 + μ, 12 − μ, 12 + ν − ρ, 12 − ν − ρ ET II 412(54) [Re α > 0, Re β > 0]     1 exp − λet M k,μ λet − λ dt 2   Γ(2μ + 1) Γ 12 + k − μ + s = W −k− 12 s,μ− 12 s (λ) Γ(s + 1)

Re μ > − 21 , Re s > Re(μ − k) − 12 ET I 216(15)

α−1 x + βx−1

 μ− 12 e−st et − 1

837



7.64 Combinations of confluent hypergeometric and trigonometric functions 7.641

12

7.64211 7.643



∞

0



∞

0



∞

1. 0

 2.

3.

4.

7.644 1.11

∞

cos(ax) 1 F 1 (ν + 1; 1; ix) 1 F 1 (ν + 1; 1; −ix) dx



2

cos(2xy) 1 F 1 a; c; −x

4ν − 12 x2

x e

sin(bx) 1 F 1

2

[0 < a < 1] ;

=0

[1 < a]

[−1 < Re ν < 0]  1 1 Γ(c) 2α−1 −y2  |y| dx = π 2 e Ψ c − 12 , a + 12 ; y 2 2 Γ(a)



1

  = −a−1 sin(νπ) P ν 2a−2 − 1

− 2ν; 2ν + 1;

 0

∞

x

e

sin(bx) M 3ν,ν

2

2x







 dx =

EH I 285(12)

  π 4ν − 1 b2 b c 2 1 F 1 12 − 2ν; 1 + 2ν; 12 b2 2

ET I 115(5) b > 0, Re ν > − 14

  π 2ν−1 − 1 b2 b e 4 M 3ν,ν 12 b2 2 0

b > 0, Re ν > − 14   ∞  1 2   π −2ν−1 1 b2 −2ν−1 14 x2 b x e cos(bx) W 3ν,ν 2 x dx = e 4 W 3ν,ν 12 b2 2 0

Re ν < 14 , b > 0   ∞  1 2   π −2ν 1 b2 −2ν 14 x2 b x e sin(bx) W 3ν−1,ν 2 x dx = e 4 W 3ν−1,ν 12 b2 2 0

Re ν < 12 , b > 0 2ν−1 − 14 x2

1

1 2 2x

ET II 402(4)

dx =

ET I 116(10)

ET I 61(7)

ET I 116(9)

 2     1 1 1 1 Γ(3 − 2μ) a  exp − W ρ,σ a2 , x−μ− 2 e− 2 x sin 2ax 2 M k,μ (x) dx = π 2 ak+μ−1  1 2 Γ 2 +k+μ 2ρ = k − 3μ + 1, 2σ = k + μ − 1 [a > 0, Re(k + μ) > 0] ET II 403(10)

838

Confluent Hypergeometric Functions

 2.

3.

 1 1 c Γ(1 + μ + ρ) Γ (1 − μ + ρ)   xρ−1 sin cx 2 e− 2 x W k,μ (x) dx = Γ 3 −k+ρ 0   2 c2 3 3 × 2 F 2 1 + μ + ρ, 1 − μ + ρ; , − k + ρ; − 2 2 4 [Re ρ > |Re μ| − 1] ET II 407(28)  ∞  1 1 xρ−1 sin cx 2 e 2 x W k,μ (x) dx 0    1 c2  12 + μ − ρ, 12 − μ − ρ π2 22    G 23 = 1  4  12 , −k − ρ, 0 Γ 2 − k + μ Γ 12 − k − μ

ET II 407(29) c > 0, Re ρ > |Re μ| − 1, Re(k + ρ) < 12

5.

∞

     1 1 Γ 12 + μ + ρ Γ 12 − μ + ρ −2x 2 x cos cx e W k,μ (x) dx = 0  Γ(1 − k + ρ) 1 1 c2 1 + μ + ρ, − μ + ρ; , 1 − k + ρ; − × 2F 2 2 2

4 2 Re ρ > |Re μ| − 12 ET II 407(30)  ∞  1 1 xρ−1 cos cx 2 e 2 x W k,μ (x) dx 0    1 2  1 + μ − ρ, 1 − μ − ρ c π2  2    G 22 = 1 2 23 4  0, −k − ρ, 12 Γ 2 − k + μ Γ 12 − k − μ

ET II 407(31) c > 0, Re ρ > |Re μ| − 12 , Re(k + ρ) < 12 

4.

7.651

∞

ρ−1

7.65 Combinations of confluent hypergeometric functions and Bessel functions 7.651



∞

1. 0

 2. 0

∞

J ν (xy) M − 12 μ, 12 ν (ax) W

1 1 2 μ, 2 ν

(ax) dx   2  1 μ  2 − 1 Γ(ν + 1) 2 2  a + a = ay −μ−1  1 1 a + y2 2 + y 1 Γ 2 − 2μ + 2ν

y > 0, Re ν > −1, Re μ < 12 , Re a > 0 ET II 85(19)

M k, 12 ν (−iax) M −k, 12 ν (−iax) J ν (xy) dx ae− 2 (ν+1)πi [Γ(1 + ν)]2 1    y −1−2k Γ 2 + k + 12 ν Γ 12 − k + 12 ν   − 1   1 2k   1 2k   a + a2 − y 2 2 × a2 − y 2 2 + a − a2 − y 2 2 1

=

=0



a > 0,

Re ν > −1,

|Re k|
0, Re ν > −1, Re μ < 14 , Re a > 0, Re b > 0 ET II 87(29)

7.66 Combinations of confluent hypergeometric functions, Bessel functions, and powers 7.661



∞

1.

  1 1 2 2 2 2 Γ(1 + 2μ) y y k k  P μ− 1 1+ 2 Q μ− 1 1+ 2 = e−ikπ  1 2 2 a a Γ 2 +μ+k

1 3 ET II 18(44) y > 0, Re a > 0, Re μ > − 2 , Re k < 4

0



∞

2. 0

 0

 0

 5. 0

−1

x

  1 1 y2 2 y2 2 1 k −k 1+ 2 P μ− 1 1+ 2 W k,μ (ax) W −k,μ (ax) J 0 (xy) dx = π cos(μπ) P μ− 1 2 2 2 a a

1 y > 0, Re a > 0, |Re μ| < 2

ET II 18(45)

∞

3.

4.

x−1 W k,μ (ax) M −k,μ (ax) J 0 (xy) dx

x2μ−ν W k,μ (ax) M −k,μ (ax) J ν (xy) dx Γ(2μ + 1)  = 22μ−ν+2k a2k y ν−2μ−2k−1  Γ ν − k − μ + 12   1 1 y2 1 − k, 1 − k, − k + μ; 1 − 2k, − k − μ + ν; − 2 × 3F 2 2 2 2 a

y > 0, Re μ > − 12 , Re a > 0, Re(2μ + 2k − ν) < 12 ET II 85(20)

∞

x2ρ−ν W k,μ (iax) W k,μ (−iax) J ν (xy) dx      −1   1 1 y 2  12 , 0, 12 − μ, 12 + μ 1 2ρ−ν ν−2ρ−1 − 2 24 −k+μ Γ −k−μ =2 Γ y π  G 44 2 2 a2  ρ + 12 , −k, k, ρ − ν + 12

ET II 86(23)a y > 0, Re a > 0, Re ρ > |Re μ| − 1, Re(2ρ + 2k − ν) < 12

∞

x2ρ−ν W k,μ (ax) M −k,μ (ax) J ν (xy) dx



y > 0,

Re a > 0,

   22ρ−ν Γ(2μ + 1) ν−2ρ−1 23 y 2  12 , 0, 12 − μ, 12 + μ = 1 1 y  G 44 a2  ρ + 12 , −k, k, ρ − ν + 12 π2 Γ 2 − k + μ

ET II 86(21)a Re ρ > −1, Re(ρ + μ) > −1, Re(2ρ + 2k + ν) < 12

840

Confluent Hypergeometric Functions



∞

6. 0

x2ρ−ν W k,μ (ax) W −k,μ (ax) J ν (xy) dx Γ(ρ + 1 + μ) Γ(ρ + 1 − μ) Γ(2ρ + 2) ν −ν−1 −2ρ−1     y 2 a Γ 32 + k + ρ Γ 32 − k + ρ Γ(1 + ν)   3 3 y2 3 × 4 F 3 ρ + 1, ρ + , ρ + 1 + μ, ρ + 1 − μ; + k + ρ, − k + ρ, 1 + ν; − 2 2 2 2 a [y > 0, Re ρ > |Re μ| − 1, Re a > 0] ET II 86(22)a

=

7.662



∞

1. 0

 0

∞

3. 0



−1

x

M −μ, 14 ν

1

2

2x



W μ, 14 ν

1

2

2x



      Γ 1 + 12 ν  I 1 ν−μ 14 y 2 K 1 ν+μ 14 y 2 J ν (xy) dx =  1 1 4 4 Γ 2 + 4ν − μ [y > 0,

∞

2.



7.662

x−1 M α−β, 14 ν−γ

−1

x



2

M k,0 iax



1

2 2x



W α+β, 14 ν+γ

1

2 2x

Re ν > −1]



J ν (xy) dx       Γ 1 + 12 ν − 2γ −2  y M α−γ, 1 ν−β 12 y 2 W α+γ, 1 ν+β 12 y 2 =  1 4 4 Γ 1 + 2 ν − 2β

1 ET II 86(25) y > 0, Re β < 8 , Re ν > −1, Re(ν − 4γ) > −2



2

M k,0 −iax



π K 0 (xy) dx = 16





Jk

y2 8a

2

4. 0

−1

x



2

M k,μ iax





2

M k,μ −iax



K 0 (xy) dx = ay

−2





+ Yk

[a > 0]

∞

ET II 86(24)

y2 8a

2 

2



ET II 152(83)

  iy iy 2 W −μ,k − [Γ(2μ + 1)] W −μ,k 4a

4a a > 0, Re y > 0, Re μ > − 21 2



ET II 152(84)

7.663



∞

1. 0



∞

2. 0

 3. 0

∞

  2   y  1, b 22ρ Γ(b) 21 x2ρ 1 F 1 a; b; −λx2 J ν (xy) dx = G Γ(a)y 2ρ+1 23 4λ  12 + ρ + 12 ν, a, 12 + ρ − 12 ν ET II 88(6) y > 0, −1 − Re ν < 2 Re ρ < 12 + 2 Re a, Re λ > 0 ν+1

x

 1F 1

   1 1 2 2ν−a+ 2 Γ(a + 1) 2a−ν−1 − 1 y2 1 2 4 y 2a − ν; a + 1; − x J ν (xy) dx = e K a−ν− 12 y 1 2 4 π 2 Γ(2a − ν)

ET II 87(1) y > 0, Re ν > −1, Re(4a − 3ν) > 12

    1 y2 1+a+ν 1+a+ν ; − x2 J ν (xy) dx = y a−1 1 F 1 a; ;− xa 1 F 1 a; 2 2 2 2

y > 0, Re a > − 21 , Re(a + ν) > −1

ET II 87(2)

7.664

Confluent hypergeometric functions, Bessel functions, and powers



∞

4. 0

  xν+1−2a 1 F 1 a; 1 + ν − a; − 21 x2 J ν (xy) dx =



∞

5. 0

 6. 0

 

−1 2λ−2 − 1 y2 x 1 F 1 λ; 1; −x2 J 0 (xy) dx = 22λ−1 Γ(λ) y e 4

xν+1 1 F 1 =



∞

7. 0

7.664



∞

1. 0



1   π 2 Γ(1 + ν − a) −2a+ν+ 12 2a−ν−1 − 14 y2 2 y e I a− 12 14 y 2 Γ(a)

y > 0, Re a − 1 < Re ν < 4 Re a − 12 ET II 87(3)

[y > 0, ∞

∞

2. 0

 a; b; −λx2 J ν (xy) dx 21−a Γ(b) 1

1

Γ(a)λ 2 a+ 2 ν

y

2

a−2 − y 8λ

e

 y2 , 2k = a − 2b + ν + 2, 2μ = a − ν − 1 W k,μ 4λ

y > 0, −1 < Re ν < 2 Re a − 12 , Re λ > 0 ET II 88(4) 

  22b−2a−ν−1 Γ(b) −a 2a−2b+ν λ y x2b−ν−1 1 F 1 a; b; −λx2 J ν (xy) dx = Γ(a − b+ ν + 1)  y2 × 1 F 1 a; 1 + a − b + ν; − 4λ  

y > 0, 0 < Re b < 34 + Re a + 12 ν , Re λ > 0

xW

1 2 ν,μ

ET II 18(46)

a x

W − 12 ν,μ

a x

ET II 88(5)

    1 1 1 1 K ν (xy) dx = 2ay −1 K 2μ (2ay) 2 e 4 iπ K 2μ (2ay) 2 e− 4 iπ

ET II 152(85) [Re y > 0, Re a > 0]     2 2 W − 12 ν,μ J ν (xy) dx x W 12 ν,μ x x    1  1   1  



K 2μ 2y 2 = −4y −1 sin μ − 12 ν π J 2μ 2y 2 + cos μ − 12 ν π Y 2μ 2y 2

Re (ν ± 2μ) > −1]

ET II 87(27)

    2 2 W − 1 ν,μ Y ν (xy) dx x x 2 0   1  1   1    



cos μ − 12 ν π J 2μ 2y 2 − sin μ − 12 ν π Y 2μ 2y 2 K 2μ 2y 2 = 4y −1

y > 0, |Re μ| < 14 ET II 117(48)      ∞     1 1 2 2 4 Γ(1 + 2μ)y −1  J 2μ 2y 2 K 2μ 2y 2 M 12 ν,μ J ν (xy) dx =  1 1 x W − 12 ν,μ x x Γ + ν + μ 0 2 2

y > 0, Re ν > −1, Re μ > − 41 

4.

Re λ > 0]



[y > 0, 3.

841

∞

xW

1 2 ν,μ

ET II 86(26)

842

Confluent Hypergeometric Functions



∞

5. 0

7.665



∞

1. 0

 x W − 12 ν,μ

ia x



  ia W − 12 ν,μ − J ν (xy) dx x         −1 1 1 K μ (2iay) 2 K μ (−2iay) 2 = 4ay −1 Γ 12 + μ + 12 ν Γ 12 − μ + 12 ν

y > 0, Re a > 0, |Re μ| < 12 , Re ν > −1 ET II 87(28)

 1   1 x− 2 J ν ax 2 K 12 ν−μ 12 x M k,μ (x) dx =



∞

2. 0



∞

0

 2  2 Γ(2μ + 1) a a   1 1 1 1 1 1 M W 1 2 (k−μ), 2 k− 4 ν 2 (k+μ), 2 k+ 4 ν 2 2 a Γ k + ν + 1 2

a > 0, Re k > − 14 , Re μ > − 12 , Re ν > −1 ET II 405(18)

  1 1  1 x 2 c+ 2 c −1 Ψ(a, c; x) 1 F 1 (a ; c ; −x) J c+c −2 2(xy) 2 dx =

7.666

7.665

1 1  Γ (c ) y 2 c+ 2 c −1 Ψ (c − a , c + c − a − a ; y) 1 F 1 (a ; a + a ; −y) Γ (a + a )

Re c > 0, 1 < Re (c + c ) < 2 Re (a + a ) + 12 EH I 287(23)

      1 1 1 1 1 x 2 c− 2 1 F 1 a; c; −2x 2 Ψ a, c; 2x 2 J c−1 2(xy) 2 dx = 2−c

 Γ(c) a− 1 c− 1  1 c−2a 1 y 2 2 1 + (1 + y) 2 (1 + y)− 2 Γ(a)

Re c > 2, Re(c − 2a) < 12 EH I 285(13)

7.67 Combinations of confluent hypergeometric functions, Bessel functions, exponentials, and powers 7.671



∞

1. 0



 3 xk− 2 exp − 12 (a + 1)x K ν 12 ax M k,ν (x) dx 1   π 2 Γ(k) Γ(k + 2ν) −1   1 2 F 1 k, k + 2ν; 2ν + 1; −a k+ν a Γ k+ν + 2 [Re a > 0, Re k > 0, Re(k + 2ν) > 0] ET II 405(17)

=

 2. 0

∞



 3 x−k− 2 exp − 12 (a − 1)x K μ 12 ax W k,μ (x) dx =

  π Γ(−k) Γ(2μ − k) Γ(−2μ − k) 1  1  1  22k+1 ak−ν 2 F 1 −k, 2μ − k; −2k; 1 − a−1 Γ 2 −k Γ 2 +μ−k Γ 2 −μ−k [Re a > 0, Re k < 2 Re μ < − Re k] ET II 408(36)

7.672

7.672

Confluent hypergeometric functions, Bessel functions, exponentials, and powers



∞

1.

∞

2.

  2 1 x2ρ e− 2 ax W k,μ ax2 J ν (xy) dx

    Γ 1 + μ + 12 ν + ρ Γ 1 − μ + 12 ν + ρ 2−ν−1 − 1 ν−ρ− 1 ν 2y   a 2 = Γ(ν + 1) Γ 32 − k + 12 ν + ρ   y2 1 , × 2 F 2 λ + μ, λ − μ; ν + 1, − k + λ; − 2 4a

  λ = 1 + 12 ν + ρ y > 0, Re a > 0, Re ρ ± μ + 12 ν > −1 ET II 85(16)

0



  2 1 x2ρ e− 2 ax M k,μ ax2 J ν (xy) dx

   Γ(2μ + 1) 2ρ −2ρ−1 21 y 2  12 − μ, 12 + μ 2 y =   G 23 4a  12 + ρ + 12 ν, k, 12 + ρ − 12 ν Γ μ + k + 12 

 y > 0, −1 − Re 12 ν + μ < Re ρ < Re k − 14 , Re a > 0 ET II 83(10)

0



∞

3. 0

  2 1 x2ρ e 2 ax W k,μ ax2 J ν (xy) dx =



4.

12



∞

0



∞

5. 0

1

1

y > 0,

2

x2λ+ 2 e− 4 x M k,μ

1

1

2

22ρ y −2ρ−1    Γ 2 +μ − k Γ 12 − μ − k  y 2  12 − μ, 12 + μ 22 × G 23  4a  12 + ρ + 12 ν, −k, 12 + ρ − 12 ν 

 ET II 85(17) |arg a| < π, −1 − Re 12 ν ± μ < Re ρ < − 14 − Re k

x2λ+ 2 e 4 x W k,μ

1

2 2x

1

2 2x





   2λ y −1/2 Γ(2μ + 1) 31 y 2  −μ − λ, μ − λ, l 1  G 34 = 2  h, κ, k − λ − 12 , l Γ 2 +k+μ 1 1 h = 4 + 2 ν, κ = 14 − 12 ν, l = − 14 − 12 ν ET II 116(45) y > 0, Re(k − λ) > 0, Re (2λ + 2μ ± ν) > − 52

Y ν (xy) dx

∞

6. 0

7. 0

  y 2  −μ − λ, μ − λ, l y −1/2 , 2  h, κ, − 21 − k − λ, l h = 14 + 12 ν, κ = 14 − 12 ν, l = − 14 − 12 ν ET II 117(47) Re(k + λ) < 0, Re (2λ ± 2μ ± ν) > − 52

    −1 32 = 2 Γ 12 − k + μ Γ 12 − k − μ G 34



1

Y ν (xy) dx

λ



843

∞

y > 0,



   

1 2 x−1/2 e− 2 x M 12 ν− 14 , 12 ν+ 14 x2 J ν (xy) dx = (2ν + 1)2−ν y ν−1 1 − Φ 12 y

y > 0, Re ν > − 12

ET II 82(1)

   

1 2 Γ(ν + 2)y ν  1 − Φ 12 y x−1 e− 2 x M 12 ν+ 12 , 12 ν+ 12 x2 J ν (xy)dx =  3 ν Γ ν+2 2 [y > 0, Re ν > −1]

ET II 82(2)

844

Confluent Hypergeometric Functions



∞

8. 0



∞

9. 0



∞

10. 0

e

− 14 x2

 M k, 12 ν

1

 2−k Γ(ν + 1) 2k−1 − 1 y2 1 2 y x J ν (xy) dx =  e 2 2 Γ k + 12 ν + 12 y > 0, Re ν > −1,

2

xν−2μ e− 4 x M k,μ

1

2

xν−2μ e 4 x W k,±μ



Re k
0, −1 < Re ν < 2 Re(k + μ) − 12 ET II 83(9)

1

2 2x



J ν (xy) dx =

  Γ(1 + ν − 2μ) β−μ k+μ− 3 − 1 y2 2e 4 y M α,β 12 y 2 2 Γ(1 + 2β) 2α = 12 + k + ν − 3μ, 2β = 12 − k + ν − μ [y > 0,



7.672

Re ν > −1,

Re(ν − 2μ) > −1] ET II 84(14)

∞

11. 0

1

2

xν−2μ e− 4 x W k,±μ

1

2 2x



J ν (xy) dx   Γ(1 + ν − 2μ) 21 ( 12 +k−3μ+ν ) μ−k− 3 1 y2 2 e4 1 2 y W α,β 12 y 2 , Γ 2 +μ−k 2β = k − μ + ν + 12 2α = k + 3μ −ν − 12 ,  Re(ν − 2μ) > −1, Re k − μ + 12 ν < − 41 ET II 84(15)

= 

∞

12. 0

1

y > 0,

2

x2μ−ν e− 4 x M k,μ

Re ν > −1,

1

2 2x



J ν (xy) dx =

 13. 0

∞

1

2

x2μ−ν e− 4 x M k,μ

1

2 2x



Γ

  1 1 Γ(2μ + 1) 3 1 2 1  2 2 ( 2 −k+3μ−ν ) y k−μ− 2 e− 4 y M α,β 12 y 2 + k − μ + ν 2 1 k + 3μ − ν,  2β =− 21 + k − μ + ν 2α = 2 + 1 ET II 83(8) y > 0, − 2 < Re μ < Re k + 12 ν − 14

Y ν (xy) dx 3

1

2

= π −1 2μ+β y k−μ− 2 e− 4 y Γ (2μ + 1)      Γ(2μ − ν − 1) M α,β 12 y 2 × Γ 12 − k − μ cos[(ν − 2μ)π] Γ(2β + 1)   1 2 − sin[(ν + k − μ)π] W α,β 2 y y > 0,

2α = 3μ − ν + k + 12 , 2β = μ − ν − k + 12 1 −1 < 2 Re μ < Re(2k + ν) + 2 , Re(2μ − ν) > −1 ET II 116(44)

7.673

Confluent hypergeometric functions, Bessel functions, exponentials, and powers



∞

14. 0

1

2

x2μ+ν e− 4 x M k,μ

1

2 2x



845

Y ν (xy) dx 3

= π −1 2μ+β y k−μ− 2 Γ(2μ + 1)  1  1 2  − 1 y2 Γ(2μ + ν + 1)  ×Γ 2 −μ−k e 4 cos(2μπ)  3 M α,β 2 y Γ μ+ν −k+ 2   1 2 + sin[(μ − k)π] W α,β 2 y y > 0, 

∞

15. 0

2α = 3μ + ν + k + 12 , 2β = μ + ν − k + 12 1 −1 < 2 Re μ < Re(2k − ν) + 2 , Re(2μ + ν) > −1 ET II 116(43)

  2 1 1 1 1 3 x2μ+ν e− 2 ax M k,μ ax2 K ν (xy) dx = 2μ−k− 2 a 4 − 2 (μ+ν+k) y k−μ− 2   2 y y2 W κ,m , 8a 4a 2κ = −3μ − ν − k − 12 , 2m = μ + ν − k + 12 1 Re a > 0, Re μ > − 2 , Re(2μ + ν) > −1 ET II 152(82) 

× Γ(2μ + 1) Γ(2μ + ν + 1) exp

Re y > 0, 7.673 1.

10



∞

0

 √  1 1 e− 2 ax x 2 (μ−ν−1) M κ, 12 μ (ax) J ν 2 bx dx   κ−1 − 1+μ 4 1 b b 2 a− 2 (μ+1−ν) Γ(1 + μ)e− 2a = a   b × M 1 (κ−ν−1)+ 3 (1+μ), κ+ν − 1+μ 2 4 2 4 a    ν −μ 3 Re(1 + μ) > 0, Re κ + >− , 2 4

 2. 0

∞

1   1+μ κ+ν − Γ 1+ 2 4  Im b = 0

BU 128(12)a

b  √  1 1 1 Γ (ν + 1 ∓ μ) e 2a  a  12 (κ+1)+ 14 (1∓μ)  1±μ  e 2 ax x 2 (ν−1∓μ) W κ, 12 μ (ax) J ν 2 bx dx = a− 2 (ν+1∓μ) b Γ 2 −κ   b × W 12 (κ+1−ν)− 34 (1∓μ), 12 (κ+ν)+ 14 (1∓μ) a     3 ν ∓μ Re + κ < , Re ν > −1 BU 128(13) 2 4

846

Confluent Hypergeometric Functions

7.674  1.

∞

0

2.

7.674

    1 xρ−1 e− 2 κ J λ+ν ax1/2 J λ−ν ax1/2 W k,μ (x) dx     1 2λ  1 Γ 2 + λ + μ + ρ Γ 12 + λ − μ + ρ 2a = Γ (1 +  λ + ν) Γ(1 + λ − ν) Γ(1 + λ − k + ρ)

× 4 F 4 1 + λ, 12 + λ, 12 + λ + μ + ρ, 12 + λ − μ + ρ; 1 + λ + ν,  1 + λ − ν, 1 + 2λ, 1 + λ − k + ρ; −a2

|Re μ| < Re(λ + ρ) + 12 ET II 409(37)  ∞     1 xρ−1 e− 2 κ I λ+ν ax1/2 K λ−ν ax1/2 W k,μ (x) dx 0    π −1/2 24 2  0, 12 , 12 + μ − ρ, 12 − μ − ρ a = G 45  λ, ν, −λ, −ν, k − ρ 2

1 |Re μ| < Re(λ + ρ) + 2 , |Re μ| < Re(ν + ρ) + 12 ET II 409(38)

Combinations of Struve functions and confluent hypergeometric functions 7.675  1.

∞

0

2λ+ 12 − 14 x2

x

e

M k,μ

2.12



1

2

2x



Re(2λ + 2μ + ν) > − 72 ,

1

1

2

x2λ+ 2 e− 4 x W k,μ

1

2 2x



Hν (xy) dx

   + 12 ν + λ + μ Γ 74 + 12 ν + λ − μ     π y =2 Γ ν + 32 Γ 94 + λ − k − 12 ν   7 ν 3 3 9 ν y2 7 ν × 3 F 3 1, + + λ + μ, + + λ − μ; , ν + , + λ − k + ; − 4 2 2 2 4 2 2 4 2

Re(2λ + ν) > 2|Re μ| − 72 , y > 0 ET II 171(43) 1 1 4 −λ− 2 ν

∞

3. 0

1

 0

1

2

x2λ+ 2 e 4 x W k,μ

1

2 2x



−1/2 ν+1 Γ

Hν (xy) dx

7 4

  y 2  l, −μ − λ, μ − λ 2  l, −k − λ − 12 , h, κ 1 1 h = 4 + 2 ν, κ = 14 − 12 ν, l = 34 + 12 ν Re (2k + 2λ + ν) < − 21 , Re(k + λ) < 0 ET II 172(46)a

    −1 −1/2 23 = 2λ Γ 12 − k + μ Γ 12 − k − μ y G 34

y > 0, 4.

Re(k − λ) > 0,

ET II 171(42) ∞

0



   l, −μ − λ, μ − λ   l, k − λ − 1 , h, κ 2 κ = 14 − 12 ν, l = 34 + 12 ν

y > 0, Re(2λ − 2k + ν) < − 21

 2 2−λ Γ(2μ + 1) y 22   Hν (xy) dx = 1/2 G 34 2 y Γ 12 + k + μ 1 1 h = 4 + 2 ν,

∞

Re(2λ + ν) > 2|Re μ| − 72 ,



  y    1 2 1 2 e 2 x W − 12 ν− 12 , 12 ν x2 Hν (xy) dx = 2−ν−1 y ν πe 4 y 1 − Φ 2 [y > 0,

Re ν > −1]

ET II 171(44)

7.681

Confluent hypergeometric functions and other special functions

847

7.68 Combinations of confluent hypergeometric functions and other special functions Combinations of confluent hypergeometric functions and associated Legendre functions 7.681



∞

1. 0

 1 x M k,μ (x) dx 1+2 x−1/2 (a + x)μ e− 2 x P −2μ ν a     1 sin(νπ) Γ(2μ + 1) Γ k − μ + ν + 12 Γ k − μ − ν − 12 e 2 a W ρ,σ (a), =− π Γ(k) ρ = 12 − k + μ,  σ = 12 + ν 1 |arg a| < π, Re μ > − , Re(k − μ) > Re ν + 1  ET II 403(11) 2



∞

 1 x M k,μ (x) dx 1+2 x−1/2 (a + x)−μ e− 2 x P −2μ ν a    1 Γ(2μ + 1) Γ k + μ + ν + 12 Γ k + μ − ν − 12 e 2 a   W 12 −k−μ, 12 +ν (a) = Γ k + μ + 12 Γ(2μ + ν + 1) Γ(2μ − ν) 

 |arg a| < π, Re μ > − 21 , Re(k + μ) > Re ν + 12  ET II 403(12)

∞

 1 1 1 1 x W k,ν (x) dx x− 2 − 2 μ−ν (a + x) 2 μ e− 2 x P μk+ν− 3 1 + 2 2 a Γ(1 − μ − 2ν) 1 1 1 1  a− 4 + 2 k− 2 ν e 2 a W ρ,σ (a) = 3 Γ 2 −k−μ−ν 2ρ = 12 + 2μ + ν − k, 2σ = k + 3ν − 32

2. 0



2

3. 0

[|arg a| < π, 

1

1

1



1

x− 2 − 2 μ−ν (a + x)− 2 μ e− 2 x P μk+μ+ν− 3 1 + 2 2

0

x a =

W k,ν (x) dx Γ

1 1 1 1 Γ(1 − μ − 2ν) 3  a− 2 + 2 k− 2 ν e 2 a W ρ,σ (a) − k − μ − ν 2 2σ = k + 2μ + 3ν − 32 2ρ = 12 − k + ν,

[|arg a| < π,

5. 0

Re(μ + 2ν) < 1] ET II 407(32)

∞

4.



Re μ < 1,

Re μ < 1,

Re(μ + 2ν) < 1] ET II 408(33)

∞

1

1

1

1



1

xμ− 4 k− 2 ν− 2 (a + x) 2 ν e− 2 x Q νμ−k+ 3 1 + 2

x

M k,μ (x) dx a   eνπi Γ(1 + 2μ − ν) Γ(1 + 2μ) Γ 52 − k + μ + ν 1 (κ+2μ−2ν+5) 1 a   a4 e 2 W ρ,σ (a) = 2 Γ 12 + k + μ 2σ = k − 3μ − 32 2ρ = 12 − k − μ + 2ν,

|arg a| < π, Re μ > − 12 , Re(2μ − ν) > −1 2

ET II 404(14)

848

7.682

Confluent Hypergeometric Functions



∞

1. 0



∞

2. 0



∞

3. 0



∞

4. 0

1

x−1/2 e− 2 x P −2μ ν

  x 1/2 1+ M k,μ (x) dx a   1   Γ(2μ + 1) Γ k + 12 ν Γ k − 12 ν − 12 e 2 a      W 3 −k, 1 + 1 ν (a) = 2μ 1/4  1 1 1 1 4 4 2 2 a Γ k + μ + 2 1Γ μ + 21ν + 2 Γ μ − 12 ν

ET II 404(13) |arg a| < π, Re k > 2 Re ν − 2 , Re k > − 2 Re ν

  x 1/2 1+ M k,μ (x) dx  1a  Γ 2 − ν Γ(1 + 2μ) Γ(k + μ + ν) 1 a 1   e 2 W ρ,σ (a), = e(1−k+μ−ν)πi 2μ−k−ν a 2 (k+μ−1) Γ k + μ + 12 ρ = 12 − k − 12 ν, σ = μ + 12 ν 1 |arg a| < π, Re μ > − 2 , Re(k + μ + ν) > 0 ET II 404(15)

1

1

x 2 (k+μ+ν)−1 (a + x)−1/2 e− 2 x Q 1−k+μ−ν k−μ−ν−1

1

 x 1/2 M k,μ (x) dx a   1 2(μ−ν)πi 2μ−2ν−1 12 (k+μ−1) 12 a Γ(2μ + 1) Γ(ν + 1) Γ k + μ − 2ν − 2 W ρ,σ (a), 2 a e =e Γ k + μ + 12 2ρ = 1 − k + μ − 2ν, 2σ = k − μ − 2ν − 2 |arg a| < π, Re μ > − 12 , Re ν > −1, Re(k + μ − 2ν) > 12 ET II 404(16)

1

xν− 2 e− 2 x Q 2μ−2ν 2k−2ν−3

− 12 − 12 μ−ν − 12 x

x

e



1+

P μ2k+μ+2ν−3



 x  12 1+ W k,ν (x) dx a 2μ Γ(1 − μ − 2ν) − 1 + 1 k− 1 ν 1 a  a 2 2 2 e 2 W ρ,σ (a), = 3 Γ 2 −k−μ−ν 2ρ = 1 − k + μ + ν, 2σ = k + μ + 3ν − 2 [|arg a| < π,

5.8

 0

∞

7.682

Re μ < 1,

  x 1/2 − 12 − 12 μ−ν −1/2 − 12 x μ W k,ν (x) dx x (a + x) e P 2k+μ+2ν−2 1 + a μ 2 Γ(1 − μ − 2ν) − 12 + 12 k− 12 ν 12 a a = 3 e W ρ,σ (a), 2ρ = μ + ν − k, Γ 2 −k−μ−ν [|arg a| < π, Re μ > 0,

Re(μ + 2ν) < 1] ET II 408(34)

2σ = k + μ + 3ν − 1 Re ν > 0]

ET II 408(35)

A combination of confluent hypergeometric functions and orthogonal polynomials  1 μ−α 1 8 7.683 e− 2 ax xα (1 − x) 2 −1 Lα n (ax) M k− 1+α , μ−α−1 [a(1 − x)] dx 0

2

2

Γ(μ − α) Γ(1 + n + α) − 1+α a 2 M k+n, μ2 (a) Γ(1 + μ) n! Re(μ − α) > 0, n = 0, 1, 2, . . .] BU 129(14b) =

[Re a > −1,

7.711

Parabolic cylinder functions

849

A combination of hypergeometric and confluent hypergeometric functions    ∞ λ ρ−1 − 12 x dx x e M γ+ρ,β+ρ+ 12 (x) 2 F 1 α, β; γ; − 7.684 x 0 Γ(α + β + 2ρ) Γ(2β + 2ρ) Γ(γ) 1 β+ρ− 1 1 λ 2 e2 = λ2 W k,μ (λ); Γ(β) Γ(β + γ + 2ρ) 1 1 k = 2 − α − 2 β − ρ, μ = 12 β + ρ [|arg λ| < π, Re(β + ρ) > 0, Re (α + β + 2ρ) > 0, Re γ > 0] ET II 405(19)

7.69 Integration of confluent hypergeometric functions with respect to the index   1 (aβ)1/2 exp − (α + β) 7.691 sech(πx) W ix,0 (α) W −ix,0 (β) dx = 2 α+β 2 −∞ i∞ 7.692 Γ(−a) Γ(c − a)Ψ(a, c; x)Ψ(c − a, c; y) da = 2πi Γ(c)Ψ(c, 2c; x + y) 

7.693

∞

ET II 414(61) EH I 285(15)

−i∞



∞

Γ(ix) Γ(2k + ix) W k+ix,k− 12 (α) W −k−ix,k− 12 (β) dx

1. −∞

= 2π

1/2

k

Γ(2k)(aβ) (α + β)

1 2 −2k

 K

2k− 12

a+β 2



ET II 414(62)



i∞

2.

Γ −i∞

1 2

       + ν + μ + x Γ 12 + ν + μ − x Γ 12 + ν − μ + x Γ 12 + ν − μ − x × M μ+ix,ν (α) M μ−ix,ν (β) dx 1

7.69411



2

2π(aβ)ν+ 2 [Γ(2ν + 1)] Γ(2ν + 2μ + 1) Γ(2ν − 2μ + 1) M 2μ,2ν+ 12 (α + β) = (α + β)2ν+1 Γ(4ν + 2)

1 Re ν > |Re μ| − 2 ET II 413(59) ∞

−∞

e−2ρxi Γ

1 2

   + ν + ix Γ 12 + ν − ix M ix,ν (α) M ix,ν (β) dx      1 αβ sech ρ = π αβ [Γ(2ν + 1)]2 sech ρ exp − (α + β) tanh ρ J 2ν 2

|Im ρ| < 12 π, Re ν > − 12

7.7 Parabolic Cylinder Functions 7.71 Parabolic cylinder functions 7.711



∞

1. −∞

D n (x) D m (x) dx = 0 = n!(2π)1/2

[m = n] [m = n] WH

850

Parabolic Cylinder Functions





1 1     ∓ 1 1   1  Γ 12 − 12 μ Γ − 12 ν Γ 2 − 2ν Γ −2μ [when the lower sign is taken, Re μ > Re ν] BU 11 117(13a), EH II 122(21) 1

∞

π2 2 (μ+ν+1) D μ (±t) D ν (t) dt = μ−ν

2. 0



7.721

∞

2

[D ν (t)] dt = π

3. 0

1/2 −3/2 ψ

2

1 2

   − 12 ν − ψ − 12 ν Γ(−ν)

BU 117(13b)a, EH II 122(22)a

7.72 Combinations of parabolic cylinder functions, powers, and exponentials 7.721



∞

1

1. −∞

2

1

2

e− 4 x (x − z)−1 D n (x) dx = ±ie∓nπi (2π)1/2 n!e− 4 z D −n−1 (∓iz)

[The upper or lower sign is taken according as the imaginary part of z is positive or negative.] 

∞

2. 1

xν (x − 1)

1 1 2 μ− 2 ν−1

   μ ν (x − 1)2 a2 μ−ν D μ (ax) dx = 2μ−ν−2 a 2 − 2 −1 Γ D ν (a) exp − 4 2 

[Re(μ − ν) > 0] 7.722



∞

1. 0

 2. 3.11 7.723

∞

1. 0



1/2 − 12 μ− 12 ν

0

 3. 0



1

2

e− 4 x xμ−1 D −ν (x) dx =

WH

[Re μ > 0]

EH II 122(20)

[Re ν > −1]

ET II 395(2)

 π 1/2  −1 1 2 1 2 e− 4 x xν x2 + y 2 D ν (x) dx = Γ(ν + 1)y ν−1 e 4 y D −ν−1 (y) 2 [Re y > 0, Re ν > −1] EH II 121(18)a, ET II 396(6)a

∞

2.

7.724

3 2 1 1 1 e− 4 x xν D ν+1 (x) dx = 2− 2 − 2 ν Γ(ν + 1) sin (1 − ν)π 4

π 2 Γ(μ)  1 1 Γ 2 μ + 2 ν + 12 0    ∞ 1 − 34 x2 ν − 12 ν πν e x D ν−1 (x) dx = 2 Γ(ν) sin 4 0



ET II 395(4)a

[Re ν > −1] ∞

WH

 −1/2 1 2 1 2 e− 4 x xν−1 x2 + y 2 D ν (x) dx = y ν−1 Γ(ν)e 4 y D −ν (y) [Re y > 0,

1

Re ν > 0]

ET II 396(7)

 λ−1 a2 x2 Γ(λ) Γ(2ν) λ−1 a2 2 x2ν−1 1 − x2 e 4 D −2λ−2ν (ax) dx = e 4 D −2ν (a) Γ(2λ + 2ν) [Re λ > 0,

∞

−∞

e−

(x−y)2 2μ

Re ν > 0]

  y2 1 2 1 e 4 x D ν (x) dx = (2πμ)1/2 (1 − μ) 2 ν e 4−4μ D ν y(1 − μ)−1/2

ET II 395(3)a

[0 < Re μ < 1] EH II 121(15)

7.731

7.725 1.

2.

3.

4.

5.

6.

7.726

Parabolic cylinder and trigonometric functions

√   π 1/2 √p + 1 − 1ν+1 e (2t) e D −ν−2 2t dt = 2 (ν + 1)pν+1 0 [Re ν > −1] MO 175   √  ∞ ν √   π 1/2 p+1−1 ν−1 t √ e−pt (2t) 2 e− 2 D −ν 2t dt = 2 pν p + 1 0 [Re ν > −1] MO 175  ∞ √  3       n −n− 2 b + 12 e−bx D 2n+1 2x dx = (−2)n Γ n + 32 b − 12 0

Re b > − 21 ET I 210(3)  ∞ √  1 √ −1 −bx       n −n− 2 b + 12 x e D 2n 2x dx = (−2)n Γ n + 12 b − 12 0

Re b > − 21 ET I 210(5)  ν  ∞  √  √ 1 1  x− 2 (ν+1) e−sx D ν x dx = π 1 + 12 + 2s 1 0 4 +s

ET I 210(7) Re s > − 41 , Re ν < 1    ∞ ν   β 21−β− 2 π 1/2 Γ(β) ν β ν +β+1 z−k −β 2 F  , ; ; (z + k) e−zt t−1+ 2 D −ν 2(kt)1/2 dt =  1 2 2 2 z+k Γ 2 ν + 12 β + 12 0   z Re(z + k) > 0, Re > 0 k



∞



∞

∞

eixy−

(1+λ)x2 4

e 2 x e−bx μ+ 12

(ex − 1)

− ν2

(2t) 0

    1/2  (1+λ)y2 1 D ν x(1 − λ)1/2 dx = (2π)1/2 λ 2 ν e− 4λ D ν i λ−1 − 1 y

1

∞

0



− 2t

EH II 121(11)

−∞

7.727

ν−1 2

−pt



7.728

851

e−pt e

 exp −

2 − q8t

a 1 − e−x

 D ν−1

q √ 2t



 D 2μ

 dt =

EH II 121(16)

 √  √  2 a √ dx = e−a 2b+μ Γ(b + μ) D −2b 2 a −x 1−e [Re a > 0, Re b > − Re μ]

 π  12 2

[Re λ > 0]

ET I 211(13)

p

1 2 ν−1

e−q

√

p

MO 175

7.73 Combinations of parabolic cylinder and hyperbolic functions 7.731  1.

2.

    3 cosh(2μx) exp − (a sinh x)2 D 2k (2a cosh x) dx = 2k− 2 π 1/2 a−1 W k,μ 2a2 0 2

Re a > 0 ET II 398(20)  ∞     Γ(μ − k) Γ(−μ − k) cosh(2μx) exp (a sinh x)2 D 2k (2a cosh x) dx = W k+ 12 ,μ 2a2 5 k+ 2 a Γ(−2k) 0

2 Re k + |Re μ| < 0 |arg a| < 3π 4 , ∞

ET II 398(21)

852

Parabolic Cylinder Functions

7.741

7.74 Combinations of parabolic cylinder and trigonometric functions 7.741



∞

1. 0



∞

2. 0

 3. 4.

5.

7.742

∞

    1 2 i √ 2 2 sin(bx) [D −n−1 (ix)] − [D −n−1 (−ix)] dx = (−1)n+1 π 2πe− 2 b Ln b2 n!

1



2

e− 4 x sin(bx) D 2n+1 (x) dx = (−1)n

[b > 0]

ET I 115(3)

[b > 0]

ET I 115(1)

π 2n+1 − 1 b2 e 2 b 2



π 2n − 1 b2 b e 2 [b > 0] 2 0  ∞   √ 

 1 2 1 1 2 e− 4 x sin(bx) D 2ν− 12 (x) − D 2ν− 12 (−x) dx = 2π sin ν − 14 π b2ν− 2 e− 2 b 0

Re ν > 14 , b > 0 √  ∞ 1 1 1 2   1 2 2 4 −2ν πb2ν− 2 e− 4 b 

 e− 2 x cos(bx) D 2ν− 12 (x) + D 2ν− 12 (−x) dx = cosec ν + 14 π 0

Re ν > 14 , b > 0 

∞

1. 0



∞

2. 0



∞

3. 0



∞

4. 0

1

2

e− 4 x cos(bx) D 2n (x) dx = (−1)n

x2ρ−1 sin(ax)e−

x2ρ−1 sin(ax)e

x2ρ−1 cos(ax)e

ET I 115(2)

ET I 61(4)

x2 4

x2 4

x2ρ−1 cos(ax)e−

1 Γ (2ρ + 1) D 2ν (x) dx = 2ν−ρ− 2 π 1/2 a Γ(ρ − ν + 1)   3 a2 1 × 2 F 2 ρ + , ρ + 1; , ρ − ν + 1; − 2 2 2

Re ρ > − 21    2ρ−ν−2 22 a2  12 − ρ, 1 − ρ D 2ν (x) dx =  G Γ(−2ν) 23 2  −ρ − ν, 12 , 0 a > 0, Re ρ > − 21 ,

ET I 60(2)

x2 4

x2 4

ET II 396(8)

Re(ρ + ν)
0]   a2  12 − ρ, 1 − ρ  2  −ρ − ν, 0, 12 a > 0, Re ρ > 0,

ET II 396(10)a

Re(ρ + ν)
0] ET I 115(4)  √  √  − 2x D 2ν− 12 2x dx √   2ν  √ π sin ν − 14 π 1 + 1 + b2 √ =− 1 1 + b2 b2ν+ 2 [b > 0] ET I 60(3)

7.75 Combinations of parabolic cylinder and Bessel functions 7.751 1. 2.

3.



  y 2 [D n (ax)]2 J 1 (xy) dx = (−1)n−1 y −1 D n [y > 0] a 0  ∞ y  y D n+1 J 0 (xy) D n (ax) D n+1 (ax) dx = (−1)n y −1 D n a a 0

y > 0, |arg a| < 14 π  ∞ J 0 (xy) D ν (x) D ν+1 (x) dx = 2−1 y −1 [D ν (−y) D ν+1 (y) − D ν+1 (−y) D ν (y)] ∞

ET II 20(24)

ET II 17(42) ET II 397(17)a

0

7.752



∞

1. 0



ET II 76(1), MO 183

∞

2. 0



∞

3. 0

 0

5. 0

  1 2 1 1 2 xν e 4 x D 2ν−1 (x) J ν (xy) dx = 2 2 −ν π sin(νπ)y −ν Γ(2ν)e 4 y K ν 14 y 2

y > 0, − 12 < Re ν < 12 1

2

xν+1 e− 4 x D 2ν (x) J ν (xy) dx =

∞

1

2

xν e− 4 x D 2ν+1 (x) J ν (xy) dx =

1

2

ET II 77(4)

1 2 1 sec(νπ)y ν−1 e− 4 y [D 2ν+1 (y) − D 2ν+1 (−y)] 2

[y > 0, ∞

4. 

1 1 2 1 2 xν e− 4 x D 2ν−1 (x) J ν (xy) dx = − sec(νπ)y ν−1 e− 4 y [D 2ν−1 (y) − D 2ν−1 (−y)] 2

y > 0, Re ν > − 12

Re ν > −1]

1 2 1 sec(νπ)e− 4 y y ν [D 2ν (y) + D 2ν (−y)] 2

y > 0, Re ν > − 12 1

ET II 78(13)

ET II 77(5)

2

xν+1 e− 4 x D 2ν+2 (x) J ν (xy) dx = − 12 sec(νπ)y ν e− 4 y [D 2ν+2 (y) + D 2ν+2 (−y)] [Re ν > −1,

y > 0]

ET II 78(16)

854

Parabolic Cylinder Functions



∞

6. 0



∞

7. 0



∞

8. 0

 9.

10.

7.752

  1 2 1 2 xν+1 e 4 x D 2ν+2 (x) J ν (xy) dx = π −1 sin(νπ) Γ(2ν + 3)y −ν−2 e 4 y K ν+1 14 y 2

y > 0, −1 < Re ν < − 65   1 2 1 2 xν e− 4 x D −2ν (x) J ν (xy) dx = 2−1/2 π 1/2 y −ν e− 4 y I ν 14 y 2

y > 0, Re ν > − 12 1

2

1

Re ν > − 12 ,

2

xν e 4 x D −2ν (x) J ν (xy) dx = y ν−1 e 4 y D −2ν (y)

y>0

ET II 78(19)

ET II 77(8)



ET II 77(9), EH II 121(17) ∞

1

2

1

2

xν e 4 x D −2ν−2 (x) J ν (xy) dx = (2ν + 1)−1 y ν e 4 y D −2ν−1 (y) 0

y > 0, Re ν > − 12 ET II 77(10)     1  ∞ 2μ− 2 Γ ν + 12 y ν y2 1 1 2 2 ; ν − μ + 1; − ν + xν e− 4 a x D 2μ (ax) J ν (xy) dx = F 1 1 Γ(ν − μ + 1)a1+2ν 2 2a2 0

y > 0, |arg a| < 14 π, Re ν > − 21 

∞

11. 0



∞

12. 0



∞

13. 0



∞

14. 0

 15. 0

∞

1

xν e 4 a

2

x2

D 2μ (ax) J ν (xy) dx =

1

xν+1 e− 4 a

1

xν+1 e 4 a

1

1

2

xλ+ 2 e 4 a

1

x2

x2

2

2

2

x2

Γ

1





+ ν a2k 2m+μ y22 y2 1  μ+ 3 e 4a W k,m 4a2 Γ 2 −μ y 2 2k = 12 + μ − ν, 2m = 12 + μ + ν  

y > 0, |arg a| < 14 π, − 12 < Re ν < Re 12 − 2μ 2

D 2μ (ax) J ν (xy) dx =

D 2μ (ax) J ν (xy) dx =

ET II 77(11)



Γ

μ

2  Γ ν 3 2

ET II 78(12)





  3 y2 3 ν + ;ν − μ+ ;− 2 2 2 2a

y > 0, |arg a| < 14 π, Re ν > −1

Γ ν + 32 y ν  1F 1 − μ + 32 a2ν+2

ET II 79(23)





1 2 +m+μ



+ν 2 a2k+1 y22 y2 4a W e k,m Γ(−μ)y μ+2 2a2 2k = μ − ν − 1, 2m = μ + ν + 1

y > 0, |arg a| < 34 π, −1 < Re ν < − 21 − 2 Re μ 1

1

2λ− 2 μ π − 2 22 D μ (ax) J ν (xy) dx = 3 G 23 Γ(−μ)y λ+ 2 y > 0, |arg a| < 34 π, 1

ET II 79(24)

  y 2  12 , 1  μ 3 λ−ν 2a2  34 + λ+ν 2 , − 2 , 4 + 2

Re μ < − Re λ < Re ν + 32 ET II 80(26)



2

xν+1 e 4 x D −2ν−1 (x) J ν (xy) dx = (2ν + 1)y ν−1 e 4 y D −2ν−2 (y)

y > 0, Re ν > − 12

ET II 79(20)

7.754

Parabolic cylinder and Bessel functions



∞

16. 0

 0

0



∞

1. 0



2

2

1

1

2. 0



∞

1. 0

ν

x e

1 2 2 4a x

−1

3 3 4 ν+ 4

D 12 ν− 12 (ax) Y ν (xy) dx = −π 2 y > 0,

2

1

xν− 2 e−(x+a) I ν− 12 (2ax) D ν (2x) dx =

a

−ν −1

y

ET II 79(21)

2

3

xν− 2 e−(x+a) I ν− 32 (2ax) D ν (2x) dx =

1

y2 4a2

ET II 79(22)



y2 2a2



W − 12 ν− 12 , 12 ν

− 21 < Re ν < 23 ET II 115(39)

1 1 −1/2 π Γ(ν)aν− 2 D −ν (2a) 2

Re ν > 0]

ET II 397(12)

3 1 −1/2 π Γ(ν)aν− 2 D −ν (2a) 2

Re ν > 1]

ET II 397(13)

2

xν e− 4 x {[1 ∓ 2 cos(νπ)] D 2ν−1 (x) − D 2ν−1 (−x)} J ν (xy) dx 2

= ±y ν−1 e− 4 y {[1 ∓ 2 cos(νπ)] D 2ν−1 (y) − D 2ν−1 (−y)}

y > 0, Re ν > − 12 ET II 76(2, 3) ∞

2. 0

1

2

xν e− 4 x {[1 ∓ 2 cos(νπ)] D 2ν+1 (x) − D 2ν+1 (−x)} J ν (xy) dx 1





Re ν > −1]

Γ(ν + 1)e

|arg a| < 34 π,

1



2

2

[Re a > 0, 7.754

4y

Re ν > −1]

[Re a > 0, ∞

1

xν+1 e 4 x D −2ν−3 (x) J ν (xy) dx = y ν e 4 y D −2ν−3 (y) [y > 0,

∞

18.

7.753

1

2

xν+1 e− 4 x D −2ν−3 (x) J ν (xy) dx = 2−1/2 π 1/2 y −ν−2 e− 4 y I ν+1 [y > 0,

∞

17. 

1

855

2

= ∓y ν e− 4 y {[1 ∓ 2 cos(νπ)] D 2ν (y) + D 2ν (−y)}

y > 0, Re ν > − 12 ET II 77(6, 7) ∞

3. 0

1

2

xν+1 e− 4 x {[1 ± 2 cos(νπ)] D 2ν (x) + D 2ν (−x)} J ν (xy) dx 1

2

= ±y ν−1 e− 4 y {[1 ± 2 cos(νπ)] D 2ν+1 (y) − D 2ν+1 (−y)}  4. 0

[y > 0, ∞

1

Re ν > −1]

ET II 78(14, 15)

2

xν+1 e− 4 x {[1 ∓ 2 cos(νπ)] D 2ν+2 (x) + D 2ν+2 (−x)} J ν (xy) dx 1

2

= ±y ν e− 4 y {[1 ∓ 2 cos(νπ)] D 2ν+2 (y) + D 2ν+2 (−y)} [y > 0,

Re ν > −1]

ET II 78(17, 18)

856

7.755 1.12

Parabolic Cylinder Functions



∞

0

x−1/2 D ν

√  √  ax D −ν−1 ax J 0 (xy) dx =2



∞

2. 0

3.

12



∞

0



∞

0

 0

1

x1/2 D − 12 −ν ae 4 πi x1/2



πa

−1/2

P

1 1 2 ν+ 4 − 14



4y 2 1+ 2 a

1/2

− 1 ν− 1 P − 21 4 4

[y > 0, Re a > 0]   1 D − 12 −ν ae− 4 πi x1/2 J ν (xy) dx



4y 2 1+ 2 a

1/2

ET II 17(43)

2ν   −1  2 −1/2  1/2 a + 2y Γ ν + 12 −a = 2−ν π 1/2 y −ν−1 a2 + 2y

y > 0, Re a > 0, Re ν > − 21 ET II 80(27)

 1    1 D − 12 −ν ae 4 πi x−1/2 D − 12 −ν ae− 4 πi x−1/2 J ν (xy) dx

    −1 exp −a(2y)1/2 = 21/2 π 1/2 y −1 Γ ν + 12

y > 0, Re a > 0, Re ν > − 21 ET II 80(28)a

4.

5.



−3/2

7.755

∞

    x1/2 D ν− 12 ax−1/2 D −ν− 12 ax−1/2 Y ν (xy) dx

  x1/2 D ν− 12 ax−1/2 D −ν− 12

      = y −3/2 exp −ay 1/2 sin ay 1/2 − 12 ν − 12 π

y > 0, |arg a| < 14 π ET II 115(40)     ax−1/2 K ν (xy) dx = 2−1 y −3/2 π exp −a(2y)1/2

ET II 151(81) Re y > 0, |arg a| < 14 π

Combinations of parabolic cylinder and Struve functions  ∞ 1 2 x−ν e− 4 x [D μ (x) − D μ (−x)] Hν (xy) dx 7.756 0       23/2 Γ 12 μ + 12 μ+ν 1 1 1 2 1 1 y μπ 1 F 1 μ + ; μ + ν + 1; − y sin = 1 2 2 2 2 2 Γ 2 μ + ν + 1

y > 0, Re(μ + ν) > − 23 , Re μ > −1 ET II 171(41)

7.773

Integration of a parabolic cylinder function

857

7.76 Combinations of parabolic cylinder functions and confluent hypergeometric functions 7.761



∞

1. 0



∞

2. 0

  1 2 1 e 4 t t2c−1 D −ν (t) 1 F 1 a; c; − pt2 dt 2     1 1 1 π 1/2 Γ(2c) Γ 12 ν − c + a  F a, c + ; a + + ν; 1 − p = c+ 1 ν  1   2 2 2 2 2 Γ 2 ν Γ a + 12 + 12 ν [|1 − p| < 1, Re c > 0, Re ν > 2 Re(c − a)] EH II 121(12)   1 2 1 e 4 t t2c−2 D −ν (t) 1 F 1 a; c; − pt2 dt 2     1 1 π 1/2 Γ(2c − 1) Γ 12 ν + 12 − c + a     F a, c − ; a + ν; 1 − p = c+ 1 ν− 1 2 2 Γ 12 + 12 ν Γ a + 12 ν 2 2 2

1 |1 − p| < 1, Re c > 2 , Re ν > 2 Re(c − a) − 1 EH II 121(13)

7.77 Integration of a parabolic cylinder function with respect to the index 

7.771

0

∞

  1  π 1/2 β 2 cos a cos(ax) D x− 12 (β) D −x− 12 (β) dx = exp − 2 cos a 2 =0



|a| < 12 π

|a| > 12 π ET II 298(22)

7.772 1.



− 12 +i∞

⎡ ⎣

− 12 −i∞

ν  1   1  tan 12 ϕ 4 iπ ξ D −e e 4 iπ η D ν −ν−1 cos 12 ϕ ⎤  ν  1   1  cot 12 ϕ dν D −ν−1 e 4 iπ ξ D ν −e 4 iπ η ⎦ + 1 sin νπ sin 2 ϕ  

= −2i(2π)1/2 exp − 14 i ξ 2 − η 2 cos ϕ − 12 iξη sin ϕ EH II 125(7)

 − 12 +i∞  2.

− 12 −i∞

ν  1   1  dν tan 12 ϕ 4 iπ ζ D −e e 4 iπ η D ν −ν−1 sin νπ cos 12 ϕ  1   1    iπ 1 ζ cos 2 ϕ + η sin 12 ϕ D −1 e 4 iπ η cos 12 ϕ − ζ sin 12 ϕ = −2i D 0 e 4

EH II 125(8)

7.773



c+i∞

1. c−i∞

1

D ν (z)tν Γ(−ν) dν = 2πie− 4 z

2

−zt− 12 t2

 c < 0,

|arg t|
0; j = 1, . . . , m; Re σ > 0, ⎡ ⎤   p q $ $ 1 ⎦ > −1, Re ⎣ aj − bj + (p − q) ρ − 2 2 j=1 j=1

or p 0;

j = 1, . . . , m;

Re σ > 0 ET II 417(1)

       a1 , . . . , ap  a1 , . . . , ap , ρ m+1,n   αx dx = Γ(σ) α x−ρ (x − 1)σ−1 G mn G p+1,q+1 pq  b1 , . . . , bq  ρ − σ, b1 , . . . , bq 1 where

 3.

(or p ≤ q for |α| < 1) ,

∞

• • • • •

p + q < 2(m  + n) 1  |arg α| < m + n − 2 p − 12 q π Re (ρ − σ − aj ) > −1; j = 1, . . . , n Re σ > 0 either   p + q ≤ 2(m + n), |arg α| ≤ m + n − 12 p − 12 q π, Re (ρ − σ − aj ) > −1; j = 1, . . . , n; Re σ > 0, ⎡ ⎤   p q $ $ 1 ⎦ > −1, Re ⎣ aj − bj + (q − p) ρ − σ + 2 2 j=1 j=1 or q 1) ,

Re (ρ − σ − aj ) > −1;

Re σ > 0 ET II 417(2)

/m /n    ∞  a1 , . . . , ap Γ (bj + ρ) j=1 Γ (1 − aj − ρ) j=1 ρ−1 mn /p α−ρ dx = /q x G pq αx  b , . . . , b Γ (1 − b − ρ) Γ (a + ρ) 1 q j j 0 j=m+1  j=n+1  p + q < 2(m + n), |arg α| < m + n − 12 p − 12 q π, − min Re bj < Re ρ < 1 − max Re aj

 4.

j = 1, . . . , n;

1≤j≤m

1≤j≤n

ET II 418(3)a, ET I 337(14)

Meijer’s and MacRobert’s Functions (G and E)

860

5.

12



∞

0

ρ−1

x

−σ

(x + β)

mn G pq

7.812

       a1 , . . . , ap  1 − ρ, a1 , . . . , ap β ρ−σ m+1,n+1   αx  dx = αβ  G b1 , . . . , bq Γ(σ) p+1,q+1 σ − ρ, b1 , . . . , bq

where • • • • • •

p + q < 2(m   + n) |arg α| < m + n − 12 p − 12 q π |arg β| < π Re (ρ + bj ) > 0, j = 1, . . . , m Re (ρ − σ + aj ) < 1, j = 1, . . . , n either p ≤ q,

p + q ≤ 2(m + n),

  |arg α| ≤ m + n − 12 p − 12 q π,

|arg β| < π

Re (ρ + bj ) > 0, j = 1, . . . , m, Re (ρ − σ + aj ) < 1, j = 1, . . . , n, ⎡ ⎤   p q $ $ 1 ⎦ > 1, aj − bj − (q − p) ρ − σ − Re ⎣ 2 j=1 j=1 or

  1 1 p ≥ q, p + q ≤ 2(m + n), |arg α| ≤ m + n − p − q π, |arg β| < π, 2 2 Re (ρ + bj ) > 0, j = 1, . . . , m, Re (ρ − σ + aj ) < 1, j = 1, . . . , n, ⎡ ⎤   p q $ $ 1 ⎦>1 Re ⎣ aj − bj + (p − q) ρ − 2 j=1 j=1

ET II 418(4)

7.812



1. 0

1

 z  xβ−1 (1 − x)γ−β−1 E a1 , . . . , ap : ρ1 , . . . , ρq : m dx x = Γ(γ − β)mβ−γ E (a1 , . . . , ap+m : ρ1 , . . . , ρq+m : z) β+k−1 γ+k−1 , ρq+k = , k = 1, . . . , m ap+k = m m [Re γ > Re β > 0, m = 1, 2, . . .] ET II 414(2)

 2. 0

∞

xρ−1 (1 + x)−σ E [a1 , . . . , ap : ρ1 , . . . , ρq : (1 + x)z] dx = Γ(ρ) E (a1 , . . . , ap , σ − ρ : ρ1 , . . . , ρq , σ : z) [Re σ > Re ρ > 0]

ET II 415(3)

G and E and the elementary functions

7.815



∞

3. 0

−β s−1

(1 + x)

x

mn G pq



861

      1 − s, a1 , . . . , ap ax  a1 , . . . , ap m,n+1  dx = Γ(β − s) G p+1,q+1 a  1 + x  b1 , . . . , bq b1 , . . . , bq , 1 − β − min Re bk < Re s < Re β,

1 ≤ k ≤ m;

(p + q) < 2(m + n),    1 1 |arg a| < m + n − 2 p − 2 q π ET I 338(19)

7.813  1.

∞

0



−ρ −βx

x

e

mn G pq



      a1 , . . . , ap α  ρ, a1 , . . . , ap m,n+1 ρ−1  αx  dx = β G p+1,q b1 , . . . , bq β  b1 , . . . , bq    p + q < 2(m + n), |arg α| < m + n − 12 p − 12 q π,  |arg β| < 12 π, Re (bj − ρ) > −1, j = 1, . . . , m ET II 419(5)

∞

2. 0

       4α  0, 12 , a1 , . . . , ap m,n+2 2  a1 , . . . , ap −1/2 −1 αx dx = π e−βx G mn β G p+2,q pq  b1 , . . . , bq β 2  b1 , . . . , bq    p + q < 2(m + n), |arg α| < m + n − 12 p − 12 q π,  1 1 |arg β| < 2 π, Re bj > − 2 ; j = 1, . . . , m ET II 419(6)

7.814  1.

∞

0

xβ−1 e−x E (a1 , . . . , ap : ρ1 , . . . , ρq : xz) dx    = π cosec(βπ) E a1 , . . . , ap : 1 − β, ρ1 , . . . , ρq : e±iπ z −z

2.

−β



E a1 + β, . . . , ap + β : 1 + β, ρ1 + β, . . . , ρl + β : e

±iπ

z

 

[p ≥ q + 1, Re (ar + β) > 0, r = 1, . . . , p, |arg z| < π. The formula holds also for p < q + 1, provided the integral converges]. ET II 415(4)  ∞   xβ−1 e−x E a1 , . . . , ap : ρ1 , . . . , ρq : x−m z dx 0

 Re β > 0, 7.815  1. 0

∞

ap+k

1 1 1   = (2π) 2 − 2 m mβ− 2 E a1 , . . . , ap+m : ρ1 , . . . , ρq : m−m z  β+k−1 , k = 1, . . . , m; m = 1, 2, . . . = ET II 415(5) m

       √ −1 m,n+1 4α  0, a1 , . . . , ap , 12 2  a1 , . . . , ap  αx dx = sin(cx) G mn πc G p+2,q pq  b1 , . . . , bq c2  b 1 , . . . , b q   p + q < 2(m + n), |arg α| < m + n − 12 p − 12 q π,

c > 0, Re bj > −1, j = 1, 2, . . . , m, Re aj < 12 , j = 1, . . . , n ET II 420(7)

Meijer’s and MacRobert’s Functions (G and E)

862



∞

2. 0

cos(cx) G mn pq

7.821

       4α  12 , a1 , . . . , ap , 0 m,n+1 2  a1 , . . . , ap 1/2 −1 αx  dx = π c G p+2,q b1 , . . . , bq c2  b 1 , . . . , b q   p + q < 2(m + n), |arg α| < m + n − 12 p − 12 q π,

c > 0, Re bj > − 12 , j = 1, . . . , m, Re aj < 12 , j = 1, . . . , n ET II 420(8)

7.82 Combinations of the functions G and E and Bessel functions 7.821  1.

∞

0

       a1 , . . . , ap  ρ − 1 ν, a1 , . . . , ap , ρ + 1 ν  √  m,n+1 2 2   αx dx = α x−ρ J ν 2 x G mn G p+2,q pq  b1 , . . . , bq  b1 , . . . , bq    p + q < 2(m + n), |arg α| < m + n − 12 p − 12 q π  3 1 − 4 + max Re aj < Re ρ < 1 + 2 Re ν + min Re bj 1≤j≤n



∞

2.

1≤j≤m

ET II 420(9)

    a1 , . . . , ap  √   αx dx x−ρ Y ν 2 x G mn pq  b1 , . . . , bq

    ρ − 1 ν, ρ + 1 ν, a , . . . , a , ρ + 1 + 1 ν 1 p  2 2 2 2 = α  b1 , . . . , bq , ρ + 12 + 12 ν    p + q < 2(m + n), |arg α| < m + n − 12 p − 12 q π,  − 34 + max Re aj < Re ρ < min Re bj + 12 |Re ν| + 1

0

m,n+2 G p+3,q+1

1≤j≤n



1≤j≤m

ET II 420(10)

∞

3. 0

       a1 , . . . , ap  ρ − 1 ν, ρ + 1 ν, a1 , . . . , ap  √  mn 1 m,n+2 −ρ 2 2   dx = G p+2,q α  x K ν 2 x G pq αx  b1 , . . . , bq 2 b1 , . . . , bq    p + q < 2(m + n), |arg α| < m + n − 12 p − 12 q π,  Re ρ < 1 − 12 |Re ν| + min Re bj 1≤j≤m

ET II 421(11)

7.822  1. 0

∞

2ρ

x

       22ρ 4λ  h, a1 , . . . , ap , k m,n+1 2  a1 , . . . , ap λx  dx = 2ρ+1 G p+2,q b1 , . . . , bq y y 2  b1 , . . . , bq h = 12 − ρ − 12 ν, k = 12 − ρ + 12 ν      p + q < 2(m + n), |arg λ| < m + n − 12 p − 12 q π, Re bj + ρ + 12 ν > − 12 ,  j = 1, 2, . . . , m, Re (aj + ρ) < 34 , j = 1, . . . , n, y > 0

J ν (xy) G mn pq

ET II 91(20)

G and E and Bessel functions

7.823



∞

2. 0

1/2

x

Y ν (xy) G mn pq

863

    2  a1 , . . . , ap λx  dx b1 , . . . , bq



n+2,m = (2λ)−1/2 y −1/2 G q+1,p+3

+ 12 ν, k = 14 − 12 ν, l = − 14 − 12 ν   p + q < 2(m + n), |arg λ| < m + n − 12 p − 12 q π, y > 0,    3 Re aj < 1, j = 1, . . . , n, Re bj ± 12 ν > − , j = 1, . . . , m 4 h=





∞

3.

1 4

  y 2  12 − b1 , . . . , 12 − bq , l  4λ  h, k, 12 − a1 , . . . , 12 − ap , l

ET II 119(56)

    1/2 mn 2  a1 , . . . , ap dx x K ν (xy) G pq λx  b1 , . . . , bq

  y 2  12 − b1 , . . . , 12 − bq λ y =2  4λ  h, k, 12 − a1 , . . . , 12 − ap h = 14 + 12 ν, k = 14 − 12 ν    Re y > 0, p + q < 2(m + n), |arg λ| < m + n − 12 p − 12 q π ,  Re bj > 12 |Re ν| − 34 , j = 1, . . . , m

0

−3/2 −1/2 −1/2



n+2,m G q,p+2

ET II 153(90)

7.823



∞

1. 0

  xβ−1 J ν (x) E a1 , . . . , ap : ρ1 , . . . , ρq : x−2m z dx 0

= (2π)−m (2m)β−1 exp 12 π (β − ν − 1) i E a1 , . . . , ap+2m : ρ1 , . . . , ρq : (2m)−2m ze−mπi

1

+ exp − 21 π(β − ν − 1)i E a1 , . . . , ap+2m : ρ1 , . . . , ρq : (2m)−2m zemπi , β + ν + 2k − 2 β − ν + 2k − 2 , ap+m+k = , m = 1, 2, . . . , ; k = 1, . . . , m ap+k = 2m 2m

Re(β + ν) > 0, Re (2ar m − β) > − 32 , r = 1, . . . , p ET II 415(7)

 2. 0

∞

  xβ−1 K ν (x) E a1 , . . . , ap : ρ1 , . . . , ρq : x−2m z dx

ap+k

= (2π)1−m 2β−2 mβ−1

× E a1 , . . . , ap+2m : ρ1 , . . . , ρq : (2m)−2m z , β + ν + 2k − 2 β − ν + 2k − 2 , ap+m+k = , k = 1, 2, . . . , m = 2m 2m [Re β > |Re ν|, m = 1, 2, . . .] ET II 416(8)

Meijer’s and MacRobert’s Functions (G and E)

864

7.824



∞

1.

    2  a1 , . . . , ap λx dx x1/2 Hν (xy) G mn pq  b1 , . . . , bq

  y 2  l, 12 − b1 , . . . , 12 − bq = (2λy)  4λ  l, 12 − a1 , . . . , 12 − ap , h, k 1 ν 3 ν 1 ν h= + , k= − , l= + 4 2 4 2 4 2    p + q < 2(m + n), |arg λ| < m + n − 12 p − 12 q π, y > 0,   3 1  5 Re aj < min 1, 4 − 2 ν , j = 1, . . . , n, Re (2bj + ν) > − 2 , j = 1, . . . , m

0



−1/2



7.824

∞

2. 0

 √  x−ρ Hν 2 x G mn pq



n+1,m+1 G q+1,p+3

ET II 172(47)

   a1 , . . . , ap  αx  dx b1 , . . . , bq

  ρ −  α ρ −

 − 12 ν, a1 , . . . , ap , ρ + 12 ν, ρ − 12 ν = − 12 ν, b1 , . . . , bq    p + q < 2(m + n), |arg α| < m + n − 12 p − 12 q π ,    ν −1 3 1 3 + max Re aj < Re ρ < min Re bj + 2 Re ν + 2 max − , Re 1≤j≤n 1≤j≤m 4 2 m+1,n+1 G p+3,q+1

1 2 1 2

ET II 421(12)

7.83 Combinations of the functions G and E and other special functions 

7.831

1

∞

−ρ

x

σ−1

(x − 1)

    a1 , . . . , ap  αx  dx F (k + σ − ρ, λ + σ − ρ; σ; 1 − b1, . . . , bq   a1 , . . . , ap , k + λ + σ − ρ, ρ m+2,n  = Γ(σ) G p+2,q+2 α  k, λ, b1 , . . . , bq x) G mn pq

where

ET II 421(13)

⎡

⎤   p q $ $ 1 ⎦ 1 • Re ⎣ aj − bj + (q − p) k + >− , 2 2 j=1 j=1 ⎡ ⎤   p q $ $ 1 ⎦ > −1 • Re ⎣ aj − bj + (q − p) λ + 2 2 j=1 j=1 • either   p + q < 2(m + n), |arg α| < m + n − 12 p − 12 q π, Re σ > 0, Re k ≥ Re λ > Re aj − 1, j = 1, . . . , n,

G and E and other special functions

7.832

865

or   p + q ≤ 2(m + n), |arg α| ≤ m + n − 12 p − 12 q π, Re σ > 0, Re k ≥ Re λ > Re aj − 1, j = 1, . . . , n,  7.832

∞

  1 xβ−1 e− 2 x W κ,μ (x) E a1 , . . . , ap : ρ1 , . . . , ρq : x−m z dx 0   1 1 1 = (2π) 2 − 2 m mβ+κ− 2 E a1 , . . . , ap+2m : ρ1 , . . . , ρq+m : m−m z , ap+k =

β + k + μ − 12 , m

ap+m+k =

β − μ + k − 12 β−κ+k , ρq+k = , k = 1, . . . , m m m

1 Re β > |Re μ| − 2 , m = 1, 2, . . . ET II 416(10)

Table of Integrals, Series, and Products. http://dx.doi.org/10.1016/B978-0-12-384933-5.00008-4 c 2015 Elsevier Inc. All rights reserved. Copyright 

8–9 Special Functions 8.1 Elliptic Integrals and Functions 8.11 Elliptic integrals 8.110 1.

 Every integral of the form

   R x, P (x) dx, where P (x) is a third- or fourth-degree polyno-

mial, can be reduced to a linear combination of integrals leading to elementary functions and the following three integrals:    √ dx 1 − k 2 x2 dx  √  dx, , , 2 2 2 2 2 1 − x (1 − x ) (1 − k x ) (1 − nx ) (1 − x2 ) (1 − k 2 x2 ) which are called respectively elliptic integrals of the first, second, and third kind in the Legendre normal form. The results of this reduction for the more frequently encountered integrals are given√in formulas 3.13–3.17. The number k is called the modulus ∗ of these integrals, the number k  = 1 − k 2 is called the complementary modulus, and the number n is called the parameter of the integral of the third kind. BY (110.04) 2.

3.11 4.∗

∗ The

By means of the substitution x = sin ϕ, elliptic integrals can be reduced to the normal trigonometric forms     dϕ dϕ 2 2  1 − k sin ϕ dϕ, BY (110.04) ,   2 2 2 1 − n sin ϕ 1 − k sin ϕ 1 − k 2 sin2 ϕ The results of reducing integrals of trigonometric functions to normal form are given in 2.58– 2.62. π in the 8.110 2 formuElliptic integrals from 0 to 1 in the 8.110 1 formulation (or from 0 to 2 lation) are called complete elliptic integrals. Take note that in mathematical software, and elsewhere, the notation for elliptic integrals is often modified by replacing the parameter k 2 that is used here by k.

quantity k is sometimes called the module of the functions.

867

868

Elliptic Integrals and Functions

8.110

8.111 Notations: 

1 − k 2 sin2 ϕ;

k =

 1 − k2 ;

k 2 < 1.

1.

Δϕ =

2.

The elliptic integral of the first kind:  sin ϕ  ϕ dα dx   = F (ϕ, k) = . 2 2 (1 − x ) (1 − k 2 x2 ) 0 0 1 − k 2 sin α

3.

The elliptic integral of the second kind:  ϕ  2 2 E (ϕ, k) = 1 − k sin α dα = 0

4.11

0

FI II 135

ϕ



BY (110.04) sin ϕ

sin α dα x dx F (ϕ, k) − E (ϕ, k)   = = 2 2 k (1 − x2 ) (1 − k 2 x2 ) 0 0 1 − k 2 sin α

   π/2 dx b b π i  F arcsin √ arctan  = , 2 2 2 2 2 2 2|a| a a +b +1 0 a + sin x a + sin x

D(ϕ, k) =

6.∗

√ 1 − k 2 x2 √ dx 1 − x2

The elliptic integral of the third kind:  sin ϕ  ϕ dα dx  = Π(ϕ, n, k) =   2 2 2 2 (1 − nx ) (1 − x2 ) (1 − k 2 x2 ) 0 0 1 − n sin α 1 − k sin α 

5.

sin ϕ

2

2

[a and b are real] 7.

∗

Carlson has introduced a notation for the elliptic functions that preserves certain symmetries; see http://dlmf.nist.gov/19. The Carlson elliptic integrals are:  dt 1 ∞  7.1 RF (x, y, z) = 2 0 (t + x)(t + y)(t + z)  ∞ dt 3  7.2 RJ (x, y, z, p) = 2 0 (t + p) (t + x)(t + y)(t + z)  dt 1 ∞  7.3 RC (x, y) = RF (x, y, y) = 2 0 (t + y) (t + x)  dt 3 ∞  7.4 RD (x, y, z) = RJ (x, y, z, z) = 2 0 (t + z) (t + x)(t + y)(t + z) RC and RJ are interpreted as Cauchy principal values when the last argument is negative. Each function has the value unity when all of its arguments are unity. Use of Carlson’s notation for elliptical integrals reduces the number of separate cases that need to be tabulated.

8.113

8.∗

Elliptic integrals

The incomplete elliptic integrals can be written in Carlson’s notation as follows: 8.1

F (φ, k) = sin φRF (cos2 φ, 1 − k 2 sin2 φ, 1)

8.2

E(φ, k) = sin φRF (cos2 φ, 1 − k 2 sin2 φ, 1) 1 − k 2 sin3 φRD (cos2 φ, 1 − k 2 sin2 φ, 1) 3 Π(φ, n, k) = sin φRF (cos2 φ, 1 − k 2 sin2 φ, 1) 1 + n sin3 φRJ (cos2 φ, 1 − k 2 sin2 φ, 1, 1 − n sin2 φ) 3

8.3

9.∗

869

The complete elliptic integrals (i.e., the incomplete elliptic integrals with φ = in Carlson’s notation as follows: 9.1 9.2 9.3

π 2)

can be written

K (k) = RF (0, 1 − k 2 , 1) 1 K E(k) = RF (0, 1 − k 2 , 1) − k 2 RD (0, 1 − k 2 , 1) 3 1 2 Π(n, k) = RF (0, 1 − k , 1) + nRJ (0, 1 − k 2 , 1, 1 − n) 3

8.112 Complete elliptic integrals π  , k = K  (k  ) 1. K(k) = F 2 π  , k = E  (k  ) 2. E(k) = E 2 π  3. K  (k) = F , k  = K (k  ) 2 π  , k  = E (k  ) 4. E  (k) = E 2 π  K − E ,k = 5. D=D 2 k2 In writing complete elliptic integrals, the modulus k, which acts as an independent variable, is often omitted and we write     K (≡ K(k)) , K  ≡ K  (k) , E (≡ E(k)) , E  ≡ E  (k) Series representations 8.113 1.

π K= 2

  2 2 2    1 1 1 1·3 (2n − 1)!! π 2 4 2n 2 1+ F , ; 1; k k + k + ··· + k + . . . = 2 2·4 2n n! 2 2 2 FI II 487, WH 499

870

Elliptic Integrals and Functions

π K= 1 + k

2.

8.114

 2  2   2  2  4 2n  1 1 − k 1 − k 1 − k 1·3 (2n − 1)!! 1+ + + ··· + + ... 2 1 + k 2·4 1 + k 2n n! 1 + k DW

2   2   2 4 4 4 1 2 1·3 2 k 2 + − k 4 ln  − ln  − K = ln  + k 2 k 1·2 2·4 k 1·2 3·4 2   4 2 2 2 1·3·5 k 6 + . . . ln  − + − − 2·4·6 k 1·2 3·4 5·6

3.

DW

See also 8.197 1 and 8.197 2. 8.114 1.

2.

3.

12

π E= 2

1 12 · 32 1 − 2 k2 − 2 2 k4 − · · · − 2 2 ·4

(1 + k  ) π E= 4



(2n − 1)!! 2n n!

2

  k 2n π 1 1 − . . . = F − , ; 1; k 2 2n − 1 2 2 2 WH 518, FI II 487

1 1+ 2 2



 2

1−k 1 + k

12 + 2 2 2 ·4



 4

1−k 1 + k

 + ···+

(2n − 3)!! 2n n!

2 

1 − k 1 + k

  4 4 1 1 1 12 · 3 2 2 ln  − k + 2 ln  − − k 4 E= 1 + 2 k 1 · 2 2 ·4 k 1·2 3·4 4 2 1 12 · 32 · 5 2 ln  − − − k 6 + . . . + 2 2 2 ·4 ·6 k 1·2 3·4 5·6

2n



+ ... DW

DW

   2  2 2 1 1 · 3 (2n − 1)!! 1 2 n 2 2(n−1) 8.11512 D = π + k + ··· + k + ... 1 2 3 2·4 2n − 1 2n n!  1 3 π , ; 2; k 2 = F 4 2 2

 1  arccos 1 − k 2 sin2 ϕ  n  +R , where dϕ = n 2 − k 2 8.116  n 2 − 1 1 − n2 sin2 ϕ 0 n      1 1 6 k 2 k 4 1 1 p+ −1 + p + 1 + 2 R= + 3  2  2 n 3 16 4 n n    7 1 1 1 3 1 5 k 6 − − + 2 + 4 + p+ + 16 16 n 2 6 n 3 8 n n  37 21 1 9 5 1 1 1 14 15k 8 − − + + ..., − + p + + + + 256 144 40n 2 8 n 3 24 20n 2 n 4 n 4 3n 6 4 2 p = ln  , k  = 4e−p , k 2 = 1 − k 2 , n = 1 − n2 k 

ZH 43(158)

π 2

ZH 44(163)

ZH 44(163)

8.119

Elliptic integrals

871

Trigonometric series 8.117 For small values of k and ϕ, we may use the series  2 2 2·4 2 4 a2 sin ϕ + . . . , 1. F (ϕ, k) = Kϕ − sin ϕ cos ϕ a0 + a1 sin ϕ + π 3 3·5 2  2 (2n − 1)!! k 2n a0 = K − 1; an = an−1 − π 2n n!  2 2 2·4 2 4 12 b2 sin ϕ + . . . , 2. E (ϕ, k) = Eϕ − sin ϕ cos ϕ b0 + b1 sin ϕ + π 3 3·5 2  k 2n 2 (2n − 1)!! b0 = 1 − E, bn = bn−1 − n π 2 n! 2n − 1

where

ZH 10(19)

where

ZH 27(86)

8.118 For k close to 1, we may use the series 1.

2.

 2  2·4   2 4 a0 − a1 tan ϕ + a tan ϕ − . . . , where 3 3·5 2 2  2 (2n − 1)!! k 2n ZH 10(23) a0 = K − 1; an = an−1 − π 2n n! ϕ π  2 + E (ϕ, k) = (K − E ) ln tan π 2 2    2  2·4  1  tan ϕ  2 4 b1 − b2 tan ϕ + b3 tan ϕ − . . . + 1 − cos ϕ 1 − k 2 sin ϕ , + cos ϕ 3 3·5 sin ϕ  ϕ π  tan ϕ 2 + − F (ϕ, k) = K ln tan π 2 4 cos ϕ

where b0

2   2n − 1 2 (2n − 3)!!     k 2n = (K − E ) , bn = bn−1 − n−1 π 2 (n − 1)! 2n

ZH 27(90)

For the expansion of complete elliptic integrals in Legendre polynomials, see 8.928. 8.119 Representation in the form of an infinite product: 1.

K(k) =

∞ π  (1 + kn ) , 2 n=1

where  2 1 − kn−1  kn = ; 2 1 + 1 − kn−1 1−

See also 8.197.

k0 = k

FI II 166

872

Elliptic Integrals and Functions

8.121

8.12 Functional relations between elliptic integrals 8.121 1.

F (−ϕ, k) = − F (ϕ, k)

JA

2.

E (−ϕ, k) = − E (ϕ, k)

JA

3.

F (nπ ± ϕ, k) = 2n K(k) ± F (ϕ, k)

JA

4.

E (nπ ± ϕ, k) = 2n E(k) ± E (ϕ, k)

JA

8.122 E(k) K  (k) + E  (k) K(k) − K(k) K  (k) = 8.123 1. 2. 3. 4.

1 ∂F = 2 ∂k k



k sin ϕ cos ϕ E − k 2 F − k 1 − k 2 sin2 ϕ

π 2

FI II 691, 791

MO 138, BY (710.07)

E(k) K(k) d K(k) = − dk kk 2 k E−F ∂E = ∂k k E(k) − K(k) d E(k) = dk k

FI II 691 MO 138 FI II 690

8.124 1.

2.

The functions K and K satisfy the equation   d du kk 2 − ku = 0. dk dk The functions E and E − K satisfy the equation  du 2 d k + ku = 0. k dk dk

8.125 1.

 F

ψ, 

2.

3.

4.

WH 499, WH 502

E

1 − k 1 + k

1 − k ψ, 1 + k



= (1 + k  ) F (ϕ, k)

=

[tan(ψ − ϕ) = k  tan ϕ]

WH

MO 130

2 1 − k  [E (ϕ, k) + k F (ϕ, k)] − sin ψ 1 + k 1 + k

[tan(ψ − ϕ) = k  tan ϕ] √

  (1 + k) sin ϕ 2 k = (1 + k) F (ϕ, k) sin ψ = F ψ, 1+k 1 + k sin2 ϕ √

   1 sin ϕ cos ϕ 2 k 2 sin2 ϕ = 2E(ϕ, k) − k 2 F (ϕ, k) + 2k 1 − k E ψ, 1+k 1+k 1 + k sin2 ϕ   (1 + k) sin ϕ sin ψ = 1 + k sin2 ϕ

MO 131

MO 131

8.129

Functional relations between elliptic integrals

873

8.126 In particular,  1 + k 1 − k K(k) = 1. K  1+k 2  1 1 − k = 2. E [E(k) + k  K(k)] 1 + k 1 + k √

2 k = (1 + k) K(k) 3. K 1+k √

 1  2 k = 2 E(k) − k 2 K(k) 4. E 1+k 1+k

MO 130 MO 130

MO 130

MO 130

8.12712 k1

sin ϕ1

cos ϕ1

F (ϕ1 , k1 )

sin ϕ Δϕ

cos ϕ Δϕ

k  F (ϕ, k)

E (ϕ1 , k1 ) 1  E (ϕ, k) − k



k k

k

k

−i tan ϕ

sec ϕ

−i F (ϕ, k)

i [E (ϕ, k) − F (ϕ, k) − Δϕ tan ϕ]

1 k

k sin ϕ

Δϕ

k F (ϕ, k)

 1 E (ϕ, k) − k 2 F (ϕ, k) k

1 k

−ik  tan ϕ

Δϕ cos ϕ

−ik  F (ϕ, k)

k ik

−ik sin ϕ Δϕ

1 Δϕ

−ik F (ϕ, k)

i

k2 sin ϕ cos ϕ Δϕ

 i  E (ϕ, k) − k 2 F (ϕ, k) − Δϕ tan ϕ  k  2 i ϕ cos ϕ E (ϕ, k) − F (ϕ, k) − k sinΔϕ k (see 8.111 1)

8.128 In particular,  k 1. K i  = k  K(k) k  k 2.12 K  i  = k  [K (k  ) − i K(k)] k    1 = k K(k) + i K  (k) 3. K k

MO 131

[Im(k) < 0]

MO 130

[Im(k) < 0]

MO 130

[Im(k) < 0]

MO 130

For integrals of elliptic integrals, see 6.11–6.15. For indefinite integrals of complete elliptic integrals, see 5.11. 8.129 Special values: √

√

  2  √  1 dt π 2 2 1 1  √ 1. K sin =K =K = 2 = √ Γ MO 130 4 4 2 2 4 π 4 1−t 0  √  √ √  π = K 2.12 K  tan 2 − 1 = 2K 2−1 MO 130 8

874

3.12

4.

5. 6. 7. 8.∗

Elliptic Integrals and Functions

8.130

√ √

   6− 2 33/4 1 π √ π √ = 3 K sin = 3K = 7/3 Γ3 K sin 12 12 4 3 2 π

√   π π 2− 2 √ = K = 2 K tan2 K  tan2 8 8 2+ 2 √  3−1 π √ = K sin 12 2 2  √ 2  π 3 + kK E= 12K 3  √ π 3 2  kK E = + 4K 3 √

√

   2 2 1 1 1 3 π  =E =E = √ Γ2 + √ Γ2 E sin 4 2 2 4 4 8 π 2 π 

MO 130

MO 130

8.13 Elliptic functions 8.130 Definition and general properties. A single-valued function f (z) of a complex variable, which is not a constant, is said to be elliptic if it has two periods 2ω1 and 2ω2 , that is

1.

f (z + 2mω1 + 2nω2 ) = f (z)

[m, n integers]

The ratio of the periods of an analytic function cannot be a real number. For an elliptic function f (z), the z-plane can be partitioned into parallelograms—the period parallelograms—the vertices of which are the points z0 + 2mω1 + 2nω2 . At corresponding points of these parallelograms, the function f (z) has the same value. ZH 117, SI 299 2.

Suppose that α is the angle between the sides a and b of one of the period parallelograms. Then,  a   a a ω1 a π cos α + i sin π cos α τ= = eiα , q = eiπτ = e− b π sin α cos ω2 b b b

3.

The derivative of an elliptic function is also an elliptic function with the same periods. SM III 598

4.

12

A non-constant elliptic function has a finite number of poles in a period parallelogram: it has at least two simple poles or one second-order pole in such a parallelogram. Suppose that these poles lie at the points a1 , a2 , . . . , an and that their orders are α1 , α2 , . . . , αn . Suppose that the zeros of an analytic function that occur in a single parallelogram are b1 , b2 , . . . , bm and that the orders of the zeros are β1 , β2 , . . . , βm , respectively. Then, γ = α1 + α2 + · · · + αn = β1 + β2 + · · · + βm

ZH 118

The number γ representing this sum is called the order of the elliptic function. 5.

The sum of the residues of an elliptic function with respect to all the poles belonging to a period parallelogram is equal to zero.

6.

The difference between the sum of all the zeros and the sum of all the poles of an elliptic function that are located in a period parallelogram is equal to one of its periods.

8.145

7.

Jacobian elliptic functions

875

Every two elliptic functions with the same periods are related by an algebraic relationship. GO II 151

8.

7

A non-constant single-valued function which is not constant cannot have more than two periods. GO II 147

9.

An elliptic function of order γ assumes an arbitrary value γ times in a period parallelogram. SM 601, SI 301

8.14 Jacobian elliptic functions 8.141 Consider the upper limit ϕ of the integral  ϕ dα  u= 0 1 − k 2 sin2 α as a function of u. Using the notation ϕ = am u we call this upper limit the amplitude. The quantity u is called the argument, and its dependence on ϕ is written u = arg ϕ 8.142 The amplitude is an infinitely-many-valued function of u and has a period of 4Ki. The branch points of the amplitude correspond to the values of the argument u = 2mK + (2n + 1)K i, ZH 67–69 where m and n are arbitrary integers (see also 8.151). 8.143 The first two of the following functions snu = sin ϕ = sin am u, cn u = cos ϕ = cos am u,  dϕ dn u = Δϕ = 1 − k 2 sin2 ϕ = du are called, respectively, the sine-amplitude and the cosine-amplitude while the third may be called the delta amplitude. All these elliptic functions were exhibited by Jacobi and they bear his name. SI 16 The Jacobian elliptic functions are doubly-periodic functions and have two simple poles in a period parallelogram. ZH 69 8.144  snu dt  1. u= SI 21(23) 2 (1 − t ) (1 − k 2 t2 ) 0  cn u dt  2. u= SI 21(23) 2 ) (k  2 + k 2 t2 ) (1 − t 1  dn u dt  3. u= SI 21(23) 2 (1 − t ) (t2 − k 2 ) 1 8.145 Power series representations: 1.11

1 + k 2 3 1 + 14k 2 + k 4 5 1 + 135k 2 + 135k 4 + k 6 7 u + u − u 3! 5! 7! 2 4 6 8 1 + 1228k + 5478k + 1228k + k 9 u − ... + 9! [|u| < |K |]

snu = u −

ZH 81(97)

876

Elliptic Integrals and Functions

2.

cn u = 1 −

3.

dn u = 1−

8.146

1 2 1 + 4k 2 4 1 + 44k 2 + 16k 4 6 1 + 408k 2 + 912k 4 + 64k 6 8 u + u − u + u − ... 2! 4! 6! 8!  [|u| < |K |] ZH 81(98)

      k 2 2 k 2 4 + k 2 4 k 2 16 + 44k 2 + k 4 6 k 2 64 + 912k 2 + 408k 4 + k 6 8 u + u − u + u − ... 2! 4! 6! 8!  [|u| < |K |] ZH 81(99)

am u

      k 2 3 k 2 4 + k 2 5 k 2 16 + 44k 2 + k 4 7 k 2 64 + 912k 2 + 408k 4 + k 6 9 u − u + u − ... =u− u + 3! 5! 7! 9!  [|u| < |K |] LA 380(4)

4.

  πK  8.146 Representation as a trigonometric series or a product q = e− K = eπiτ ∗ ∞

1.

11

2π  q n− 2 πu snu = sin(2n − 1) kK n=1 1 − q 2n−1 2K 1

∞

2.11

cn u =

2π  q n− 2 πu cos(2n − 1) kK n=1 1 + q 2n−1 2K

3.

dn u =

2π  q n π nπu + cos 2K K n=1 1 + q 2n K

4.11

am u =

WH 511a, ZH 84(108)

1

WH 511a, ZH 84(109)

∞

5.

6.

7.

8. 9.11

10.

∞  πu nπu 1 qn sin +2 2n 2K n1+q K n=1   ∞  q 2n−1 π 1 1 πu = sin(2n − 1) πu + 4 snu 2K sin 2K 1 − q 2n−1 2K n=1   ∞ 2n−1  π 1 1 πu n q =  (−1) cos(2n − 1) πu + 4 cn u 2k K cos 2K 1 + q 2n−1 2K n=1   ∞  π 1 nπu qn =  1+4 (−1)n cos 2n dn u 2k K 1+q K n=1   ∞ 2n  π πu snu nπu n q =  tan +4 (−1) sin cn u 2k K 2K 1 + q 2n K n=1 1 ∞ snu q n− 2 2π  πu =−  (−1)n sin(2n − 1) 2n−1 dn u kk K n=1 1+q 2K   ∞  q 2n π πu cn u πnu = cot −4 sin 2n snu 2K 2K 1+q K n=1

WH 511a, ZH 84(110)

WH 511a

LA 369(3)

LA 369(3)

LA 369(3)

LA 369(4)

LA 369(4)

LA 369(5)

 1 πu   expansions 1–22 are valid in every strip of the form Im  < π Im τ . The expansions 23–25 are valid in an 2K 2 arbitrary bounded portion of u. ∗ The

8.146

11.

12.

13.

14.

15.

16.

17.

18.

19.

Jacobian elliptic functions 1 ∞ q n− 2 2π  cn u πu =− (−1)n cos(2n − 1) dn u kK n=1 1 − q 2n−1 2K   ∞  q 2n−1 π 1 dn u πu = sin(2n − 1) πu − 4 snu 2K sin 2K 1 + q 2n−1 2K n=1   ∞ 2n−1  π dn u πu 1 n q = (−1) cos(2n − 1) πu − 4 cn u 2K cos 2K 1 − q 2n−1 2K n=1   ∞  qn π πu cn u dn u nπu = cot −4 sin n snu 2K 2K 1+q K n=1

 ∞  qn π πu snu dn u nπu = tan +4 sin cn u 2K 2K 1 + (−1)n q n K n=1

∞ q 2n−1 4π 2  snu cn u πu = 2 sin(2n − 1) dn u k K n=1 1 − q 2(2n−1) K   ∞ n  π πu snu nπu n q = tan +4 (−1) sin cn u dn u 2 (1 − k 2 ) K 2K 1 − qn K n=1   ∞  (−1)n q n π πu cn u nπu = cot −4 sin n n snu dn u 2K 2K 1 + (−1) q K n=1   ∞  q 2(2n−1) π 1 dn u πu = +4 sin(2n − 1) snu cn u K sin πu K 1 − q 2(2n−1) K n=1

20.11 ln snu = ln

∞  1 qn nπu πu 2K + ln sin −4 sin2 n π 2K n 1 + q 2K n=1

21.

ln cn u = ln cos

22.

ln dn u = −8

∞  1 nπu qn πu −4 sin2 n n 2K n 1 + (−1) q 2K n=1

∞ 

πu 1 q 2n−1 sin2 (2n − 1) 2(2n−1) 2n − 1 2K 1 − q n=1

√ ∞ 4n 24q πu  1 − 2q 2n cos πu K +q 23.11 snu = √ sin πu 2n−1 2K n=1 1 − 2q cos K + q 4n−2 k √ √ ∞ 4n 2 k 4 q πu  1 + 2q 2n cos πu K +q 24. cn u = √ cos 4n−2 2K n=1 1 − 2q 2n−1 cos πu k K +q 25.

26.12

∞ 4n−2 √  1 + 2q 2n−1 cos πu K +q  dn u = k πu 1 − 2q 2n−1 cos K + q 4n−2 n=1   1 ∞ 2 πu  1 + k2 (2n + 1) π 2 2πq n+ 2 sin(2n + 1) 2K 2 sn u = − 3 3 2 2n+1 2k 2k 4K K (1 − q ) n=0   u    < Im τ Im 2K

877

LA 369(5)

LA 369(6)

LA 369(6)

LA 369(7)

LA 369(7)

LA 369(7)

LA 369(8)

LA 369(8)

LA 369(8)

LA 369(2)

LA 369(2)

LA 369(2)

WH 508a, ZH 86(145)

WH 508a, ZH 86(146)

WH 508a, ZH 86(147)

MO 147

878

Elliptic Integrals and Functions

8.147

∞

27.

8.147 1.

2.

3.

π2 K − E 2π 2  nq 2n cos nπu 1 2 πu K = + − 2 cosec sn2 u 4K2 2K K K n=1 1 − q 2n   u  1   < Im τ Im 2K 2

MO 148

∞  1 π π 2kK n=−∞ sin [u − (2n − 1)iK ] 2K ∞ (−1)n πi  cn u = 2kK n=−∞ sin π [u − (2n − 1)iK ] 2K ∞  πi (−1)n dn u = π 2K n=−∞ tan [u − (2n − 1)iK ] 2K

snu =

MO 149

MO 150

MO150

8.148 The Weierstrass expansions of the functions snu, cn u, dn u: C D B cn u = , dn u = , snu = , A A A where ∞ 

A=1− C=

∞ 

(−1)n+1 an+1

n=1

(−1)n cn

n=0

u2n+2 (2n + 2)!

B=

u (2n)! 2n

D=

ZH 82–83(105,106,107) ∞  n=0 ∞ 

(−1)n bn

u2n+1 (2n + 1)!

(−1)n dn

u2n (2n)!

n=0

and     a2 = 2k 2 , a3 = 8 k 2 + k 4 , a4 = 32 k 2 + k 6 + 68k 4 ,     a6 = 512 k 2 + k 10 + 3008 k 4 + k 8 + 5400k 6, . . .

    a5 = 128 k 2 + k 8 + 480 k 4 + k 6 ,

  b1 = 1 + k 2 , b2 = 1 + k 4 + 4k 2 , b3 = 1 + k 6 + 9 k 2 + k 4 ,       b4 = 1 + k 8 + 16 k 2 + k 6 − 6k 4 , b5 = 1 + k 10 + 25 k 2 + k 8 − 494 k 4 + k 6 ,     b6 = 1 + k 12 + 36 k 2 + k 10 − 5781 k 4 + k 8 − 12184k 6, . . . b0 = 1,

c0 = 1,

c1 = 1,

c2 = 1 + 2k 2 ,

c3 = 1 + 6k 2 + 8k 4 ,

c5 = 1 + 20k 2 + 348k 4 + 448k 6 + 128k 8, d0 = 1,

d1 = k ,

d2 = 2k + k ,

2

2

4

c4 = 1 + 12k 2 + 60k 4 + 32k 6 ,

c6 = 1 + 30k 2 + 2372k 4 + 4600k 6 + 2880k 8 + 512k 10,

d3 = 8k + 6k + k , 2

4

6

d5 = 128k + 448k + 348k + 20k + k , 2

4

6

8

10

d6 = 512k 2 + 2880k 4 + 4600k 6 + 2372k 8 + 30k 10 + k 12 ,

...

d4 = 32k + 60k + 12k + k , 2

4

6

8

...

8.152

Jacobian elliptic functions

879

8.15 Properties of Jacobian elliptic functions and functional relationships between them 8.151 The periods, zeros, poles, and residues of Jacobian elliptic functions: 1.12 Periods

Zeros

Poles

Residues

snu

4mK + 2nK i

2mK + 2nK i

2mK + (2n + 1)K i

(−1)m k1

cn u

4mK + 2n (K + K i)

(2m + 1) K + 2nK i

2mK + (2n + 1)K i

(−1)m+n−1 ki

dn u

2mK + 4nK i

(2m + 1)K + (2n + 1)K i

2mK + (2n + 1)K i

(−1)n−1 i SM 630, ZH 69–72

2. u∗ = u + K snu∗ =

cn u dn u

cn u∗ = −k  dn u∗ = k 

snu dn u

u + iK

u + K + iK

u + 2K

u + 2iK

u + 2K + 2iK

1 k snu

1 dn u k cn u

− snu

snu

− snu

ik  k cn u

− cn u

− cn u

cn u

snu cn u

dn u

− dn u

− dn u

−

1 dn u

i dn u k snu

−i

cn u snu

−

ik 

SM 630

3. u∗ = 0

−u

snu∗ = 0

− snu

cn u∗ = 1

cn u

dn u∗ = 1

dn u

SI 19, SI 18(13), WH,

1 2K

1 √ 1 + k √ k √ 1 + k √ k

(K + iK ) √ √ 1+k+i 1−k √ 2k √ (1 − i) k  √ 2k √ √  √ k 1 + k − i 1 − k √ 2 1 2

WH

1  2 iK

u + 2mK + 2nK i

i √ k

(−1)m snu

√ 1+k √ k

(−1)m+n cn u

√ 1+k

(−1)n dn u

WH

WH

dn (u1 , k1 )

ku

1 k

k sn(u, k)

dn(u, k)

cn(u, k)

iu

k

i

sn(u, k) cn(u, k)

1 cn(u, k)

dn(u, k) cn(u, k)

k u

i

k k

k

sn(u, k) dn(u, k)

cn(u, k) dn(u, k)

1 dn(u, k)

iku

i

k k

ik

sn(u, k) dn(u, k)

1 dn(u, k)

cn(u, k) dn(u, k)

sn(u, k) cn(u, k)

dn(u, k) cn(u, k)

1 cn(u, k)

cn(u, k) dn(u, k) 1 + k sn2 (u, k)

1 − k sn2 (u, k) 1 + k sn2 (u, k)

1 − (1 + k  ) sn2 (u, k) dn(u, k) √ dn(u, k) − k  √ 1 − k  

1 − (1 − k  ) sn2 (u, k) dn(u, k)  √  √ 1 + k1 dn(u, k) + k   [1 + dn(u, k)] [k  + dn(u, k)]

1 k √ 2 k 1+k

ik  u (1 + k)u (1 + k  ) u  √ 2 1 + k 2

u

1 − k 1 + k √ 2 1 − k √ 1 + k

ik 

(1 + k) sn(u, k) 1 + k sn2 (u, k) (1 + k  )

sn(u, k) cn(u, k) dn(u, k)

k 2 sn(u, k) cn(u, k) √ k1 [1 + dn(u, k)] [k  + dn(u, k)] ×

Elliptic Integrals and Functions

cn (u1 , k1 )

880

sn (u1 , k1 )

Transformation formulas

k1

8.15212

u1

2(1+k ) [1+dn(u,k)][k +dn(u,k)]

JA 8.152

8.156

8.153 1. 2. 3. 4. 5. 6. 7.11

8.12

9.11

Jacobian elliptic functions

sn(u, k  ) cn (u, k  ) 1 cn(iu, k) = cn (u, k  ) sn(iu, k) = i

dn (u, k  ) cn (u, k  )   sn(u, k) = k −1 sn ku, k −1   cn(u, k) = dn ku, k −1   dn(u, k) = cn ku, k −1   √   2 , k 1 + k 2 −1/2 sn u 1 + k 1   √ sn(u, ik) = √ 1 + k 2 dn u 1 + k 2 , k (1 + k 2 )−1/2 dn(iu, k) =

881

SI 50(64) SI 50(65) SI 50(65)

  1/2  −1/2  cn u 1 + k 2 , k 1 + k2   cn(u, ik) = 1/2 −1/2 dn u (1 + k 2 ) , k (1 + k 2 ) dn(u, ik) =

1   1/2 −1/2 dn u (1 + k 2 ) , k (1 + k 2 )

Functional relations 8.154 1. 2. 3.

1 − cn 2u 1 + dn 2u cn 2u + dn 2u cn 2 u = 1 + dn 2u dn 2u + k 2 cn 2u + k 2 dn 2 u = 1 + dn 2u sn2 u =

MO 146 MO 146 MO 146

4.

sn2 u + cn 2 u = 1

SI 16(9)

5.

dn u + k sn u = 1

SI 16(9)

2

2

2

8.155 1. 2.

sn2 u cn 2 u 1 − dn 2u = k2 1 + dn 2u dn 2 u 2 sn u dn 2 u 1 − cn 2u = 1 + cn 2u cn 2 u

MO 146 MO 146

8.156 1.

sn(u ± v) =

snu cn v dn v ± snv cn u dn u 1 − k 2 sn2 u sn2 v

SI 46(56)

882

Elliptic Integrals and Functions

8.157

cn u cn v ∓ snu snv dn u dn v 1 − k 2 sn2 u sn2 v dn u dn v ∓ k 2 snu snv cn u cn v dn (u ± v) = 1 − k 2 sn2 u sn2 v

cn (u ± v) =

2. 3. 8.157 1. 2. 3.12 8.158 1. 2. 3.8

  1 1 − dn u u 1 − cn u = ± = ± sn 2 k 1 + cn u 1 + dn u   cn u + dn u 1 − dn u k u =± cn = ± 2 1 + dn u k dn u − cn u   cn u + dn u 1 − cn u u  = ±k dn = ± 2 1 + cn u dn u − cn u

SI 46(57) SI 46(58)

SI 47(61), SU 67(15)

SI 48(62), SI 67(16) SI 48(63), SI 67(17)

d snu = cn u dn u du d cn u = − snu dn u du d dn u = −k 2 dn u cn u du

SI 21(21) SI 21(21) SI 21(21)

8.159 Jacobian elliptic functions are solutions of the following differential equations: 1. 2. 3.

 d snu = (1 − sn2 u) (1 − k 2 sn2 u) du  d cn u = − (1 − cn 2 u) (k 2 + k 2 cn 2 u), du  d dn u = − (1 − dn 2 u) (dn 2 u − k  2 ) du

SI 21(22) SI 21(22) SI 21(22)

For the indefinite integrals of Jacobi’s elliptic functions, see 5.13.

8.16 The Weierstrass function ℘(u) 8.160 The Weierstrass elliptic function ℘(u) is defined by

  1 1 1 , SI 307(6) 1. ℘(u) = 2 + 2 − 2 u (u − 2mω1 − 2nω2 ) (2mω1 + 2nω2 ) m,n  where the symbol means that the summation is made over all combinations of integers m and n except for the combination m = n = 0; 2ω1 and 2ω2 are the periods of the function ℘(u). Obviously,  ω1 2. ℘ (u + 2mω1 + 2nω2 ) = ℘(u) and Im = 0, ω2

The Weierstrass function ℘(u)

8.164

3.

883

 1 d ℘(u) = −2 3, du m,n (u − 2mω1 − 2nω2 ) where the summation is made over all integral values of m and n. The series 8.160 1 and 8.160 3 converge everywhere except at the poles, that is, at the points 2mω1 + 2nω2 (where m and n are integers).

4.

The function ℘(u) is a doubly- periodic function and has one second-order pole in a period parallelogram. SI 306

8.161 The function ℘(u) satisfies the differential equation  1.

2.

d ℘(u) du where g2 = 60

2 = 4 ℘3 (u) − g2 ℘(u) − g3 , 

−4

(mω1 + nω2 )

m,n

;

g3 = 140

SI 142, 310, WH



−6

(mω1 + nω2 )

WH, SI 310

m,n

The functions g2 and g3 are called the invariants of the function ℘(u).  ∞  ∞ dz dz   8.162 u = = , 3 4 (z − e1 ) (z − e2 ) (z − e3 ) 4z − g2 z − g3 ℘(u) ℘(u) where e1 , e2 , and e3 are the roots of the equation 4z 3 − g2 z − g3 = 0; that is, g2 g3 e1 + e2 + e3 = 0, e1 e2 + e2 e3 + e3 e1 = − , e1 e2 e3 = SI 142, 143, 144 4 4 8.163 ℘ (ω1 ) = e1 , ℘ (ω1 ) + ω2 = e2 , ℘ (ω2 ) = e3 . Here, it is assumed that if e1 , e2 , and e3 lie on a straight line in the complex plane, e2 lies between e1 and e3 . 8.164 The number Δ = g23 − 27g32 is called the discriminant of the function ℘(u). If Δ > 0, all roots e1 , e2 , and e3 of the equation 4z 3 − g2 z − g3 = 0 (where g2 and g3 are real numbers) are real. In this case, the roots e1 , e2 , and e3 are numbered in such a way that e1 > e2 > e3 . 1.

If Δ > 0, then



 e3 dz dz   , ω2 = i , 3 4z − g2 z − g3 g3 + g2 z − 4z 3 −∞ e1 where ω1 is real and ω2 is a purely imaginary number. Here, the values of the radical in the ω2 will be positive. integrand are chosen in such a way that ω1 and i If Δ < 0, the root e2 of the equation 4z 3 − g2 z − g3 = 0 is real and the remaining two roots (e1 and e3 ) are complex conjugates. Suppose that e1 = α + iβ, and e3 = α − iβ. In this case, it is convenient to take  ∞  ∞ dz dz   ω = and ω  = 3 3 4z − g2 z − g3 4z − g2 z − g3 e1 e3 as basic semiperiods. ω1 =

2.

∞

In the first integral, the integration is taken over a path lying entirely in the upper half-plane and in the second over a path lying entirely in the lower-half plane. SI 151(21, 22)

884

Elliptic Integrals and Functions

8.165

8.165 Series representation: ℘(u) =

1.

g3 u4 g 2 u6 g2 u2 3g2 g3 u8 1 + + 4 2 + ... + + 4 2 2 u 4·5 4·7 2 ·3·5 2 · 5 · 7 · 11

WH

8.166 Functional relations ℘ (u) = − ℘ (−u)  2 1 ℘ (u) − ℘ (v) 2. ℘(u + v) = − ℘(u) − ℘(v) + 4 ℘(u) − ℘(v)  g2 g3 2 8.167 ℘ (u; g2 , g3 ) = μ ℘ μu; 4 , 6 μ μ ℘(u) = ℘(−u),

1.

SI 163(32)

(the formula for homogeneity) SI 149(13)

The special case: μ = i. 1.

℘ (u; g2 , g3 ) = − ℘ (iu; g2 , −g3 )

8.168 An arbitrary elliptic function can be expressed in terms of the elliptic function ℘(u) having the same periods as the original function and its derivative ℘ (u). This expression is rational with respect to ℘(u) and linear with respect to ℘ (u). 8.169 A connection with the Jacobian elliptic functions. For Δ > 0 (see 8.164 1).  u cn 2 (u, k) 12 1. = e1 + (e1 − e3 ) 2 ℘ √ sn (u, k) e1 − e3 dn 2 (u, k) = e2 + (e1 − e3 ) 2 sn (u, k) 1 = e3 + (e1 − e3 ) 2 sn (u, k) SI 145(5), ZH 120(197–199)a

2.

3.

4. 5.

6.11

K iK ω1 = √ , ω2 = √ , e1 − e3 e1 − e3 where   e2 − e3 e1 − e2  , k = k= e1 − e3 e1 − e3 For Δ < 0 (see 8.164 2)

 1 + cn(2u, k) u ; = e2 + 9α2 + β 2 ℘  4 1 − cn(2u, k) 9α2 + β 2 K − iK K + iK ω =  , ω  =  , 4 2 9α2 + β 2 9α2 + β 2 where   1 1 3e2 3e2  −  +  k= ; k = 2 2 2 4 9α + β 2 4 9α2 + β 2

SI 154(29)

SI 145(7)

SI 147(12)

SI 153(28)

SI 147

For Δ = 0, all the roots e1 , e2 , and e3 are real and if g2 g3 = 0, two of them are equal to each other. If e1 = e2 = e3 , then

The functions ζ(u) and σ(u)

8.176

7.

8.

  3g3 9g3 9g3 2 ℘(u) = − coth u − g2 2g2 2g2 If e1 = e2 = e3 , then 1 3g3 9g3    + ℘(u) = − 2g2 2g2 sin2 u 9g3

885

SI 148

SI 149

2g2

9.

If g2 = g3 = 0, then e1 = e2 = e3 = 0, and 1 ℘(u) = 2 u

SI 149

8.17 The functions ζ(u) and σ(u) 8.171 Definitions: 1. 2.

 1 ℘(z) − 2 dz z 0   u  1 ℘(z) − 2 dz σ(u) = u exp z 0 1 ζ(u) = − u



u

8.172 Series and infinite-product representation

 1 1 1 u 1. ζ(u) = + + + 2 u m,n u − 2mω1 − 2nω2 2mω1 + 2nω2 (2mω1 − 2nω2 ) 

  u u u2 1− exp 2. σ(u) = u + 2 2mω1 + 2nω2 2mω1 + 2nω2 2 (2mω1 + 2nω2 ) m,n

SI 181(45) SI 181(46)

SI 307(8)

SI 308(9)

8.173

g3 u5 g 2 u7 3g2 g3 u9 g2 u3 − 2 − 4 2 2 − 4 − ··· · 3 · 5 2 · 5 · 7 2 · 3 · 5 · 7 2 · 5 · 7 · 9 · 11 g3 u7 g 2 u9 g2 g3 u11 g2 u5 − 3 − 9 22 − 7 2 2 − ··· 2.12 σ(u) = u − 4 2 · 3 · 5 2 · 3 · 5 · 7 2 · 3 · 5 · 7 2 · 3 · 5 · 7 · 11  ∞  πu ζ (ω1 ) π πu π  ω2 cot 8.174 ζ(u) = u+ cot + + nπ ω1 2ω1 2ω1 2ω1 n=1 2ω1 ω1   ω2 πu − nπ + cot 2ω1 ω1 ∞ ζ (ω1 ) π πu 2π  q 2n πnu = u+ cot + sin ω1 2ω1 2ω1 ω1 n=1 1 − q 2n ω1 1.

ζ(u) = u −

22

SI 181(49) SI 181(49)

MO 154 MO 155

Functional relations and properties 8.175 ζ(u) = − ζ(−u), 8.176 1.

σ(u) = − σ(−u)

ζ (u + 2ω1 ) = ζ(u) + 2ζ (ω1 )

SI 181

SI 184(57)

886

Elliptic Integrals and Functions

8.177

2.

ζ (u + 2ω2 ) = ζ(u) + 2ζ (ω2 )

SI 184(57)

3.

σ (u + 2ω1 ) = − σ(u) exp {2 (u + ω1 ) ζ (ω1 )} .

SI 185(60)

4.

σ (u + 2ω2 ) = − σ(u) exp {2 (u + ω2 ) ζ (ω2 )} . π ω2 ζ (ω1 ) − ω1 ζ (ω2 ) = i 2

SI 185(60)

5.

SI 186(62)

8.177 1 ℘ (u) − ℘ (v) 2 ℘(u) − ℘(v)

1.

ζ(u + v) − ζ(u) − ζ(v) =

2.

℘(u) − ℘(v) = −

3.

ζ(u − v) + ζ(u + v) − 2ζ(u) =

SI 182(53)

σ(u − v) σ (u + v) σ 2 (u) σ 2 (v)

SI 183(54)

℘ (u) ℘(u) − ℘(v)

SI 182(51)

8.178 1.

ζ (u; ω1 , ω2 ) = t ζ (tu; tω1 , tω2 )

MO 154

2.8

σ (u; ω1 , ω2 ) = t−1 σ (tu; tω1 , tω2 )

MO 156

For the indefinite integrals of Weierstrass elliptic functions, see 5.14.

8.18–8.19 Theta functions 8.180 Theta functions are defined as the sums (for |q| < 1) of the following series: 1.

ϑ4 (u) =

∞ 

2

(−1)n q n e2nui = 1 + 2

n=−∞

ϑ1 (u) =

2.

3.

11

ϑ2 (u) = ϑ3 (u) =

2

(−1)n q n cos 2nu

∞ ∞  1 2 1 2 1  (−1)n q (n+ 2 ) e(2n+1)ui = 2 (−1)n+1 q (n− 2 ) sin(2n − 1)u i n=−∞ n=1 ∞ 

1 2 q (n+ 2 ) e(2n+1)ui = 2

∞ 

∞ 

1

WH

2

q (n− 2 ) cos(2n − 1)u

WH

n=1 2

q n e2nui = 1 + 2

n=−∞

WH

n=1

n=−∞

4.

∞ 

∞ 

2

q n cos 2nu

WH

n=1

The notations ϑ(u, q) and ϑ (u | τ ), where τ and q are related by q = eiπτ , are also used. Here, q is called the nome of the theta function and τ its parameter. 8.181 Representation of theta functions in terms of infinite products 1.

ϑ4 (u) =

∞  

1 − 2q 2n−1 cos 2u + q 2(2n−1)



1 − q 2n



SI 200(9), ZH 90(9)

n=1

2.

ϑ3 (u) =

∞   n=1

1 + 2q 2n−1 cos 2u + q 2(2n−1)



1 − q 2n



SI 200(9), ZH 90(9)

8.186

Theta functions

887

∞

3. 4.8

   √ 1 − 2q 2n cos 2u + q 4n 1 − q 2n ϑ1 (u) = 2 4 q sin u √ ϑ2 (u) = 2 4 q cos u

n=1 ∞ 



1 + 2q 2n cos 2u + q 4n

  1 − q 2n

SI 200(9), ZH 90(9)

SI 200(0), ZH 90(9)

n=1

Functional relations and properties 8.182 Quasiperiodicity. Suppose that q = eπτ i (Im τ > 0). Then, theta functions that are periodic functions of u are called quasiperiodic functions of τ and u. This property follows from the equations 1.

ϑ4 (u + π) = ϑ4 (u)

SI 200(10)

2.

1 ϑ4 (u + τ π) = − e−2iu ϑ4 (u) q

SI 200(10)

3.

ϑ1 (u + π) = − ϑ1 (u)

SI 200(10)

4.

1 ϑ1 (u + τ π) = − e−2iu ϑ1 (u) q

SI 200(10)

5.

ϑ2 (u + π) = − ϑ2 (u)

SI 200(10)

6.

ϑ2 (u + τ π) =

7.

ϑ3 (u + π) = ϑ3 (u)

8.

ϑ3 (u + τ π) =

8.183  1. ϑ4 u +  2. ϑ1 u +  3. ϑ2 u +  4. ϑ3 u +  5. ϑ4 u +  6. ϑ1 u +  7. ϑ2 u +  8. ϑ3 u +



1 2π  1 2π  1 2π  1 2π

1 −2iu e ϑ2 (u) q

SI 200(10) SI 200(10)

1 −2iu e ϑ3 (u) q

SI 200(10)

= ϑ3 (u)

WH

= ϑ2 (u)

WH

= − ϑ1 (u)

WH

= ϑ4 (u)  −1/4 −iu 1 e ϑ1 (u) 2 πτ = iq  −1/4 −iu 1 e ϑ4 (u) 2 πτ = iq  −1/4 −iu 1 e ϑ3 (u) 2 πτ = q  −1/4 −iu 1 e ϑ2 (u) 2 πτ = q

WH WH WH WH WH

8.184 Even and odd theta functions 1.

ϑ1 (−u) = − ϑ1 (u)

WH

2.

ϑ2 (−u) = ϑ2 (u)

WH

3.

ϑ3 (−u) = ϑ3 (u)

WH

4.

ϑ4 (−u) = ϑ4 (u)

8.185 8.1867

ϑ44 (u)

ϑ42 (u)

WH

ϑ41 (u)

ϑ43 (u)

+ = + WH Considering the theta functions as functions of two independent variables u and τ , we have

888

Elliptic Integrals and Functions

8.187

∂ 2 ϑk (u | τ ) ∂ ϑk (u | τ ) =0 [k = 1, 2, 3, 4] +4 WH 2 ∂u ∂τ 8.187 We denote the partial derivatives of the theta functions with respect to u by a prime and consider them as functions of the single argument u. Then, πi

1.

ϑ1 (0) = ϑ2 (0) ϑ3 (0) ϑ4 (0)

2.

ϑ 1 (0) ϑ1 (0)

=

ϑ2 (0) ϑ2 (0)

ϑ3 (0)

+

ϑ3 (0)

WH

+

ϑ4 (0)

WH

ϑ4 (0)

8.18812 ϑ1 (u) ϑ2 (u) ϑ3 (u) ϑ4 (u) = 12 ϑ1 (2u) ϑ2 (0) ϑ3 (0) ϑ4 (0) 8.189 The zeros of the theta functions: πτ π 1.8 ϑ4 (u) = 0 for u = 2m + (2n − 1) 2 2 πτ π 10 2. ϑ1 (u) = 0 for u = 2m + 2n 2 2 πτ π 3. ϑ2 (u) = 0 for u = (2m − 1) + 2n 2 2 πτ π 4. ϑ3 (u) = 0 for u = (2m − 1) + (2n − 1) 2 2

WH

SI 201 SI 201 SI 201

[m and n are integers or zero]

SI 201

For integrals of theta functions, see 6.16. 8.191 Connections with the Jacobian elliptic functions:  K K For τ = i K , i.e. for q = exp −π K ,

1.

2.

3.

 πu  ϑ 1 1 H (u) 1 = √  2K snu = √ πu k ϑ4 k Θ (u) 2K  πu    k  ϑ2 2K k  H 1 (u)  πu  = cn u = k ϑ4 k Θ(u) 2K  πu  √ Θ1 (u) √ ϑ3  dn u = k   2K πu  = k Θ(u) ϑ4 2K

SI 206(22), SI 209(35)

SI 207(23), SI 209(35)

SI 207(24), SI 209(35)

8.192 Series representation of H , H 1 , Θ, Θ1 .  the functions   In these formulas, q = exp −π KK . 1.

2.

3.

Θ(u) = ϑ4 H (u) = ϑ1

 πu  2K  πu 

Θ1 (u) = ϑ3

2K

=1+2

2

(−1)n q n cos

n=1

=2

 πu  2K

∞ 

∞ 

(−1)n+1

n=1

=1+2

∞ 

2

 πu 2 4 q (2n+1) sin(2n − 1) 2K

q n cos

n=1

nπu K

nπu K

SI 207(25), SI 212(42)

SI 207(25), SI 212(43)

SI 207(25), SI 212(45)

8.197

H 1 (u) = ϑ2

4.

 πu  2K

=2

Theta functions

889

∞   πu 4 q (2n−1)2 cos(2n − 1) 2K n=1

SI 207(25), SI 212(44)

8.193 Connections with the Weierstrass elliptic functions  √   √  ⎡ ⎡  √  ⎤2 ⎤2 ⎤2 H 1 u λ H  (0) Θ1 u λ H  (0) Θ u λ H  (0)  √  ⎦ λ = e2 + ⎣  √  ⎦ λ = e3 + ⎣  √ ⎦ λ ℘(u) = e1 + ⎣ H 1 (0) H u λ Θ1 (0) H u λ Θ(0) H u λ ⎡

1.12

SI 235(77,78)

 √   u λ H √ η1 u ζ(u) = + λ  √  ω1 H u λ   H u√λ  2 1 η1 u σ(u) = √ exp 2ω1 H  (0) λ

2.

3.

SI 234(73)

SI 234(72)

where λ = e1 − e3 ;

η1 = ζ (ω1 ) = −

ω1 λ H  (0) 3 H  (0)

SI 236

8.194 The connection with elliptic integrals: 1.12 2.12

Θ (0) Θ (u) + SI 228(65) Θ(0) Θ(u)     u   dϕ sna Θ (a) 1 Θ(u − a) 2 2 =u+ u + ln Π am u, k sn a, k = 2 2 2 cn a dn a Θ(a) 2 Θ(u + a) 0 1 − k sn a sn ϕ E (am u, k) = u − u

SI 228(65)

   q-series and products, q = exp −π KK  2 ∞  2 π π 1+2 qn = K = Θ2 (K) 8.195 2 2 n=1 ∞  2 (−1)n+1 n2 q n  2 2π n=1 Θ (0) = K− 8.196 E = K − K ∞  Θ(0) K 2 1+2 (−1)n q n

2.

SI 219

SI 230(67)

n=1

8.197 1.

(cf. 8.197 1)

∞ 



2K = ϑ3 (0) π n=1  ∞  2n−1 2 kK 1 ) ( = ϑ2 (0) q 2 = 2π 2 n=1

1+2

q

n2

=

(cf. 8.195)

WH

WH

890

3.

4.

5.

6.

Elliptic Integrals and Functions

4 ∞  1 + q 2n √  4 q =k 1 + q 2n−1 n=1 4 ∞   1 − q 2n−1 = k 2n−1 1 + q n=1 2 ∞  √ K √  1 − q 2n 24q =2 k 2n−1 1−q π n=1  ∞ 2  √ K 1 − q 2n = 2 k 2n 1+q π n=1

8.198

1.

√ 1 1 − k √ = λ= 2 1 + k

∞ 

q (2n+1)

n=0

1+2

∞ 

q

8.198

SI 206(17, 18)

SI 206(19, 20)

WH

WH

2

[for 0 < k < 1, we have 0 < λ < 12 ]

2

WH

4n

n=1

The series 2.

q = λ + 2λ5 + 15λ9 + 150λ13 + 1707λ17 + . . .

WH

is used to determine q from the given modulus k. 8.19910

6.

Identities involving products of theta functions         ϑ1 (x, q) ϑ1 (y, q) = ϑ3 x + y, q 2 ϑ2 x − y, q 2 − ϑ2 x + y, q 2 ϑ3 x − y, q 2         ϑ1 (x, q) ϑ2 (y, q) = ϑ1 x + y, q 2 ϑ4 x − y, q 2 + ϑ4 x + y, q 2 ϑ1 x − y, q 2         ϑ2 (x, q) ϑ2 (y, q) = ϑ2 x + y, q 2 ϑ3 x − y, q 2 + ϑ3 x + y, q 2 ϑ2 x − y, q 2         ϑ3 (x, q) ϑ3 (y, q) = ϑ3 x + y, q 2 ϑ3 x − y, q 2 + ϑ2 x + y, q 2 ϑ2 x − y, q 2         ϑ3 (x, q) ϑ4 (y, q) = ϑ4 x + y, q 2 ϑ4 x − y, q 2 − ϑ1 x + y, q 2 ϑ1 x − y, q 2         ϑ4 (x, q) ϑ4 (y, q) = ϑ3 x + y, q 2 ϑ3 x − y, q 2 − ϑ2 x + y, q 2 ϑ2 x − y, q 2

7.

ϑ1 (x + y) ϑ1 (x − y) ϑ24 (0) = ϑ23 (x) ϑ22 (y) − ϑ22 (x) ϑ23 (y) = ϑ21 (x) ϑ24 (y) − ϑ24 (x) ϑ21 (y)

LW 8(1.4.16)

8.

ϑ2 (x + y) ϑ2 (x − y) ϑ24 (0) = ϑ24 (x) ϑ22 (y) − ϑ21 (x) ϑ23 (y) = ϑ22 (x) ϑ24 (y) − ϑ23 (x) ϑ21 (y)

LW 8(1.4.17)

9.

ϑ3 (x + y) ϑ3 (x − y) ϑ24 (0)

LW 8(1.4.18)

10.

ϑ4 (x + y) ϑ4 (x − y) ϑ24 (0) = ϑ24 (x) ϑ24 (y) − ϑ21 (x) ϑ21 (y)

1. 2. 3. 4. 5.

=

ϑ24 (x) ϑ23 (y) − ϑ21 (x) ϑ22 (y)

LW 8(1.4.8) LW 8(1.4.9) LW 8(1.4.10) LW 8(1.4.11) LW 8(1.4.12)

LW 8(1.4.15)

11.

ϑ4 (x + y) ϑ4 (x − y) ϑ24 (0)

ϑ24 (x) ϑ24 (y) − ϑ21 (x) ϑ21 (y)

LW 9(1.4.19)

12.

ϑ1 (x + y) ϑ1 (x − y) ϑ23 (0) = ϑ21 (x) ϑ23 (y) − ϑ23 (x) ϑ21 (y) = ϑ24 (x) ϑ22 (y) − ϑ22 (x) ϑ24 (y)

LW 9(1.4.23)

13.

ϑ2 (x + y) ϑ2 (x − y) ϑ23 (0) = ϑ22 (x) ϑ23 (y) − ϑ24 (x) ϑ21 (y) = ϑ23 (x) ϑ22 (y) − ϑ21 (x) ϑ24 (y)

LW 9(1.4.24)

14.

ϑ3 (x + y) ϑ3 (x − y) ϑ23 (0)

ϑ22 (x) ϑ22 (y) + ϑ24 (x) ϑ24 (y)

LW 9(1.4.25)

15.

ϑ4 (x + y) ϑ4 (x − y) ϑ23 (0) = ϑ21 (x) ϑ22 (y) + ϑ23 (x) ϑ24 (y) = ϑ22 (x) ϑ21 (y) + ϑ24 (x) ϑ23 (y)

LW 9(1.4.26)

16.

ϑ1 (x + y) ϑ1 (x − y) ϑ22 (0)

LW 9(1.4.30)

=

=

=

ϑ23 (x) ϑ23 (y) − ϑ22 (x) ϑ22 (y)

=

ϑ23 (x) ϑ24 (y) − ϑ22 (x) ϑ21 (y)

LW 7(1.4.7)

ϑ21 (x) ϑ21 (y) + ϑ23 (x) ϑ23 (y) ϑ21 (x) ϑ22 (y) − ϑ22 (x) ϑ21 (y)

=

=

=

ϑ24 (x) ϑ23 (y) − ϑ23 (x) ϑ24 (y)

8.199(3)

Theta functions

891

17.

ϑ2 (x+y) ϑ2 (x−y) ϑ22 (0) = ϑ22 (x) ϑ22 (y)−ϑ21 (x) ϑ21 (y) = ϑ23 (x) ϑ23 (y)−ϑ24 (x) ϑ24 (y)

LW 10(1.4.31)

18.

ϑ3 (x+y) ϑ3 (x−y) ϑ22 (0)

ϑ22 (x) ϑ23 (y)+ϑ21 (x) ϑ24 (y)

LW 10(1.4.32)

19.

ϑ4 (x+y) ϑ4 (x−y) ϑ22 (0) = ϑ24 (x) ϑ22 (y)+ϑ23 (x) ϑ21 (y) = ϑ21 (x) ϑ23 (y)+ϑ22 (x) ϑ24 (y)

LW 10(1.4.33)

20.

ϑ23 (x) ϑ23 (0)

+

ϑ22 (x) ϑ22 (0)

LW 11(1.4.49)

21.

ϑ24 (x) ϑ23 (0)

+

ϑ23 (x) ϑ24 (0)

LW 11(1.4.50)

22.

ϑ24 (x) ϑ22 (0) = ϑ21 (x) ϑ23 (0) + ϑ22 (x) ϑ24 (0)

LW 11(1.4.51)

23.

ϑ23 (x) ϑ22 (0)

LW 11(1.4.52)

24.8

ϑ43 (x) = ϑ42 (0) + ϑ44 (0)

8.199(2)10 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

=

=

ϑ24 (x) ϑ24 (0)

=

ϑ21 (x) ϑ22 (0)

=

ϑ21 (x) ϑ24 (0)

ϑ23 (x) ϑ22 (y)+ϑ24 (x) ϑ21 (y)

+

=

ϑ22 (x) ϑ23 (0)

Derivatives of ratios of theta functions

d (ϑ1 / ϑ4 ) = ϑ24 (0) ϑ2 (x) ϑ3 (x)/ ϑ24 (x) dx d (ϑ2 / ϑ4 ) = − ϑ23 (0) ϑ1 (x) ϑ3 (x)/ ϑ24 (x) dx d (ϑ3 / ϑ4 ) = − ϑ22 (0) ϑ1 (x) ϑ2 (x)/ ϑ24 (x) dx d (ϑ1 / ϑ3 ) = ϑ23 (0) ϑ2 (x) ϑ4 (x)/ ϑ23 (x) dx d (ϑ2 / ϑ3 ) = − ϑ24 (0) ϑ1 (x) ϑ4 (x)/ ϑ23 (x) dx d (ϑ1 / ϑ2 ) = ϑ22 (0) ϑ3 (x) ϑ4 (x)/ ϑ22 (x) dx d (ϑ4 / ϑ1 ) = − ϑ24 (0) ϑ2 (x) ϑ3 (x)/ ϑ21 (x) dx d (ϑ4 / ϑ2 ) = ϑ23 (0) ϑ1 (x) ϑ3 (x)/ ϑ22 (x) dx d (ϑ4 / ϑ3 ) = ϑ22 (0) ϑ1 (x) ϑ2 (x)/ ϑ23 (x) dx d (ϑ3 / ϑ1 ) = − ϑ23 (0) ϑ2 (x) ϑ4 (x)/ ϑ21 (x) dx d (ϑ3 / ϑ2 ) = ϑ24 (0) ϑ1 (x) ϑ4 (x)/ ϑ22 (x) dx d (ϑ2 / ϑ1 ) = − ϑ22 (0) ϑ3 (x) ϑ4 (x)/ ϑ21 (x) dx

8.199(3)10

LW 11(1.4.53)

Derivatives of theta functions

1.

∞  q 2n d ln ϑ1 (u) = cot u + 4 sin 2u du 1 − 2q 2n cos 2u + q 4n n=1

2.

∞  d q 2n ln ϑ2 (u) = − tan u − 4 sin 2u du 1 + 2q 2n cos 2u + q 4n n=1

LW 19(1.9.3) LW 19(1.9.6) LW 19(1.9.7) LW 19(1.9.8) LW 19(1.9.9) LW 19(1.9.10) LW 19(1.9.11) LW 20(1.9.12) LW 20(1.9.13) LW 20(1.9.14) LW 20(1.9.15) LW 20(1.9.16)

892

The Exponential Integral Function and Functions Generated by It

3.12

∞  d q 2n−1 ln ϑ3 (u) = −4 sin 2u 2n−1 du 1 + 2q cos 2u + q 4n−2 n=1

4.

∞  q 2n−1 d ln ϑ4 (u) = 4 sin 2u du 1 − 2q 2n cos 2u + q 4n−2 n=1

5.

∞  d2 ln ϑ (u) = − sech2 {i(u + nπτ )} 2 du2 n=−∞

8.199(4)

8.199(4) Relations among theta function arguments   2  z 1 z ϑ1 −  − 1.∗ (−iτ )1/2 ϑ1 (z|τ ) = −i exp iπτ τ τ   2  z 1 z ϑ4 −  − (−iτ )1/2 ϑ2 (z|τ ) = exp 2.∗ iπτ τ τ   2  z 1 z ϑ3 −  − (−iτ )1/2 ϑ3 (z|τ ) = exp 3.∗ iπτ τ τ   2  z  1 z 1/2 ∗ ϑ2 −  − (−iτ ) ϑ4 (z|τ ) = exp 4. iπτ τ τ

8.2 The Exponential Integral Function and Functions Generated by It 8.21 The exponential integral function Ei(x) 8.211 1. 2.11





∞ −t

x

et dt = li (ex ) [Re x > 0] −x t −∞ t  −ε −t  ∞ −t   x t e e e dt + dt = PV dt Ei(x) = − lim ε→0+ t t −x ε −∞ t Ei(x) = −

e

dt =

[x > 0] 3.7 8.212 1.8

2.7 3. 4.

Ei(x) =

1 2

{Ei(x + i0) + Ei(x − i0)} 

x −t

−1 dt 0  t x = C + e−x ln x + e−t ln t dt

Ei(−x) = C + ln x +

ET I 386

[x > 0]

NT 11(1)

[x > 0]

NT 11(10)

0

  ∞ −t e dt 1 + 2 x 0 (x − t)    ∞ −t e dt 1 −x Ei(−x) = e − + 2 x 0 (x + t)  1 dt Ei (±x) = ±e±x x ± ln t 0

Ei(x) = ex



e

[x > 0]

[x > 0]

(cf. 8.211 1)

[x > 0]

(cf. 8.211 1)

[x > 0]

(cf. 8.211 1)

LA 281(28)

8.214

5. 6. 7.8 8.

9. 10. 11. 12. 13. 14. 15. 16.

The exponential integral function Ei(x)

Ei (±xy) = ±e

±xy

Ei (±x) = −e±x 





0 ∞

0

∞

e−xt dt y∓t

e−it dt t ± ix

[x > 0]

ty−1 dt 0 x + ln t  1 y−1 t Ei(−xy) = −e−xy dt x 0  − ln t   1 tx−1 −1 −xy −1 =x e 2 dt − y 0 (y − ln t)  ∞ 1 dt x Ei(x) = e 2 1 x − ln t t  ∞ 1 dt Ei(−x) = −e−x 2 1 x + ln t t  ∞ t cos t + x sin t Ei(−x) = −e−x dt t2 + x2 0  ∞ t cos t − x sin t −x Ei(−x) = −e dt t2 + x2 0  t 2 ∞ cos t arctan dt Ei(−x) = π 0 t x −x  ∞ x cos t − t sin t 2e Ei(−x) = ln t dt π 0 t2 + x2  2ex ∞ x cos t + t sin t Ei(x) = 2 ln x − ln t dt π 0 t2 + x2  ∞ Ei(−x) = −x e−tx ln t dt

Ei(xy) = exy

[Re y > 0,

893

x > 0]

NT 19(11) NT 23(2, 3)

1

LA 282(44)a LA 282(45)a

[x > 0,

y > 0]

LA 283(47)a

[x > 0]

LA 283(48)

[x > 0]

LA 283(48)

[x > 0]

NT 23(6)

[x < 0]

NT 23(6)

[Re x > 0]

NT 25(13)

[x > 0]

NT 26(7)

[x > 0]

NT 27(8)

[x > 0]

NT 32(12)

1

See also 3.327, 3.881 8, 3.916 2 and 3, 4.326 1, 4.326 2, 4.331 2, 4.351 3, 4.425 3, 4.581. For integrals of the exponential integral function, see 6.22–6.23, 6.78. Series and asymptotic representations 8.213 1.

li(x) = C + ln (− ln x) +

∞ k  (ln x) k=1

2.

li(x) = C + ln ln x +

∞ k  (ln x) k=1

8.214 1.

k · k!

Ei(x) = C + ln(−x) +

k · k!

∞  xk k · k! k=1

[0 < x < 1] [x > 1]

[x < 0]

NT 3(9)

NT 3(10)

894

The Exponential Integral Function and Functions Generated by It

Ei(x) = C + ln x +

2.

∞  xk k · k!

8.215

[x > 0]

k=1

Ei(x) − Ei(−x) = 2x

3.

∞  k=0

8.215

7

x2k (2k + 1)(2k + 1)! 

 n ez  k! Ei(z) = + Rn (z) z zk

[x > 0]

  |Rn (z)| = O |z|−n−1

k=0

−n−1

[Re z ≤ 0] |arg(−z)|  ≤ π − δ; δ > 0 small] |Rn (z)| ≤ (n + 1)!|z| , 1 1 kn nx + Ei(nx) − Ei(−nx) = e + 3 3 , nx n2 x2 n x where x = x sign Re(x), kn = O(1), and n → ∞ NT 39(15) [z → ∞,

8.2167

NT 39(13)

8.217 Functional relations:

 ∞    x sin t ex Ei (−x ) − e−x Ei (x ) = −2 NT 24(11) dt 2 2  0∞ t + x  x cos t 4 [x = x sign Re x] NT 27(9) = ln t dt − 2e−x ln x 2 2 π 0 t +x   ∞   t cos t 4 ∞ t sin t −x  ex Ei (−x ) + e−x Ei (x ) = −2 dt = 2e ln x − ln t dt 2 2 π 0 t2 + x2 0 t +x NT 24(10), NT 27(10) [x = x sign Re x]     ∞ 1 t x− x 2 1 cos t = arctan Ei(−x) − Ei − dt x π 0 t 1 + t2

1.

2.

3.

Ei(−αx) Ei(−βx) − ln(αβ) Ei[−(α + β)x] = e−(α+β)x

4.

 0

[Re x > 0] ∞ −tx

e

ln[(α + t)(β + t)] dt t+α+β

NT 25(14) NT 32(9)

See also 3.723 1 and 5, 3.742 2 and 4, 3.824 4, 4.573 2. • For a connection with a confluent hypergeometric function, see 9.237. • For integrals of the exponential integral function, see 5.21, 5.22, 5.23, 6.22, and 6.23. 8.218 Two numerical values: 1.

Ei(−1) = −0.219 383 934 395 520 273 665 . . .

NT 89

2.

Ei(1) = 1.895 117 816 355 936 755 478 . . .

NT 89

8.219∗ 1.∗ 2.∗ 3.∗

Definite integrals of exponential functions  ∞ π2 Ei2 (x)e−2x dx = 4 0  ∞ π2 Ei2 (−x)e2x dx = 4 0  ∞ Ei(x) Ei(−x) dx = 0 0

8.234

The sine integral and cosine integrals

895

8.22 The hyperbolic sine integral shi x and the hyperbolic cosine integral chi x 8.221 1. 2.11



π  sinh t dt = −i + si(ix) t 2 0  x cosh t − 1 dt chi x = C + ln x + t 0

shi x =

x

(see 8.230 1)

EH II 146(17) EH II 146(18)

8.23 The sine integral and the cosine integral: si x and ci x 8.230 1.10 2.10



sin t π dt = − + Si(x), where Si(x) = t 2 x  x  ∞ cos t cos t − 1 dt = C + ln x + dt ci(x) = − t t x 0

8.231 1.

∞

si(x) = −

 si(xy) = −

∞

x

 2.

ci(xy) = − 

3.

si(x) = −

∞

x π/2



x 0

sin t dt t

NT 11(3)

[ci(x) is also written Ci(x)]

NT 11(2)

sin ty dt t

NT 18(7)

cos ty dt t

NT 18(6)

e−x cos t cos (x sin t) dt

NT 13(26)

0

8.232 1.

∞

si(x) = −

π  (−1)k+1 x2k−1 + 2 (2k − 1)(2k − 1)!

NT 7(4)

k=1

2.7

∞ 

ci(x) = C + ln(x) +

(−1)k

k=1

x2k 2k(2k)!

NT 7(3)

8.233 2.

ci(x) ± i si(x) = Ei (±ix)   ci(x) − ci xe±πi = ∓πi

NT 7(5)

3.

si(x) + si(−x) = −π

NT 7(7)

1.

8.234 1.7

 Ei(−x) − ci(x) =

π/2

0

2.

2

NT 6a

e−x cos ϕ sin (s sin ϕ) dϕ



2

[ci(x)] + [si(x)] = −2 0

π/2

exp (−x tan ϕ) ln cos ϕ dϕ sin ϕ cos ϕ

NT 13(27)

[Re x > 0]

(see also 4.366) NT 32(11)

See also 3.341, 3.351 1 and 2, 3.354 1 and 2, 3.721 2 and 3, 3.722 1, 3, 5 and 7, 3.723 8 and 11, 4.338 1, 4.366 1.

896

The Exponential Integral Function and Functions Generated by It

8.235

8.235 1.12

lim (xρ si(x)) = 0,

x→+∞

lim si(x) = −π,

2.

x→−∞

lim (xρ ci(x)) = 0

x→+∞

[ρ < 1]

lim ci(x) = ±πi

NT 38(5) NT 38(6)

x→−∞

• For integrals of the sine integral and cosine integral, see 6.24–6.26, 6.781, 6.782, and 6.783. • For indefinite integrals of the sine integral and cosine integral, see 5.3.

8.24 The logarithm integral li(x) 8.240



x

dt = Ei (ln x) [x < 1] ln t 0   1−ε  x dt dt + = Ei (ln x) [x > 1] li(x) = lim ε→0 ln t 0 1+ε ln t $   #  li exp −xe±πi = Ei −xe±iπ = Ei (x ∓ i0) = Ei(x) ± iπ = li (ex ) ± iπ

1.

li(x) =

2. 3.

[x > 0]

JA

JA

JA, NT 2(6)

Integral representations 8.241 1.12



ln x t

li(x) = −∞

 li(x) = x

2.

3.12

e 1 dt = x ln ln − t x



∞ − ln x

e−t ln t dt

[0 < x < 1]

LA 281(33)

1

dt ln x + ln t 0  1 dt x +x = 2 ln x 0 (ln x + ln t)  ∞ 1 dt =x 2 ln x − ln t t 1  x t a dt li (ax ) = −∞ t

LA 280(22) LA 280(29)

[x < 1]

LA 280(30)

[x > 0]

For integrals of the logarithm integral, see 6.21

8.25 The probability integral Φ(x), the Fresnel integrals S (x), C (x), the error function erf (x), and the complementary error function erfc(x) 8.250 Definition: 11

1. 2.

 x 2 2 Φ(x) = erf(x) = √ e−t dt π 0  x 2 sin t2 dt S (x) = √ 2π 0

(called the error function)

8.252

3. 4.11 5.

6.

7. 8.

9.

The probability integral, Fresnel integrals and error functions



2 C (x) = √ 2π

x

cos t2 dt

0

erfc(x) = 1 − erf(x) (called the complementary error function)  ∞ −(p+x)y  √  e sin a x dx 0 π(p + x)   √ √ √ 1 −a√p a 1 a√p a = − sinh (a p) + e Φ Φ √ − py + e √ + py 2 2 y 2 2 y  2 √   ∞ −(p+x)y  √  p −a√p e a √ 1 a cos a x dx = √ exp − − py − e Φ √ − py πy a y 0 π(p + x)  4y 2 √ p √p √ a √ √ e Φ + √ + py − p cosh (a p) 2 2 y [Re p > 0, a, b are real] 2 √    p  p     π p Φ √ exp −x2 Φ(p − x) dx = exp −x2 erf(p − x) dx = 2 2 0 0  p  p     x2 exp −x2 Φ(p − x) dx = x2 exp −x2 erf(p − x) dx 0  2  2 √0   x p π p p Φ √ erf √ = − √ Φ − 4 2 2 2 2 2 √ √  (a−b)/ 2  (b−a)/ 2    2  √  2  √ √ Φ b Φ a exp −x 2 − x dx + exp −x 2 − x dx = π Φ(a) Φ(b) √ √ (a+b)/ 2

(a+b)/ 2

Integral representations 8.251 1. 2. 3.

1 Φ(x) = √ π



1 S (x) = √ 2π

2. 3.

2y Φ(xy) = √ π

x2 −t

e √ dt t

0



1 C (x) = √ 2π

8.252 1.

897

(see also 3.361 1)

x2

sin t √ dt t

x2

cos t √ dt t

0



0

 0

x

2 2

e−t

y

dt

 x   2y sin t2 y 2 dt S (xy) = √ 2π 0  x   2y C (xy) = √ cos t2 y 2 dt 2π 0

898

The Exponential Integral Function and Functions Generated by It

4.

5.12

6.8

 ∞ −t2 y2  2  2 2 e ty dt 2 Re y > 0 Φ(xy) = 1 − √ e−x y √ π t2 + x2 0 2 2  2  2x −x2 y2 ∞ e−t y dt Re y > 0 e =1− 2 2 π t +x 0  2 2  ∞   2 2 y 4xiey /4x −y √ −Φ = Φ e−t y sin(ty) dt 2xi 2xi π 0  ∞ y  2 2 2 −y 2 /4x2 = 1 − √ xe Φ e−t x −ty dt 2x π 0

NT 19(11)a NT 19(13)a

  Re x2 > 0

NT 28(3)a

  Re x2 > 0

NT 27(1)a

See also 3.322, 3.362 2, 3.363, 3.468, 3.897, 6.511 4 and 5. 8.2538 Series representations:  ∞ x2k−1 3 2 2  2 −x2 11 1. erf(x) = √ e x F 1 1; ; x = √ (−1)k+1 2 (2k − 1)(k − 1)! π π k=1 ∞ k 2k+1  2 2 x 2 = √ e−x π (2k + 1)!! k=0    2 3 2 5 3 1 2 7 5 1 2 2 2 12 x sin x F 1, ; ; − x − x cos x F 1, ; ; − x 2. S (x) = √ 4 4 4 3 4 4 4 2π ∞ 2  (−1)k x4k+3 √ 2π k=0 (2k + 1)!(4k + 3) 

∞ ∞   (−1)k 22k x4k+1 (−1)k 22k+1 x4k+3 2 2 2 − cos x sin x =√ (4k + 1)!! (4k + 3)!! 2π k=0 k=0    2 7 5 1 2 5 3 1 2 2 3 2 2 12 3. C (x) = √ x sin x F 1, ; ; − x − x cos x F 1, ; ; − x 4 4 4 4 4 4 2π 3 ∞ k 4k+1  (−1) x 2 = √ 2π k=0 (2k)!(4k + 1) 

∞ ∞   2 (−1)k 22k+1 x4k+3 (−1)k 22k x4k+1 2 2 + cos x sin x =√ (4k + 3)!! (4k + 1)!! 2π k=0

8.253

NT 7(9)a

NT 10(11)a

NT 8(14)a

NT 10(13)a

NT 8(13)a

NT 10(12)a

k=0

For the expansions in Bessel functions, see 8.515 2, 8.515 3. Asymptotic representations  n    −z 2  (2k − 1)!! e −2n−2 k , (−1) + O |z| 8.25412 Φ(z) = 1 − √ k πz (2z 2 ) k=0

[z → ∞, where |Rn |
0 small] NT 37(10)a

[x → ∞]

MO 127a

8.258

2.

12

The probability integral, Fresnel integrals and error functions

1 1 sin2 x + O C (x) = + √ 2 2πx



1 x2

[x → ∞]

8.256 Functional relations:    z 2 i z 2 Φ √ =√ eit dt 1. C (z) + i S (z) = 2 2π 0 i  z  √ 2 2 1 e−it dt 2. C (z) − i S (z) = √ Φ z i = √ 2i 2π 0 12

3.

4.12



 1 2  cos u + sin2 u + cos u C (u) + sin u S (u) = 2 2

2

5.11

C (x) −

1 2

2

 + S (x) −

1 2

2

[Re u ≥ 0]  ϕ√ 2 π/2 exp −x2 tan ϕ sin 2 cos ϕ dϕ = π 0 sin 2ϕ 

MO 127a

  ∞ 2 e−2ut sin2 t dt π 0

[Re u ≥ 0]   ∞  2  1 2  2 cos u − sin2 u − cos u S (u) − sin2 u C (u) = e−2ut cos2 t dt 2 π 0 

899



(see also 6.322)

NT 28(6)a

NT 28(5)a

NT 33(18)a

• For a connection with a confluent hypergeometric function, see 9.236. • For a connection with a parabolic cylinder function, see 9.254. 8.257 1. 2.

  xρ S (x) − 12 = 0    lim xρ C (x) − 12 = 0 lim



x→+∞ x→+∞

[ρ < 1]

NT 38(11)

[ρ < 1]

NT 38(11)

1 x→+∞ 2 1 lim C (x) = x→+∞ 2

3.

lim S (x) =

4.

NT 38(12)a NT 38(12)a

• For integrals of the probability integral, see 6.28–6.31. • For integrals of Fresnel’s sine integral and cosine integral, see 6.32. 8.25810 1.

Integrals involving the complementary error function     ∞ 2 1 1 + 2 arctan − arccos erfc2 (x)e−βx dx = √ β 1+β βπ 0



 2. 0

∞

2

x erfc2 (x)e−βx dx =

1 2β





√ 4 arctan 1 + β √ 1− π 1+β

[β > 0]

[β > 0]

900

The Exponential Integral Function and Functions Generated by It

4. 5.12





√ 4 arctan 1 + β √ 1− x erfc (x)e π 1+β 0 √ 

arctan 1 + β 1 1 − + 3 βπ (1 + β) (β 2 + 2β + 2) (1 + β) 2 [β > 0]    ∞ √  1 + 32 β 1 [β > 0] x erfc x e−βx dx = 2 1 − 3 β (1 + β) 2 0

√   ∞ √  −βx √ 1 arctan β 1 1 x erfc x e dx = √ − 3 π β(1 + β) β2 0 

3.

8.259

∞

3

−βx2

2

1 dx = 2β 2

[β > 0] 8.259∗ 1.

2.

Integrals involving the error function and an exponential function  ∞  √

a p π −px2 [Re p > 0] , a, b real Φ  e Φ(a + bx) dx = p b2 + p −∞    ∞ √

a p ab2 1 π a2 p 2 −px2 − Φ  x e Φ(a + bx) dx = exp − 2p p b2 + p b2 + p p (b2 + p)3/2 −∞ 

∞

3. −∞

2

∂n ∂pn

n

x2n e−px Φ(a + bx) dx = (−1)



π Φ p



[Re p > 0, √  a p  b2 + p

a, b are real]

[n = 0, 1, . . . ,

Re p > 0,

a, b are real]

8.26 Lobachevskiy’s function L(x) 8.260 Definition:

 L(x) = −

x

ln cos t dt

LO III 184(10)

0

For integral representations of the function L(x), see also 3.531 8, 3.532 2, 3.533, and 4.224. 8.261 Representation in the form of a series: ∞ 1 sin 2kx L(x) = x ln 2 − (−1)k−1 LO III 185(11) 2 k2 k=1

8.262 Functional relationships: 1.

L(−x) = − L(x)

2.

L(π − x) = π ln 2 − L(x)

3.

L(π + x) = π ln 2 + L(x)     L(x) − L π2 − x = x − π4 ln 2 −

4.

 π −2 ≤ x ≤

π 2



LO III 185(13) LO III 286

1 2

L

π 2

 − 2x

 0≤x
0]

(Euler)

FI II 777(6)

0

2.

Generalization:  1 Γ(z) = − (−t)z−1 e−t dt 2i sin πz C for z not an integer. The contour C is shown in the drawing.

WH

Γ(z) is an analytic function z with simple poles at the points z = −l (for l = 0, 1, 2,. . . ) to which (−1)l . Γ(z) satisfies the relation Γ(1) = 1. correspond to residues WH, MO 1 l! Integral representations 8.311 Γ(z) = 8.312 1.

e2πiz − 1 

1

Γ(z) = 0

2.

z



∞

Γ(z) = x

4.

5.

Γ(z) =

Γ(z) =

(0+)

∞

e−t tz−1 dt

[Re z > 0]

e−xt tz−1 dt

a

2az e sin πz

∞

2

e−at



1 + t2

[Re z > 0, z− 12

1 2 sin πz

Γ(y) = xy e−iβy

Γ(z) =

bz 2 sin πz

FI II 778

Re x > 0]

FI II 779(8)

cos [2at + (2z − 1) arctan t] dt

0



[a > 0] ∞

WH

 z 2 e−t tz−1 1 + t2 2 {3 sin [t + z arccot(−t)] + sin [t + (z − 2) arccot(−t)]} dt

0



[arccot denotes an obtuse angle] ∞

  ty−1 exp −xte−iβ dt

0

6.

MO 2

z−1  1 ln dt t 0

3.



1



∞

−∞

ebti (it)z−1 dt

 x, y, β real, [b > 0,

x > 0,

y > 0,

0 < Re z < 1]

|β|
0,

0 < Re z < 1]

Re z > −1]

NH 152(5)

[Re z > 0]

NH 173(7)

[Re z > 0]

NH 145(14)

 λ > 0,

0

12.

b ≥ 0,

  exp zt − et dt

11.12 Γ(z) cos αz = λz

12

[a > 0,

e−t (t − z)tz−1 ln t dt

0

10.

8.313

⎢e ⎢ ⎢ ⎢ ⎣

−t

= vu

−

n 

kt

(−1)

k=0 tz+1

z+1 v



∞

k

 λ > 0,

Re z > 0,

−

π π −1,

−

π π 0,

MO 2

Re v > 0,

Re z > −1]

0

JA, MO 7a ∞

e−t tz−1 dt +

∞ 

(−1)k k!(z + k) n=0

 1 i = (−t)−z e−t dt Γ(z) 2π C  ∞ ebti 2πe−ab bz−1 dt = z Γ(z) −∞ (a + it)  ∞ −bti  e dt = 0 Re a > 0, z (a + it) −∞

[For the contour C see 8.310 2]

b < 0,

Re z > 0,

|arg(a + it)| < 12 π



8.325

Representation of the gamma function as series and products

ea 1 = a1−z Γ(z) π

3.



π/2

cos (a tan θ − zθ) cosz−2 θ dθ

903

[Re z > 1]

NH 157(14)

0

See also 3.324 2, 3.326, 3.328, 3.381 4, 3.382 2, 3.389 2, 3.433, 3.434, 3.478 1, 3.551 1, 2, 3.827 1, 4.267 7, 4.272, 4.353 1, 4.369 1, 6.214, 6.223, 6.246, 6.281.

8.32 Representation of the gamma function as series and products 8.321 Representation in the form of a series: 1.6

Γ(z + 1) = 

∞ 

ck z k

k=0

n

c0 = 1,

cn+1 =

k+1 sk+1 cn−k k=0 (−1)

n+1

 ;

s1 = C,

sn = ζ(n) for n ≥ 2,

|z| < 1 NH 40(1, 3)

∞ 

1 dk z k = Γ(z + 1) n  k=0 (−1)k sk+1 dn−k ; d0 = 1, dn+1 = k=0 n+1

2.11

 s1 = C,

sn = ζ(n) for n ≥ 2

NH 41(4, 6)

Infinite-product representation ∞

1  ez/k [Re z > 0] z 1+ z k=1  k  z ∞ 1  1 + k1 = [Re z > 0] z 1 + kz k=1 n nz  k = lim [Re z > 0] n→∞ z z+k k=1 ∞     2k z −z 7 8.323 Γ(z) = 2z e B 2k−1 z, 12 k=1  z 1 + ∞ Γ  2 2k √ 8.3247 Γ(1 + z) = 4z π k=1 8.322 Γ(z) = e−Cz 11

8.325 1. 2.11 3.7

  ∞   γ Γ(α) Γ(β) γ = 1− 1+ Γ(α + γ) Γ(β − γ) α+k β+k k=0  ∞   eCx Γ(z + 1) x = ex/k 1− [z = 0, −1, −2, . . . ; Γ(z − x + 1) z+k k=1  √ ∞   z  z π     1 + 1 − = 2k − 1 2k Γ 1 + z2 Γ 12 − z2 k=1

SM 269

WH

SM 267(130)

NH 98(12)

MO 3

NH 62(2)

Re z > 0,

Re(z − x) > 0]

MO 2

904

Euler’s Integrals of the First and Second Kinds

8.326

8.326 1.12

[Γ(x)]2   ∞   Γ(x) 2  y2 Γ(2x)  = =  1 + B(x + iy, x − iy) Γ(x + iy)  (x + k)2 k=0

2.11

−iCy

Γ(x + iy) xe = Γ(x) x + iy

∞  n=1

exp 1+

 iy  n iy x+n

[x, y are real,

x = 0, −1, −2, . . .] LO V, NH 63(4)

[x, y are real,

x = 0, −1, −2, . . .] MO 2

8.327 Asymptotic representation for large arguments:   √  −5  1 1 139 571 z− 12 −z + 1. Γ(z) ∼ z e 2π 1 + − − +O z 12z 288z 2 51840z 3 2488320z 4 [|arg z| < π]

2.

3.

For z real and positive, the remainder of the series is less than the last term that is retained.  n n  n n √ √ n! ∼ 2πn or equivalently Γ(n + 1) ∼ 2πn e e [n → ∞] AS 6.1.38  1 1 1 1 1 1 ln z − z + ln(2π) + − ln Γ(z) ∼ z − + − + ... 2 2 12z 360z 3 1260z 5 1680z 7 [z → ∞,

8.328 1. 2.

π

1

lim |Γ(x + iy)|e 2 |y| |y| 2

−x

|y|→∞

lim

|z|→∞

=

√ 2π

Γ(z + a) −a ln z e =1 Γ(z)

8.331 1.

Γ(x + 1) = x Γ(x)

2.

Γ(x + a) = (x + a − 1) Γ(x + a − 1) =

Γ(x + a + 1) (x + a)

Γ(x − a) = (x − a − 1) Γ(x − a − 1) =

Γ(x − a + 1) (x − a)

|arg z| < π]

[x and y are real]

8.33 Functional relations involving the gamma function

3.

WH

AS 6.1.38

MO 6 MO 6

8.335

8.332

Functional relations involving the gamma function

π y sinh πy  1  π Γ + iy 2 = 2 cosh πy 2

|Γ(iy)| =

1. 2.

Γ(1 + ix) Γ(1 − ix) =

[y is real]

4.

πx sinh xπ

[x and y are real] n

8.333 [Γ(n + 1)] = G(n + 1)

MO 3

[y is real]

[x is real]   2π 2 x2 + y 2 Γ(1 + x + iy) Γ(1 − x + iy) Γ (1 + x − iy) Γ(1 − x − iy) = cosh 2yπ − cos 2xπ

3.

905

n 

LO V

LO V

kk ,

k=1

where n is a natural number and   ∞    z z n z(z + 1) C 2   z2 2 − z 1+ exp −z + G(z + 1) = (2π) exp − 2 2 n 2n n=1

8.334 1. 2. 3.

 z n   1 n  1 − = −z k Γ −z exp 2πki n k=1 k=1 1  1  π Γ 2 +x Γ 2 −x = cos πx π Γ(1 − x) Γ(x) = sin πx n 

WH

∞



[n = 2, 3 . . .]

MO 2

FI II 430

Special cases 8.3357

Γ(nx) = (2π)

1−n 2

1

nnx− 2

n−1  k=0

 k Γ x+ n

1.∗

  22x−1 Γ(2x) = √ Γ(x) Γ x + 12 π

2.

Γ(3x) =

3.

4.10

[product theorem]

[doubling formula]

1     33x− 2 Γ(x) Γ x + 13 Γ x + 23 2π n−1  k  k (2π)n−1 Γ 1− = Γ n n n k=1   ∞  Γ2 n − 12 1 1 1 25 1 1 2 2  1  = 4 + 16 + 256 + 1024 + 65536 + · · · = π − n=0 4 (n!) Γ

2

FI II 782a, WH

WH

906

8.336

Euler’s Integrals of the First and Second Kinds

8.336

 ∞   yz + xi yz − xi z+1 Γ(1 + z) = (2i) y Γ 1 + Γ − e−tx sinz (ty) dt 2y 2y 0 [Re(yi) > 0, Re(x − yzi) > 0]

12

NH 133(10)

• For a connection with the psi function, 8.361 1. • For a connection with the beta function, see 8.384 1. • For integrals of the gamma function, see 8.412 4, 8.414, 9.223, 9.242 3, 9.242 4. 8.337 1. 2.



2 Γ (x) < Γ(x) Γ (x)

[x > 0]

For x > 0, min Γ(1 + x) = 0.88560 . . . is attained when x = 0.46163 . . .

MO 1 JA

Particular values 8.338 1. 2. 3. 4.

Γ(1) = Γ(2) = 1   √ Γ 12 = π   √ Γ − 12 = −2 π     4 ∞  (4k − 1)2 (4k + 1)2 − 1 1 2 = 16π Γ 4 [(4k − 1)2 − 1] (4k + 1)2

MO 1a

k=1

5.

8  k=1

 3  640 π k = 6 √ Γ 3 3 3

WH

8.339 For n a natural number 1. 2.12 3. 4. 5.

Γ(n) = (n − 1)! √ √   π π (2n)! 1 Γ n + 2 = n (2n − 1)!! = 2n 2 2 n! √ n   2 π Γ 12 − n = (−1)n (2n − 1)!!  2       4p − 12 4p2 − 32 . . . 4p2 − (2n − 1)2 Γ p + n + 12 =  22n Γ p − n + 12 Γ(n + k) = (n + k − 1)! =

6.

Γ(n + k + 1) (n + k)

[n + k ≥ 0, 1, . . .]

Γ(n − k) = (n − k − 1)! =

Γ(n − k + 1) (n − k)

[n − k ≥ 0, 1, . . .]

WA 221

8.343

The logarithm of the gamma function

907

8.34 The logarithm of the gamma function 8.341 Integral representation:  −tz  ∞ 1 1 1 e 1 1 1. ln Γ(z) = z − ln z − z + ln 2π + − + t dt 2 2 2 t e −1 t 0 

2.11

3. 4.

5.

7.

WH

√ arctan 1 dt ln Γ(z) = z ln z − z − ln z + ln 2π + 2 2 e2πt − 1 0    w du Re z > 0 and arctan w = is taken over a rectangular path in the w-plane 2 0 1+u   ∞  −zt − e−t e dt −t [Re z > 0] ln Γ(z) = + (z − 1)e −t 1−e t 0   ∞ (1 + t)−z − (1 + t)−1 dt (z − 1)e−t + ln Γ(z) = ln(1 + t) t 0

ln Γ(x) =

ln π − ln sin πx 1 + 2 2

 0

∞

t z

[Re z > 0]   sinh 2 − x t dt − (1 − 2x)e−t t t sinh 2 1

 dt tz − t − t(z − 1) ln Γ(z) = t − 1 t ln t 0   ∞ −tz −t −e dt e −t (z − 1)e + ln Γ(z) = −t 1 − e t 0 

6.

[Re z > 0] ∞

1



WH

WH

WH

[0 < x < 1]

WH

[Re z > 0]

WH

[Re z > 0]

NH 187(7)

See also 3.427 9, 3.554 5. 8.342 Series representations: 1.11

ln Γ(z + 1)    ∞  1+z 1 1 − ζ(2k + 1) 2k+1 πz  ln − ln + (1 − C) z + = z 2 sin πz 1−z 2k + 1 k=1 ∞  zk = −Cz + (−1)k ζ(k) k

[|z| < 1] NH 38(16, 12)

k=2

2.

ln Γ(1 + x) =

∞  x2n+1 πx 1 ln − Cx − ζ(2n + 1) 2 sin πx 2n + 1 n=1

[|x| < 1] 8.343 1.

∞  √ ln Γ(x) = ln 2π + n=1



NH 38(14)

 1 1 cos 2nπx + (C + ln 2nπ) sin 2nπx 2n nπ [0 < x < 1]

FI III 558

908

Euler’s Integrals of the First and Second Kinds

ln Γ(z) = z ln z − z −

2.

8.344

∞ ∞  √ m 1 1 1  ln z + ln 2π + 2 2 m=1 (m + 1)(m + 2) n=1 (z + n)m+1

[|arg z| < π] 8.3447

MO 9

Asymptotic expansion for large values of |z|: n−1

ln Γ(z) = z ln z − z −

 √ B2k 1 ln z + ln 2π + + Rn (z), 2 2k(2k − 1)z 2k−1 k=1

where |Rn (z)| < For integrals of ln Γ(x), see 6.44.

|B2n | 2n−1 cos2n−1

2n(2n − 1)|z|

1 2

arg z



MO5

8.35 The incomplete gamma function 8.350 Definition:  x e−t tα−1 dt 1. γ(α, x) = 0

 2.11

Γ(α, x) =

∞

x

3.

Γ(z, 0) = Γ(z)

4.

Γ(a, ∞) = 0

5.

γ(a, 0) = 0

8.351

[Re α > 0]

e−t tα−1 dt

EH II 133(1), NH 1(1) EH II 133(2), NH 2(2), LE 339

x−α γ(α, x) is an analytic function with respect to α and x Γ(α)

1.

γ ∗ (α, x) =

2.

Another definition of Γ(α, x), that is also suitable for the case Re α ≤ 0: γ(α, x) =

xα xα −x e Φ (1, 1 + α; x) = Φ(a, 1 + a; −x) α α

EH II 133(5)

EH II 133(3)

3.

For fixed x, Γ(α, x) is an entire function of α. For non-integral α, Γ(α, x) is a multiple-valued function of x with a branch point at x = 0.

4.

A second definition of Γ(α, x): Γ(α, x) = xα e−x Ψ(1, 1 + α; x) = e−x Ψ(1 − α, 1 − α; x)

8.352 Special cases: 1.





γ(1 + n, x) = n! 1 − e

−x

n  xm m! m=0

EH II 133(4)

 [n = 0, 1, . . .] EH II 136(17, 16), NH 6(11)

2.

Γ(1 + n, x) = n!e−x

n 

m

x m! m=0

[n = 0, 1, . . .]

EH II 136(16, 18)

8.354

The incomplete gamma function

909

m

3.12

Γ(−n, x) =

 xk−n−1 (−1)n+1 Ei(−x) − e−x n! (−n)k

[Re x > 0,

n = 1, 2, . . .]

k=1

n−1 

xm m! m=0 

4.

Γ(n, x) = (n − 1)!e−x

5.12

n−2  m! (−1)n−1 −x Γ(0, x) − e Γ(−n + 1, x) = (−1)m m+1 (n − 1)! x m=0



[n = 1, 2, . . .]

n−1 

xm m! m=0



[n = 2, 3, . . .]



6.

γ(n, x) = (n − 1)! 1 − e−x

7.

  n−k−1  (−1)n−k −z m m! Γ(0, x) − e Γ(−n + k, x) = (−1) m+1 (n − k)! x m=0

[n = 1, 2, . . .]

[n − k ≥ 1, 8.353 Integral representations:  π 1.12 γ(α, x) = xα cosec πα ex cos θ cos (αθ + x sin θ) dθ

[x = 0,

k = 0, 1, . . .]

Re α > 0,

α = 1, 2, . . .]

0

1

γ(α, x) = x 2 α

2.



EH II 137(2) ∞

 √  1 e−t t 2 α−1 J α 2 xt dt

0

3.12

Γ(α, x) =

e−x xα Γ(1 − α)



∞ −t −α

e t dt x+t

0

[Re α > 0] [Re α < 1,

EH II 138(4)

x > 0] EH II 137(3), NH 19(12)

4. 5.

1 2α

−x 

2x e Γ(1 − α)  α −xy Γ(α, xy) = y e Γ(α, x) =

∞

0 ∞

 √  1 e−t t− 2 α K α 2 xt dt

[Re α < 1]

EH II 138(5)

e−ty (t + x)α−1 dt

0

[Re y > 0,

x > 0,

Re α > 1]

(See also 3.936 5, 3.944 1–4)

NH 19(10)

For integrals of the gamma function, see 6.45. 8.354 Series representations: 1.

γ(α, x) =

∞  (−1)n xα+n n!(α + n) n=0

EH II 135(4)

910

2.

Euler’s Integrals of the First and Second Kinds

Γ(α, x) = Γ(α) −

∞  (−1)n xα+n n!(α + n) n=0

8.355

[α = 0, −1, −2, . . .] EH II 135(5), LE 340(2)

3.

Γ(α, x) − Γ (α, x + y) = γ(α, x + y) − γ(α, x) ∞  (−1)k [1 − e−y ek (y)] Γ(1 − α + k) = e−x xα−1 xk Γ(1 − α) k=0

ek (x) =

4.

1

γ(α, x) = Γ(α)e−x x 2 α

n  √  1 (−1)m x 2 n I n+α 2 x m! n=0 m=0 ∞ 

k  xm m! m=0

[x = 0,

[|y| < |x|] α = 0,

EH II 139(2)

−1, −2, . . .] EH II 139(3)

5.12

8.355

Γ(α, x) = e−x xα

∞ 

Lα n (x) n +1 n=0

[x > 0,

Γ(α, x) γ(α, y) = e−x−y (xy)α

α > −1]

EH II 140(5)

∞ 

n! Γ(α) α Lα n (x) Ln (y) (n + 1) Γ(α + n + 1) n=0 [y > 0, x ≥ y,

α = 0, −1, . . .] EH II 139(4)

8.356 Functional relations: 1.11

γ(α + 1, x) = α γ(α, x) − xα e−x

EH II 134(2)

α −x

2.

Γ(α + 1, x) = α Γ(α, x) + x e

EH II 134(3)

3.

Γ(α, x) + γ(α, x) = Γ(α)

EH II 134(1)

4. 5.

d Γ(α, x) d γ(α, x) =− = xα−1 e−x dx dx n−1  xα+s Γ(α, x) Γ(α + n, x) = + e−x Γ(α + n) Γ(α) Γ(α + s + 1) s=0

6.11

Γ(α) Γ(α + n, x) − Γ(α + n) Γ(α, x) = Γ(α + n) γ(α, x) − Γ(α) γ(α + n, x)

7.

Γ(a + k, x) = (a + k − 1) Γ(a + k − 1, x) + xa+k−1 e−x  1  Γ(a + k + 1, x) − xa+k e−x = a+k

8.

Γ(a − k, x) = (a − k − 1) Γ(a − k − 1, x) + xa−k−1 e−x  1  Γ(a − k + 1, x) − xa−k e−x = a−k

9.12

γ(a + k, x) = (a + k − 1) γ(a + k − 1, x) − xa+k−1 e−x  1  γ(a + k + 1, x) + xa+k e−x = a+k

EH II 135(8)

NH 4(3) NH 5

The psi function ψ(x)

8.361

10.

911

γ(a − k, x) = (a − k − 1) γ(a − k − 1, x) − xa−k−1 e−x  1  γ(a − k + 1, x) + xa−k e−x = a−k

8.357 Asymptotic representation for large values of |x|: M−1     (−1)m Γ(1 − α + m) −M α−1 −x + O |x| e 1. Γ(α, x) = x xm Γ(1 − α)   m=0 3π M = 1, 2, . . . |x| → ∞, − 3π 2 < arg x < 2 ,

EH II 135(6), NH 37(7), LE 340(3)

8.358 Representation as a continued fraction: e−x xα Γ(α, x) = 1−α x+ 1 1+ 2−α x+ 2 1+ 3−α x+ 1 + ... 8.359 Relationships with other functions: 1. 2. 3. 4.11

EH II 136(13), NH 42(9)

Γ(0, x) = − Ei(−x)  1 = − li(x) Γ 0, ln x  √  √ Γ 12 , x2 = π − π Φ(x)  √  γ 12 , x2 = π Φ(x)

EH II 143(1) EH II 143(2) EH II 147(2) EH II 147(1)

8.36 The psi function ψ(x) 8.360 Definition: 1.

ψ(x) =

d ln Γ(x) dx

8.361 Integral representations:  ∞  −t e−zt e d ln Γ(z) 1.8 = − dt ψ(z) = dz t 1 − e−t 0   ∞ dt 1 −t e − 2. ψ(z) = z (1 + t) t 0  ∞ t dt 1 −2 3. ψ(z) = ln z − 2 2 2πt − 1) 2z 0 (t + z ) (e  1 tz−1 1 − dt 4. ψ(z) = − ln t 1 − t 0

[Re z > 0]

NH 183(1), WH

[Re z > 0]

NH 184(7), WH

[Re z > 0]

WH

[Re z > 0]

WH

912

Euler’s Integrals of the First and Second Kinds

 5.

ψ(z) = 0

 6.

ψ(z) =

∞ −t

− e−zt dt − C, 1 − e−t

e

∞

#

(1 + t)−1 − (1 + t)−z

0



7. 8.

$ dt − C, t

1 z−1

−1 dt − C t − 1 0    ∞ 1 1 − dt ψ(z) = ln z + e−tz t 1 − e−t 0 ψ(z) =

t

8.362

[Re z > 0]

WH

[Re z > 0]

WH

[Re z > 0]

FI II 796, WH

[Re z > 0]

MO 4

See also 3.244 3, 3.311 6, 3.317 1, 3.457, 3.458 2, 3.471 14, 4.253 1 and 6, 4.275 2, 4.281 4, 4.482 5. For integrals of the psi function, see 6.46, 6.47. Series representation 8.362 1.

2.

1 1 − x+k k+1 k=0 ∞  1 1 = −C − + x x k(x + k) k=1   ∞   1 1 − ln 1 + ψ(x) = ln x − x+k x+k ψ(x) = −C −

∞  

k=0

∞

3.

 π2 (x − 1) − (x − 1) ψ(x) = −C + 6

k=1



1 1 − k+1 x+k

FI II 799(26), KU 26(1)

FI II 495

MO 4

k−1 

1 x+n n=0

NH 54(12)

8.363 1.

ψ(x + 1) = −C +

∞ 

(−1)k ζ(k)xk−1

NH 37(5)

k=2

2.

3.

∞  π x2 1 − cot πx − − C + [1 − ζ(2k + 1)] x2k 2x 2 1 − x2 k=1  ∞  1 1 − ψ(x) − ψ(y) = y+k x+k

ψ(x + 1) =

k=0

4.

5.

(see also 3.219, 3.231 5, 3.311 7, 3.688 20, 4.253 1, 4.295 37)

k=0

NH 99(3)

∞ 

2yi y 2 + (x + k)2 k=0   ∞  q 1 p = −C + − ψ q k + 1 p + kq ψ(x + iy) − ψ(x − iy) =

NH 38(10)

(see also 3.244 3)

NH 29(1)

The psi function ψ(x)

8.365

913

6.8

q+1   2 −1   π pπ kπ 2kpπ p = −C − ln(2q) − cot +2 ln sin cos ψ q 2 q q q k=1 [q = 2, 3, . . . , p = 1, 2, . . . , q − 1]

7.

  ∞  ∞  1 p−1 p −ψ =q ψ n−1 q q (p + kq) n=2

MO 4, EH I 19(29) NH 59(3)

k=0

ψ (n) (x) = (−1)n+1 n!

8.

∞  k=0

1 = (−1)n+1 n! ζ(n + 1, x) (x + k)n+1

NH 37(1)

Infinite-product representation 8.364 ψ(x)

1 1 e− x+k x+k k=0 ∞  y y Γ(x + y)  e− x+k 1+ = Γ(x) x+k

∞   1+ =x

1.

e

2.

ey ψ(x)

NH 65(12)

NH 65(11)

k=0

See also 8.37. • • • •

For a connection with Riemann’s zeta function, see 9.533 2. For a connection with the gamma function, see 4.325 12 and 4.352 1. For a connection with the beta function, see 4.253 1. For series of psi functions, see 8.403 2, 8.446, and 8.447 3 (Bessel functions), 8.761 (derivatives of associated Legendre functions with respect to the degree), 9.153, 9.154 (hypergeometric function), 9.237 (confluent hypergeometric function). • For integrals containing psi functions, see 6.46–6.47. 8.365 Functional relations: 1. 2. 3.

1 ψ(x + 1) = ψ(x) + x  x x+1 −ψ = 2 β(x) ψ 2 2 ψ(x + n) = ψ(x) +

n−1  k=0

4.

JA

(cf. 8.37 0)

1 x+k

GA 154(64)a

n  1 ψ(n + 1) = −C + k

MO 4

k=1

5. 6.

lim [ψ(z + n) − ln n] = 0

MO 3

n→∞

n−1  1 k + ln n ψ(nz) = ψ z+ n n k=0

[n = 2, 3, 4, . . .]

MO 3

914

7.

Euler’s Integrals of the First and Second Kinds

ψ(x − n) = ψ(x) −

n  k=1

8. 9. 10.

8.366

1 x−k

ψ(1 − z) = ψ(z) + π cot πz     ψ 12 + z = ψ 12 − z + π tan πz     ψ 34 − n = ψ 14 + n + π

GA 155(68)a JA

[n = 0,

±1,

±2, . . .]

8.366 Particular values 1. 2. 3.

ψ(1) = −C   ψ 12 = −C − 2 ln 2 = −1.963 510 026 . . .   n  1 1  − ln 2 ψ 2 ± n = −C + 2 2k − 1 1

10. 11.

ψ  (n) =

5. 6. 7. 8. 9.

ψ

4

GA 155a JA

k=1

π − 3 ln 2 2   π ψ 34 = −C + − 3 ln 2 2  1 π 1 3 ψ 3 = −C − − ln 3 2 3 2    π 1 3 ψ 23 = −C + − ln 3 2 3 2 π2 = 1.644 934 066 848 . . . ψ  (1) = 6   π2 = 4.934 802 200 5 . . . ψ  12 = 2 ψ  (−n) = ∞

4.

(cf. 8.367 1)

= −C −

GA 157a GA 157a GA 157a GA 157a JA JA

[n is a natural number]

JA

[n is a natural number]

JA

n   π2 1 −4 +n = 2 (2k − 1)2

[n is a natural number]

JA

n   π2 1 +4 −n = 2 (2k − 1)2

[n is a natural number]

JA

n−1

π2  1 − 6 k2 k=1

12.

13.

ψ ψ

1 2

1 2

k=1

k=1

8.367 Euler’s constant (also denoted by γ): 1. 2. 3.12

C = − ψ(1) = 0.577 215 664 90 . . .  n−1 1 − ln n C = lim n→∞ k k=1   1 C = lim ζ(x) − x→0 x−1

FI II 319, 795 FI II 801a

FI II 804

The function β(x)

8.371

915

Integral representations:  ∞ 4. C=− e−t ln t dt

FI II 807

0

 1 dt C=− ln ln FI II 807 t 0   1 1 1 + dt C= DW ln t 1 − t 0   ∞ dt 1 cos t − C=− MO 10 1+t t 0   ∞ 1 dt sin t − C=1− MO 10 t 1 + t t 0   ∞ dt 1 −t e − C=− FI II 795, 802 1 + t t 0   ∞ dt 1 [K ≥ 1] e−t − C=− k 1+t t 0   ∞ 1 1 C= − dt DW et − 1 tet 0  ∞ −t  1   dt e − dt 1 − e−t C= FI II 802 t t 0 1 See also 8.361 5–8.361 7, 3.311 6, 3.435 3 and 4, 3.476 2, 3.481 1 and 2, 3.951 10, 4.283 9, 4.331 1, 4.421 1, 4.424 1, 4.553, 4.572, 6.234, 6.264 1, 6.468. 

5. 6. 7. 8. 9. 10. 11. 12.

13.

1

Asymptotic expansions C =

n−1  k=1

14.

∗

 C=− 1

15.∗

 C=−

1 1 1 1 1 1 − ln n + + − + − + ... k 2n 12n2 120n4 252n6 240n8

θ B2r+2 B2r 1 + ···+ 2r n2r 2 (r + 1) n2r+2 ∞

[0 < θ < 1]

FI II 827

ln(ln t) dt t2

∞

−∞

t exp(t − et ) dt

8.37 The function β(x)    x  1 x+1 ψ −ψ 8.370 β(x) = 2 2 2 8.371 Integral representations:  1 x−1 t dt 1.3 β(x) = 0 1+t Definition:

NH 16(13)

[Re x > 0]

WH

916

Euler’s Integrals of the First and Second Kinds

8.372



2. 3.

∞ e−xt β(x) = dt −t 0 1+e  ∞ −xt  e x+1 = dt β 2 0 cosh t

[Re x > 0]

MO 4

[Re x > −1]

See also 3.241 1, 3.251 7, 3.522 2 and 4, 3.623 2 and 3, 4.282 2, 4.389 3, 4.532 1 and 3. Series representation 8.372 1.7 2.7 3.8

β(x) = β(x) = β(x) =

∞  (−1)k k=0 ∞ 

x+k 1 (x + 2k)(x + 2k + 1)

k=0 ∞ 

1 2

k=0

k! 1 x(x + 1) . . . (x + k) 2k

8.373 1.6

β(x + 1) = ln 2 +

[−x ∈ N]

NH 37, 101(1)

[−x ∈ N]

NH 101(2)

[−x ∈ N]

[β has simple poles at x = −n with residue (−1)n ]

∞ 

  (−1)k 1 − 2−k ζ(k + 1)xk

[|x| < 1]

NH 246(7)

NH 37(5)

k=1

2.6

∞      π 1 1 − + 1 − 1 − 2−2k ζ(2k + 1) x2k − 2x 2 sin πx 1 − x2

β(x + 1) = ln 2 − 1 +

k=1

[0 < |x| < 2; 8.374

dn β(x) = (−1)n n! dxn

∞  k=0

(−1)k (x + k)n+1

x = ±1]

[−x ∈ N]

NH 38(11) NH 37(2)

8.375 Representation in the form of a finite sum: 1.6

q−1 

 2  π (2k + 1)π p p(2k + 1)π = ln sin β − cos q 2 sin pπ q 2q q

k=0

2.

β(n) = (−1)n+1 ln 2 +

n−1  k=1

[q = 2, 3, . . . , p = 1, 2, 3, . . . , q − 1]

NH 23(9)

(−1)k+n+1 k

Functional relations  2n  x+k = (2n + 1) β(x) (−1)k β 8.376 2n + 1 k=0 n    8.377 β 2k x = ψ (2n x) − ψ(x) − n ln 2 k=1

(see also 8.362 5–7)

NH 19 NH 20(10)

The beta function (Euler’s integral of the first kind): B(x, y)

8.380

917

8.38 The beta function (Euler’s integral of the first kind): B(x, y) Integral representation 8.380 1.

B (x, y) =



1

0 

tx−1 (1 − t)y−1 dt∗ 1

=2  2.

B(x, y) = 2 

3. 4. 5. 6.

7.

 y−1 t2x−1 1 − t2 dt

π/2

10. 11.

[Re x > 0,

0 ∞

 ∞ tx−1 t2x−1 dt = 2 dt [Re x > 0, x+y 2 x+y 0 (1 + t) 0 (1 + t )  1 (1 + t)2x−1 (1 − t)2y−1 B(x, y) = 22−y−x dt [Re x > 0, x+y (1 + t2 ) −1  1 x−1  ∞ x−1 t + ty−1 t + ty−1 B(x, y) = dt = dt [Re x > 0, x+y (1 + t)x+y 0 (1 + t) 1  1  1 x−1 (1 + t) B(x, y) = x+y−1 (1 − t)y−1 + (1 + t)y−1 (1 − t)x−1 2 0 B(x, y) =

B (x, y) = z y (1 + z)x

[Re x > 0,



FI II 774(1)

t

B(x, y) = z y (1 + z)x



π/2

(1 − t)y−1 dt (t + z)x+y [Re x > 0, Re y > 0,

Re y > 0]

KU 10

Re y > 0]

FI II 775

Re y > 0]

MO 7

Re y > 0]

BI (1)(15)

dt Re y > 0]

BI (1)(15)

1 x−1

0

9.

Re y > 0]

sin2x−1 ϕ cos2y−1 ϕ dϕ

0

8.

[Re x > 0,

0

cos2x−1 ϕ sin2y−1 ϕ x+y

(z + cos2 ϕ) [Re x > 0,

0 > z > −1,

Re(x + y) < 1]

NH 163(8)

0 > z > −1,

Re(x + y) < 1]

NH 163(8)

dϕ

Re y > 0,

See also 3.196 3, 3.198, 3.199, 3.215, 3.238 3, 3.251 1–3, 11, 3.253, 3.312 1, 3.512 1 and 2, 3.541 1, 3.542 1, 3.621 5, 3.623 1, 3.631 1, 8, 9, 3.632 2, 3.633 1, 4, 3.634 1, 2, 3.637, 3.642 1, 3.667 8, 3.681 2.  1  1   1 (1 − t)x−1 1 2 x−1 √ B(x, x) = 2x−2 dt 1−t dt = 2x−1 2 2 t 0 0 See 8.384 4, 8.382 3, and also 3.621 1, 3.642 2, 3.665 1, 3.821 6, 3.839 6.  ∞ cosh 2yt 1−x B(x + y, x − y) = 4 dt [Re x > |Re y|, Re x > 0] MO 9 cosh2x t 0  1   y  y x−1 y−1 =z B x, (1 − tz ) t dt Re z > 0, Re > 0, Re x > 0 z z 0 FI II 787a

∗ This

equation is used as the definition of the function B(x, y).

918

8.381 1.

Euler’s Integrals of the First and Second Kinds



3.

1−x−y

dt 2π (a + b) = x y (x + y − 1) B(x, y) −∞ (a + it) (b − it) [a > 0,

 2.

∞

8.381

b > 0;

x and y are real,

x + y > 1]

MO 7

b > 0;

x and y are real,

x + y > 1]

MO 7

∞

dt =0 x (b − it)y (a − it) −∞

B(x + iy, x − iy) = 2

1−2x

αe

[a > 0, −2iγy



∞

e2iαyt dt 2x −∞ cosh (αt − γ)

[y, α, γ are real,

α > 0;

Re x > 0] MI 8a

4.

For an integral representation of ln B(x, y), see 3.428 7.  2x+y−1 (x + y − 1) π/2 1 = cos[(x − y)t] cosx+y−2 t dt B (x, y) π 0  2x+y−2 (x + y − 1) π   = cos[(x − y)t] sinx+y−2 t dt π cos (x − y) π2 0  2x+y−2 (x + y − 1) π   = sin[(x − y)t] sinx+y−2 t dt π sin (x − y) π2 0

NH 158(5)a NH 159(8)a NH 159(9)a

Series representation 8.382 1.

2.

∞ (y − 1) . . . (y − n) 1 [y > 0] (−1)n y WH y n=0 n!(x + n)         ∞ √ 1 − 1 − 2−2k ζ(2k + 1) 2k+1 tan πx 1+x 1+x 1 1 2 , ln 2π + ln − ln + x ln B 2 2 2 x 1−x 2k + 1

B(x, y) =

k=0

3.

  ∞ (2k − 1)!! 1 1 1 = + B z, k 2 2 k! z + k z

[|x| < 2]

NH 39(17)

(see also 8.384 and 8.380 9)

WH

k=1

8.383 Infinite-product representation: (x + y + 1) B(x + 1, y + 1) =

∞  k(x + y + k) (x + k)(y + k)

[x,

y = −1,

−2, . . .]

MO 2

k=1

8.384 Functional relations involving the beta function: Γ(x) Γ(y) = B(y, x) Γ(x + y)

1.

B(x, y) =

2.

B(x, y) B(x + y, z) = B(y, z) B(y + z, x)

FI II 779 MO 6

8.403

3.

Definitions ∞ 

B(x, y + k) = B(x − 1, y)

k=0

4. 5. 6.

919

B(x, x) = 21−2x B

1

WH



2, x

(see also 8.380 9 and 8.382 3) FI II 784

 1



π B(x, x) B x + 12 , x + 2 = 4x−1 2 x   n+m−1 n+m−1 1 =m =n B(n, m) n−1 m−1

WH

[m and n are natural numbers]

For a connection with the psi function, see 4.253 1.

8.39 The incomplete beta function Bx (p, q) 

8.3917 Bx (p, q) = 8.392 I x (p, q) =

x

tp−1 (1 − t)q−1 dt =

0 Bx (p, q)

xp 2 F 1 (p, 1 − q; p + 1; x) p

ET I 373 ET II 429

B(p, q)

8.4–8.5 Bessel Functions and Functions Associated with Them 8.40 Definitions 8.401 Bessel functions Z ν (z) are solutions of the differential equation  d2 Z ν 1 dZν ν2 + KU 37(1) + 1 − 2 Zν = 0 dz 2 z dz z Special types of Bessel functions are what are called Bessel functions of the first kind J ν (z), Bessel functions of the second kind Y ν (z) (also called Neumann functions and often written N ν (z)), and Bessel (2) functions of the third kind H (1) ν (z) and H ν (z) (also called Hankel’s functions). ∞ zν  z 2k [|arg z| < π] (−1)k 2k KU 55(1) 8.402 J ν (z) = ν 2 2 k! Γ(ν + k + 1) k=0

8.403 1.

Y ν (z) =

1 [cos νπ J ν (z) − J −ν (z)] sin νπ

[for non-integer ν,

|arg z| < π] KU 41(3)

920

Bessel Functions and Functions Associated with Them

8.404

z  (n − k − 1)!  z 2k−n − 2 k! 2 n−1

π Y n (z) = 2 J n (z) ln

2.

k=0

−

∞ 

(−1)k

k=0

 z n+2k 1 [ψ(k + 1) + ψ(k + n + 1)] k! (k + n)! 2

 z  n−1  (n − k − 1)!  z 2k−n = 2 J n (z) ln + C − 2 k! 2

KU 43(10)

k=0

  n+2k  n+k n ∞ k  z n 1   1  1  (−1)k z2 1 − + − 2 n! k k! (k + n)! m m=1 m m=1 k=1 k=1 [n + 1 a positive integer, |arg z| < π] KU 44, WA 75(3)a

8.404 1.

Y −n (z) = (−1)n Y n (z)

[n is a natural number]

KU 41(2)

2.

J −n (z) = (−1)n J n (z)

[n is a natural number]

KU 41(2)

8.4057 1.

H (1) ν (z) = J ν (z) + i Y ν (z)

KU 44(1)

2.

H (2) ν (z) = J ν (z) − i Y ν (z)

KU 44(1)

In all relationships that hold for an arbitrary Bessel function Z ν (z), that is, for the functions J ν (z), (2) Y ν (z), and linear combinations of them, for example, H (1) ν (z) and H ν (z), we shall write simply the letter Z instead of the letters J, Y , H (1) , and H (2) . Modified Bessel functions of imaginary argument I ν (z) and K ν (z) 8.406 1. 2.

 π  π I ν (z) = e− 2 νi J ν e 2 i z  3  3 I ν (z) = e 2 πνi J ν e− 2 πi z

 π −π < arg z ≤ 2  π < arg z ≤ π 2

WA 92 WA 92

For integer ν, I n (z) = i−n J n (iz)

3. 8.407 1.8 2.8

πi π νi (1)  1 πi  e 2 H ν ze 2 2 −πi − π2 νi (2)  − 12 πi  e K ν (z) = H −ν ze 2 K ν (z) =

KU 46(1)

  −π < arg z ≤ 12 π  1  − 2 π < arg z ≤ π

For the differential equation defining these functions, see 8.494.

WA 92(8)

Integral representations of the functions Jν (z) and Nν (z)

8.411

921

8.41 Integral representations of the functions Jν (z) and Nν (z) 8.411 1.

11

2.

 1 π −niθ+iz sin θ e dθ J n (z) = 2π −π π 1 = cos (nθ − z sin θ) dθ [n = 0, 1, 2, . . .] π 0   1 π 2 π/2 J 2n (z) = cos 2nθ cos (z sin θ) dθ = cos 2nθ cos (z sin θ) dθ π 0 π 0 

3.11

4.

5.

6.

7. 8.

9.

10. 11.

12.

[n an integer] π

WH

WA 30(7)

1 sin(2n + 1)θ sin (z sin θ) dθ π 0  π/2 2 WA 30(6) [n an integer] sin(2n + 1)θ sin (z sin θ) dθ = π 0  z ν  π/2 2    J ν (z) = 2  sin2ν θ cos (z cos θ) dθ Γ ν + 12 Γ 12 0   Re ν > − 12 WH  z ν  π   2  J ν (z) = sin2ν θ cos (z cos θ) dθ Re ν > − 12   1 0 Γ ν + 12 Γ 2  z ν  π/2 2  cos (z sin θ) cos2ν θ dθ J ν (z) =   1 −π/2 1 Γ ν+2 Γ 2   KU 65(5), WA 35(4)a Re ν > − 12  z ν  π     2    J ν (z) =  e±iz cos ϕ sin2ν ϕ dϕ Re ν + 12 > 0 WH 1 1 Γ ν+2 Γ 2 0  z ν  1  ν− 12   2     1 − t2 J ν (z) = cos zt dt Re ν > − 12 KU 65(6), WH 1 1 Γ ν + 2 Γ 2 −1  x −ν  ∞  1  sin xt J ν (x) = 2  1 2   1  − 2 < Re ν < 12 , x > 0 MO 37 1 dt ν+ Γ 2 − ν Γ 2 1 (t2 − 1) 2  z ν  1  ν− 12   2  1 J ν (z) =  eizt 1 − t2 dt Re ν > − 12 WA 34(3) 1 Γ ν + 2 Γ 2 −1   2 ∞ νπ  cosh νt dt J ν (x) = sin x cosh t − WA 199(12) π 0 2    π/2 cosν− 12 θ sin z − νθ + 1 θ 2 2ν+1 z ν  1 e−2z cot θ dθ J ν (z) =  2ν+1 1 Γ ν+2 Γ 2 0 sin θ     π |arg z| < , Re ν + 12 > 0 WH 2 J 2n+1 (z) =

922

Bessel Functions and Functions Associated with Them

13.

10

14.

1 J ν (z) = π

J ν (z) =

e



π

0

±νπi

π

sin νπ cos (νθ − z sin θ) dθ − π 

π

0



∞

8.412

e−νθ−z sinh θ dθ

0



[Re z > 0] ∞



WA 195(4)

cos (νθ + z sin θ) dθ − sin νπ e−νθ+z sinh θ dθ 0  for π2 < |arg z| < π, with the upper sign taken for |arg z| >

π 2



and the lower sign taken for |arg z| < − π2 WH

8.412

    1 π z t− dt |arg z| < t exp 2 t 2 −∞   (0+) zν z2 dt J ν (z) = ν+1 t−ν−1 exp t − 2 πi −∞ 4t  ∞ z ν  (−1)k z 2k (0+) t −ν−k−1 J ν (z) = ν+1 et dt 2 πi 22k k! −∞ k=0  i∞  x ν+2t Γ(−t) 1 J ν (x) = dt [Re ν ≥ 0, x > 0] 2πi −i∞ Γ(ν + t + 1) 2   z ν   (1+,−1−) Γ 12 − ν 2 ν− 12  1 2 t −1 J ν (z) = cos(zt) dt 2πi Γ 2 A 

1 J ν (z) = 2πi

1. 2. 3.12

4.

5.7



(0+)

−ν−1

WH, WA 195(2)

WA 195(1)

WA 195(1)

WA 214(7)

ν = 12 , 32 , . . . ; The pointA falls to the right of the point t = 1,  and arg(t − 1) = arg(t + 1) = 0 at the point A

6.8

J ν (z) =

1 2π



WH π+∞i

−π+∞i

e−iz sin θ+iνθ dθ

[Re z > 0]

The path of integration being taken around the semi-infinite strip y ≥ 0, −π ≤ x ≤ π.    ∞ Jν z2 + ζ2 1 8 8.413 = eζ cos t cos (z sin t − νt) dt ν π(z + ζ)ν (z 2 − ζ 2 ) 2 0 ∞

− sin νπ 0



∞



− 12 +i∞

J 0 (t) Γ(−t) 2t 1 dt = x dt t 4π − 12 −i∞ t Γ(1 + t) See 3.715 2, 9, 10, 13, 14, 19–21, 3.865 1, 2, 4, 3.996 4.

8.414

2x



exp (−z sinh t − ζ cosh t − νt) dt [Re(z + ζ) > 0]

MO 40

[x > 0]

MO 41

(1)

(2)

Integral representations of the functions Hν (z) and Hν (z)

8.421

923

• For an integral representation of J 0 (z) see 3.714 2, 3.753 2, 3, and 4.124. • For an integral representation of J 1 (z) see 3.697, 3.711, 3.752 2, and 3.753 5. 8.415 1.

Y 0 (x) =



1

0

arcsin t 4 √ sin(xt) dt − 2 π 1 − t2



∞

1

√   ln t + t2 − 1 √ sin(xt) dt t2 − 1 [x > 0]

 x −ν  ∞ cos xt Y ν (x) = −2  1 2   1  1 dt Γ 2 − ν Γ 2 1 (t2 − 1)ν+ 2

2.

Y ν (x) = −

3. 4.

4 π2

8

5.

6.12

1 Y ν (z) = π

2 π 

x>0



KU 89(28)a, MO 38



0

 1 − 2 < Re ν < 12 ,

MO 37

∞



cos x cosh t −

0 π

νπ  2

cosh νt dt

1 sin (z sin θ − νθ) dθ − π



∞

[−1 < Re ν < 1,

x > 0]

WA 199(13)



 eνt + e−νt cos νπ e−z sinh t dt

0

[Re z > 0] WA 197(1)    z ν  π/2 ∞ 2 2 2ν    Y ν (z) =  sin (z sin θ) cos θ dθ − e−z sinh θ cosh2ν θ dθ Γ ν + 12 Γ 12 0 0   WA 181(5)a Re ν > − 12 , Re z > 0    π2 1 cosν− 2 θ cos z − νθ + 12 θ −2z cot θ 2ν+1 z ν    e Y ν (z) = −  dθ Γ ν + 12 Γ 12 0 sin2ν+1 θ     |arg z| < π2 , Re ν + 12 > 0 WA 186(8)

For an integral representation of Y 0 (z), see 3.714 3, 3.753 4, 3.864. See also 3.865 3.

8.42 Integral representations of the functions Hν(1) (z) and Hν(2) (z) 8.421 1.

2.

νπi  ∞ e− 2 eix cosh t−νt dt πi −∞ νπi  ∞ 2e− 2 = eix cosh t cosh νt dt πi 0

H (1) ν (x) =

νπi 2



∞ e e−ix cosh t−νt dt πi −∞ νπi  ∞ 2e 2 =− e−ix cosh t cosh νt dt πi 0

H (2) ν (x) = −

[−1 < Re ν < 1,

x > 0]

WA 199(10)

[−1 < Re ν < 1,

x > 0]

WA 199(11)

924

3.

12

4.12

5.

6. 7.

8.

9.

10.

11. 8.422

Bessel Functions and Functions Associated with Them

H (1) ν (z)

2ν+1 iz ν    =−  Γ ν + 12 Γ 12

H (2) ν (z) =

2ν+1 iz ν     Γ ν + 12 Γ 12

2.12



π/2

0

π/2

t 1 cosν− 2 tei(z−νt+ 2 ) exp (−2z cot t) dt sin2ν+1 t   Re ν > − 12 , Re z > 0

ν− 12

cos

0

z−νt+ 2t

te−i( sin2ν+1 t

)

WA 168(5)

exp (−2z cot t) dt  Re ν > − 12 ,

Re z > 0



WA 168(6)

 −ν  ∞  1  2i x2 eixt (1) 1  H ν (x) = − √ − 2 < Re ν < 12 , x > 0 WA 87(1) 1 dt ν+ π Γ 2 − ν 1 (t2 − 1) 2  −ν  ∞  1  2i x2 e−ixt (2) 1  H ν (x) = √ − 2 < Re ν < 12 , x > 0 WA 187(2) 1 dt ν+ π Γ 2 − ν 1 (t2 − 1) 2     1 i − 12 iνπ ∞ 1 e iz t + t−ν−1 dt H (1) (z) = − exp ν π 2 t 0 [0 < arg z < π; or arg z = 0 and − 1 < Re ν < 1] MO 38     z2 i − 1 iνπ ν ∞ 1 2 e ix t + t−ν−1 dt H (1) (xz) = − z exp ν π 2 t 0   π π 0 < arg z < , x > 0, Re ν > −1; or arg z = , x > 0 and − 1 < Re ν < 1 2 2     xν exp i xz − π ν − π  ∞  ν− 12 1 2 it (1) 2 4   1+ H ν (xz) = tν− 2 e−xt dt πz 2z Γ ν + 12  0 Re ν > − 21 , − 12 π < arg z < 32 π, x > 0  z ν  ∞ −2ie−iνπ (1) 2   H ν (z) = √ eiz cosh t sinh2ν t dt π Γ ν + 12  0 0 < arg z < π, Re ν > − 12 or arg z = 0 and − (1)

H 0 (x) = −

i π

 √ exp i x2 + t2 √ dt x2 + t2 −∞



∞

1 2

[x > 0]

  z ν  (1+) −ν  ν− 12 2   H (1) (z) = eizt t2 − 1 dt [−π < arg z < 2π] ν 1 πi Γ 2 1+∞i   z ν   (−1−) Γ 12 − ν  ν− 12 2   H (2) (z) = eizt t2 − 1 dt ν 1 πi Γ 2 −1+∞i [−2π < arg z < π] Γ

1.



8.422

< Re ν
0]

WA 197(2)a

[Re z > 0]

WA 197(3)a

The path of integration for 8.423 1 is shown in the left hand drawing and for 8.423 2 in the right hand drawing.

8.424 1.

2.

H (1) ν (z) J ν (ζ) =

H (2) ν (z) J ν (ζ) =

1 πi i π



γ+i∞

exp 0

 0

γ−i∞

    z2 + ζ2 zζ dt 1 t− Iν 2 t t t

[γ > 0,     z2 + ζ2 zζ dt 1 t− Iν exp 2 t t t

Re ν > −1,

|ζ| < |z|]

MO 45

[γ > 0,

Re ν > −1,

|ζ| < |z|]

MO 45

8.43 Integral representations of the functions Iν (z) and Kν (z) The function I ν (z) 8.431 1.

I ν (z) =

2.

I ν (z) =

3.

I ν (z) =

4.

I ν (z) =

 z ν  1  ν− 12 ±zt 2     1 − t2 e dt 1 1 Γ ν + 2 Γ 2 −1  z ν  1  ν− 12 2     1 − t2 cosh zt dt 1 1 Γ ν + 2 Γ 2 −1  z ν  π 2     e±z cos θ sin2ν θ dθ Γ ν + 12 Γ 12 0  z ν  π 2     cosh (z cos θ) sin2ν θ dθ Γ ν + 12 Γ 12 0

    Re ν + 12 > 0

WA 94(9)

    Re ν + 12 > 0

WA 94(9)

    Re ν + 12 > 0

WA 94(9)

    Re ν + 12 > 0

WA 94(9)

926

5.

Bessel Functions and Functions Associated with Them

1 I ν (z) = π



π

0

e

z cos θ

sin νπ cos νθ dθ − π

 0

∞

e−z cosh t−νt dt  |arg z| ≤

π 2,

8.432

Re ν > 0



WA 201(4)

See also 3.383 2, 3.387 1, 3.471 6, 3.714 5. For an integral representation of I 0 (z) and I 1 (z), see 3.366 1, 3.534 3.856 6. The function K ν (z) 8.432 1.



∞

K ν (z) =

 |arg z|
− 12 , Re z > 0; or Re z = 0 and −  z ν  1   ∞  ν− 12 Γ 2  K ν (z) = 2  e−zt t2 − 1 dt 1 Γ ν+2 1    Re ν + 12 > 0,  ∞ 1 K ν (x) = cos (x sinh t) cosh νt dt cos νπ 0 2    Γ ν + 12 (2z)ν ∞ cos xt dt   K ν (xz) = ν+ 12 xν Γ 12 0 (t2 + z 2 )

|arg z|
0,

|arg z|
0 WA 203(15) 2 2 tν+1 4 0     z2 zν ∞ x t+ t−ν−1 dt K ν (xz) = exp − 2 0 2 t   |arg z| < π4 or |arg z| = π4 and Re ν < 1 MO 39 

8.

9.

K ν (xz) =

π xν e−xz   2z Γ ν + 12

 0

∞

e

−xt ν− 12

t

ν− 12  t 1+ dt 2z  |arg z| < π,

√  x ν  ∞ exp −x√t2 + z 2  π  √ K ν (xz) =  t2ν dt 2 + z2 2z Γ ν + 12 t 0  Re ν > − 21 ,

Re z > 0,

Re



Re ν > − 12 , x > 0

t2 + z 2 > 0,

x>0

 MO 39

 MO 39

8.443

Series representation

927

See also 3.383 3, 3.387 3, 6, 3.388 2, 3.389 4, 3.391, 3.395 1, 3.471 9, 3.483, 3.547 2, 3.856, 3.871 3, 4, 7.141  √ 5.  ∞   3 2x x √ 8.433 K 13 = √ cos t3 + xt dt KU 98(31), WA 211(2) x 0 3 3 For an integral representation of K 0 (z), see 3.754 2, 3.864, 4.343, 4.356, 4.367.

8.44 Series representation The function J ν (z) 8.440

J ν (z) =

∞  z ν 

2

8.441 Special cases: 1.

J 0 (z) =

∞ 

k=0

(−1)k

k=0

J 13 (z) =

3.

Γ

1 4 3

 3

[|arg z| < π]

WH 358 a

z 2k 2

22k (k!)

J 1 (z) = − J 0 (z) =

2.

 z 2k (−1)k k! Γ(ν + k + 1) 2

∞

z  (−1)k z 2k 2 22k k!(k + 1)! k=0

z 2

∞  k=0

 √ 2k z 3 (−1) 2k 2 k! · 1 · 4 · 7 · · · · · (3k + 1) k

   √ 2k ∞  z 3 1 3 2 1+ (−1)k 2k J − 13 (z) =  2  z 2 k! · 2 · 5 · 8 · · · · · (3k − 1) Γ 3

4.

k=1

For the expansion of J ν (z) in Laguerre polynomials, see 8.975 3. 8.442 1.

 μ+ν+2m ∞  (−1)m 12 z (μ + ν + m + 1)m J ν (z) J μ (z) = m! Γ(μ + m + 1) Γ(ν + m + 1) m=0   2k   az ν bz μ b2 k az (−1) F −k, −ν − k; μ + 1; ∞ 2  2 2 a2 J ν (az) J μ (bz) = Γ(μ + 1) k! Γ(ν + k + 1)

7

2.12

k=0

The function Y ν (z)

∞  z ν  z 2k 1 cos νπ 8.443 Y ν (z) = (−1)k 2k sin νπ 2 2 k! Γ (ν + k + 1) k=0  ∞  z −ν  z 2k k − (−1) 2k 2 2 k! Γ(k − ν + 1) k=0 [ν = an integer] For ν + 1 a natural number, see 8.403 2.; for ν a negative integer see 8.404 1 11

(cf. 8.403 1)

MO 28

928

Bessel Functions and Functions Associated with Them

8.444

8.444 Special cases, 1.

2.11

∞ k   z  (−1)k  z 2k  1 π Y 0 (z) = 2 J 0 (z) ln + C − 2 2 2 2 m (k!) m=1 k=1  z 2k−1

∞ (−1)k+1 k−1  2 z   z  1 1 2 +2 π Y 1 (z) = 2 J 1 (z) ln + C − − − 2 z 2 k!(k − 1)! k m m=1

KU 44

k=2

The functions I ν (z) and K n (z) 8.445 I ν (z) =

∞ 

 z ν+2k 1 k! Γ(ν + k + 1) 2

k=0 n−1 

8.4468 K n (z) =

1 2

(−1)k

k=0

n+1

+(−1)

WH 372a

(n − k − 1)!  n−2k k! z2

∞  k=0

 z n+2k 

 z 1 1 ln − ψ(k + 1) − ψ(n + k + 1) k!(n + k)! 2 2 2 2

l

 n+l ∞  z n+2l  1  1 1 1 n+1 n 2 + = (−1) I n (z) ln z + C + (−1) 2 2 l!(n + l)! k k l=0

+

k=1

WA 95(15)

k=1

n−1 1  (−1)l (n − l − 1)!  z 2l−n 2 l! 2 l=0

[n + 1 is a natural number] MO 29

8.447 Special cases: 1.

I 0 (z) =

∞  k=0

2.

I 1 (z) =

 z 2k 2 2

(k!)

I 0 (z)

=

∞  k=0

 z 2k+1 2

k!(k + 1)! ∞

3.

K 0 (z) = − ln

 z 2k z I 0 (z) + 2 ψ(k + 1) 2 22k (k!) k=0

WA 95(14)

8.451

Asymptotic expansions of Bessel functions

929

8.45 Asymptotic expansions of Bessel functions 8.451 For large values of |z| ∗ n−1

     Γ ν + 2k + 12 2 π π   (−1)k  + R1  cos z ∓ ν − 1. J ±ν (z) = πz 2 4 (2z)2k (2k)! Γ ν − 2k + 12 k=0 n−1     Γ ν + 2k + 32 π   (−1)k π  + R2  − sin z ∓ ν − 2 4 (2z)2k+1 (2k + 1)! Γ ν − 2k − 12 k=0

2.

(see 8.339 4)

WA 222(1, 3)

n−1     Γ ν + 2k + 12 π π   (−1)k  + R1  sin z ∓ ν − Y ±ν (z) = 2 4 (2z)2k (2k)! Γ ν − 2k + 12 k=0 n−1     Γ ν + 2k + 32 π   (−1)k π  + R2  + cos z ∓ ν − 2 4 (2z)2k+1 (2k + 1)! Γ ν − 2k − 12 

11

[|arg z| < π]

2 πz

k=0

[|arg z| < π]

(see 8.339 4)

WA 222(2, 4, 5)



n−1      2 i(z− π2 ν− π4 )  (−1)k Γ ν + k + 12 (−1)n Γ ν + n + 12  + θ1    e πz (2iz)k k! Γ ν − k + 12 (2iz)n n! Γ ν − n + 12 k=0   (see 8.339 4) WA 221(5) Re ν > − 12 , |arg z| < π



n−1      Γ ν + k + 12 Γ ν + n + 12 2 −i(z− π2 ν− π4 )  1 1  + θ2    e πz (2iz)k k! Γ ν − k + 12 (2iz)n n! Γ ν − n + 12 k=0   (see 8.339 4) WA 221(6) Re ν > − 12 , |arg z| < π

12

H (1) ν (z)

4.12

H (2) ν (z)

5.

For indices of the form ν = 2n−1 (where n is a natural number), the series 8.451 terminate. In 2 this case, the closed formulas 8.46 are valid for all values.   ∞ ez  (−1)k Γ ν + k + 12   I ν (z) ∼ √ 2πz k=0 (2z)k k! Γ ν − k + 12   ∞     Γ ν + k + 12 exp −z ± ν + 12 πi  1   √ + (2z)k k! Γ ν − k + 12 2πz k=0

3.

=

=

[The + sign is taken for − 21 π < arg z < 32 π, the − sign for − 32 π < arg z < 12 π]†  6.11

K ν (z) =

π −z e 2z

n−1  k=0



 1



 1

Γ ν +k+ 2 Γ ν +n+ 2 1  + θ3    (2z)k k! Γ ν − k + 12 (2z)n n! Γ ν − n + 12



(see 8.339 4)

(see 8.339 4)] WA 226(2,3)

WA 231, 245(9)

An estimate of the remainders of the asymptotic series in formulas 8.451: ∗ An

estimate of the remainders in formulas 8.451 is given in 8.451 7 and 8.451 8. contradiction that this condition contains at first glance is explained by the so-called Stokes phenomenon (see Watson, G.N., A Treatise on the Theory of Bessel Functions, 2nd Edition, Cambridge Univ. Press, 1944, page 201). † The

930

Bessel Functions and Functions Associated with Them

7.

8.12

      Γ ν + 2n + 12     |R1 | <  1  (2z)2n (2n)! Γ ν − 2n + 2        Γ ν + 2n + 32     |R2 | <  1  (2z)2n+1 (2n + 1)! Γ ν − 2n − 2 

For − π2 < arg z < 32 π, ν real, and n +

|θ1 | < π 2,ν

real, and n +

1 2

ν 1 − 2 4

n≥

ν 3 − 2 4



 WA 231

 WA 231

WA 245

1, if Im z ≥ 0 |sec (arg z)|, if Im z ≤ 0

> |ν|

|θ2 |


> |ν|

1 2

For − 23 π < arg z
π can be greater in absolute value than the first discarded term. “Approximation by tangents” 8.45211 For large values of the index (where the argument is less than the index). Suppose that x > 0 and ν > 0. Let us set ν/x = cosh α. Then, for large values of ν, the following expansions are valid:   ν  exp (ν tanh α − να)  5 1 1 3 √ 1. Jν ∼ coth α − coth α 1+ cosh α ν 8 24 2νπ tanh α   231 1155 1 9 2 4 6 coth α − coth α + coth α + . . . + 2 ν 128 576 3456 WA 269(3)

8.453

Asymptotic expansions of Bessel functions

 2.

Yν

931

  exp (να − ν tanh α) 5 1 1 ν  3 π ∼− coth α − coth α 1− cosh α ν 8 24 ν tanh α  2  9 1 231 1155 coth2 α − coth4 α + coth6 α + . . . + 2 ν 128 576 3456 WA 270(5)

8.453 For large values of the index (where the argument is greater than the index). Suppose that x > 0 and ν > 0. Let us set ν/x = cos β. Then, for large values of ν, the following expansions are valid: ⎧⎡ ⎛  ⎨ 2 1 ⎝ 9 cot2 β + 231 cot4 β ⎣1 − 1. J ν (ν sec β) ∼ νπ tan β ⎩ ν 2 128 576 ⎞ ⎤  π 1155 cot6 β ⎠ + . . .⎦ cos ν tan β − νβ − + 3456 4 ⎫    ⎬  π 5 1 1 + cot β + cot3 β − . . . sin ν tan β − νβ − ν 8 24 4 ⎭

2.

⎧⎡ ⎛  ⎨ 2 1 ⎝ 9 cot2 β + 231 cot4 β ⎣1 − Y ν (ν sec β) ∼ νπ tan β ⎩ ν 2 128 576 ⎞ ⎤  π 1155 cot6 β ⎠ + . . .⎦ sin ν tan β − νβ − + 3456 4 ⎫ ⎡ ⎤    1 1 π ⎬ 5 − ⎣ cot β + cot3 β − . . .⎦ cos ν tan β − νβ − ν 8 24 4 ⎭ 

3.

H (1) ν (ν sec β) ∼

H (2) ν (ν sec β) ∼

WA 271(5)



exp νi (tan β − β) − π4 i 5 i 1 π cot β + cot3 β 1− ν 8 24 ν tan β 2   9 1 231 1155 cot2 β + cot4 β + cot6 β + . . . − 2 ν 128 576 3456 

4.





WA 271(4)



WA 271(1)



exp −νi (tan β − β) + π4 i 5 i 1 π cot β + cot3 β 1+ ν 8 24 ν tan β 2   1 231 1155 9 cot2 β + cot4 β + cot6 β + . . . − 2 ν 128 576 3456



WA 271(2) 1

Formulas 8.453 are not valid when |x − ν| is of a size comparable to x 3 . For arbitrary small (and also large) values of |x − ν|, we may use the following formulas:

932

Bessel Functions and Functions Associated with Them

8.454

8.454 Suppose that x > 0 and ν > 0, we set  w=

x2 − 1; ν2

Then,       w3 1 π (1) ν 3 +ν w− − arctan w i H 1 w +O 3 6 3 3 |ν|        w3 1 π w (2) ν 3 (2) − arctan w i H 1 w +O MO 34 2. H ν (x) = √ exp − − ν w − 3 6 3 3 |ν| 3    √ 1 1 is then less than 24 2 . The absolute value of the error O |ν| ν 8.455 For x real and ν a natural number (ν = n), if n 1, the following approximations are valid: 

 3 2 2(n − x) [2(n − x)] 1 √ K 13 J n (x) ≈ 1.7 π 3x 3 x [n > x] (see also 8.433) 1.

w H (1) ν (x) = √ exp 3

1 2 ≈ e 3 πi 2





2 (n − x) (1) H1 3 3x

3

i [2(n − x)] 2 √ 3 x

WA 276(1)



[n > x] 1 ≈√ 3



2(x − n) 3x

 J 13

3

{2(x − n)} 2 √ 3 x



 + J − 13

3

{2(x − n)} 2 √ 3 x



MO 34

(see also 8.441 3, 8.441 4) WA 276(2)

 2.

12

Y n (x) ≈ −

2(x − n) 3x

 J − 13

3

{2(x − n)} 2 √ 3 x



 − J 13

3

{2(x − n)} 2 √ 3 x



[x > n] An estimate of the error in formulas 8.455has not yet  been achieved. ∞ 1  Γ ν + k + (2k − 1)!! 2 2   [|arg z| < π] 8.45611 J 2ν (z) + Y 2ν (z) ≈ πz 2k z 2k k! Γ ν − k + 12

WA 276(3)

(see also 8.479 1)

k=0

WA 250(5)

8.457

2 J 2ν (x) + J 2ν+1 (x) ≈ πx

[x  |ν|]

WA 223

8.464

Bessel functions of order equal to an integer plus one-half

933

8.46 Bessel functions of order equal to an integer plus one-half The function J ν (z) 8.461 1.11

 J n+ 12 (z) =

⎧ ⎪

n2  2 ⎨  π   (−1)k (n + 2k)! sin z − n (2z)−2k πz ⎪ 2 (2k)!(n − 2k)! ⎩ k=0

⎫ n−1 ⎪

 2  ⎬  k (−1) (n + 2k + 1)! π −(2k+1) (2z) + cos z − n ⎪ 2 (2k + 1)!(n − 2k − 1)! ⎭ 

k=0

[n + 1 is a natural number]  2.

J −n− 12 (z) =

(cf. 8.451 1)

KU 59(6), WA 66(2)

⎧ ⎪

n2   2 ⎨ π   (−1)k (n + 2k)! cos z + n ⎪ πz ⎩ 2 (2k)!(n − 2k)!(2z)2k k=0

⎫ ⎪

 ⎬   k (−1) (n + 2k + 1)! π − sin z + n 2 (2k + 1)!(n − 2k − 1)!(2z)2k+1 ⎪ ⎭ n−1 2

k=0

[n + 1 is a natural number] 8.462 1.

2.

1 J n+ 12 (z) = √ 2πz

eiz

n  i−n+k−1 (n + k)! k=0

1 J −n− 12 (z) = √ 2πz

k!(n − k)!(2z)k

+ e−iz

(cf. 8.451 1)

n  (−i)−n+k−1 (n + k)! k=0

eiz



k!(n − k)!(2z)k [n + 1 is a natural number] 

KU 58(7), WA 67(5)

KU 59(6), WA 66(1)

n n   in+k (n + k)! (−i)n+k (n + k)! −iz + e k!(n − k)!(2z)k k!(n − k)!(2z)k k=0 k=0 [n + 1 is a natural number]

KU 59(7), WA 67(4)

8.463 1.

 n n+ 12

J n+ 12 (z) = (−1) z



2.

J −n− 12 (z) = z

n+ 12

 2 dn sin z π (z dz)n z   n d cos z

2 n π (z dz)

8.464 Special cases:  2 sin z 1. J 12 (z) = πz  2 cos z 2. J − 12 (z) = πz

z

KU 58(4)

KU 58(5)

DW DW

934

Bessel Functions and Functions Associated with Them

8.465



3. 4. 5.8 6.

 2 sin z − cos z πz z   2 cos z  − sin z − J − 32 (z) = πz z    2 3 3 J 52 (z) = − 1 sin z − cos z πz z2 z     2 3 3 sin z + J − 52 (z) = − 1 cos z πz z z2 J 32 (z) =

DW

DW DW

DW

The function Y n+ 12 (z) 8.465 1.

Y n+ 12 (z) = (−1)n−1 J −n− 12 (z)

JA

2.

Y −n− 12 (z) = (−1)n J n+ 12 (z)

JA

(1,2)

The functions H n− 1 (z), I n+ 12 (z), K n+ 12 (z) 2

8.466 1.

 (1) H n− 1 (z) 2

=

n−1

2 −n iz  (n + k − 1)! 1 i e (−1)k πz k!(n − k − 1)! (2iz)k k=0

(cf. 8.451 3)



2.

8.467

n−1 

2 n −iz (n + k − 1)! 1 i e (cf. 8.451 4) πz k!(n − k − 1)! (2iz)k k=0   n n   (−1)k (n + k)! (n + k)! 1 z n+1 −z I ±(n+ 1 ) (z) = √ e ± (−1) e 2 k!(n − k)!(2z)k k!(n − k)!(2z)k 2πz (2)

H n− 1 (z) = 2

k=0

 8.468 K n+ 12 (z) = 8.469 Special cases:  1. 2. 3. 4.

π −z e 2z

n  k=0

2 cos z Y 12 (z) = − πz  2 Y − 12 (z) = sin z πz  π −z e K ± 12 (z) = 2z  2 eiz (1) H 1 (z) = 2 πz i

(n + k)! k!(n − k)!(2z)k

k=0

(cf. 8.451 5) (cf. 8.451 6)

KU 60a KU 60

WA 95(13)

MO 27

8.474

Functional relations

935



5. 6. 7.

2 e−iz 2 πz −i  2 iz (1) e H − 1 (z) = 2 πz  2 −iz (2) e H − 1 (z) = 2 πz (2)

H 1 (z) =

MO 27 MO 27 MO 27

8.47–8.48 Functional relations 8.4718 1.

Recursion formulas: z Z ν−1 (z) + z Z ν+1 (z) = 2ν Z ν (z)

KU 56(13), WA 56(1), WA 79(1), WA 88(3)

d Z ν (z) KU 56(12), WA 56(2), WA 79(2), We 88(4) dz Sonin and Nielsen, in their construction of the theory of Bessel functions, defined Bessel functions as analytic functions of z that satisfy the recursion relations 8.471. Z denotes J, N , H (1) , H (2) or any linear combination of these functions, the coefficients in which are independent of z and ν. 8.472 Consequences of the recursion formulas for Z defined as above: d 1. z Z ν (z) + ν Z ν (z) = z Z ν−1 (z) KU 56(11), WA 56(3), WA 79(3), WA 88(5) dz d 2. z Z ν (z) − ν Z ν (z) = −z Z ν+1 (z) KU 56(10), WA 56(4), WA 79(4), WA 88(6) dz m  d 3. (z ν Z ν (z)) = z ν−m Z ν−m (z) KU 56(8), WA 57(5), WA 89(9) z dz m   −ν  d z Z ν (z) = (−1)m z −ν−m Z ν+m (z) 4. WA 89(10), Ku 55(5), WA 57(6) z dz 2.

Z ν−1 (z) − Z ν+1 (z) = 2

5.

Z −n (z) = (−1)n Z n (z)

[n is a natural number]

(cf. 8.404)

8.473 Special cases: 2 1. J 2 (z) = J 1 (z) − J 0 (z) z 2 2. Y 2 (z) = Y 1 (z) − Y 0 (z) z 2 (1,2) (1,2) (1,2) 3. H 2 (z) = H 1 (z) − H 0 (z) z d J 0 (z) = − J 1 (z) 4. dz d Y 0 (z) = − Y 1 (z) 5. dz d (1,2) (1,2) H (z) = − H 1 (z) 6. dz 0 8.47412 Each of the pairs of functions J ν (z) and J −ν (z) (for ν = 0, ±1, ±2,. . . ), J ν (z) and Y ν (z), and (2) H (1) ν (z) and H ν (z), which are solutions of equation 8.401, and also the pair I ν (z) and K ν (z) is a pair of linearly independent functions. The Wronskians of these pairs are, respectively,

936

Bessel Functions and Functions Associated with Them

8.475

2 2 4i 1 sin νπ, , − , − KU 52(10, 11, 12), WA 90(1, 4) πz πz πz z 8.4756 The functions J ν (z), and Y ν (z), H (1,2) (z), I ν (z), K ν (z) with the exception of J n (z) and I n (z) ν for n an integer are non-single-valued : z = 0 is a branch point for these functions. The branches of these functions that lie on opposite sides of the cut (−∞, 0) are connected by the relations 8.476   1. J ν emπi z = emνπi J ν (z) WA 90(1)  mπi  −mνπi 2. Yν e z =e Y ν (z) + 2i sin mνπ cot νπ J ν (z) WA 90(3)  mπi  3. Y −ν e z = e−mνπi Y −ν (z) + 2i sin mνπ cosec νπ J ν (z) WA 90(4)  mπi  mνπi 4. Iν e z =e I ν (z) WA 95(17)   sin mνπ I ν (z) 5. K ν emπi z = e−mνπi K ν (z) − iπ [ν not an integer] WA 95(18) sin νπ  mπi  −νπi sin mνπ J ν (z) e 6. H (1) z = e−mνπi H (1) ν ν (z) − 2e sin νπ sin mνπ (2) sin(1 − m)νπ (1) H ν (z) − e−νπi H ν (z) = sin νπ sin νπ WA 95(5)

9.

 mπi  νπi sin mνπ e H (2) z = e−mνπi H (2) J ν (z) ν ν (z) + 2e sin νπ sin mνπ (1) sin(1 + m)νπ (2) H ν (z) + eνπi H ν (z) = sin νπ sin νπ [m an integer]   (2) eiπ z = − H −ν (z) = −e−iπν H (2) H (1) ν ν (z)   (1) e−iπ z = − H −ν (z) = −eiπν H (1) H (2) ν ν (z)

10.8

H (2) ν (z) = H ν (z)

7.

8.

(1)

WA 90(6) MO 26 MO 26 MO 26

8.477 1.

J ν (z) Y ν+1 (z) − J ν+1 (z) Y ν (z) = −

2.12

I ν (z) K ν+1 (z) + J ν+1 (z) K ν (z) =

2 πz

WA 91(12)

1 z

WA 95(20)

See also 3.863. • For a connection with Legendre functions, see 8.722. • For a connection with the polynomials C λn (t), see 8.936 4. • For a connection with a confluent hypergeometric function, see 9.235. 8.478 For ν > 0 and x > 0, the product   x J 2ν (x) + Y 2ν (x) , considered as a function of x, decreases monotonically, if ν >

1 2

and increases monotonically if 0 < ν < 12 . MO 35

8.486

8.479 1.11 2.

Functional relations

 1 π 2 1 √ J ν (x) + Y 2ν (x) ≥ > 2 2 2 x x −ν |J n (nz)| ≤ 1

937

  x ≥ ν ≥ 12

MO 35

    z exp √1 − z 2    √  < 1, n a natural number   1 + 1 − z2 

MO 35

Relations between Bessel functions of the first, second, and third kinds Y −ν (z) − Y ν (z) cos νπ = H (1) ν (z) − i Y ν (z) sin νπ   1 (2) H (1) = H (2) ν (z) + i Y ν (z) = ν (z) + H ν (z) 2

8.481 J ν (z) =

J ν (z) cos νπ − J −ν (z) = i J ν (z) − i H (1) 8.482 Y ν (z) = ν (z) sin νπ   i (1) H (2) = i H (2) ν (z) − i J ν (z) = ν (z) − H ν (z) 2

(cf. 8.403 1, 8.405)

WA 89(1), JA

(cf. 8.403 1, 8.405)

WA 89(3), JA

8.483 1. 2.

Y −ν (z) − e−νπi Y ν (z) J −ν (z) − e−νπi J ν (z) = = J ν (z) + i Y ν (z) i sin νπ sin νπ Y −ν (z) − eνπi Y ν (z) eνπi J ν (z) − J −ν (z) = = J ν (z) − i Y ν (z) H (2) (z) = ν i sin νπ sin νπ (cf. 8.405)

H (1) ν (z) =

WA 89(5)

WA 89(6)

8.484 1.

H −ν (z) = eνπi H (1) ν (z)

2.

(2) H −ν (z)

(1)

8.4857

WA 89(7)

= e−νπi H (2) ν (z)

K ν (z) =

π I −ν (z) − I ν (z) 2 sin νπ

WA 89(7)

[ν not an integer]

(see also 8.407) WA 92(6)

8.486 Recursion formulas for the functions I ν (z) and K ν (z) and their consequences: 1.

z I ν−1 (z) − z I ν+1 (z) = 2ν I ν (z)

2.

I ν−1 (z) + I ν+1 (z) = 2

3.

z

4. 5.

d I ν (z) dz

d I ν (z) + ν I ν (z) = z I ν−1 (z) dz d z I ν (z) − ν I ν (z) = z I ν+1 (z) dz m  d {z ν I ν (z)} = z ν−m I ν−m (z) z dz

WA 93(1) WA 93(2) WA 93(3) WA 93(4) WA 93(5)

938

Bessel Functions and Functions Associated with Them

 6. 7.12

d z dz

m

# −ν $ z I ν (z) = z −ν−m I ν+m (z)

I −n (z) = In (z)

10. 11.

K ν−1 (z) + K ν+1 (z) = −2

12.

z

9.

13. 14. 15.

K −ν (z) = K ν (z)

17.

K 2 (z) =

19.

1.

2.

3.

4.12

WA 93(1)

d K ν (z) dz

WA 93(2) WA 93(3) WA 93(4) WA 93(5)

WA 93(6) WA 93(8)

2 K 1 (z) + K 0 (z) z

d K 0 (z) = − K 1 (z) dz ∞  ∂ J ν (z)  z (z/2)ν+1  (z/2)n J n+1 (z) = ln − ψ(ν + 1) J ν (z) + ∂ν 2 Γ(ν + 1) n=0 n!(ν + n + 1)2

8.486(1)7

WA 93(8)

WA 93(7)

d K ν (z) + ν K ν (z) = −z K ν−1 (z) dz d z K ν (z) − ν K ν (z) = −z K ν+1 (z) dz m  d {z ν K ν (z)} = (−1)m z ν−m K ν−m (z) z dz m  # −ν $ d z K ν (z) = (−1)m z −ν−m K ν+m (z) z dz

16.

18.

WA 93(6)

[n a natural number]

2 I 2 (z) = − I1 (z) + I 0 (z) z d I 0 (z) = I 1 (z) dz z K ν−1 (z) − z K ν+1 (z) = −2ν K ν (z)

8.12

8.486(1)

WA 93(7) LUKE 360

Differentiation with respect to order

  ν+2k ∞ ψ(ν + k + 1) 1 1 k z − z (−1) 2 2 k! Γ(ν + k + 1) k=0  ν = n or n + 12 ,  −ν+2k   ∞ ψ(−ν + k + 1) 1 ∂ J −ν (z) 1 = − J −ν (z) ln z + z (−1)k ∂ν 2 2 k! Γ(−ν + k + 1) k=0  ν = n or n + 12 , ∂ J ν (z) = J ν (z) ln ∂ν



∂ J ν (z) ∂ J −ν (z) ∂ Y ν (z) = cot πν − cosec πν − π cosec πν Y ν (z) ∂ν ∂ν ∂ν  ν = n or n + 12 , ν+2k   ∞  ∂ I ν (z) ψ(ν + k + 1) 1 1 = I ν (z) ln z − z ∂ν 2 2 k! Γ(ν + k + 1) k=0

 n integer

MS 3.1.3

 n integer

MS 3.1.3

 n integer

MS 3.1.3 MS 3.1.3

8.486(1)

5.

12

6.

7.

Functional relations

939

  ∂ K ν (z) ∂ I −ν (z) ∂ I ν (z) 1 = −π cot πν K ν (z) + π cosec πν − ∂ν 2 ∂ν ∂ν MS 3.1.3 [ν = n, n integer]     k−n n−1 J k (z) 1  12 z ∂ J ν (z) 1 [n = 0, 1, . . .] = π (±1)n Y n (z)±(±1)n n! MS 3.2.3 ∂ν 2 2 k!(n − k) ν=±n k=0     n−1  1 z k−n Y k (z) ∂ Y ν (z) 1 n n 1 2 [n = 0, 1, . . .] = − π (±1) J n (z) ± (±1) n! ∂ν 2 2 k!(n − k) ν=±n k=0

MS 3.2.3

 8.

∂ I ν (z) ∂ν

 ν=±n

  n−1  (−1)k 1 z k−n I k (z) n+1 n1 2 = (−1) K n (z) ± (−1) n! 2 k!(n − k)

[n = 0, 1, . . .]

k=0

MS 3.2.3

 9.

10.

(−1)n

 11.11

12. 13. 14. 15. 16. 17. 18. 19.

∂ K ν (z) ∂ν 

 ν=±n

1 = ± n! 2

 ∂ I ν (z) ∂ν ν=n

∂ K ν (z) ∂ν

 = ν=n

1 n! 2

  n−1  1 z k−n K k (z) 2

k=0

[n = 0, 1, . . .]

k!(n − k)

k−n 1 z I k (z) n−1 (−1) 1  2 = − K n (z) + n! 2 k!(n − k) 

k

k=0

n−1  k=0

 1 k−n K k (z) 2z k!(n − k)

[n = 0, 1, . . .]

AS 9.6.44

[n = 0, 1, . . .]

AS 9.6.45

Special cases   ∂ J ν (z) = 12 π Y 0 (z) ∂ν ν=0   ∂ Y ν (z) = − 12 π J 0 (z) ∂ν ν=0   ∂ I ν (z) = − K 0 (z) ∂ν ν=0   ∂ K ν (z) =0 ∂ν ν=0    −1/2 ∂ J ν (x) = 12 πx [sin x Ci(3x) − cos x Si(2x)] ∂ν ν= 12    −1/2 ∂ J ν (x) = 12 πx [cos x Ci(2x) + sin x Si(2x)] ∂ν ν=− 12    −1/2 ∂ Y ν (x) = 12 πx {cos x Ci(2x) + sin x [Si(2x) − π]} ∂ν ν= 12    −1/2 ∂ Y ν (x) = − 12 πx {sin x Ci(2x) − cos x [Si(2x) − π]} ∂ν ν=− 1 2

MS 3.2.3

MS 3.2.3 MS 3.2.3 MS 3.2.3 MS 3.2.3 MS 3.3.3

MS 3.3.3

MS 3.3.3

MS 3.3.3

940

Bessel Functions and Functions Associated with Them

 20.

12

∂ I ν (x) ∂ν

8.487

 ν=± 12

 x  e Ei(−2x) ∓ e−x Ei(2x) = (2πx)    ∞  x ±1/2  (x2 /4)k 1 x − Ψ 1 ± + k ln = 2 2 2 k! Γ(1 ± 12 + k) k=0       ∞ ∞   2 k x π  x 1/2  (x2 /4)k x π x −1/2  (x /4) 1 3 + k − ln + + k − ln Ψ Ψ = 2 2 2 2 2 2 2 2 k! Γ( 12 + k) k! Γ( 32 + k) −1/2

k=0

 21.

∂ K ν (x) ∂ν

k=0

MS 3.3.3

 ν=± 12

 π  12 =∓ ex Ei(−2x) 2x

MS 3.3.3

8.487 Continuity with respect to the order∗ : 1. 2. 3.

lim Y ν (z) = Y n (z)

[n an integer]

WA 76

lim H (1,2) (z) = H (1,2) (z) ν n

[n an integer]

WA 183

lim K ν (z) = K n (z)

[n an integer]

WA 92

ν→n ν→n ν→n

8.49 Differential equations leading to Bessel functions See also 8.401 8.491 1. 2. 3. 4. 5. 6. 7.

 ν2 1 d (zu ) + β 2 − 2 u = 0 z dz z   νγ 2    1 d  γ−1 2 (zu ) + βγz u=0 − z dz z   2 α2 − ν 2 γ 2  1 − 2α  u + βγz γ−1 + u=0 u + z z2     4ν 2 γ 2 − 1  γ−1 2 u=0 u + βγz − 4z 2  4ν 2 − 1 u=0 u + β 2 − 4z 2  1 − 2α  α2 − ν 2 u + β2 + u=0 u + z z2 

m

u + bz u = 0

u = Z ν (βz)

JA

u = Z ν (βz γ )

JA

u = z α Z ν (βz γ )

JA

u=

√ z Z ν (βz γ )

JA

u=

√ z Z ν (βz)

JA

u = z α Z ν (βz)

JA

√ 1 u = z Z m+2



√ 2 b m+2 z 2 m+2 JA 111(5)

∗ The

continuity of the functions J ν (z) and I ν (z) follows directly from the series representations of these functions.

8.494

8. 9. 10. 11. 12.

8.492

Differential equations leading to Bessel functions

 1  ν2 2 u + u +4 z − 2 u=0 z z  ν2 1 1 1− u=0 u + u + z 4z z 1−ν  1u u + =0 u + z 4z

  u + e2z − ν 2 u = 0

2.

u +

1. 2. 8.494 1. 2. 3. 4.12 5. 6.

7.

8.



z 2 u + (2α − 2βν + 1)zu + β 2 γ 2 z 2β

1.

8.493

  u = Z ν z2



u + β 2 γ 2 z 2β−2 u = 0

u = Zν

√  z

WA 111(7)

√  z   1 γz β u = z 1/2 Z 2β ν

WA 111(9)a WA 110(3)

 + α(α − 2βν) u = 0

  u = z βν−α Z ν γz β

u = Z ν (ez )   u = z Z ν e1/z

e2/z − ν 2 u=0 z4

WA 112(21)

WA 112(22) WA 112(22)



 1 ν2 u + u − 1 + 2 u = 0 z z    ν 2 1  1  + u=0 u + u − z z 2z  1 1 − ν2 u = 0 u + u + 2 z 4  2ν + 1 2ν + 1  u + − k u − ku = 0 z 2z

JA

u = cosec z Z ν (z)

JA

u = Z ν (iz) = C1 I ν (z) + C2 K ν (z)

JA

 √  u = Z ν 2i z

JA

iz 2  ikz −ν 21 kz Zν u=z e 2 √  ν u = z2 Zν i z

1−ν  1u u − =0 z 4z u u ± √ = 0 z

√ −z ze 2 Z ν



u=

 3 √ z Z 23 43 z 4 ,

u=

  √ 3 z Z 23 43 iz 4

u=

 3 √ z Z 13 23 z 2 ,

u=

  √ 3 z Z 13 23 iz 2

u ± zu = 0  ν(ν + 1) 2 u=0 u − c + z2

u = sec z Z ν (z)

u=

u +



WA 111(6)

u = z2 Zν

 2 1 ν tan z  − 2 tan z u − u=0 u + + z z2 z   2 1 ν cot z + 2 cot z u − u=0 u + − z z2 z 

941

u=

√ z Z ν+ 12 (icz)

JA JA WA 111(8)

WA 111(10)

WA 111(10)

WA 108(1)

942

Bessel Functions and Functions Associated with Them

2ν  u − c2 u = 0 z

9.

u −

10.

u − c2 z 2ν−2 u = 0

8.495 1. 2. 3. 4.

1

u = z ν+ 2 Z ν+ 12 (icz)  c  √ u = z Z 2ν1 i z ν ν

 1  ν2 u + u + i− 2 u=0 z z   2 i 1 ν u=0 u + ± ∓ 2i u − z z2 z 1 u + u + seiα u = 0 z  1  iα u + se + 2 u = 0 4z 

8.496

8.495

d2 dz 2

2.

 2 16 d u 8 d 5 z − z5u = 0 2 2 dz dz

3.

 2 d 12 d u z − z6u = 0 dz 2 dz 2

JA

u = e±iz Z ν (z)

JA

u=

u=

√  i sze 2 α

JA

√  √ i z Z0 sze 2 α

JA

 √ 3 12  √  Z 2 2 z + Z 2 2i z z 

2

WA 109(5, 6)

 √ u = Zν z i

u = Z0

 d2 u z 4 2 − z 2u = 0 dz

1.

WA 109(3, 4)

u = z −7/10 Z 56

WA 122(7)



 3

5 5 3z

 + Z 56

3 5 5 3 iz



WA 122(8)

2

4.

2    3 u = z −4 Z 10 2z −1/2 + Z 10 2iz −1/2

WA 122(9)

 4 ν − 4ν 2 d4 u 2 d3 u 2ν 2 + 1 d2 u 2ν 2 + 1 du + + − + − 1 u = 0, dz 4 z dz 3 z 2 dz 2 z 3 dz z4 u = A1 J ν (z) + A2 Y ν (z) + A3 I ν (z) + A4 K ν (z), where A1 , A2 , A3 , A4 are constants

MO 29

8.51–8.52 Series of Bessel functions 8.511 Generating functions for Bessel functions:  ∞ ∞   k  1 1 −k t− z = J 0 (z) + t + (−t) J k (z) = J k (z)tk 1. exp 2 t k=1

2.

3.

 1 exp t − t

z=

∞ 

k=−∞

 tk J k (z)

[|z| < |t|]

tm J m (z)

KU 119(12) WA 40

m=−∞

k=−∞

exp (±iz sin ϕ) = J 0 (z) + 2

∞ 



∞  k=1

J 2k (z) cos 2kϕ ± 2i

∞  k=0

J 2k+1 (z) sin(2k + 1)ϕ

KU 120(13)

8.514

Series of Bessel functions

943



4.

∞ π  (2k + 1)ik J k+ 12 (z) P k (cos ϕ) 2z k=0 ∞  = ik J k (z)eikϕ

exp (iz cos ϕ) =

k=−∞

= J 0 (z) + 2  5.

The series

i iz cos 2ϕ e π 

√

MO 27

ik J k (z) cos kϕ

MO 27

k=1 2z cos ϕ

−∞

2

e−it dt =

∞

 1 1 J 0 (z) + e 4 kπi J k (z) cos kϕ 2 2

MO 28

k=1

J k (z)

8.512 1.



∞ 

WA 401(1)

J 0 (z) + 2

∞ 

J 2k (z) = 1

WA 44

k=1

2.

3.

∞  z n  (n + 2k)(n + k − 1)! J n+2k (z) = k! 2 k=0  ∞  (4k + 1)(2k − 1)!! 2z J 2k+ 12 (z) = 2k k! π

[n = 1, 2, . . .]

WA 45

k=0

8.513 Notation: In formulas 8.513

1.

∞ 

(2k)2p J 2k (z) =

k=1

2.

∞ 

p  k=0

(p)

Qk =

  k p m (k − 2m)

m=0

(2p)

Q2k z 2k

(2k + 1)2p+1 J 2k+1 (z) =

k=0

3.

k−1

 2  (−1)m

p  k=0

(2p+1)

Q2k+1 z 2k+1

In particular: ∞   1 z + z3 (2k + 1)3 J 2k+1 (z) = 2

2k k! [p = 1, 2, 3, . . .]

WA 46(1)

[p = 0, 1, 2, 3, . . .]

WA 46(2)

WA 47(4)

k=0

4.

∞  k=1

5.

∞ 

(2k)2 J 2k (z) =

1 2 z 2

2k(2k + 1)(2k + 2) J 2k+1 (z) =

k=1

8.514 1.

∞  k=0

(−1)k J 2k+1 (z) =

sin z 2

WA 47(4)

1 3 z 2

WA 47(4)

WH

944

2.

Bessel Functions and Functions Associated with Them

J 0 (z) + 2

∞ 

(−1)k J 2k (z) = cos z

8.515

WH

k=1

3.

∞ 

(−1)k+1 (2k)2 J 2k (z) =

k=1

4.

∞ 

z sin z 2

(−1)k (2k + 1)2 J 2k+1 (z) =

k=0

5.

J 0 (z) + 2

∞ 

WA 32(9)

z cos z 2

WA 32(10)

J 2k (z) cos 2kθ = cos (z sin θ)

KU 120(14), WA 32

k=1

6.

∞ 

J 2k+1 (z) sin(2k + 1)θ =

k=0

7. 8.∗

∞  k=0 ∞  k=1

9.∗

J 2k+1 (x) =

1 2



sin (z sin θ) 2

KU 120(15), WA 32

x

J 0 (t) dt

[x is real]

WA 638

0

 (−1)k π 1 z J2k (z) = − N0 (z) + ln + C J0 (z) 2k 8 4 2

∞   (−1)k (2k + 1) π 1 z 1 J2k+1 (z) = − N1 (z) + ln + C − 1 J1 (z) − J0 (z) 2 (2k + 1) − 1 8 4 2 4z k=1

8.515 1.

 k ν  ∞  (−1)k tk 2z + t z J ν+k (z) = J ν (z + t) k! 2z z+t

AD (9140)

k=0

2.

3.

∞  k=1 ∞  k=0

8.516

  J 2k− 12 x2 = S (x)

MO 127a

  J 2k+ 12 x2 = C (x)

MO 127a

∞  (2n + 2k)(2n + k − 1)!

k!

k=0

The series 8.517 1.



∞  k=1

2.

∞ 

ak J k (kx) and

(−1)k J k (kz) = −

∞  k=1

J 2k (2kz) =

z 2(1 + z)

z2 2 (1 − z 2 )

WA 47

ak J k (kx)

z J k (kz) = 2(1 − z)

k=1

3.



2n

J 2n+2k (2z sin θ) = (z sin θ)

    z exp √1 − z 2    √  0, 0 ≤ t 0, t > 1, 2mπ < x(t − 1) < 2(m + 1)π, 2nπ < x(t + 1) < 2(n + 1)π, m + 1 and n + 1 are natural numbers. 8.525 1.

2.

∞ 

n  1 1  (−1)k J 0 (kx) cos kxt = − + 2 2 k=1 l=m+1 x2 − [(2l − 1)π − tx] ∞  k=1

(−1)k J 0 (kx) sin kxt =

MO 61

m 

n 1 1 1  + 2π l 2 l=1 l=1 [(2l − 1)π − tx] − x2 ⎧ ⎫ ∞ ⎨  1 1 ⎬  − + ⎩ 2lπ ⎭ 2 l=1 [(2l − 1)π + tx] − x2 ⎧ ⎫ ∞ ⎨  1 1 ⎬  − − ⎩ 2lπ ⎭ 2 l=n+1 [(2l − 1)π − tx] − x2

MO 61

948

3.

Bessel Functions and Functions Associated with Them ∞ 

8.526

n x 1 1 1 C + ln + π 4π 2π l l=1 m  1  − l=1 [(2l − 1)π − tx]2 − x2 ⎧ ⎫ ∞ ⎨  1 1 ⎬  − − ⎩ 2lπ ⎭ 2 l=1 [(2l − 1)π + tx] − x2 ⎧ ⎫ ∞ ⎨  1 1 ⎬  − − ⎩ 2lπ ⎭ 2 l=n+1 [(2l − 1)π − tx] − x2

(−1)k Y 0 (kx) cos kxt = −

k=1

MO 61

In formulas 8.525, x > 0, t > 1, (2m − 1)π < x(t − 1) < (2m + 1)π, (2n − 1)π < x(t + 1) < (2n + 1)π, m and n are natural numbers. 8.526 1.

2.

∞ x π 1 π C + ln + √ K 0 (kx) cos kxt = + 2 4π 2 2x 1 + t2 k=1 l=1 

∞ 1 π 1  + − 2 2lπ x2 + (2lπ + tx)2 l=1 ∞ 

1  − 2 2 2lπ x + (2lπ − tx) 1



⎧ ⎫ ∞ ∞ ⎨ ⎬   π  1 x 1 1  C + ln + (−1)k K 0 (kx) cos kxt = − ⎩ 2 4π 2 2lπ ⎭ 2 k=1 l=1 x2 + [(2l − 1)π − xt] ⎧ ⎫ ∞ 1 1 ⎬ π ⎨  − + ⎩ 2 2lπ ⎭ l=1 x2 + [(2l − 1)π + xt]2 [x > 0, t real] (see also 8.66)

MO 61

MO 62

8.53 Expansion in products of Bessel functions “Summation theorems”  8.53012 Suppose that r > 0, ρ > 0, ϕ > 0, and R = r2 + ρ2 − 2rρ cos ϕ; that is, suppose that r, ρ, and R are the sides of a triangle such that the angle between the sides r and ρ is equal to ϕ. Suppose also that ρ −1, all its zeros are real. WA 526, 530 A Bessel function Z ν (z) has no multiple zeros except possibly the coordinate origin. WA 528 8.542 All zeros of the function Y 0 (z) with positive real parts are real. WA 531 8.543 If −(2s + 2) < ν < −(2s + 1), where s is a natural number or 0, then J ν (z) has exactly 4s + 2 complex roots, two of which are purely imaginary. If −(2s + 1) < ν < −2s, where s is a natural number, then the function J ν (z) has exactly 4s complex zeros none of which are purely imaginary. WA 532 8.544 If xν and xν are, respectively, the smallest positive zeros of the functions J ν (z) and J ν (z) for ν > 0, then xν > ν and xν > ν. Suppose also that yν is the smallest positive zero of the function Y ν (z). Then, xν < yν < xν . WA 534, 536 Suppose that zν,m (for m = 1, 2, 3, . . . ) are the zeros of the function z −ν J ν (z), numbered in order of the absolute value of their real parts. Here, we assume that ν = −1, −2, −3, . . . . Then, for arbitrary z

8.551

Struve functions

951

 z ν

J ν (z) =

∞   z2 1− 2 Γ(ν + 1) m=1 zν,m 2

WA 550

8.5458 The number of zeros of the function z −ν J ν (z) that occur between the imaginary axis and the line on which   Re z = m + 12 Re ν + 14 π, WA 497 is exactly m. 8.546 For ν ≥0, the number of zeros of the function K ν (z) that occur in the region Re z − 12 WA 358(1)  z ν  π/2 2 2 2ν   2. Lν (z) = √ sinh (z cos ϕ) (sin ϕ) dϕ π Γ ν + 12 0   Re ν > − 12 WA 360(11)

952

Bessel Functions and Functions Associated with Them

8.552

8.552 Special cases: 1.6

2.6

  z n−2m−1  n−1

 2 Γ m + 1 2 1 2   − En (z) Hn (z) = [n = 1, 2, . . .] π m=0 Γ n + 12 − m   z −n+2m+1  n−1

 2 Γ n − m − 1 2 1 2  − E−n (z) H−n (z) = (−1)n+1 π m=0 Γ m + 32   z −2m+n− 12

3.

EH II 40(66), WA 337(1)

[n = 1, 2, . . .]

EH II 40(67), WA 337(2)

[n = 0, 1, . . .]

EH II 39(64)

 1 n 1  Γ m+ 2 2 Hn+ 12 (z) = Y n+ 12 (z) + π m=0 Γ(n + 1 − m) n

4.

H−(n+ 1 ) (z) = (−1) J n+ 12 (z)

[n = 0, 1, . . .]

EH II 39(65)

5.

L−(n+ 1 ) (z) = I n+ 12 (z) [n = 0, 1, . . .] 2 √ 2 H 12 (z) = √ (1 − cos z) πz  1/2   z 1/2  2 2 cos z  1+ 2 − sin z + H 32 (z) = 2π z πz z

EH II 39(65)

6. 7.

2

8.553 Functional relations:   1. Hν zeimπ = eiπ(ν+1)m Hν (z)

[m = 1, 2, 3, . . .]

d ν [z Hν (z)] = z ν Hν−1 (z) dz   −1  d  −ν z Hν (z) = 2−ν π −1/2 Γ ν + 32 3. − z −ν Hν+1 (z) dz  z ν   −1 Γ ν + 32 4. Hν−1 (z) + Hν+1 (z) = 2νz −1 Hν (z) + π −1/2 2  ν   −1  −1/2 z Γ ν + 32 5. Hν−1 (z) − Hν+1 (z) = 2 Hν (z) − π 2 8.554 Asymptotic representations:   ξ −2m+ν−1  1 Γ m + p−1   2 1  2 ν−2p−1   Hν (ξ) = Y ν (ξ) + + O |ξ| 1 π m=0 Γ ν+ 2 −m [|arg ξ| < π] For the asymptotic representation of Y ν (ξ), see 8.451 2. 8.555 The differential equation for Struve functions:  ν+1   2 1 4 z2 2   2   z y + zy + z − ν y = √ π Γ ν + 12 2.

EH II 39, WA 364(3)

WA 364(3)

WA 362(5) WA 358 WA 359 WA 359(5) WA 359(6)

EH II 39(63), WA 363(2)

WA 359(10)

8.566

Thomson functions

953

8.56 Thomson functions and their generalizations berν (z ), beiν (z ), herν (z ), heiν (z ), kerν (z ), keiν (z ) 8.561  3  1. berν (z) + i beiν (z) = J ν ze 4 πi   3 2. berν (z) − i beiν (z) = J ν ze− 4 πi 8.562 1.12 2.12

 3  ze 4 πi herν (z) + i heiν (z) = H (1) ν   − 34 πi ze herν (z) − i heiν (z) = H (1) ν

WA 96(6) WA 96(6)

(see also 8.567)

WA 96(7)

(see also 8.567)

WA 96(7)

8.563 1. 2.12

ber0 (z) ≡ ber(z); bei0 (z) ≡ bei(z) π π ker(z) ≡ − hei0 (z); kei(z) ≡ her0 (z) 2 2

WA 96(8) WA 96(8)

For integral representations, see 6.251, 6.536, 6.537, 6.772 4, 6.777. Series representation 8.564 1.

ber(z) =

∞  (−1)k z 4k k=0

2.

bei(z) =

∞ 

24k [(2k)!]

(−1)k z 4k+2 2

24k+2 [(2k + 1)!]  ∞ 2k   z 4k 1 π 2 ker(z) = ln − C ber(z) + bei(z) + (−1)k 2 4k z 4 m 2 [(2k)!] m=1 k=0

3.

WA 96(3)

2

WA 96(4)

WA 96(9)a, DW

k=1

4.

 ∞ 2k+1   1 z 4k+2 π 2 kei(z) = ln − C bei(z) − ber(z) + (−1)k 2 z 4 24k+2 [(2k + 1)!] m=1 m

WA 96(10)a, DW

k=0

8.565 ber2ν (z) + bei2ν (z) =

∞  k=0

2ν+4k

(z/2) k! Γ(ν + k + 1) Γ(ν + 2k + 1)

WA 163(6)

Asymptotic representation 8.566 1. 2. 3.

eα(z) ber(z) = √ cos β(z) 2πz

 |arg z|
0, 1, m = 0

WA 591(12)

WA 593(9)

2

S ν− 32 , 12

2, m > 0, 1, m = 0

(−1)m ε2m+1 J 2m+1 (z) [n ≥ 0] ,

n

εm =

w 2

WA 593(10)

8.579 Functional relations: 1. 2.

 z 2 ∂ U ν (w, z) = U ν−1 (w, z) + U ν+1 (w, z) ∂w w  z 2 ∂ V ν (w, z) = V ν+1 (w, z) + 2 V ν−1 (w, z) ∂w w 2

WA 593(2) WA 593(4)

Anger and Weber functions Jν (z) and Eν (z)

8.582

The function U ν (w, z) is a particular solution of the differential equation  w ν−2 ∂2U z2U 1 ∂U + − = J ν (z) ∂z 2 z ∂z w2 z

3.

The function V ν (w, z) is a particular solution of the differential equation  w −ν ∂2V z2V 1 ∂V + 2 = − J −ν+2 (z) 2 ∂z z ∂z w z

4.

957

WA 592(2)

WA 592(3)

8.58 Anger and Weber functions Jν (z) and Eν (z) 8.580 Definitions: 1.

The Anger function Jν (z): Jν (z) =

1 π

The Weber function Eν (z):

2.

Eν (z) =

1 π



π

cos (νθ − z sin θ) dθ

WA 336(1), EH II 35(32)

sin (νθ − z sin θ) dθ

WA 336(2), EH II 35(32)

0



π

0

8.581 Series representations: 1.

 2n ∞ (−1)n z2 νπ      Jν (z) = cos 2 n=0 Γ n + 1 + 12 ν Γ n + 1 − 12 ν  2n+1 ∞ (−1)n z2 νπ      + sin 2 n=0 Γ n + 32 + 12 ν Γ n + 32 − 12 ν EH II 36(36), WA 337(3)

 2n (−1)n z2 νπ     Eν (z) = sin 2 n=0 Γ n + 1 + 12 ν Γ n + 1 − 12 ν  2n+1 ∞ (−1)n z2 νπ      − cos 2 n=0 Γ n + 32 + 12 ν Γ n + 32 − 12 ν ∞ 

2.

EH II 36(37), WA 338(4)

8.582 Functional relations: 1.6

2 Jν (z) = Jν−1 (z) − Jν+1 (z)

2.6

2 Eν (z) = Eν−1 (z) − Eν+1 (z)

3.

6

4.6

Jν−1 (z) + Jν+1 (z) = 2νz

−1

EH II 36(40), WA 340(2) EH II 36(41), WA 340(6) −1

Jν (z) − 2(πz)

sin(νπ)

Eν−1 (z) + Eν+1 (z) = 2νz −1 Eν (z) − 2(πz)−1 (1 − cos νπ)

EH II 36(42), WA 340(1) EH II 36(43), WA 340(5)

958

Bessel Functions and Functions Associated with Them

8.583

8.583 Asymptotic expansions: ⎡     p−1 1+ν  Γ n + 1−ν sin νπ n 2n Γ n+ 2 2 6 ⎣  1−ν  z −2n 1. Jν (z) = J ν (z) + (−1) 2 1+ν πz Γ Γ 2 2 n=0 ⎤     p−1 1 1      Γ n + 1 + ν Γ n + 1 − ν −2p 2     2 z −2n−1 + ν O |z|−2p−1 ⎦ −ν + O |z| (−1)n 22n 1 1 Γ 1 + ν Γ 1 − ν 2 2 n=0 [|arg z| < π] 2.

EH II 37(47), WA 344(1)

Eν (z) = − Y ν (z)

 p−1      1+ν 1−ν   1 + cos(νπ)  −2p n 2n Γ n + 2  Γ n +  2 −2n z (−1) 2 + O |z| − πz Γ 1−ν Γ 1+ν 2 2 n=0  p−1      1 1   ν −2n−1 ν (1 − cos νπ)  −2p−1 n 2n Γ n +1 + 2 ν  Γ n + 1 − 2  z − (−1) 2 + O |z| zπ Γ 1 + 12 ν Γ 1 − 12 ν n=0 WA344(2), EH II 37(48)

For the asymptotic expansion of J ν (z) and Y ν (z), see 8.451. 8.584 The Anger and Weber functions satisfy the differential equation  ν2  −1  y + z y + 1 − 2 y = f (ν, z), z z−ν sin νπ for Jν (z) where f (ν, z) = πz 2 1 and f (ν, z) = − 2 [z + ν + (z − ν) cos νπ] for Eν (z) πz

WA 341(9), EH II 37(44) EH II 37(45), WA 341(10)

8.59 Neumann’s and Schl¨ afli’s polynomials: O n (z) and S n (z) 8.590 Definition of Neumann’s polynomials 1.

n2  1  n(n − m − 1)!  z 2m−n−1 O n (z) = 4 m=0 m! 2

[n ≥ 1]

WA 299(2), EH II 33(6)

2.

O −n (z) = (−1)n O n (z)

[n ≥ 1]

WA 303(8)

3.

O 0 (z) =

4. 5.

1 z 1 O 1 (z) = 2 z 4 1 O 2 (z) = + 3 z z

WA 299(3), EH II 33(7) EH II 33(7) EH II 33(7)

In general, O n (z) is a polynomial in z −1 of degree n + 1. 8.591 Functional relations: 1.

O 0 (z) = − O 1 (z)

2.

2 O n (z)

= O n−1 (z) − O n+1 (z)

EH II 33(9), WA 301(3)

[n ≥ 1]

EH II 33(10), WA 301(2)

8.597

3.

4.12 5.

Mathieu’s equation

959

   π 2 (n − 1) O n+1 (z) + (n + 1) O n−1 (z) − 2z −1 n2 − 1 O n (z) = 2nz −1 sin n 2 [n ≥ 1] EH II 33(11), WA 301(1)   2  2  π nz O n−1 (z) − n − 1 O n (z) = (n − 1)z O n (z) + n sin n EH II 33(12), WA 303(4) 2    π 2 nz O n+1 (z) − n2 − 1 O n (z) = −(n + 1)z O n (z) + n sin n EH II 33(13), WA 303(5)a 2

8.592 The generating function:

∞  1 = J 0 (ξ)z −1 + 2 J n (ξ) O n (z) z−ξ n=1

[|ξ| < |z|]

8.593 The integral representation: √ √ n  n  ∞ u + u2 + z 2 + u − u2 + z 2 e−u du O n (z) = 2z n+1 0 See also 3.547 6, 8, 3.549 1, 2. 8.594 The inequality −n−1

1

EH II 32(1), WA 298(1)

EH II 32(3), WA 305(1)

2

|O n (z)| ≤ 2n−1 n!|z| e 4 |z| [n > 1] EH II 33(8), WA 300(8) 8.595 Neumann’s polynomial O n (z) satisfies the differential equation    π 2 dy  2 π 2 d2 y z 2 2 + 3z + n sin n EH II 33(14), WA 303(1) + z + 1 − n2 y = z cos n dz dz 2 2 8.596 Schl¨afli’s polynomials S n (z). These are the functions that satisfy the formulas 1. 2.

3.

S 0 (z) = 0

EH II 34(18), WA 312(2)



  1 π 2 2zOn (z) − 2 cos n S n (z) = n 2 n

 2   (n − m − 1)! z 2m−n = m! 2 m=0

[n ≥ 1]

EH II 34(19), WA 312(3)

[n ≥ 1]

EH II 34(18)

S −n (z) = (−1)n+1 S n (z)

WA 313(6)

8.597 Functional relations: 1.

S n−1 (z) + S n+1 (z) = 4 O n (z)

WA 313(7)

Other functional relations may be obtained from 8.591 by replacing O n (z) with the expression for S n (z) given by 8.596 2.

8.6 Mathieu Functions 8.60 Mathieu’s equation  d2 y  + a − 2k 2 cos 2z y = 0, 2 dz

k2 = q

MA

960

Mathieu Functions

8.610

8.61 Periodic Mathieu functions 8.610 In general, Mathieu’s equation 8.60 does not have periodic solutions. If k is a real number, there exist infinitely many eigenvalues a, not identically equal to zero, corresponding to the periodic solutions y(z) = y(2π + z), If k is nonzero, there are no other linearly independent periodic solutions. Periodic solutions of Mathieu’s equations are called Mathieu’s periodic functions or Mathieu functions of the first kind, or, more simply, Mathieu functions. 8.611 Mathieu’s equation has four series of distinct periodic solutions: 1.

ce2n (z, q) =

∞  r=0

2.

ce2n+1 (z, q) =

(2n)

A2r cos 2rz

∞  r=0

3.

se2n+1 (z, q) =

∞  r=0

4.

se2n+2 (z, q) =

∞  r=0

5.

MA

(2n+1)

A2r+1 cos(2r + 1)z

MA

(2n+1)

B2r+1 sin(2r + 1)z

MA

(2n+2)

B2r+2 sin(2r + 2)z

MA

The coefficients A and B depend on q. The eigenvalues a of the functions ce2n , ce2n+1 , se2n , se2n+1 are denoted by a2n , a2n+1 , b2n , b2n+1 .

8.612 The solutions of Mathieu’s equation are normalized so that  2π y 2 dx = π 8.613 1. 2. 3.

MO 65

0

1 lim ce0 (x) = √ 2 lim cen (x) = cos nx

q→0

[n = 0]

q→0

lim sen (x) = sin nx

MO 65

q→0

(2n)

(2n+1)

(2n+1)

(2n+2)

8.62 Recursion relations for the coefficients A2r , A2r+1 , B2r+1 , B2r+2 8.621 1. 2. 3.

(2n)

aA0

(2n)

− qA2

=0   (2n) (2n) (2n) =0 (a − 4)A2 − q A4 + 2A0    (2n)  (2n) (2n) a − 4r2 A2r − q A2r+2 + A2r−2 = 0

MA MA

[r ≥ 2]

MA

8.630

Mathieu functions with a purely imaginary argument

961

8.622 1. 2.

(2n+1)

(2n+1)

− qA3 =0 (a − 1 − q)A1     (2n+1) (2n+1) (2n+1) =0 a − (2r + 1)2 A2r+1 − q A2r+3 + A2r−1

MA

[r ≥ 1]

MA

8.623 1. 2.

(2n+1)

(2n+1)

− qB3 =0 (a − 1 + q)B1     (2n+1) (2n+1) (2n+1) =0 a − (2r + 1)2 B2r+1 − q B2r+3 + B2r−1

MA

[r ≥ 1]

MA

8.624 1. 2.11

(2n+2)

(2n+2)

− qB4 =0 (a − 4)B2     (2n+2) (2n+2) (2n+2) =0 a − 4r2 B2r − q B2r+2 + B2r−2

MA

[r ≥ 2]

MA

8.625 We can determine the coefficients A and B from equations 8.612, 8.613 and 8.621-8.624 pro(2n) vided a is known. Suppose, for example, that we need to determine the coefficients A2r for the function ce2n (z, q). From the recursion formulas, we have    a −q 0 0 0 . . .  −2q a − 4 −q 0 0 . . .   0 −q a − 16 −q 0 . . .  ST 1.  =0  0 0 −q a − 36 −q     0 0 0 −q a − 64    .. .. .. . .   . . . . For given q in equation 8.625 1, we may determine the eigenvalues 2.12

a = a0 , a2 , a4 , . . .

[|a0 | ≤ |a2 | ≤ |a4 | ≤ . . .]

3.

If we now set a = a2n , we can determine the coefficients A2r from the recursion formulas 8.621 up to a proportionality coefficient. This coefficient is determined from the formula ∞   2  2 (2n) (2n) 2 A0 A2r + = 1, MA

(2n)

r=1

which follows from the conditions of normalization.

8.63 Mathieu functions with a purely imaginary argument 8.630 If, in equation 8.60, we replace z with iz, we arrive at the differential equation 1.11

d2 y + (−a + 2q cosh 2z) y = 0 dz 2

We can find the solutions of this equation if we replace the argument z with iz in the functions cen (z, q) and sen (z, q). The functions obtained in this way are called associated Mathieu functions of the first kind and are denoted as follows:

962

Mathieu Functions

Ce2n (z, q),

1. 8.631

Ce2n (z, q) =

1.

Ce2n+1 (z, q), ∞  r=0

Ce2n+1 (z, q) =

2.

Se2n+1 (z, q) =

(2n)

∞ 

∞  r=0

Se2n+2 (z, q) =

4.

Se2n+2 (z, q)

A2r cosh 2rz

r=0

3.

Se2n+1 (z, q),

8.631

∞  r=0

MA

(2n+1)

A2r+1 cosh(2r + 1)z

MA

(2n+1)

B2r+1 sinh(2r + 1)z

MA

(2n+2)

B2r+2 sinh(2r + 2)z

MA

8.64 Non-periodic solutions of Mathieu’s equation Along with each periodic solution of equation 8.60, there exists a second non-periodic solution that is linearly independent. The non-periodic solutions are denoted as follows: fe2n (z, q), fe2n+1 (z, q), ge2n+1 (z, q), ge2n+2 (z, q) Analogously, the second solutions of equation 8.630 1 are denoted by Fe2n (z, q),

Fe2n+1 (z, q),

Ge2n+1 (z, q),

8.65 Mathieu functions for negative q 8.651 If we replace the argument z in equation 8.60 with±

π

2 d2 y + (a + 2q cos 2z) y = 0 dz 2 This equation has the following solutions:

8.652 1. 2. 3. 4. 5. 6. 7. 8.

1

 − z, q   ce2n+1 (z, −q) = (−1)n se2n+1 12 π − z, q   se2n+1 (z, −q) = (−1)n ce2n+1 12 π − z, q   se2n+2 (z, −q) = (−1)n se2n+2 12 π − z, q   fe2n (z, −q) = (−1)n+1 fe2n 12 π − z, q   fe2n+1 (z, −q) = (−1)n ge2n+1 12 π − z, q   ge2n+1 (z, −q) = (−1)n fe2n+1 12 π − z, q   ge2n+2 (z, −q) = (−1)n ge2n+2 12 π − z, q

ce2n (z, −q) = (−1)n ce2n

2π

Ge2n+2 (z, q)

 ± z , we get the equation MA

MA MA MA MA MA MA MA MA

8.662

Representation of Mathieu functions

963

π i + z in equation 8.630 1 we get the equation 2 d2 y − (a + 2q cosh z) y = 0 dz 2

8.653 Analogously, if we replace z with

It has the following solutions: 8.654 1. 2. 3. 4. 5. 6.11 7.

11

8.11

 i + z, q 2   n+1 Ce2n+1 (z, −q) = (−1) i Se2n+1 12 πi + z, q   Se2n+1 (z, −q) = (−1)n+1 i Ce2n+1 12 πi + z, q   Se2n+2 (z, −q) = (−1)n+1 Se2n+2 12 πi + z, q   Fe2n (z, −q) = (−1)n Fe2n 12 πi + z, q   Fe2n+1 (z, −q) = (−1)n+1 i Ge2n+1 12 πi + z, q   Ge2n+1 (z, −q) = (−1)n+1 i Fe2n+1 12 πi + z, q   Ge2n+2 (z, −q) = (−1)n+1 Ge2n+2 12 πi + z, q Ce2n (z, −q) = (−1)n Ce2n

π

MA MA MA MA MA MA MA MA

8.66 Representation of Mathieu functions as series of Bessel functions 8.661 1.

ce2n (z, q) = =

2.

π

2,q (2n) A0

ce2n

ce2n (0, q) (2n) A0

ce2n+1 (z, q) = −



∞ 

r=0 ∞  r=0

ce2n+1

(−1)r A2r J 2r (2k cos z) (2n)

(−1)r A2r I 2r (2k sin z)

π

(2n)

2,q (2n+1) kA1



∞  r=0

(−1)r A2r+1 J 2r+1 (2k cos z) (2n+1)

∞  ce2n+1 (0, q) (2n+1) cot z (−1)r (2r + 1)A2r+1 I 2r+1 (2k sin z) kA1 (2n + 1) r=0 π  ∞  se2n+1 2 , q (2n+1) se2n+1 (z, q) = tan z (−1)r (2r + 1)B2r+1 J 2r+1 (2k cos z) (2n+1) kB1 r=0 ∞ se2n+1 (0, q)  (2n+1) = (−1)r B2r+1 I 2r+1 (2k sin z) (2n+1) kB1 r=0 π  ∞   − se2n+2 2 , q (2n+2) se2n+2 (z, q) = tan z (−1)r (2r + 2)B2r+2 J 2r+2 (2k cosz ) (2n+2) 2 k B2 r=0 ∞   se2n+2 (0, q) (2n+2) = cot z (−1)r (2r + 2)B2r+2 I 2r+2 (2k sin z) (2n+2) 2 k B2 r=0

=

3.

4.

8.662 1.

MA MA

MA MA

MA MA

MA MA

∞

fe2n (z, q) = −

     π fe2n (0, q)  (2n) π  (−1)r A2r Im J r keiz Y r ke−iz 2 ce2n 2 , q r=0

MA

964

2.

Mathieu Functions

fe2n+1 (z, q) =

8.663

πk fe2n+1 (0, q)   2 ce2n+1 π2 , q ∞           (2n+1) (−1)r A2r+1 Im J r keiz Y r+1 ke−iz + J r+1 keiz Y r ke−iz × r=0

MA

3.

πk ge2n+1 (0, q)   2 se2n+1 π2 , q ∞           (2n+1) (−1)r B2r+1 Re J r keiz Y r+1 ke−iz − J r+1 keiz Y r ke−iz ×

ge2n+1 (z, q) = −

r=0

MA

4.

πk 2 ge2n+2 (0, q)   2 se2n+2 12 π, q ∞           (−1)r Re J k keiz Y r+2 ke−iz − J r+2 keiz Y r ke−iz ×

ge2n+2 (z, q) = −

r=0

MA

The expansions of the functions Fen and Gen as series of the functions Y ν are denoted, respectively, by Feyn and Geyn and the expansions of these functions as series of the functions K ν are denoted, respectively, by Fekn and Gekn . 8.663 1.12

∞

Fey2n (z, q) =

=

ce2n (0, q)  (2n)

A0

π

2,q (2n) A0

ce2n

r=0



(2n)

A2r Y 2r (2k sinh z)

∞ 

k 2 = q,

|sinh z| > 1,

Re z > 0

 MA

r

(−1)

r=0

ce2n (0, q) ce2n =  2 (2n) A0



(2n) A2r

Y 2r (2k cosh z) [|cosh z| > 1]

π

 ∞  2,q r=0

(2n) (−1)r A2r



J r ke

 −z

MA

Y r (kez ) MA

8.663

2.

Representation of Mathieu functions

Fey2n+1 (z, q) =

965

∞ ce2n+1 (0, q) coth z  (2n+1) (2r + 1)A2r+1 Y 2r+1 (2k sinh z) , kA1 (2n + 1) r=0

k 2 = q,

=−

ce2n+1

π

2,q (2n+1) kA1



∞ 

∞ 

(−1)

r=0

r=0

(−1)r A2r+1

(2n+1)

Re z > 0] MA

r

(2n+1) A2r+1

Y 2r+1 (2k cosh z) [|cosh z| > 1]

ce2n+1 (0, q) ce2n+1 =− 2  (2n+1) k A1 ×

[|sinh z| > 1,



π

2,q



MA

     J r ke−z Y r+1 (kez ) + J r+1 ke−z Y r (kez ) MA

∞

3.12

Gey2n+1 (z, q) =

=

se2n+1 (0, q)  (2n+1)

kB1

se2n+1

π

2,q

(2n+1)

kB1

r=0

(2n+1)

B2r+1 Y 2r+1 (2k sinh z) [|sinh z| > 1,

 tanh z

se2n+1 (0, q) se2n+1 = 2  (2n+1) k B1

∞ 

MA

(−1)r (2r +

r=0

π

Re z > 0]

2,q

(2n+1) 1)B2r+1

Y 2r+1 (2k cosh z) [|cosh z| > 1]



∞  r=0

MA r

(−1)

(2n+1) B2r+1

      × J r ke−z Y r+1 (kez ) − J r+1 ke−z Y r (kez ) MA

966

4.

Mathieu Functions

Gey2n+2 (z, q) =

se2n+2 (0, q)

coth z

(2n+2) k 2 B2

=−

se2n+2

∞  r=0

π

2,q

(2n+2)

k 2 B2

 tanh z

8.664

(2n+2)

(2r + 2)B2r+2 Y 2r+2 (2k sinh z) [|sinh z| > 1,

∞  r=0

Re z > 0] MA

(−1)r (2r + 2)B2r+2 Y 2r+2 (2k cosh z) (2n+2)

[|cosh z| > 1] MA

∞ se2n+2 (0, q) se2n+2 π ,q  (2 ) (2n+2) = (−1)r B2r+2  2 (2n+2) r=0 k 2 B2

      × J r ke−z Y r+2 (kez ) − J r+2 ke−z Y r (kez ) MA

8.664 1.

∞

Fek2n (z, q) =

ce2n (0, q)  (2n) πA0

r=0

(−1)r A2r K 2r (−2ik sinh z) (2n)

k 2 = q,

[|sinh z| > 1,

Re z > 0] MA

2.

3.

4.

Fek2n+1 (z, q) =

Gek2n+1 (z, q) = Gek2n+2 (z, q) =

ce2n+1 (0, q) (2n+1) πkA1

π

2,q (2n+1) πkB1

se2n+1 se2n+2

π

coth z

∞  r=0

(2n+1)

k2 = q

[|sinh z| > 1,

Re z > 0] MA

 tanh z

∞  r=0



2,q (2n+2) 2 πk B2

(−1)r (2r + 1)A2r+1 K 2r+1 (−2ik sinh z)

tanh z

∞  r=0

(2n+1)

(2r + 1)B2r+1 K 2r+1 (−2ik cosh z) (2n+2)

(2r + 2)B2r+2 K 2r+2 (−2ik cosh z)

MA

MA

8.67 The general theory If iμ is not an integer, the general solution of equation 8.60 can be found in the form 8.671 ∞ ∞   c2r e2rzi + Be−μz c2r e−2rzi 1. y = Aeμz r=−∞

MA

r=−∞

The coefficients c2r can be determined from the homogeneous system of linear algebraic equations 2.11

c2r + ξ2r (c2r+2 + c2r−2 ) = 0, where

r = . . . , −2, −1, 0, 1, 2, . . . ,

MA

8.700

Introduction

ξ2r =

3.7

The condition that this  · · ·  · ξ−4 1  · 0 ξ−2 Δ (iμ) =  0 0 · · 0 0  · · ·

967

q 2

(2r − iμ) − a

system be compatible yields an equation that μ must satisfy:  · · · · · · ξ−4 0 0 0 0 · 1 ξ−2 0 0 0 · =0 ξ0 1 ξ0 0 0 · 0 ξ2 1 ξ2 0 · · · · · · ·

MA

4.

This equation can also be written in the form  √ π a , where Δ(0) is the value that is assumed by the determinant cosh μπ = 1 − 2Δ(0) sin2 2 of the preceding article if we set μ = 0 in the expressions for ξ2r .

5.

If the pair (a, q) is such that |cosh μπ| < 1, then μ = iβ, Im β = 0, and the solution 8.671 1 is bounded on the real axis.

6.

If |cosh μπ| > 1, μ may be real or complex and the solution 8.671 1 will not be bounded on the real axis.

7.

If cosh μπ = ±1, then iμ will be an integer. In this case, one of the solutions will be of period π or 2π (depending on whether n is even or odd). The second solution is non-periodic (see 8.61 and 8.64).

8.7–8.8 Associated Legendre Functions 8.70 Introduction 8.700 An associated Legendre function is a solution of the differential equation     2 μ2 du 2 d u 1. 1−z + ν(ν + 1) − u = 0, − 2z dz 2 dz 1 − z2 in which ν and μ are arbitrary complex constants. This equation is a special case of (Riemann’s) hypergeometric equation (see 9.151). The points +1, −1, ∞ are, in general, its singular points, specifically, its ordinary branch points. We are interested, on the one hand, in solutions of the equation that correspond to real values of the independent variable z that lie in the interval [−1, 1] and, on the other hand, in solutions corresponding to an arbitrary complex number z such that Re z > 1. These are multiple-valued in the z-plane. To separate these functions into single-valued branches, we make a cut along the real axis from −∞ to +1. We are also interested in those solutions of equation 8.700 1 for which ν or μ or both are integers. Of special significance is the case in which μ = 0.

968

Associated Legendre Functions

8.701

8.701 In connection with this, we shall use the following notations: The letter z will denote an arbitrary complex variable; the letter x will denote a real variable that varies over the interval [−1, +1]. We shall sometimes set x = cos ϕ, where ϕ is a real number. We shall use the symbols P μν (z), Q μν (z) to denote those solutions of equation 8.700 1, that are singlevalued and regular for |z| 1. When these functions cannot be unrestrictedly extended without violating their single-valuedness we make a cut along the real axis to the left of the point z = 1. The values of the functions P μν (z) and Q μν (z) on the upper and lower boundaries of that portion of the cuts lying between the points −1 and +1 are denoted respectively by P μν (x ± i0) , Q μν (x ± i0) The letters n and m denote natural numbers or zero. The letters ν and μ denote arbitrary complex numbers unless the contrary is stated. The upper index will be omitted when it is equal to zero. That is, we set P 0ν (z) = P ν (z), Q 0ν (z) = Q ν (z) The linearly independent functions  μ  z+1 2 1 1−z μ F −ν, ν + 1; 1 − μ; 8.702 P ν (z) = Γ(1 − μ) z − 1  2  z+1 arg = 0, if z is real and greater than 1 and MO 80, WH z−1     μ2 −ν−μ−1 eμπi Γ(ν + μ + 1) Γ 12  2 3 1 ν +μ+2 ν +μ+1   , ; ν + ; z 8.703 Q μν (z) = − 1 z F 2 2 2 z2 2ν+1 Γ ν + 32   2 [arg z − 1 = 0 when z is real and greater than 1; arg z = 0 when z is real and greater than zero] which are solutions of the differential equation 8.700 1, are called associated Legendre functions (or spherical functions) of the first and second kinds respectively. They are uniquely defined, respectively, in the intervals |1 − z| 1 with the portion of the real axis that lies between −∞ and +1 excluded. They can be extended by means of hypergeometric series to the entire z-plane where the above-mentioned cut was made. These expressions for P μν (z) and Q μν (z) lose their meaning when 1 − μ and ν + 32 are nonpositive integers respectively. MO 80 When z is a real number lying on the interval [−1, +1], so that (z = x = cos ϕ), we take the following functions as linearly independent solutions of the equation  8.704 P μν (x) =

1

1

e 2 μπi P μν (cos ϕ + i0) + e− 2 μπi P μν (cos ϕ − i0) EH I 143(1)  μ2  1+x 1 1−x = F −ν, ν + 1; 1 − μ; EH I 143(6) Γ(1 − μ) 1 − x 2  1  1 8.705 Q μν (x) = 12 e−μπi e− 2 μπi Q μν (x + i0) + e 2 μπi Q μν (x − i0) EH I 143(2)   π Γ (ν + μ + 1) P μν (x) cos μπ − P −μ (x) = (cf. 8.732 5) 2 sin μπ Γ(ν − μ + 1) ν If μ = ±m is an integer, the last equation loses its meaning. In this case, we get the following formulas by passing to the limit: 8.706 1. 2.11

1 2

  m dm m Qm 1 − x2 2 Q (x) ν (x) = (−1) dxm ν Γ(ν − m + 1) m Q (x) Q −m ν (x) = Γ(ν + m + 1) ν

(cf. 8.752 1)

EH I 149(7) EH I 144(18)

8.711

Integral representations

969

The functions Q μν (z) are not defined when ν + μ is equal to a negative integer. Therefore, we must exclude the cases when ν + μ = −1, −2, −3, . . . for these formulas. The functions P ±μ Q ±μ P ±μ Q ±μ ν (±z) , ν (±z) , −ν−1 (±z) , −ν−1 (±z) are linearly independent solutions of the differential equation for ν + μ = 0, ±1, ±2, . . . . 8.707 Nonetheless, two linearly independent solutions can always be found. Specifically, for ν ± μ not an integer, the differential equation 8.700 1 has the following solutions: 1.

P ±μ ν (±z) ,

Q ±μ ν (±z) ,

P ±μ −ν−1 (±z) ,

Q ±μ −ν−1 (±z)

respectively, for z = x = cos ϕ, 2.

P ±μ ν (±x) ,

Q ±μ ν (±x) ,

P ±μ −ν−1 (±x) ,

Q ±μ −ν−1 (±x)

If ν ± μ is not an integer, the solutions 3.

P μν (z),

Q μν (z), respectively, and P μν (x),

Q μν (x)

are linearly independent. If ν ±μ is an integer but μ itself is not an integer, the following functions are linearly independent solutions of equation 8.700 1: 4.

P μν (z),

μ P −μ ν (z), respectively, and P ν (x),

P −μ ν (x)

If μ = ±m, ν = n, or ν = −n − 1, the following functions are linearly independent solutions of equation 8.700 1 for n ≥m: 5.

Pm n (z),

m Qm n (z), respectively, and P n (x),

Qm n (x),

and for n − 12 ,

|arg (z ± 1)| < π

 MO 88

 ν  (ν + 1)(ν + 2) . . . (ν + m) 2 − 1 cos ϕ z + Pm (z) = z cos mϕ dϕ ν π 0  π ν(ν − 1) . . . (ν − m + 1) cos mϕ dϕ = (−1)m √  ν+1 π 0 z + z 2 − 1 cos ϕ     π π (cf. 8.822 1) SM 483(15), WH |arg z| < , arg z + z 2 − 1 cos ϕ = arg z for ϕ = 2 2   2  μ2 ∞ √ sinh2μ t dt eμπi Γ(ν + μ + 1)  z Q μν (z) = π μ  − 1 √  ν+μ+1 2 Γ μ + 12 Γ(ν − μ + 1) 0 z + z 2 − 1 cosh t [Re (ν ± μ) > −1, |arg (z ± 1)| < π] (cf. 8.822 2) MO 88 π

970

Associated Legendre Functions

Q μν (z)

4.

 5.

1

12 −1

6.∗



1

−1

eμπi Γ(ν + 1) = Γ(ν − μ + 1)



P 2m l (x) P l (x) =

∞

0

P 2l (x) P 0l (x) dx = −

8.712

cosh μt dt √ ν+1 z + z 2 − 1 cosh t [Re(ν + μ) > −1, ν = −1, −2, −3, . . . , 

|arg (z ± 1)| < π]

WH, MO 88

1 l(l − 1) l! = −2 (l − 2)! 2l + 1 2l + 1

2l! (−1)m 2l + 1 (l − 2m)!

 ν  μ2 1  eμπi Γ(ν + μ + 1)  2 z 1 − t2 (z − t)−ν−μ−1 dt − 1 ν+1 2 Γ(ν + 1) −1 [Re(ν + μ) > −1, Re μ > −1, |arg (z ± 1)| < π] (cf. 8.821 2)

8.71212 Q μν (z) =

8.713 1.

MO 88a, EH I 155(5)a

 1

     ∞ eμπi Γ μ + 1 π −(ν+ 12 )t μ   t dt cos ν + e dt 2 2 √ Q μν (z) = z2 − 1 2 − cos νπ μ+ 12 μ+ 12 2π 0 (z − cos t) 0 (z + cosh t)   Re μ > − 21 , Re(ν + μ) > −1, |arg (z ± 1)| < π

MO 89



2.

P −μ ν (z)

μ  ∞ z2 − 1 2 sinh2ν+1 t = ν dt 2 Γ(μ − ν) Γ(ν + 1) 0 (z + cosh t)ν+μ+1 [Re z > −1, |arg (z ± 1)| < π, Re(ν + 1) > 0, 

3.

P −μ ν (z)

=

8.714 1.

2.

  μ  ∞   cosh ν + 12 t dt 2 Γ μ + 12 z 2 − 1 2 π Γ(ν + μ + 1) Γ(μ − ν) 0 (z + cosh t)μ+ 12 [Re z > −1, |arg (z ± 1)| < π, Re(ν + μ) > −1,

 P μν

(cos ϕ) =

P −μ ν (cos ϕ) =

2 sinμ ϕ   π Γ 12 − μ

 0

ϕ

  cos ν + 12 t dt (cos t − cos ϕ) μ

2μ

 0 < ϕ < π,

μ+ 12

Γ(2μ + 1) sin ϕ Γ(μ + 1) Γ(ν + μ + 1) Γ(μ − ν)

Re(μ − ν) > 0]

MO 89

Re(μ − ν) > 0]

MO 89

Re μ
−1, Re(μ − ν) > 0] MO 89

3.

Q μν (cos ϕ) =

μ

1 Γ(ν + μ + 1) sin ϕ   2μ+1 Γ(ν − μ + 1) Γ μ + 12    ∞ sinh2μ t sinh2μ t dt × ν+μ+1 + ν+μ+1 (cos ϕ + i sin ϕ cosh t) (cos ϕ − i sin ϕ cosh t) 0   Re(ν + μ + 1) > 0, Re(ν − μ + 1) > 0, Re μ > − 12

MO 89

8.721

4.

Asymptotic series

P μν (cos ϕ) =

8.715 1.

P μν

i Γ(ν + μ + 1) sinμ ϕ   2μ Γ (ν − μ + 1) Γ μ + 12    ∞ sinh2μ t sinh2μ t dt × − (cos ϕ + i sin ϕ cosh t)ν+μ+1 (cos ϕ − i sin ϕ cosh t)ν+μ+1 0   Re (ν ± μ + 1) > 0, Re μ > − 12

  √  α cosh ν + 12 t dt 2 sinhμ α   (cosh α) = √ 1 π Γ 12 − μ 0 (cosh α − cosh t)μ+ 2 

2.

971

Q μν (cosh α) =

π eμπi sinhμ α   2 Γ 12 − μ



∞

α

ν+ 12

e −(

 α > 0,

Re μ
0,

Re μ < 12 ,

1 2



MO 89

MO 87

)t dt μ+ 12

(cosh t − cosh α)

Re(ν + μ) > −1



MO 87

See also 3.277 1, 4, 5, 7, 3.318, 3.516 3, 3.518 1, 2, 3.542 2, 3.663 1, 3.894, 3.988 3, 6.622 3, 6.628 1, 4–7, and also 8.742.

8.72 Asymptotic series for large values of |ν| 8.7216 1.12

For real values of μ, |ν|  1, |ν|  |μ|, |arg ν| < π, we have:   μπ  π   1 ∞ 1 cos (2k − 1) + ν + k + ϕ +  Γ μ+k+ 2 2 2 4 2   P μν (cos ϕ) = √ Γ(ν + μ + 1)   1 k+ 12 3 π Γ μ − k + (2 sin ϕ) k! Γ ν − k + 2 k=0 2  3 ν + μ = −1, −2, −3, . . . ; ν = − 2 , − 52 , − 27 . . . ; for π6 < ϕ
0, μ > 0 and 0 < ε ≤ ϕ ≤ π − ε MO 92

2.6

Q μν (cos ϕ) =

√

π Γ(ν + μ + 1)   μπ  π   ∞ 1 cos ν + k + 12 ϕ − (2k − 1) +  Γ μ + k + 2 4 2 × (−1)k    1 k+ 12 3 Γ μ − k + k! Γ ν + k + 2 (2 sin ϕ) 2 k=0  ν + μ = −1, −2, −3, . . . ; ν = − 23 , − 52 , − 27 , . . . ; for

π 6

0, μ > 0, 0 < ε ≤ ϕ ≤ π − ϕ EH I 147(6), MO 92

972

3.

Associated Legendre Functions

2 Γ(ν + μ + 1) cos   P μν (cos ϕ) = √ π Γ ν + 32



 π ν + 12 ϕ − + 4 √ 2 sin ϕ

μπ 2







8.722

  1 1+O ν

0 < ε ≤ ϕ ≤ π − ε,

4. 5.

|ν| 

1 ε

 MO 92

For ν > 0, μ > 0 and ν > μ, it follows from formulas 8.721 1 and 8.721 2 that     2 1 π μπ 1 cos ν + ϕ− + +O √ ν −μ P μν (cos ϕ) = νπ sin ϕ 2 4 2 ν3     π 1 π μπ 1 μ −μ cos ν + ϕ+ + O √ ν Q ν (cos ϕ) = 2ν sin ϕ 2 4 2 ν3   1 0 < ε ≤ ϕ ≤ π − ε; ν  ε

MO 92

8.722 If ϕ is sufficiently close to 0 or π that νϕ or ν(π −ϕ) is small in comparison with 1, the asymptotic formulas 8.721 become unsuitable. In this case, the following asymptotic representation is applicable for μ ≤ 0, ν 1, and small values of ϕ:  μ    1 ϕ η ϕ −μ 2 ϕ J μ+1 (η) 1. ν+ cos − J μ+2 (η) + J μ+3 (η) +O sin4 P ν (cos ϕ) = J μ (η)+sin 2 2 2 2η 6 2 where η = (2ν + 1) sin ϕ2 . In particular, it follows that  x = J μ (x) 1. lim ν μ P −μ cos ν ν→∞ ν

[x ≥ 0, μ ≥ 0]

MO 93

8.723 We can see how the functions P μν (z) and Q μν (z) behave for large |ν| and real values of z > 1.12

2.

P μν

   Γ −ν − 12 e(μ−ν)α sinhμ α 1 3 1 1 F μ + , −μ + ; ν + ; Γ(−ν − μ) (e2α − 1)μ+ 12 2 2 2 1 − e2α     1 μ Γ ν+2 e(ν+μ+1)α sinh α 1 1 1 1 F μ + , −μ + ; −ν + ; + Γ(ν − μ + 1) (e2α − 1)μ+ 12 2 2 2 1 − e2α   ν = ± 12 , ± 23 , ± 25 , . . . ; α > 12 ln 2

3 √ : 2 2

2μ (cosh α) = √ π

√ Γ(ν + μ + 1) e−(ν+μ+1)α   Q μν (cosh α) = eμπi 2μ π sinhμ α μ+ 12 −2α Γ ν + 32 (1 − e )  1 1 3 1 × F μ + , −μ + ; ν + ; 2 2 2 1 − e2α μ + ν + 1 = 0, −1, −2, . . . ;

α>

1 2

ln 2



MO 94

MO 94

See also 8.776. 8.724 For the inequalities in 8.776 1–4, ν and μ are arbitrary real numbers satisfying the inequalities ν ≥ 1, ν − μ + 1 > 0, and μ ≥ 0:   ±μ  8 Γ (ν ± μ + 1) 1   1. P ν (cos ϕ) < MO 91-92 νπ Γ(ν + 1) sinμ+ 12 ϕ

8.731

2. 3. 4. 5.8

Functional relations

 ±μ  Q (cos ϕ) < ν



973

2π Γ (ν ± μ + 1) 1 ν Γ(ν + 1) sinμ+ 12 ϕ

MO 91-92

 ±μ  1 P ν (cos ϕ) < √2 Γ (ν ± μ + 1) μ+ νπ Γ(ν + 1) sin 12 ϕ   ±μ  1 Q (cos ϕ) < π Γ (ν ± μ + 1) ν μ+ ν Γ(ν + 1) sin 12 ϕ    √   Γ n + 12 2     m 2(m+n) /n sup  t J m (t)  sin ϕ P n (cos ϕ) < Γ(n − m + 1) 0 0]

MO 82

cos νπ P μν (z)

MO 83

  Γ(ν − μ + 1) 2 cos μπ P μν (x) − sin(μπ) Q μν (x) Γ(ν + μ + 1) π

1.

3.

MO 82

MO 83

Γ(ν − μ + 1) μ Q (z) Γ(ν + μ + 1) ν

4.

6

MO 82

MO 83

  Γ(ν − μ + 1) 2 −μπi μ μ P ν (z) − e = sin μπ Q ν (z) Γ(ν + μ + 1) π

P −μ ν (z)

3.

MO 82

MO 83

 (ν − μ)(ν − μ + 1) P μν+1 (x) = (ν + μ)(ν + μ + 1) P μν−1 (x) + (2ν + 1) 1 − x2 P μ+1 (x) ν

5.

MO 82



1 − x2 P μ+1 (x) = (ν − μ + 1) P μν+1 (x) ν  (ν − μ)x P μν (x) − (ν + μ) P μν−1 (x) = 1 − x2 P μ+1 (x) ν  P μν−1 (x) − x P μν (x) = (ν − μ + 1) 1 − x2 P μ−1 (x) ν  x P μν (x) − P μν+1 (x) = (ν + μ) 1 − x2 P μ−1 (x) ν

(ν + μ + 1)x P μν (x) +

1.

7.

MO 82

2 sin[(ν + μ)π] Q μν (x) π π Q μν (−x) = − cos[(ν + μ)π] Q μν (x) − sin[(ν + μ)π] P μν (x) 2

MO 84 MO 84 MO 83, EH I 144(15)

976

Q μ−ν−1 (x) =

4.

Associated Legendre Functions

8.738

sin[(ν + μ)π] μ π cos νπ cos μπ μ Q (x) − P (x) sin[(ν − μ)π] ν sin[(ν − μ)π] ν

MO 84

8.738

   √ 1 ν +1 −ν− 1 sin ϕ P −μ− 21 (cos ϕ) 1. (i cot ϕ) = exp iπ μ − π Γ(ν + μ + 1) 2 2 2  π 0 − 12 MO 88   Γ(ν − μ + 1) 2 cos νπ P μν (cos ϕ) − sin νπ Q μν (cos ϕ) Γ(ν + μ + 1) π      2 cosecμ ϕ π cos ν + 12 (t − π) dt   = 1 π Γ μ + 12 ϕ (cos ϕ − cos t) 2 −μ   Re μ > − 12 MO 88      π cos ν + 12 (t − π) dt 2 sinμ ϕ 2 μ μ   P ν (cos ϕ) cos (ν + μ) π − Q ν (cos ϕ) sin(ν + μ)π = 1 π π Γ 12 − μ ϕ (cos ϕ − cos t)μ+ 2   Re μ < 12 MO 88 Γ(ν − μ + 1) Γ(ν + μ + 1)



cos μπ P μν

cos μπ P μν (cos ϕ) −

2 sin μπ Q μν (cos ϕ) π

 π sin2μ t dt 1 Γ(ν + μ + 1) sinμ ϕ   = μ√ 1 2 π Γ(ν − μ + 1) Γ μ + 2 0 (cos ϕ ± i sin ϕ cos t)ν−μ   Re μ > − 12 , 0 < ϕ < π MO 38

For integrals of Legendre functions, see 7.11–7.21.

8.754

Special cases and particular values

977

8.75 Special cases and particular values 8.751 1. 2.

3.8

m   + m + 1) 1 − x2 2 1−x F m − ν, m + ν + 1; m + 1; = (−1) MO 84 2m Γ(ν − m + 1)m! 2  m  Γ(ν + m + 1) z 2 − 1 2 1−z m F m − ν, m + ν + 1; m + 1; P ν (z) = MO 84 2m m! Γ(ν − m + 1) 2  3 μπi  e Γ μ+n+  μ2 1/2 −n−μ−3/2 μ + n + 52 μ + n + 32 1 2  2 μ , ; n + 2; 2 Q n+ 1 (z) = F z −1 π z 3 2 2 2 z 2n+ 2 (n + 1)!

Pm ν (x)

m Γ(ν

MO 84

8.752 1. 2.

3. 4. 5.

  m dm m 1 − x2 2 P ν (x) WH, MO 84, EH I 148(6) Pm ν (x) = (−1) dxm   1  m 1  −m m m Γ(ν − m + 1) 2 −2 P (x) = 1 − x P ν (x) = (−1) ... P ν (x)(dx)m Γ(ν + m + 1) ν x x  2 − m2 P −m ν (z) = z − 1



z



z

...

1

P ν (z)(dz)m

[m ≥ 1]

HO 99a, MO 85, EH I 149(10)a

[m ≥ 1]

MO 85, EH I 149(8)

1

 2  m2 dm Qm Q (z) ν (z) = z − 1 dz m ν   2 − m2 m z Q −m (z) = (−1) − 1 ν

z

WH, MO 85, EH I 148(5) ∞

 ...

z

∞

Q ν (z)(dz)m [m ≥ 1]

MO 85, EH I 149(9)

Special values of the indices 8.753 ϕ 1 cotμ Γ(1 − μ) 2

1.

P μ0 (cos ϕ) =

2.

P −1 ν (cos ϕ) = −

3.

Pm n (z) ≡ 0,

8.754

MO 84

d P ν (cos ϕ) 1 ν(ν + 1) dϕ

MO 84

Pm n (x) ≡ 0 for m > n

MO 85



1.

1/2 P ν− 1 2

2.

P ν− 1

1/2

2

2 cosh να (cosh α) = π sinh α  2 cos νϕ (cos ϕ) = π sin ϕ

MO 85 MO 85

978

Associated Legendre Functions

3.

−1/2 P ν− 1 2

4.

Q ν− 1

1/2

2

8.755 1. 2.



2 sin νϕ π sin ϕ ν  π e−να (cosh α) = i 2 sinh α (cos ϕ) =

ν sin ϕ 2  ν sinh α 1 P −ν (cosh α) = ν Γ(1 + ν) 2

P −ν ν

8.755

1 (cos ϕ) = Γ(1 + ν)

MO 85 MO 85



MO 85 MO 85

Special values of Legendre functions 8.756

√ 2μ π   −ν−μ+1  Γ 2 +1 Γ 2   1 μ μ+1 2 sin 2 (ν + μ)π Γ ν+μ d P ν (0) 2 +1   = √ dx π Γ ν−μ+1 2  ν+μ+1   Γ 1 μ μ−1 √ 2   (ν + μ)π Q ν (0) = −2 π sin 2 Γ ν−μ 2 +1   ν+μ  Γ 2 +1 d Q μν (0) 1 μ√   = 2 π cos (ν + μ)π dx 2 Γ ν−μ+1 2

P μν (0) =

1. 2. 3.12 4.12

 ν−μ

MO 84

MO 84

MO84

MO 84

8.76 Derivatives with respect to the order 8.761

μ2  ∞ (−ν)(1 − ν) . . . (n − 1 − ν)(ν + 1)(ν + 2) . . . (ν + n) (μ + 1)(μ + 2) . . . (μ + n)1 · 2 . . . n n=1 n  1−x × [ψ(ν + n + 1) − ψ(ν − n + 1)] 2 [ν = 0, ±1, ±2, . . . ; Re μ > −1] MO 94

1 ∂P −μ ν (x) = ∂ν Γ(μ + 1)



1−x 1+x

8.762   ∂ P ν (cos ϕ) ϕ 1. = 2 ln cos ∂ν 2 ν=0   −1 ϕ ϕ ∂P ν (cos ϕ) ϕ 2. = − tan − 2 cot ln cos ∂ν 2 2 2 ν=0   −1 1 ϕ ϕ ϕ ∂P ν (cos ϕ) 3. = − tan sin2 + sin ϕ ln cos ∂ν 2 2 2 2 ν=1 • For a connection with the polynomials C λn (x), see 8.936. • For a connection with a hypergeometric function, see 8.77.

MO 94

MO 94

MO 94

8.773

Series representation

979

8.77 Series representation For a representation in the form of a series, see 8.721. It is also possible to represent associated Legendre functions in the form of a series by expressing them in terms of a hypergeometric function. 8.771 μ   1 1−z z+1 2 1. P μν (z) = F −ν, ν + 1; 1 − μ; MO 15 z−1 Γ(1 − μ) 2 μ    ν +μ+1 3 1 eμπi Γ(ν + μ + 1) Γ 12 z 2 − 1 2 ν +μ μ 8   + 1, ;ν + ; 2 2. Q ν (z) = ν+1 F MO 15 2 z ν+μ+1 2 2 2 z Γ ν + 32 See also 8.702, 8.703, 8.704, 8.723, 8.751, 8.772. The analytic continuation for |z|  1 The formulas are consequences of theorems on the analytic continuation of hypergeometric series (see 9.154 and 9.155): 8.772   μ2 −ν−μ−1 ν +μ+1 3 1 sin(ν + μ)π Γ(ν + μ + 1)  2 ν +μ μ   + 1, ;ν + ; 2 z −1 z 1. P ν (z) = ν+1 √ F 2 2 2 z 2 π cos νπ Γ ν + 32  1 ν μ   2 Γ ν+2 1 μ−ν +1 μ−ν 1 z 2 − 1 2 z ν−μ F , ; − ν; 2 +√ π Γ(ν − μ + 1) 2 2 2 z [2ν = ±1, ±3, ±5, . . . ; |z| > 1; |arg (z ± 1)| < π] MO 85

2.12

3.

12

P μν (z) =

P μν (z)

  − ν+1  2 Γ −ν − 12 z 2 − 1 3 1 ν −μ+1 ν +μ+1 √ F , ; ν + ; 2ν+1  π Γ(−ν 2 2 2 1 − z2  − μ)  1 ν  2 ν 2 Γ ν+2 μ+ν 1 1 μ−ν z −1 2 F ,− ; − ν; +√ π Γ(ν − μ + 1) 2 2  2  1 − z2   2ν = ±1, ±3, ±5, . . . ; 1 − z 2  > 1; |arg (z ± 1)| < π

1 = Γ(1 − μ)

8.773 1.

2.

Q μν (z)

z−1 z+1

− μ2 

z+1 2

−ν

 F

z−1 −ν, −ν − μ; 1 − μ; z+1    z − 1  1

MO 86

 1−z Γ(μ) F −ν, ν + 1; 1 − μ; 2  μ2   Γ(−μ) Γ(ν + μ + 1) z − 1 1−z + F −ν, ν + 1; 1 + μ; Γ (ν − μ + 1) z+1 2 [|arg (z ± 1)| < π, |1 − z| < 2]

MO 86

=e

Q μν (z) =



MO 85

μπi

 − ν+1 π Γ(ν + μ + 1)  2 ν+μ+1 ν −μ+1 2  z −1  , ;ν + F 3 ν+1 2 2 2 Γ ν+2  ν + μ = −1, −2, −3, . . . ; |arg (z ± 1)| < π;

√

1 μπi e 2



z+1 z−1

μ2

980

Associated Legendre Functions

8.774



8.774

P μν

   sin ϕ Γ −ν − 12 −i(ν+1) π  ϕ 1 3 ϕ ν+ 12 1 2 e + μ, − μ; ν + ; sin2 tan (i cot ϕ) = F 2 2 2 2 2 2π Γ(−ν − μ)1  ν+ 12  1  sin ϕ Γ ν + 2 π ϕ 1 1 ϕ + eiν 2 cot + μ, − μ; − ν; sin2 F 2π Γ(ν − μ + 1) 2  2 2 2 2  π 2ν = ±1, ±3, ±5, . . . , 0 < ϕ < MO 86 2

8.775 1.

6

2.6

P μν (x) =

     μ (ν + μ) π Γ ν+μ+1 ν+μ+1 μ−ν 1 2 2  ν−μ  , ; ;x 1 − x2 2 F √ 2 2 2 πΓ 2 +1   ν+μ    μ  −ν +μ+1 3 2 ν + μ 2μ+1 sin 12 (ν + μ)π Γ 2 + 1   ν−μ+1  + 1, ; ;x x 1 − x2 2 F + √ π 2 2 2 Γ 2 2μ cos

1 2

   ν+μ+1   √  μ π sin 12 (ν + μ)π Γ ν +μ+1 μ−ν 1 2 μ 2 2 2   , ; ;x 1−x Q ν (x) = − 1−μ F 2 2 2 2 Γ ν−μ 1 2 + 1   ν+μ   μ   μ−ν+1 3 2 ν +μ μ √ cos 2 (ν + μ)π Γ 2 2 2 +1   + 1, ; ;x x 1−x F +2 π 2 2 2 Γ ν−μ+1 2

MO 87

MO 87

8.776 For |z| 1

1.

P μν (z)

=

      Γ −ν − 12 2ν Γ ν + 12 1 ν −ν−1 √ z + ν+1 √ z 1+O z2 π Γ(ν − μ + 1) 2 π Γ(−ν − μ) [2ν = ±1, ±3, ±5, . . . ,

  √ e Γ(μ + ν + 1) −ν−1 1 μ  z  1+O Q ν (z) = π ν+1 3 2 2 z Γ ν+2

|arg z| < π] MO 87

μπi

2.

[2ν = −3, −5, −7, . . . ;

|arg z| < π] MO 87

√ 8.777 Set ζ = z + z 2 − 1. The variable ζ is uniquely defined by this equation on the entire z-plane in which a cut is made from −∞ to +1. Here, we are considering that branch of the variable ζ for which values of ζ exceeding 1 correspond to real values of z exceeding 1. In this case,   μ  2μ Γ −ν − 12 z 2 − 1 2 3 1 1 μ 1. P ν (z) = √ + μ, ν + μ + 1; ν + ; F π Γ(−ν − μ) ζ ν+μ+1 2 2 ζ2 μ      z2 − 1 2 1 1 2μ Γ ν + 12 1 + μ, μ − ν; − ν; +√ F 2 2 ζ2 π Γ(ν − μ + 1) ζ μ−ν [2ν = ±1, ±3, ±5, . . . ; |arg(z − 1)| < π] MO 86

2.

Q μν (z)

μ μπi

=2 e

μ   √ Γ(ν + μ + 1) z 2 − 1 2 3 1 1   + μ, ν + μ + 1; ν + ; π F ζ ν+μ+1 2 2 ζ2 Γ ν + 32 [|arg(z − 1)| < π]

MO 86

8.792

Series of associated Legendre functions

981

8.78 The zeros of associated Legendre functions 8.781 The function P −μ ν (cos ϕ), considered as a function of ν has infinitely many zeros for μ ≥0. These are all simple and real. If a number ν0 is a zero of the function P −μ ν (cos ϕ), the number −ν0 − 1 is also a zero of this function. MO 91 μ 8.782 If ν and μ are both real and μ ≤0, or if ν and μ are integers, the function P ν (t) has no real zeros exceeding 1. If ν and μ are both real with ν < μ 0, but does have one such zero when sin μπ sin(μ − ν)π − 23 and ν + μ + 1 > 0, the function Q μν (t) has no real zeros exceeding 1. 8.784 The function P − 12 +iλ (z) has infinitely many zeros for real λ. All these zeros are real and greater than unity. 8.785 For n a natural number, the function P n (x) has exactly n real zeros which lie in the closed interval −1, +1. 8.786 The function Q n (z) has no zeros for which |arg(z − 1)| < π if n is a natural number. The function Q n (cos ϕ) has exactly n + 1 zeros in the interval 0 ≤ ϕ ≤ π. MO 91 8.787 The following approximate formula can be used to calculate the values of ν for which the equation P −μ ν (cos ϕ) = 0 holds for given small values of ϕ: 

   sin2 ϕ2 jμ 4μ2 − 1 1 4 ϕ 1− 1− + O sin MO 93 ν+ =− 2 2 sin ϕ2 6 jμ2 2 Here, jμ denotes an arbitrary nonzero root of the equation J μ (z) = 0 (for μ ≥ 0). If ϕ is close to π then, instead of this formula, we can use the following formulas:  2μ π−ϕ Γ(2μ + k + 1) 1. ν ≈μ+k+ [μ > 0, k = 0, 1, 2, . . .] MO 93 Γ(μ) Γ(μ + 1) Γ(k + 1) 3 1   [μ = 0, k = 0, 1, 2, . . .] 2. ν ≈k+ MO 93 2 2 ln π−ϕ

8.79 Series of associated Legendre functions 8.791 1.

          t + t2 − 1 < z + z 2 − 1

∞

 1 = (2k + 1) P k (t) Q k (z) z−t k=0

2.

8.79212

Here, t must lie inside an ellipse passing through the point z with foci at the points ±1. √ ∞  1 z − t + 1 − 2tz + t2 √ √ ln tk Q k (z) = 1 − 2tz + t2 z2 − 1 k=0 MO 78 [Re z > 1, |t| < 1]   ∞ 1 sin νπ  1 −β −β k − P −α P −α (cos ϕ) P (cos ψ) = (−1) ν ν k (cos ϕ) P k (cos ψ) π ν −k ν +k+1 k=1

[a ≥ 0,

β ≥ 0,

ν real,

−π < ϕ ± ψ < π]

MO 94

982

Associated Legendre Functions

8.793 P −μ ν (cos ϕ) =

8.793

 ∞ 1 1 sin νπ  − P −μ (−1)k k (cos ϕ) π ν −k ν +k+1

[μ ≥ 0,

0 < ϕ < π]

k=0

MO 94

Addition theorems 8.794 1.11

P ν (cos ψ1 cos ψ2 + sin ψ1 sin ψ2 cos ϕ) = P ν (cos ψ1 ) P ν (cos ψ2 ) + 2

∞ 

k (−1)k P −k ν (cos ψ1 ) P ν (cos ψ2 ) cos kϕ

k=1

= P ν (cos ψ1 ) P ν (cos ψ2 ) + 2 [0 ≤ ψ1 < π, 2.12

0 ≤ ψ2 < π,

∞  Γ(ν − k + 1) k=1

Γ(ν + k + 1)

ψ1 + ψ2 < π,

Q ν (cos ψ1 cos ψ2 + sin ψ1 sin ψ2 cos ϕ)

8.795 1.

0 < ψ2 < π,

MO 90

∞ 

k=1

Re z2 > 0,

|arg (z1 − 1)| < π,

|arg (z2 − 1)| < π]

MO 91

 ∞    k Q ν x1 x2 − x21 − 1 x22 − 1 cos ϕ = P ν (x1 ) Q ν (x2 ) + 2 (−1)k P −k ν (x1 ) Q ν (x2 ) cos kϕ [1 < x1 < x2 ,

3.

(cf. 8.814, 8.844 1)

 ∞    2 2 (−1)k P kν (z1 ) P −k P ν z1 z2 − z1 − 1 z2 − 1 cos ϕ = P ν (z1 ) P ν (z2 ) + 2 ν (z2 ) cos kϕ [Re z1 > 0,

2.

ϕ real]

k (−1)k P −k ν (cos ψ1 ) Q ν (cos ψ2 ) cos kϕ k=1  0 < ψ1 + ψ2 < π; ϕ real (cf. 8.844 3) MO 90

= P ν (cos ψ1 ) Q ν (cos ψ2 ) + 2  π 0 < ψ1 < , 2

P kν (cos ψ1 ) P kν (cos ψ2 ) cos kϕ

 ∞    Q n x1 x2 + x21 + 1 x22 + 1 cosh α = k=n+1

k=1

ν = −1, −2, −3, . . . ,

MO 91

1 Q k (ix1 ) Q kn (ix2 ) e−kα (k − n − 1)!(k + n)! n [x1 > 0,

x2 > 0,

8.796 P ν (− cos ψ1 cos ψ2 − sin ψ1 sin ψ2 cos ϕ) = P ν (− cos ψ1 ) P ν (cos ψ2 ) + 2

∞ 

α > 0] (−1)k

k=1

× P −k ν See also 8.934 3.

ϕ real]

(− cos ψ1 ) P −k ν

[0 < ψ2 < ψ1 < π,

MO 91

Γ(ν + k + 1) Γ(ν − k + 1)

(cos ψ2 ) cos kϕ

ϕ real]

(cf. 8.844 2)

MO 91

8.813

Associated Legendre functions with integer indices

983

8.81 Associated Legendre functions with integer indices 8.810 For integer values of ν and μ, the differential equation 8.700 1. (with |ν| > |μ|) has a simple solution in the real domain, namely:   m dm m u = Pm 1 − x2 2 P n (x) n (x) = (−1) dxm m The functions P n (x) are called associated Legendre functions (or spherical functions) of the first kind. The number n is called the degree and the number m is called the order of the function P m n (x). The functions {cos mϑ P m sin mϑ P m n (cos ϕ) , n (cos ϕ)}, which depend on the angles ϕ and ϑ, are also called Legendre functions of the first kind, or, more specifically, tesseral harmonics for m < n and sectoral harmonics for m = n. These last functions are periodic with respect to the angles ϕ and ϑ. Their periods are respectively π and 2π. They are single-valued and continuous everywhere on the surface of the unit sphere x21 + x22 + x23 = 1 (where x1 = sin ϕ cos ϑ, x2 = sin ϕ sin ϑ, x3 = cos ϕ) and they are solutions of the differential equation  1 ∂ ∂Y 1 ∂ 2Y sin ϕ + + n(n + 1)Y = 0 sin ϕ ∂ϕ ∂ϕ sin2 ϕ ∂ϑ2 8.811 The integral representation   ϕ   (−1)m (n + m)! 2 m− 1 −m   sin Pm (cos ϕ) = ϕ (cos t − cos ϕ) 2 cos n + 12 t dt MO 75 n 1 Γ m + 2 (n − m)! π 0 8.812 The series representation:  m (n − m)(m + n + 1) 1 − x (−1)m (n + m)!  2 2 Pm 1 − x 1− (x) = n 2m m!(n − m)! 1!(m + 1) 2   2 (n − m)(n − m + 1)(m + n + 1)(m + n + 2) 1 − x − ... MO 73 + 2!(m + 1)(m + 2) 2  m (−1)m (2n − 1)!!  (n − m)(n − m − 1) n−m−2 1 − x2 2 xn−m − x = (n − m)! 2(2n − 1)  (n − m)(n − m − 1)(n − m − 2)(n − m − 3) n−m−4 x + − ... MO 73 2 · 4(2n − 1)(2n − 3)  m 1 (−1)m (2n − 1)!!  m−n m−n+1 1 1 − x2 2 xn−m F , ; − n; 2 = MO 73 (n − m)! 2 2 2 x 8.813 Special cases: 1. 2. 3. 4. 5. 6.

1/2  P 11 (x) = − 1 − x2 = − sin ϕ   1/2 P 12 (x) = −3 1 − x2 x = − 32 sin 2ϕ   P 22 (x) = 3 1 − x2 = 32 (1 − cos 2ϕ)  1/2  2  5x − 1 = − 83 (sin ϕ + 5 sin 3ϕ) P 13 (x) = − 32 1 − x2   P 23 (x) = 15 1 − x2 x = 15 4 (cos ϕ − cos 3ϕ)   3/2 P 33 (x) = −15 1 − x2 = − 15 4 (3 sin ϕ − sin 3ϕ)

MO 73 MO 73 MO 73 MO 73 MO 73 MO 73

984

Associated Legendre Functions

8.814

Functional relations For recursion formulas, see 8.731. 8.814 P n (cos ϕ1 cos ϕ2 + sin ϕ1 sin ϕ2 cos Θ) = P n (cos ϕ1 ) P n (cos ϕ2 ) + 2 [0 ≤ ϕ1 ≤ π, 8.81512

Z n2 (ϕ, ϑ) = α0 P n2 (cos ϕ) +

n1  m=1 n2  m=1

then

 2π 0

0 ≤ ϕ2 ≤ π]

(“addition theorem”)

MO 74

If Y n1 (ϕ, ϑ) = a0 P n1 (cos ϕ) +



n  (n − m)! m P n (cos ϕ1 ) P m n (cos ϕ2 ) cos mΘ (n + m)! m=1

 dϑ

π

2π

 dϑ

0

π

(am cos mϑ + bm sin mϑ) P m n1 (cos ϕ) , (αm cos mϑ + βm sin mϑ) P m n2 (cos ϕ) ,

sin ϕ dϕ Y n1 (ϕ, ϑ) Z n2 (ϕ, ϑ) = 0,

0

sin ϕ dϕ Y n (ϕ, ϑ) P n [cos ϕ cos ψ + sin ϕ sin ψ cos(ϑ − θ)] =

0

n 

4π Y n (ψ, θ) 2n + 1

n! cos mϑ P m n (cos ϕ) (n + m)! m=1 For integrals of the functions, P m n (x), see 7.112 1, 7.122 1. 8.816 (cos ϕ + i sin ϕ cos ϑ)n = P n (cos ϕ) + 2

(−1)m

MO 75 MO 75

8.82–8.83 Legendre functions 8.820 The differential equation    d  2 du 1−z + ν(ν + 1)u = 0 (cf. 8.700 1), dz dz where the parameter ν can be an arbitrary number, has the following two linearly independent solutions:  1−z 1. P ν (z) = F −ν, ν + 1; 1; 2 1  Γ(ν + 1) Γ 2 −ν−1 ν + 2 ν + 1 2ν + 3 1   , ; ; 2 z 2. Q ν (z) = ν+1 F SM 518(137) 2 2 2 z 2 Γ ν + 32 The functions P ν (z) and Q ν (z) are called Legendre functions of the first and second kind respectively. If ν is not an integer, the function P ν (z) has singularities at z = −1 and z = ∞. However, if ν = n = 0, 1, 2, . . . , the function P ν (z) becomes the Legendre polynomial P n (z) (see 8.91) For ν = −n = −1, −2, . . . , we have P −n−1 (z) = P n (z) 3.

If ν = 0, 1, 2, . . . , the function Q ν (z) has singularities at the points z = ±1 and z = ∞. These points are branch points of the function. On the other hand, if ν = n = 0, 1, 2, . . . , the function Q n (z) is single-valued for |z| > 1 and regular for z = ∞.

8.822

4.

Legendre functions

In the right half-plane,



P ν (z) = 5.

1+z 2

ν

 z−1 F −ν, −ν; 1; z+1

985

[Re z > 0]

The function P ν (z) is uniquely determined by equations 8.820 1 and 8.820 4 within a circle of radius 2 with its center at the point z = 1 in the right half-plane.

6.

For z = x = cos ϕ, a solution of equation 8.820 is the function  ϕ P ν (x) = P ν (cos ϕ) = F −ν, ν + 1; 1; sin2 ; 2 In general,

7.

P ν (z) = P −ν−1 (z) = P ν (x) = P −ν−1 (x), for z = x

8.

The function Q ν (z) for |z| > 1 is uniquely determined by equation 8.820 2 everywhere in the z-plane in which a cut is made from the point z = −∞ to the point z = 1. By means of a hypergeometric series, the function can be continued analytically inside the unit circle. On the cut (−1 ≤ x ≤ +1) of the real axis, the function Q ν (x) is determined by the equation

9.

Q ν (x) =

1 2

[Q ν (x + i0) + Q ν (x − i0)]

HO 52(53), WH

Integral representations 8.821 1.

2.

3.

ν  (1+,z+)  2 t −1 1 dt 2πi A 2ν (t − z)ν+1 Here, A is a point on the real axis to the right of the point t = 1 and to the right of z if z is real. At the point A, we set P ν (z) =

arg(t − 1) = arg(t + 1) = 0 and [|arg(t − z)| < π] WH ν  (1−,1+)  2 t −1 1 Q ν (z) = dt 4i sin νπ A 2ν (z − t)ν+1 [ν is not an integer; the point A is at the end of the major axis of an ellipse to the right of t = 1 drawn in the t-plane with foci at the points ±1 and with a minor axis sufficiently small that the point z lies outside it. The contour begins at the point A, follows the path (1−, −1+) and returns to A; |arg z| ≤ π and |arg(z − t)| → arg z as t → 0 on the contour; arg(t + 1) = arg(t − 1) = 0 at the point A; z does not lie on the real axis between −1 and 1.] For ν = n an integer,  1  n 1 Q n (z) = n+1 1 − t2 (z − t)−n−1 dt 2 −1

8.822 1.

1 P ν (z) = π

 0

π

dϕ

√ ν+1 z + z 2 − 1 cos ϕ



1 = π

SM 517(134), WH



π



z+

ν  z 2 − 1 cos ϕ dϕ

0

3 2  Re z > 0 and arg z + z 2 − 1 cos ϕ = arg z for ϕ =

π 2

 WH

986

Associated Legendre Functions

 2.

Q ν (z) = 

∞

0

dϕ √ ν+1 , 2 z + z − 1 cosh ϕ 2

8.823



Re ν > −1;

if ν is not an integer,

z+

 3   z 2 − 1 cosh ϕ for ϕ = 0 has its principal value WH

2 8.823 P ν (cos θ) = π  8.824 Q n (z) = 2n n!



θ

  cos ν + 12 ϕ

 dϕ 2 (cos ϕ − cos θ)  ∞  ∞ (dz)n+1 (t − z)n n ... = 2 n+1 n+1 dt 2 2 1) z z  (z − 1) z (t −   n ∞ dt (−1)n dn  2 z −1 = n+1 2 (2n − 1)!! dz n z (t − 1) 

1.

2.

[Re z > 1] WH, MO 78

1

P n (t) 1 dt 2 −1 z − t See also 6.622 3, 8.842. 8.826 Fourier series: 8.825 Q n (z) =

WH

0 ∞

[|arg(z − 1)| < π]

WH, MO 78

 n! 1 n+1 2n+2 sin(n + 1)ϕ + sin(n + 3)ϕ P n (cos ϕ) = π (2n + 1)!! 1 2n + 3  1 · 3(n + 1)(n + 2) + sin(n + 5)ϕ + . . . 1 · 2(2n + 3)(2n + 5) [0 < ϕ < π] ⎡ n! ⎣ cos(n + 1)ϕ + 1 n + 1 cos(n + 3)ϕ Q n (cos ϕ) = 2n+1 (2n + 1)!! 1 2n + 3 ⎤ 1 · 3 (n + 1)(n + 2) cos(n + 5)ϕ + . . .⎦ + 1 · 2 (2n + 3)(2n + 5) [0 < ϕ < π]

MO 79

MO 79

The expressions for Legendre functions in terms of a hypergeometric function (see 8.820) provide other series representations of these functions. Special cases and particular values 8.827 1. 2. 3. 4.

1 1+x ln = arctanh x 2 1−x x 1+x −1 Q 1 (x) = ln 2 1−x  1+x 3 1 2 3x − 1 ln − x Q 2 (x) = 4 1−x 2  1+x 5 2 2 1 3 5x − 3x ln − x + Q 3 (x) = 4 1−x 2 3

Q 0 (x) =

JA JA JA JA

8.832

5. 6.

Legendre functions

987

 1 + x 35 3 55 1  35x4 − 30x2 + 3 ln − x + x 16 1−x 8 24  63 1  1 + x 49 8 63x5 − 70x3 + 15x ln − x4 + x2 − Q 5 (x) = 16 1−x 8 8 15 Q 4 (x) =

JA JA

8.828 1.

P ν (1) = 1

MO 79

 ν 1 sin νπ ν +1 √ Γ − 2. P ν (0) = − Γ 2 π3 2 2   ν +1 ν 1 Γ − 8.829 Q ν (0) = √ (1 − cos νπ) Γ 4 π 2 2 

MO 79

MO 79

Functional relationships 8.831 1. 2.

3.12

π [cos νπ P ν (x) − P ν (−x)] [ν = 0, ±1, ±2, . . .] 2 sin νπ 1 1+x − Wn−1 (x) Q n (x) = P n (x) ln [n = 0, 1, 2, . . .] , 2 1−x where 2 n−1 2  n   2(n − 2k) − 1 1 P n−2k−1 (x) = P k−1 (x) P n−k (x) Wn−1 (x) = (2k + 1)(n − k) k

Q ν (x) =

k=0

MO 76

k=1

and 4. 5.

W−1 (x) ≡ 0 (see also 8.839)  ∞  1 1 π − P k (cos ϕ) = P ν (cos ϕ) (−1)k ν−k ν+k+1 sin νπ

SM 516(131), MO 76

k=0

[ν not an integer; 6.

∞ 

(−1)k

k=0



1 1 − ν−k ν+k+1

P k (cos ϕ) P k (cos ψ) = [ν not an integer,

See also 8.521 4. 8.832  d  2 P ν (z) = (ν + 1) [P ν+1 (z) − z P ν (z)] 1. z −1 dz 2. (2ν + 1)z P ν (z) = (ν + 1) P ν+1 (z) + ν P ν−1 (z) 3. 4.



 d   Q ν (z) = (ν + 1) Q ν+1 (z) − z Q ν (z) dz (2ν + 1)z Q ν (z) = (ν + 1) Q ν+1 (z) + ν Q ν−1 (z) z2 − 1

0 ≤ ϕ < π]

MO 77

π P ν (cos ϕ) P ν (cos ψ) sin νπ

−π < ϕ + ψ < π,

−π < ϕ − ψ < π]

MO 77

WH WH WH WH

988

8.833

Associated Legendre Functions

3.

2 sin νπ Q ν (z) π 2 P ν (−z) = e−νπi P ν (z) − sin νπ Q ν (z) π Q ν (−z) = −e−νπi Q ν (z)

4.

Q ν (−z) = −eνπi Q ν (z)

1. 2.

8.834 1. 2.

P ν (−z) = eνπi P ν (z) −

πi P ν (x) 2 1 z+1 − W n−1 (z) Q n (z) = P n (z) ln 2 z−1

3. 8.836 1. 2.

[Im z < 0]

MO 77

[Im z > 0]

MO 77

[Im z < 0]

MO 77

[Im z > 0]

MO 77

Q ν (x ± i0) = Q ν (x) ∓

MO 77

(see 8.831 3)

MO 77

[sin νπ = 0]

MO 77

Q −ν−1 (cos ϕ) = Q ν (cos ϕ) − π cot νπ P ν (cos ϕ) [sin νπ = 0] π Q ν (− cos ϕ) = − cos νπ Q ν (cos ϕ) − sin νπ P ν (cos ϕ) 2

MO 77

8.835 1. Q ν (z) − Q −ν−1 (z) = π cot νπ P ν (z) 2.

8.833

  n z + 1 1 1 dn  2 z+1 − P n (z) ln z − 1 ln Q n (z) = n 2 n! dz n z−1 2 z−1   n  2 n 1 + x 1 1 d 1+x − P n (x) ln x − 1 ln Q n (x) = n n 2 n! dx 1−x 2 1−x

8.837  ϕ 1. P ν (x) = P ν (cos ϕ) = F −ν, ν + 1; 1; sin2 2  ν +1 tan νπ Γ(ν + 1) −ν−1 ν z  + 1, ;ν + 2. P ν (z) = ν+1 √ F 2 π Γ ν + 32 2 2    ν 1 1 1−ν 2ν Γ ν + 12 ν √ z , − ; − ν; F + π Γ(ν + 1) 2 2 2 z2

(cf. 8.820 6) 3 1 ; 2 z2

MO 77

MO 79 MO 79

MO 76

MO 78

See also 8.820. For integrals of Legendre functions, see 7.1–7.2. 8.838 Inequalities (0 ≤ ϕ ≤ π, ν > 1, and C0 is a number that does not depend on the values of ν or ϕ):  1 1. |P ν (cos ϕ) − P ν+2 (cos ϕ)| ≤ 2C0 MO 78 νπ    Q ν (cos ϕ) − Q ν+2 (cos ϕ) < C0 π 2. MO 78 ν With regard to the zeros of Legendre functions of the second kind, see 8.784, 8.785, and 8.786. For the expansion of Legendre functions in series of associated Legendre functions, see 8.794, 8.795, and 8.796.

8.844

Conical functions

8.839 A differential equation leading to the functions Wn−1 (see 8.831 3):  d2 Wn−1  dWn−1 d Pν + (n + 1)nWn−1 = 2 1 − x2 − 2x 2 dx dx dx

989

MO 76

8.84 Conical functions 8.840 Let us set ν = − 12 + iλ, where λ is a real parameter, in the defining differential equation 8.700 1 for associated Legendre functions. We then obtain the differential equation of the so-called conical functions. A conical function is a special case of the associated Legendre function. However, the Legendre functions P − 12 +iλ (x), Q − 12 +iλ (x) have certain peculiarities that make us distinguish them as a special class—the class of conical functions. The most important of these peculiarities is the following 8.841 The functions  2   4λ + 12 4λ2 + 32 ϕ 4λ2 + 12 2 ϕ P − 12 +iλ (cos ϕ) = 1 + + sin sin4 + . . . 2 2 2 2 2 2 4 2 are real for real values of ϕ. Also, P − 12 +iλ (x) ≡ P − 12 −iλ (x) MO 95 8.842 Integral representations:   ∞ cosh λu du cos λu du 2 ϕ 2   1. P − 12 +iλ (cos ϕ) = = cosh λπ π 0 π 2 (cos u − cos ϕ) 2 (cos ϕ + cosh u) 0  ∞  ∞ cos λu du cos λu du   2.6 Q − 12 ∓λi (cos ϕ) = ±i sinh λπ + 2 (cosh u + cos ϕ) 2 (cosh u − cos ϕ) 0 0

MO 95

MO 95

Functional relations (See also 8.73) 8.843 P − 12 +iλ (− cos ϕ) = 8.844 1.

 cosh λπ  Q − 12 +iλ (cos ϕ) + Q − 12 −iλ (cos ϕ) π

P − 12 +iλ (cos ψ cos ϑ + sin ψ sin ϑ cos ϕ)

MO 95

k ∞ (−1)k 22k P k 1  − +iλ (cos ψ) P − 1 +iλ (cos ϑ) cos kϕ

2 2 = P − 12 +iλ (cos ψ) P − 12 +iλ (cos ϑ) + 2 (4λ2 + 12 ) (4λ2 + 32 ) · · · [4λ2 + (2k − 1)2 ] k=1   (cf. 8.794 1) MO 95 0 < ϑ < π2 , 0 < ψ < π, 0 < ψ + ϑ < π

2.

P − 12 +iλ (− cos ψ cos ϑ − sin ψ sin ϑ cos ϕ)

k ∞ (−1)k 22k P k 1  − +iλ (cos ψ) P − 1 +iλ (− cos ϑ) cos kϕ

2 2 = P − 12 +iλ (cos ψ) P − 12 +iλ (− cos ϑ) + 2 (4λ2 + 1) (4λ2 + 32 ) · · · [4λ2 + (2k − 1)2 ]  k=1 (cf. 8.796) MO 95 0 < ψ < π2 < ϑ, ψ + ϑ < π

990

3.

Associated Legendre Functions

8.850

Q − 12 +iλ (cos ψ cos ϑ + sin ψ sin ϑ cos ϕ)

k ∞ (−1)k 22k P k 1  − 2 +iλ (cos ψ) Q − 12 +iλ (cos ϑ) cos kϕ   = P − 12 +iλ (cos ψ) Q − 12 +iλ (cos ϑ) + 2 2 (4λ2 + 1) (4λ2 + 32 ) · · · 4λ2 + (2k − 1) k=1   0 < ψ < π2 < ϑ, ψ + ϑ < π (cf. 8.794 2) MO 96

Regarding the zeros of conical functions, see 8.784.

8.85 Toroidal functions* 8.850 Solutions of the differential equation  m2 d2 u cosh η du 1 2 − n + 1. u = 0, + − dη 2 sinh η dη 4 sinh2 η are called toroidal functions. They are equivalent (under a coordinate transformation) to associated Legendre functions. In particular, the functions Pm Qm MO 96 n− 12 (cosh η) , n− 12 (sinh η) are solutions of equation 8.850 1. The following formulas, obtained from the formulas obtained earlier for associated Legendre functions, are valid for toroidal functions: 8.851 Integral representations:    π Γ n + m + 12 sin2m ϕ dϕ (sinh η)m m    1. P n− 1 (cosh η) =  √ 1 2 Γ n − m + 12 2m π Γ m + 12 0 (cosh η + sinh η cos ϕ)n+m+ 2   2π  1 cos mϕ dϕ (−1)m Γ n + 2   = 2π Γ n − m + 12 0 (cosh η + sinh η cos ϕ)n+ 12 

2.



MO 96



∞

cosh mt dt m Γ n +  Qm 1 n− 12 (cosh η) = (−1) 1 Γ n − m + 2 0 (cosh η + sinh η cosh t)n+ 2   η  ln coth 2 Γ n + m + 12 n− 1   = (−1)m (cosh η − sinh η cosh t) 2 cosh mt dt 1 Γ n+ 2 0 1 2

[n ≥ m]

MO 96

8.852 Functional relations: 1.12

√  2m Γ n + m + 12 π 1 sinhm (η) e−(n+m+ 2 )η Γ(n + 1)   × F m + 12 , n + m + 12 ; n + 1; e−2η

m Qm n− 1 (cosh η) = (−1)

∗ Sometimes

2

called torus functions

8.901

2.12

Introduction

P −m (cosh η) = n− 1 2

991

m   1 2−m  1 − e−2η e−(n+ 2 )η F m + 12 , n + m + 12 ; 2m + 1; 1 − e−2η Γ(m + 1) MO 96

8.853 An asymptotic representation P n− 12 (cosh η) for large values of n: 1 Γ(n)e(n− 2 )η   1 P n− 2 (cosh η) = √ 1 π Γ n + 2      2 Γ2 n + 12 1 1 η −2nη −2η ln (4e ) e , n + ; n + 1; e +A+B , × F πn! Γ(n) 2 2 where 2  (2n − 1)!! 1 1 · (2n − 1) −2η 1 1 · 3 · (2n − 1)(2n − 3) −4η 1 e e A=1+ 2 + 4 + · · · + 2n−2 e−2(n−1)η 2 1 · (n − 1) 2 1 · 2 · (n − 1)(n − 2) 2 (n − 1)!    1 1   ∞ Γ k+ 2 Γ n+k+  Γ n + 12  2  un+k + uk − vn+k− 12 − vk− 12 e−2(n+k)η B= √ Γ(n + k + 1) Γ(k + 1) π 3 Γ(n) k=1

Here, ur =

r  1 s=1

s

,

vr− 12 =

r  s=1

2 2s − 1

[r is a natural number]

MO 97

8.9 Orthogonal Polynomials 8.90 Introduction 8.901 Suppose that w(x) is a nonnegative real function of a real variable x. Let (a, b) be a fixed interval on the x-axis. Let us suppose further that, for n = 0, 1, 2, . . . , the integral  b xn w(x) dx exists and that the integral

a

 a

b

w(x) dx

is positive. In this case, there exists a sequence of polynomials p0 (x), p1 (x), . . . , pn (x), . . . , that is uniquely determined by the following conditions: 1.

pn (x) is a polynomial of degree n and the coefficient of xn in this polynomial is positive.

2.

The polynomials p0 (x), p1 (x), . . . are orthonormal; that is,

 b 0 for n = m, pn (x)pm (x)w(x) dx = 1 for n = m. a We say that the polynomials pn (x) constitute a system of orthogonal polynomials on the interval (a, b) with the weight function w(x).

992

Orthogonal Polynomials

8.902

8.902 If qn is the coefficient of xn in the polynomial pn (x), then 1.

n 

pk (x)pk (y) =

k=0

qn pn+1 (x)pn (y) − pn (x)pn+1 (y) qn+1 x−y

(Darboux–Christoffel formula) EH II 159(10)

2.11

n 

2

[pk (x)] =

k=0

 qn  pn (x)pn+1 (x) − pn (x)pn+1 (x)

EH II 159(11)

qn+1

8.903 Between any three consecutive orthogonal polynomials, there is a dependence [n = 2, 3, 4, . . .] pn (x) = (An x + Bn ) pn−1 (x) − Cn pn−2 (x) In this formula, An , Bn , and Cn are constants and qn qn qn−2 An = , Cn = 2 qn−1 qn−1 8.904 Examples of normalized systems of orthogonal polynomials: 

n+

 1 1/2 2 

Notation and name

P n (x) 1/2 (n + λ) n! 2λ Γ(λ) C λn (x) 2π Γ (2λ + n)  εn T n (x), ε0 = 1, εn = 2 for n = 1, 2, 3, . . . π −1/2 −n 2 2 π −1/4 (n!) H n (x)  1/2 Γ(n + 1) Γ(α + β + 1 + n)(α + β + 1 + 2n) P (α,β) (x) n Γ(α + 1 + n) Γ(β + 1 + n)2α+β+1 1/2  Γ(n + 1) (−1)n Lα n (x) Γ (α + n + 1)

MO 102

Interval

Weight

see 8.91

(−1, +1)

see 8.93

(−1, +1)

see 8.94

(−1, +1)

see 8.95

(−∞, ∞)

e−x

see 8.96

(−1, +1)

(1 − x)α (1 + x)β

see 8.97

(0, ∞)

xα e−x

1 

λ− 12



−1/2

1 − x2

1 − x2

2

Cf. 7.221 1, 7.313, 7.343, 7.374 1, 7.391 1, 7.414 3.

8.91 Legendre polynomials 8.910 Definition. The Legendre polynomials P n (z) are polynomials satisfying equation 8.700 1 with μ = 0 and ν = n: that is, they satisfy the differential equation 1.

2.

  d2 u du 1 − z2 + n(n + 1)u = 0 − 2z dz 2 dz This equation has a polynomial solution if, and only if, n is an integer. Thus, Legendre polynomials constitute a special type of associated Legendre function. Legendre polynomials of degree n are of the form n 1 dn  2 P n (z) = n z −1 n 2 n! dz

8.911

Legendre polynomials

993

8.911 Legendre polynomials written in expanded form:

1.

n2  1  (−1)k (2n − 2k)! n−2k z P n (z) = n 2 k!(n − k)!(n − 2k)! k=0  (2n)! n(n − 1) n−2 n(n − 1)(n − 2)(n − 3) n−4 n z z z − = + − ... 2 2(2n − 1) 2 · 4(2n − 1)(2n − 3) 2n (n!)  1 n 1−n 1 (2n − 1)!! n z F − , ; − n; 2 = n! 2 2 2 z HO 13, AD (9001), MO 69

2.

3.

 2n(2n + 1) 2 2n(2n − 2)(2n + 1)(2n + 3) 4 n (2n − 1)!! 1− z + z − ... P 2n (z) = (−1) 2n n! 2! 4!  (2n − 1)!! 1 1 F −n, n + ; ; z 2 = (−1)n n 2 n! 2 2

AD (9002), MO 69



+ 1)!! 2n(2n + 3) 3 2n(2n − 2)(2n + 3)(2n + 5) 5 z− z + z − ... P 2n+1 (z) = (−1) 2n n! 3! 5!  3 3 (2n + 1)!! z F −n, n + ; ; z 2 = (−1)n n 2 n! 2 2 n (2n

AD (9002), MO 69

⎛ 4.12

P n (cos ϕ) = 2

(2n − 1)!! ⎝ 1 n cos(n − 2)ϕ cos nϕ + (2n)!! 1 2n − 1

+

n(n − 1) 1·3 cos(n − 4)ϕ 1 · 2 (2n − 1)(2n − 3)

⎞ n(n − 1)(n − 2) 1·3·5 cos(n − 6)ϕ − . . .⎠ + 1 · 2 · 3 (2n − 1)(2n − 3)(2n − 5)   there are  n+1 2  terms; divide last term by 2 when n is even 5.

WH

(2n − 1)!! P 2n (cos ϕ) = (−1)n 2n n!   (2n)2 2n n! 2n 2n−2 2 n 2n sin cos ϕ × sin ϕ − ϕ cos ϕ + · · · + (−1) 2! (2n − 1)!! AD (9011)

6.

(2n + 1)!! cos ϕ P 2n+1 (cos ϕ) = (−1)n 2n n!   2n n! (2n)2 2n 2n−2 2 n 2n sin cos ϕ ϕ cos ϕ + · · · + (−1) × sin ϕ − 3! (2n + 1)!! AD (9012)

7.

P n (z) =

n  k=0

(−1)k (n + k)! 2

(n − k)! (k!) 2k+1

  (1 − z)k + (−1)n (1 + z)k

WH

994

Orthogonal Polynomials

8.912

8.912 Special cases: 1.

P 0 (x) = 1

JA

2.

P 1 (x) = x = cos ϕ  1 1 2 3x − 1 = (3 cos 2ϕ + 1) P 2 (x) = 2 4  1 1 3 5x − 3x = (5 cos 3ϕ + 3 cos ϕ) P 3 (x) = 2 8  1 1 4 2 35x − 30x + 3 = (35 cos 4ϕ + 20 cos 2ϕ + 9) P 4 (x) = 8 64  1 1 63x5 − 70x3 + 15x = (63 cos 5ϕ + 35 cos 3ϕ + 30 cos ϕ) P 5 (x) = 8 128  1  1 231x6 − 315x4 + 105x2 − 5 = (231 cos 6ϕ + 126 cos 4ϕ + 105 cos 2ϕ + 50) P 6 (x) = 16 512  1  429x7 − 693x5 + 315x3 − 35x P 7 (x) = 16 1 (429 cos 7ϕ + 231 cos 5ϕ + 189 cos 3ϕ + 175 cos ϕ) = 1024  1  P 8 (x) = 6435x8 − 12012x6 + 6930x4 − 1260x2 + 35 128 1 (6435 cos 8ϕ − 3432 cos 6ϕ + 2772 cos 4ϕ − 2520 cos 2ϕ + 1225) = 16384

JA

3. 4. 5. 6. 7.10 8.

9.

8.913 Integral representations:    2 π sin n + 12 t  1. P n (cos ϕ) = dt π ϕ 2 (cos ϕ − cos t) See also 3.611 3, 3.661 3, 4. 2.7

3.

Laplace integral formula: P n (z) =

1 π



π



JA JA



 C

n t2 − 1 dt, 2n (t − z)n+1

with C a simple contour containing z. 10

JA

WH

Schl¨afli’s integral formula: 1 P n (z) = 2πi

JA

n  1/2 x + x2 − 1 cos ϕ dϕ

SA 175(9)

[|x| ≤ 1]

SA 180(19)

0

Functional relations 8.914 Recurrence formulas: 1. 2.

(n + 1) P n+1 (z) − (2n + 1)z P n (z) + n P n−1 (z) = 0 

z2 − 1

 d Pn n(n + 1) = n [z P n (z) − P n−1 (z)] = [P n+1 (z) − P n−1 (z)] dz 2n + 1

WH WH

8.917

8.915 1.10

Legendre polynomials

n 

(2k + 1) P k (x) P k (y) = (n + 1)

k=0

995

P n (x) P n+1 (y) − P n (y) P n+1 (x) y−x (Christoffel summation formula) MO 70

1(1)10 .

(y − x)

n 

  (2k + 1) P k (x) Q k (y) = 1 − (n + 1) P n+1 (x) Q n (y) − P n (x) Q n+1 (y)

k=0

AS 335(8.9.2)

2.7

n−1

 2 

(2n − 4k − 1) P n−2k−1 (z) = P n (z)

(summation theorem)

MO 70

k=0

3.7

n−2

 2 

(2n − 4k − 3) P n−2k−2 (z) = z P n (z) − n P n (z)

SM 491(42), WH

k=0

4.

10

n

 2

(2n − 4k + 1)[k(2n − 2k + 1) − 2] P n−2k (z) = z 2 P n (z) − n(n − 1) P n (z)

WH

k=1

5.11

8.916 1. 2. 3. 4. 5.

 m  am−k ak an−k 2n + 2m − 4k + 1 P n+m−2k (z) = P n (z) P m (z) an+m−k 2n + 2m − 2k + 1 k=0  (2k − 1)!! , ak = k!  1 (2n − 1)!! ∓inϕ 1 ±2iϕ e , −n; − n; e P n (cos ϕ) = F 2n n! 2 2   ϕ P n (cos ϕ) = F n + 1, −n; 1; sin2 2  ϕ P n (cos ϕ) = (−1)n F n + 1, −n; 1; cos2 2  1 1 1 n 2 P n (cos ϕ) = cos ϕ F − n, − n; 1; − tan ϕ 2 2 2  ϕ ϕ P n (cos ϕ) = cos2n F −n, −n; 1; − tan2 2 2

 m≤n

AD (9036)

MO 69 MO 69 WH HO 23 HO 23, 29, WH

See also 8.911 1, 8.911 2, 8.911 3. For a connection with other functions, see 8.936 3, 8.836, 8.962 2. • For integrals of Legendre polynomials, see 7.22–7.25. • For the zeros of Legendre polynomials, see 8.785. 8.917 Inequalities: 1.

P 0 (x) < P 1 (x) < P 2 (x) < · · · < P n (x) < . . .

2.

For x > −1, P 0 (x) + P 1 (x) + · · · + P n (x) > 0.

[x > 1]

MO 71 MO 71

996

3.12

[P n (cos ϕ)]2 > √

4.

sin(2n + 1)ϕ (2n + 1) sin ϕ

8.918

[0 < ϕ < π]

MO 71

n sin ϕ|P n (cos ϕ)| ≤ 1.

MO 71

|P n (cos ϕ)| ≤ 1.

5. 6.

Orthogonal Polynomials

10

WH

Let n ≥ 2. The successive relative maxima of |P n (x)|, when x decreases from 1 to 0, form a decreasing sequence. More precisely, if μ1 , μ2 , . . . , μn/2 denote these maxima corresponding to decreasing values of x, we have 1 > μ1 > μ2 > · · · > μn/2

SZ 162(7.3.1)

7.10

Let n ≥ 2. The successive relative maxima of (sin θ)1/2 |P n (cos θ)| when θ increases from 0 to π/2, form an increasing sequence. SZ 163(7.3.2)

8.10

We have (sin θ)

1/2

|P n (cos θ)| < (2/π)1/2 n−1/2

[0 ≤ qθ ≤ qπ]

SZ 163(7.3.8)

Here the constant (2/π)1/2 cannot be replaced by a smaller one. 9.10

max (sin θ)

1/2

0≤qθ≤qπ

|P n (cos θ)| ∼ = (2/π)1/2 n− 2 1

[n → ∞]

SZ 164(7.3.12)

10.10 Stieltjes’ first theorem:  1/2 4 2 √ |P n (cos θ)| ≤ π n sin θ

[n = 1, 2, . . . , 0 < θ < π]

SA 197(8)

11.10 Stieltjes’ second theorem:

12.10 13.10 8.91810

4 |P n (x) − P n+2 (x)| < √ √ π n+2   √  d P n (x)  n   < √2  dx  π 1 − x2  12 2 |P n+1 (x) + P n (x)| < 6 (1 − x)−1/2 πn

1.

SA 199(15)

[|x| < 1,

n = 1, 2, . . .]

SA 201(18)

[|x| < 1,

n = 0, 1, . . .]

SA 201(19)

Asymptotic approximations: 

12

[|x| ≤ 1]

P n (cos θ) =

2 πn sin θ

1/2

    π 1 θ− + O n−3/2 cos n + 2 4 [ε ≤ θ ≤ π − ε, 0 < ε < π/2m]

(Laplace’s formula)

SA 208(1)

1/2      1 π 1 1 π 1 2 cos n + θ− + cot θ sin n + θ− 1− P n (cos θ) = πn sin θ 4n 2 4 8n 2 4   −5/2 +O n 

2.

12

[ε ≤ θ ≤ π − ε,

0 < ε < π/2]

(Bonnet–Heine formula)

SA 208(2)

8.923

Series of Legendre polynomials

997

8.91910 Series of products of Legendre and Chebyshev polynomials  1.

1

2 −1

T n (x) P n (x) dx =

i+j=n   1 i,j=0

−1

P i (x) P j (x) P n (x) dx

8.92 Series of Legendre polynomials 8.921 The generating function: ∞  1 = tk P k (z) √ 1 − 2tz + t2 k=0 ∞  1 = P k (z) tk+1

      |t| < min z ± z 2 − 1

SM 489(31), WH

      |t| > max z ± z 2 − 1

MO 70

k=0

8.922 1.

∞

z 2n =

 1 2n(2n − 2) . . . (2n − 2k + 2) P 0 (z) + P 2k (z) (4k + 1) 2n + 1 (2n + 1)(2n + 3) . . . (2n + 2k + 1)

MO 72

k=1

∞

2.

3.

 3 2n(2n − 2) . . . (2n − 2k + 2) P 1 (z) + P 2k+1 (z) z = (4k + 3) 2n + 3 (2n + 3)(2n + 5) . . . (2n + 2k + 3) k=1 2  ∞ 1 π (2k − 1)!! √ = (4k + 1) P 2k (x) [|x| < 1, (−1)!! ≡ 1] 2 2k k! 1 − x2 k=0 2n+1

MO 72

MO 72, LA 385(15)

4.

5.

x π √ = 2 2 1−x 

1 − x2 =

 6.10 7.12

8.923

π 2

∞ 

(4k + 3)

k=0

(2k − 1)!!(2k + 1)!! P 2k+1 (x) 22k+1 k!(k + 1)!

∞

(−1)!! ≡ 1]

LA 385(17)

[|x| < 1,

(−1)!! ≡ 1]

LA 385(18)

1  (2k − 3)!!(2k − 1)!! − P 2k (x) (4k + 1) 2k+1 2 2 k!(k + 1)! k=1

∞  1−x 1 2 = P 0 (x) − 2 P n (x) 2 3 (2n − 1)(2n + 3) n=1

1 − ρ2 (1 − 2ρx + ρ2 )3/2 arcsin x =

[|x| < 1, 

=

∞ 

(2n + 1)ρn P n (x)

[−1 ≤ x ≤ 1] [|ρ| < 1,

|x| ≤ 1]

SA 170(4)

n=0

2 ∞  π  (2k − 1)!! [P 2k+1 (x) − P 2k−1 (x)] + πx/2 2 2k k! k=1

[|x| < 1,

(−1)!! ≡ 1]

WH

998

8.924 1.

Orthogonal Polynomials

8.924

    ∞ 1 + cos nπ 1 + cos nπ  (4k + 5)n2 n2 − 22 . . . n2 − (2k)2 − P 0 (cos θ) − P 2k+2 (cos θ) 2 (n2 − 1) 2 (n2 − 12 ) (n2 − 32 ) . . . [n2 − (2k + 3)2 ] k=0 3 (1 − cos nπ) − P 1 (cos θ) 2 (n2 − 22 )     ∞ 1 − cos nπ  (4k + 3) n2 − 12 . . . n2 − (2k − 1)2 − P 2k+1 (cos θ) = cos nθ 2 (n2 − 22 ) (n2 − 42 ) . . . [n2 − (2k + 2)2 ] k=1

 2

2.

 2



AD (9060.1)

 ∞ − sin nπ sin nπ  (4k + 5)n n2 − 2 . . . n2 − (2k)2 P 0 (cos θ) − P 2k+2 (cos θ) 2 (n2 − 1) 2 (n2 − 12 ) (n2 − 32 ) . . . [n2 − (2k + 3)2 ] k=0 3 sin nπ P 1 (cos θ) + 2 (n2 − 22 )      ∞ sin nπ  (4k + 3) n2 − 12 n2 − 32 . . . n2 − (2k − 1)2 P 2k+1 (cos θ) = sin nθ + 2 (n2 − 22 ) (n2 − 42 ) . . . [n2 − (2k + 2)2 ] k=1

AD (9060.2)

3.3

n/2  2n−1 n! 2n−2k−1 (n − k − 1)!(2k − 3)!! P n (cos θ) − n P n−2k (cos θ) (2n − 4k + 1) (2n − 1)!! (2n − 2k + 1)!!k! k=1

= cos nθ AD (9061.1)

4.

8.925 1.

2.

3.

4.

n (2n − 1)!!Pn−1 (cos θ) − n+1 2n−1 (n − 1)! 2 4 sin nθ = π

∞  k=0

(2n + 2k − 1)!!(2k − 1)!! (2n + 4k + 3) P n+2k+1 (cos θ) 22k (n + k + 1)!(k + 1)!

2  2θ 4k − 1 (2k − 1)!! P 2k−1 (cos θ) = 1 − 2k 2 2 (2k − 1) k! π k=1   ∞ 2  4k + 1 (2k − 1)!! 1 2 sin θ P 2k (cos θ) = − 2k+1 2 (2k − 1)(k + 1) k! 2 π k=1   ∞ 2  k(4k − 1) (2k − 1)!! 2 cot θ P 2k−1 (cos θ) = 2k−1 2 (2k − 1) k! π k=1   ∞ 2  4k + 1 (2k − 1)!! 2 −1 P 2k (cos θ) = 22k k! π sin θ

AD (9061.2)

∞ 

AD (9062.2)

AD (9062.3)

AD (9062.4)

k=1

8.926 1.

 ∞  2 tan π−θ 1 θ θ 4 P n (cos θ) = ln = − ln sin − ln 1 + sin n sin θ 2 2 n=1

AD (9063.2)

Gegenbauer polynomials Cnλ (t)

8.930

∞ 

1 + sin θ2 1 P n (cos θ) = ln −1 n+1 sin θ2 n=1

2.

8.927

999

AD (9063.1)

∞ 

  1 cos k + 12 β P k (cos ϕ) =  2 (cos β − cos ϕ) k=0

[0 ≤ β < ϕ < π] [0 < ϕ < β < π]

=0

MO 72

8.928 1.

∞  (−1)n (4k + 1) [(2n − 1)!!]3

23n (n!)3

n=1 ∞ 

2.

(−1)n+1

n=1

P 2n (cos θ) =

(4n + 1) [(2n − 1)!!]3 (2n − 1)(2n +

2)23n

3

(n!)

4 K (sin θ) −1 π2

P 2n (cos θ) =

4 E (sin θ) 1 − π2 2

AD (9064.1)

AD (9064.2)

• For series of products of Bessel functions and Legendre polynomials, see 8.511 4, 8.531 3, 8.533 1, 8.533 2, and 8.534. • For series of products of Legendre and Chebyshev polynomials, see 8.919.

8.93 Gegenbauer polynomials Cnλ (t) 8.930 Definition. The polynomials C λn (t) of degree n are the coefficients of αn in the power-series expansion of the function ∞  −λ  1 − 2tα + α2 = C λn (t)αn WH n=0

Thus, the polynomials C λn (t) are a generalization of the Legendre polynomials. 1.10

C λ0 (t) = 1

2.10

C λ1 (t) = 2λt

3.10

C λ2 (t) = 2λ(λ + 1)t2 − λ   C λ3 (t) = 13 λ 4λ2 + 12λ + 8 t3 − 2λ(λ + 1)t     C λ4 (t) = 23 λ λ3 + 6λ2 + 11λ + 6 t4 − 2λ λ2 + 3λ + 2 t2 + 12 λ(λ + 1)   1 C λ5 (t) = 15 λ 4λ4 + 40λ3 + 140λ2 + 200λ + 96 t5     − 31 λ 4λ3 + 24λ2 + 44λ + 24 t3 + λ λ2 + 3λ + 2 t

4.10 5.11 6.10

7.10

C λ6 (t) =

 λ5 + 60λ4 + 340λ3 + 900λ2 + 1096λ + 480 t6   − 31 λ 2λ4 + 20λ3 + 70λ2 + 100λ + 48 t4     +λ λ3 + 6λ2 + 11λ + 6 t2 + 16 λ λ2 + 3λ + 2 1 45 λ



1000

Orthogonal Polynomials

8.931

8.931 Integral representation:

 π   n  1 Γ(2λ + n) Γ 2λ+1 2 t + t2 − 1 cos ϕ sin2λ−1 ϕ dϕ =√ π n! Γ(2λ) Γ(λ) 0 See also 3.252 11, 3.663 2, 3.664 4. C λn (t)

MO 99

Functional relations 8.932 Expressions in terms of hypergeometric functions:  ∗ 1 1−t Γ(2λ + n) λ F 2λ + n, −n; λ + ; 1. C n (t) = Γ(n + 1) Γ(2λ)  2 2 n 1 2 Γ(λ + n) n n 1−n t F − , ; 1 − λ − n; 2 = n! Γ(λ) 2 2 t  n 1 (−1) F −n, n + λ; ; t2 2. C λ2n (t) = (λ + n) B(λ, n + 1) 2  3 2 (−1)n 2t λ F −n, n + λ + 1; ; t 3. C 2n+1 (t) = B(λ, n + 1) 2

MO 97 MO 99 MO 99 MO 99

8.933 Recursion formulas: 1. 2. 3. 4.

(n + 2) C λn+2 (t) = 2(λ + n + 1)t C λn+1 (t) − (2λ + n) C λn (t)   λ+1 n C λn (t) = 2λ t C λ+1 (t) − C (t) n−1 n−2   λ+1 (2λ + n) C λn (t) = 2λ C λ+1 n (t) − t C n−1 (t)   n C λn (t) = (2λ + n − 1)t C λn−1 (t) − 2λ 1 − t2 C λ+1 n−2 (t)

8.934

WH WH WH

  1 −λ  1  1 − t2 2 dn  (−1)n Γ(2λ + n) Γ 2λ+1 2 λ+n− 2 2   1 − t = 2n Γ(2λ) Γ 2λ+1 n! dtn +n 2

1.

C λn (t)

2.

C λn (cos ϕ) =

n  Γ(λ + k) Γ(λ + l) k,l=0 k+l=n

3.

Mo 98

k!l! [Γ(λ)]

2

cos(k − l)ϕ

WH

MO 99

C λn (cos ψ cos ϑ + sin ψ sin ϑ cos ϕ) n Γ(2λ − 1)  22k (n − k)! [Γ(λ + k)]2 = (2λ + 2k − 1) sink ψ si nk ϑ 2 Γ(2λ + n + k) [Γ(λ)] k=0 λ− 1

 ψ, ϑ, ϕ real; 4.

lim Γ(λ) C λn (cos ϕ) =

λ→0

λ+k 2 × C λ+k (cos ϕ) n−k (cos ψ) C n−k (cos ϑ) C k  1 λ = 2 [“summation theorem”] (see also 8.794–8.796)

2 cos nϕ n

For orthogonality, see 8.904, 7.313. ∗ Equation

8.932.1 defines the generalized functions C λ n (t), where the subscript n can be an arbitrary number

WH

MO 98

Gegenbauer polynomials Cnλ (t)

8.938

1001

8.935 Derivatives: 1.

2.11

Γ(λ + k) λ+k dk λ C n−k (t) C n (t) = 2k k dt Γ(λ) In particular,

MO 99

d C λn (t) = 2λ C λ+1 n−1 (t) dt

WH

For integrals of the polynomials C λn (x) see 7.31–7.33. 8.936 Connections with other functions:   1 λ  4 − 2 12 −λ Γ(2λ + n) Γ λ + 12 1 λ 12 1 − t2 C n (t) = P λ+n− 1 (t) 1. 2 Γ(2λ) Γ(n + 1) 4 m   − 2 2 m!2m m 1 dm P n (t) m+ 1 m 1−t 2. C n−m2 (t) = = (−1) P n (t) (2m − 1)!! dtm (2m)! [m + 1 a natural number]

MO 98

MO 98, WH

3.

C 1/2 n (t) = P n (t)

4.

J λ− 12 (r sin ϑ sin α) (r sin ϑ sin α)−λ+ 2 e−ir cos ϑ cos α ∞ √ Jλ+k (r) C λk (cos ϑ) C λk (cos α) Γ(λ)   = 2  (λ + k)i−k 1 Γ λ + 2 k=0 rλ C λk (1)

5.

1

n

λ

lim λ− 2 C n2

λ→∞



n 2 2− 2 = H n (t) t λ n!

MO 99 MO 99a

See also 8.932. 8.937 Special cases and particular values: sin(n + 1)ϕ sin ϕ

1.

C 1n (cos ϕ) =

2.

C 00 (cos ϕ) = 1

MO 98

3.

C λ0 (t)

MO 98

4.

≡1  2λ + n − 1 C λn (1) ≡ n

8.938 A differential equation leading to the polynomials C λn (t): (2λ + 1)t  n(2λ + n) y − y=0 (cf. 9.174) y  + 2 t −1 t2 − 1 For series of products of Bessel functions and the polynomials C λn (x), see 8.532, 8.534.

MO 99

MO 98

WH

1002

Orthogonal Polynomials

8.93910 1. 2. 3. 4. 5.

8.939

Differentiation and Rodrigues’ formulas and orthogonality relation d λ C (t) = 2λ C λ+1 MS n−1 (t) dt n m d C λ (t) = 2m λ(λ + 1)(λ + 2) . . . (λ + m − 1) C λ+m MS n−m (t) dtm n d λ d C (t) = t C λn (t) − n C λn (t) MS dt n−1 dt d λ d C n+1 (t) = t C λn (t) + (2λ + n) C λn (t) MS dt dt  d λ  C (t) = (n + 2λ − 1) C λn−1 (t) − nt C λn (t) = (n + 2λ)t C λn (t) − (n + 1) C λn+1 (t) 1 − t2 dt n   = 2λ 1 − t2 C λ+1 n−1 (t)

5.3.2 5.3.2 5.3.2 5.3.2

MS 5.3.2

6.

 d  λ C n+1 (t) − C λn−1 (t) = 2(n + λ) C λn (t) dt

7.

C λn (t) =



MS 5.3.2

 1 −λ  n+λ− 12  (−1)n 2λ(2λ + 1)(2λ + 2) . . . (2λ + n − 1) 1 − t2 2 dn       1 − t2 1 3 1 n n dt 2 n! λ + 2 λ + 2 . . . λ + n − 2   1  1 n 2 2 −λ n  1  (−1) Γ λ + 2 Γ(n + 2λ) 1 − t d 2 n+λ− 2   1 − t = dtn 2n n! Γ(2λ) Γ n + λ + 12 [Rodrigues’ formula]

1

8. −1

λ− 12  C λn (t) C λm (t) 1 − t2 dt = 0 =

n = m

π21−2λ Γ(n + 2λ) n!(λ + n) [Γ(λ)]

MS 5.3.2

2

n=m [λ = 0]

[Orthogonality relation]

MS 5.3.2

8.94 The Chebyshev polynomials Tn (x) and Un (x) 8.940 Definition 1.

Chebyshev’s polynomials of the first kind n  n    1  T n (x) = cos (n arccos x) = x + i 1 − x2 + x − i 1 − x2 2 n    n  n−4  2  n  n−6  3 n n−2 1 − x2 + x 1 − x2 − 1 − x2 + . . . x x =x − 4 2 6 NA 66, 71

2.

Chebyshev’s polynomials of the second kind:  n+1  n+1    1 sin [(n + 1) arccos x] = √ x + i 1 − x2 − x − i 1 − x2 U n (x) = 2 x]   2i 1 − x  sin [arccos  2 n + 1 n−4  n + 1 n−2  n+1 n 2 1−x + 1 − x2 − . . . x x = x − 5 3 1

8.943

The Chebyshev polynomials

1003

Functional relations 8.941 Recursion formulas: 1.

T n+1 (x) − 2x T n (x) + T n−1 (x) = 0

2.

U n+1 (x) − 2x U n (x) + U n−1 (x) = 0

3. 4.

T n (x) = U n (x) − x U n−1 (x)   1 − x2 U n−1 (x) = x T n (x) − T n+1 (x)

5.∗

Tm±n (x) = Tm (x)Tn (x) ∓ (1 − x2 )Um−1 (x)Un−1 (x)

6.∗

Um±n (x) = Um (x)Tn (x) ± Tm+1 (x)Un−1 (x)

7.∗

Tmn (x) = Tm (x)(Tn (x))

8.∗

Umn−1 (x) = Um−1 (Tn (x))Un−1 (x)

NA 358

EH II 184(3) EH II 184(4)

For the orthogonality, see 7.343 and 8.904. 8.942 Relations with other functions:  1 1−x 1. T n (x) = F n, −n; ; 2 2 √ n− 12 1 − x2 dn  1 − x2 2. T n (x) = (−1)n n (2n − 1)!! dx n+ 12 dn  (−1)n (n + 1) 1 − x2 3. U n (x) = √ n 1 − x2 (2n + 1)!! dx  3 1−x 4.∗ Un (x) = (n + 1)F n + 2; −n; ; 2 2

MO 104

MO 104 EH II 185(15)

See also 8.962 3. 8.94310

Special cases 10.

U 0 (x) = 1

11.

U 1 (x) = 2x

T 2 (x) = 2x − 1

12.

U 2 (x) = 4x2 − 1

4.

T 3 (x) = 4x3 − 3x

13.

U 3 (x) = 8x3 − 4x

5.

T 4 (x) = 8x4 − 8x2 + 1

14.

U 4 (x) = 16x4 − 12x2 + 1

6.

T 5 (x) = 16x5 − 20x3 + 5x

15.

U 5 (x) = 32x5 − 32x3 + 6x

7.

T 6 (x) = 32x6 − 48x4 + 18x2 − 1

16.

U 6 (x) = 64x6 − 80x4 + 24x2 − 1

8.

T 7 (x) = 64x − 112x + 56x − 7x

17.

U 7 (x) = 128x7 − 192x5 + 80x3 − 8x

9.

T 8 (x) = 128x8 − 256x6 + 160x4 − 32x2 + 1

18.

U 8 (x) = 256x8 − 448x6 + 240x4 − 40x2 + 1

1.

T 0 (x) = 1

2.

T 1 (x) = x

3.

2

7

5

3

1004

Orthogonal Polynomials

8.944

8.944 Particular values: 1.

T n (1) = 1 n

2.

T n (−1) = (−1)

3.

n

T 2n (0) = (−1)

4.

T 2n+1 (0) = 0

5.

U 2n+1 (0) = 0

6.

U 2n (0) = (−1)n

8.945 The generating function: ∞

1.11

 1 − t2 = T 0 (x) + 2 T k (x)tk 2 1 − 2tx + t

[|t| < 1]

MO 104

[|t| < 1]

MO 104a, EH II 186(31)

k=1

∞

2.11

 1 = U k (x)tk 2 1 − 2tx + t k=0

8.946 Zeros. The polynomials T n (x) and U n (x) only have real simple zeros. All these zeros lie in the interval (−1, +1). √ 8.947 The functions T n (x) and 1 − x2 U n−1 (x) are two linearly independent solutions of the differential equation   d2 y dy + n2 y = 0 1 − x2 −x NA 69(58) dx2 dx 8.948 Of all polynomials of degree n with leading coefficient equal to 1, the one that deviates the least from zero on the interval [−1, +1] is the polynomial 2−n+1 T n (x). 8.94910 Differentiation and Rodrigues’ formulas and orthogonality relations 1. 2. 3. 4. 5. 6.

d T n (x) = n U n−1 (x) MS dx dm T n (x) = 2m−1 Γ(m)n C m MS n−m (x) dxm  d  T n (x) = n [T n−1 (x) − x T n (x)] = n [x T n (x) − T n+1 (x)] 1 − x2 MS dx d U n (x) = 2 C 2n−1 (x) MS dx dm m+1 U n (x) = 2m m!Cn−m (x) MS dxm  d  U n (x) = (n + 1) U n−1 (x) − nx U n (x) = (n + 2)x U n (x) − (n + 1) U n+1 (x) 1 − x2 dx

5.7.2 5.7.2 5.7.2 5.7.2 5.7.2

MS 5.7.2

7.

8.

 c 1  1 (−1)n π 1/2 1 − x2 2 dn  2 n− 2   1 − x [Rodrigues’ formula] T n (x) = dxn 2n+1 Γ n + 12 −1/2   1 (−1)n π 1/2 (n + 1) 1 − x2 dn  2 n+ 2   1 − x U n (x) = dxn 2n+1 Γ n + 32 [Rodrigues’ formula]

MS 5.7.2

MS 5.7.2

8.952

Hermite polynomials



1

9. −1



1

10. −1

−1/2  T m (x) T n (x) 1 − x2

1005

⎧ ⎪ m = n ⎨0, dx = π/2, m = n = 0 ⎪ ⎩ π, m=n=0

−1/2  U m (x) U n (x) 1 − x2 dx =

[Orthogonality relation]

MS 5.7.2

[Orthogonality relation]

MS 5.7.2

0, m = n π/8, m = n

8.95 The Hermite polynomials H n (x) 8.950 Definition

3.10

dn  −x2  e dxn or n n n H n (x) = 2n xn − 2n−1 xn−2 + 2n−2 · 1 · 3 · xn−4 − 2n−3 · 1 · 3 · 5 · xn−6 + . . . 2 4 6 H 0 (x) = 1

4.10

H 1 (x) = 2x

1. 2.

5.

10

H n (x) = (−1)n ex

2

SM 567(14)

MO 105a

H 2 (x) = 4x2 − 2

6.10

H 3 (x) = 8x3 − 12x

7.10

H 4 (x) = 16x4 − 48x2 + 12

8.10

H 5 (x) = 32x5 − 160x3 + 120x

9.10

H 6 (x) = 64x6 − 480x4 + 720x2 − 120

10.10 H 7 (x) = 128x7 − 1344x5 + 3360x3 − 1680x 11.10 H 8 (x) = 256x8 − 3584x6 + 13440x4 − 13440x2 + 1680 8.951 The integral representation:

2n H n (x) = √ π



∞

−∞

2

(x + it)n e−t dt

MO 106a

Functional relations 8.952 Recursion formulas: 1. 2.

d H n (x) = 2n H n−1 (x) dx H n+1 (x) = 2x H n (x) − 2n H n−1 (x)

SM 569(22) SM 570(23)

For the orthogonality, see 7.374 1 and 8.904. 3.

10

n H n (x) = −n H n−1 (x) + x H n (x)

MS 5.6.2

1006

4.10

Orthogonal Polynomials

8.953

H n (x) = 2x H n−1 (x) − H n−1 (x)

MS 5.6.2

8.953 The connection with other functions:  (2n)!  Φ −n, 12 ; x2 n!   (2n + 1)! x Φ −n, 32 ; x2 H 2n+1 (x) = (−1)n 2 n! H 2n (x) = (−1)n

1. 2.

MO 106a MO 106a

• For a connection with the polynomials C λn (x), see 8.936 5. • For a connection with the Laguerre polynomials, see 8.972 2 and 8.972 3. • For a connection with functions of a parabolic cylinder, see 9.253. 8.954 Inequalities: √ n! e2x n/2

n/2!

1.10

|H n (x)| ≤ 2 2 − 2 

2.10

√ 2 |H n (x)| < k n!2n/2 ex /2 ,

n

n

k ≈ 1.086435

8.955 Asymptotic representation: n n

MO 106a

x2 /2



√  cos 4n + 1x + O

SA 324



1 √ 4 n



1.

H 2n (x) = (−1) 2 (2n − 1)!!e

2.

   √ √  2 1 1 H 2n+1 (x) = (−1)n 2n+ 2 (2n − 1)!! 2n + 1ex /2 sin 4n + 3x + O √ 4 n

SM 579 SM 579

8.956 Special cases and particular values: 1.

H 0 (x) = 1

2.

H 1 (x) = 2x

3.

H 2 (x) = 4x2 − 2

4.

H 3 (x) = 8x3 − 12x

5.

H 4 (x) = 16x4 − 48x2 + 12

6.

H 2n (0) = (−1)n 2n (2n − 1)!!

7.

H 2n+1 (0) = 0

SM 570(24)

Series of Hermite polynomials 8.957 The generating function: 1.

∞ k    t H k (x) exp −t2 + 2tx = k!

SM 569(21)

k=0

∞

2.

 1 1 sinh 2x = H 2k+1 (x) e (2k + 1)! k=0

MO 106a

8.960

Jacobi’s polynomials

1007

∞

3.

 1 1 cosh 2x = H 2k (x) e (2k)!

MO 106a

k=0

4.

∞ 

e sin 2x =

(−1)k

k=0

5.

e cos 2x =

∞ 

(−1)k

k=0

1 H 2k+1 (x) (2k + 1)!

MO 106a

1 H 2k (x) (2k)!

MO 106a

8.958 “The summation theorem”: ⎛ ⎞ r

n2 r   ⎜ a2k ak xk ⎟ ⎜ ⎟  r  mk   ⎜ ⎟ ak k=1 k=1 ⎟= 5 Hn ⎜ H 1.12 (x ) mk k ⎜6 r ⎟ n! ⎜ 6 ⎟ m1 +m2 +···+mr =n k=1 mk ! 2⎠ ⎝7 ak mi ≥0

MO 106a

k=1

2.

A special case: n 2

2 H n (x + y) =

n    n k=0

k

 √   √  H n−k x 2 H k y 2

MO 107a

8.959 Hermite polynomials satisfy the differential equation 1.

2. 3.

dun d2 un + 2nun = 0; − 2x SM 566(9) dx2 dx A second solution of this differential equation is provided by the functions (A and B are arbitrary constants):   u2n = Ax Φ 12 − n; 32 ; x2 ,   MO 107 u2n+1 = B Φ − 21 − n; 12 ; x2

8.959(1)10 1. 2.

Rodrigues’ formula and orthogonality relation  n  2 d −x2 e H n (x) = (−1)n ex dxn

 ∞ 2 0 for m = n e−x H m (x) H n (x) dx = 1/2 n π 2 n! for m = n −∞

[Rodrigues’ formula]

MS 5.6.2 MS 5.6.2

8.96 Jacobi’s polynomials 8.960 Definition 1.

n   (−1)n −α −β d (1 − x) (1 − x)α+n (1 + x)β+n (1 + x) n 2n n!  dx  n n+β 1  n+α (x − 1)n−m (x + 1)m = n 2 m=0 m n−m

(x) = P (α,β) n

EH II 169(10), CO EH II 169(2)

1008

Orthogonal Polynomials

8.961

8.961 Functional relations: 1.11

P (α,α) (−x) = (−1)n P (α,α) (x) n n

2.

2(n + 1)(n + α + β + 1)(2n + α + β) P n+1 (x)   = (2n + α + β + 1) (2n + α + β)(2n + α + β + 2)x + α2 − β 2 P (α,β) (x) n

EH II 169(13) (α,β)

(α,β)

−2(n + α)(n + β)(2n + α + β + 2) P n−1 (x) EH II 169(11)

3.

 d (α,β) P (2n + α + β) 1 − x2 (x) = n[(α − β) − (2n + α + β)x] P (α,β) (x) n dx n (α,β) +2(n + α)(n + β) P n−1 (x) 

EH II 170(15)

4.11

5. 6.

m

d dxm

  1 Γ(n + m + α + β + 1) (α+m,β+m) P (α,β) (x) = m (x) P n−m n 2 Γ(n + α + β + 1)



[m = 1, 2, . . . , n]



EH II 170(17) (α,β)

n + 12 α + 12 β + 1 (1 − x) P (α+1,β) (x) = (n + α + 1) P (α,β) (x) − (n + 1) P n+1 (x) n n   (α,β) n + 12 α + 12 β + 1 (1 + x) P (α,β+1) (x) = (n + β + 1) P (α,β) (x) + (n + 1) P n+1 (x) n n x) P (α+1,β) (x) n

7.

(1 −

8.

(2n + α + β) P (α−1,β) (x) = (n + α + β) P (α,β) (x) − (n + β) P n−1 (x) n n

9.

(2n + α + β) P (α,β−1) (x) = (n + α + β) P (α,β) (x) + (n + α) P n−1 (x) n n

10.

P (α,β−1) (x) − P (α−1,β) (x) = P n−1 (x) n n

+ (1 +

x) P (α,β+1) (x) n

=

2 P (α,β) (x) n

EH II 173(32) EH II 173(33) EH II 173(34)

(α,β)

(α,β)

(α,β)

EH II 173(35) EH II 173(36) EH II 173(37)

8.962 Connections with other functions: 1.

2.

 1+x (−1)n Γ(n + 1 + β) F n + α + β + 1, −n; 1 + β; n! Γ(1 + β)  2 1−x Γ(n + 1 + α) = F n + α + β + 1, −n; 1 + α; n! Γ(1 + α)  2 n  x−1 Γ(n + 1 + α) 1 + x −n, −n − β; α + 1; = F n! Γ(1 + α)  2 x + 1  n x+1 Γ(n + 1 + β) x − 1 = F −n, −n − α; β + 1; n! Γ(1 + β) 2 x−1

P (α,β) (x) = n

P n (x) = P (0,0) (x) n

CO, EH II 170(16) EH II 170(16) EH II 170(16) EH II 170(16) CO, EH II 179(3)

2

3. 4. 5.∗

22n (n!) (− 1 ,− 1 ) P n 2 2 (x) (2n)!   Γ(n + 2ν) Γ ν + 12 ν  P (ν−1/2,ν−1/2)  C n (x) = (x) n Γ(2ν) Γ n + ν + 12 T n (x) =

U n (x) =

2n (n + 1)! ( 12 , 12 ) Pn (x) (2n + 1)!!

CO, EH II 184(5)a

MO 108a, EH II 174(4)

8.970

The Laguerre polynomials

1009

8.963 The generating function: ∞  P (α,β) (x)z n = 2α+β R−1 (1 − z + R)−α (1 + z + R)−β , n n=0

R=

 1 − 2xz + z 2

[|z| < 1] EH II 172(29)

8.964 The Jacobi polynomials constitute the unique rational solution of the differential (hypergeometric) equation   1 − x2 y  + [β − α − (α + β + 2)x]y  + n(n + α + β + 1)y = 0 EH II 169(14) 8.965 Asymptotic representation Pn (α, β)(cos θ) = #  $     cos n + 12 (α + β + 1) θ − 12 α + 14 π −3/2 + O n α+ 12  β+ 12 √  cos 12 θ πn sin 12 θ

[Im α = Im β = 0,

0 < θ < π] EH II 198(10)

8.966 A limit relationship:

  z   z −α = cos J α (z) EH II 173(41) lim n−α P (α,β) n n→∞ n 2 8.967 If α > −1 and β > −1, all the zeros of the polynomial P (α,β) (x) are simple and they lie in the n interval (−1, 1).

8.97 The Laguerre polynomials 8.970 Definition. 1 x −α dn  −x n+α  1. Lα e x e x n (x) = n! dxn n  n + α xm m = (−1) n − m m! m=0

[Rodrigues’ formula]

EH II 188(5), MO 108 MO 109, EH II 188(7)

L0n (x) = Ln (x)

2. 3.

10

4.

10

5.10 6.10 7.10

10

8.

Lα 0 (x) Lα 1 (x)

=1

= −x + α + 1  1 2 x − 2(α + 2)x + (α + 1)(α + 2) Lα 2 (x) = 2  1 α L3 (x) = − x3 − 3(α + 3)x2 + 3(α + 2)(α + 3)x − (α + 1)(α + 2)(α + 3) 6  1 α x4 − 4(α + 4)x3 + 6(α + 3) (α + 4) x2 − 4(α + 2)(α + 3)(α + 4)x L4 (x) = 24  + (α + 1)(α + 2)(α + 3)(α + 4) Lα 5 (x) =

 1 x5 − 5(α + 5)x4 + 10(α + 4)(α + 5)x3 − 10(α + 3)(α + 4)(α + 5)x2 − 120  + 5(α + 2)(α + 3)(α + 4)(α + 5)x − (α + 1)(α + 2)(α + 3)(α + 4)(α + 5)

ET I 369

1010

Orthogonal Polynomials

8.971

8.971 Functional relations: 1. 2.11 3.

 d  α α L (x) − Lα n+1 (x) = Ln (x) dx n α n Lα d α n (x) − (n + α) Ln−1 (x) Ln (x) = − Lα+1 n−1 (x) = dx x d α α x L (x) = n Lα n (x) − (n + α) Ln−1 (x) dx n α = (n + 1) Lα n+1 (x) − (n + α + 1 − x) Ln (x)

EH II 189(16) EH II 189(15), SM 575(42)a

EH II 189(12), MO 109 α x Lα+1 (x) = (n + α + 1) Lα n n (x) − (n + 1) Ln+1 (x)

4.

α = (n + α) Lα n−1 (x) − (n − x) Ln (x)

SM 575(43)a, EH II 190(23)

Lα−1 (x) n

5. 6.

(n +

=

Lα n (x)

1) Lα n+1 (x)

−

Lα n−1 (x)

− (2n + α + 1 −

SM 575(44)a, EH II 190(24)

x) Lα n (x)

+ (n +

α) Lα n−1 (x)

=0

[n = 1, 2, . . .] 7.10 8.

10

α (n + α) Lα−1 (x) = (n + 1) Lα n n+1 (x) − (n + 1 − x) Ln (x)

n Lα n (x)

= (2n + α − 1 −

x) Lα n−1 (x)

− (n + α −

MS 5.5.2

1) Lα n−2 (x) [n = 2, 3, . . .]

8.972 Connections with other functions:  n+α α Φ(−n, α + 1; x) 1. Ln (x) = n  2 x 2. H 2n (x) = (−1)n 22n n! L−1/2 n  2 x 3. H 2n+1 (x) = (−1)n 22n+1 n!x L1/2 n

MO 109, EH II 190(25, 24)

MS 5.5.2

MO 109, FI II 189(14) EH II 193(2), SM 576(47) EH II 193(3), SM 577(48)

8.973 Special cases: 1.

Lα 0 (x) = 1

2.

Lα 1 (x) = α + 1 − x  n+α α Ln (0) = n

3.

EH II 188(6)

xn n!

4.

n L−n n (x) = (−1)

5.

L1 (x) = 1 − x

6.

L2 (x) = 1 − 2x +

x2 2

EH II 188(6) EH II 189(13) MO 109

MO 109

8.977

The Laguerre polynomials

1011

8.974 Finite sums: 1.

n 

 α  (n + 1)! m! α α α Ln (x) Lα Lα m (x) Lm (y) = n+1 (y) − Ln+1 (x) Ln (y) Γ(m + α + 1) Γ(n + α + 1)(x − y) m=0 EH II 188(9)

2.12

n 

Γ(α − β + m) β Ln−m (x) = Lα n (x) Γ(α − β)m! m=0 n 

3.

MO 110, EH II 192(39)

α+1 Lα (x) m (x) = Ln

EH II 192(38)

β α+β+1 Lα (x + y) m (x) Ln−m (y) = Ln

EH II 192(41)

m=0

4.11

n  m=0

8.975 Arbitrary functions: 1.

(1 − z)−α−1 exp

2.

e−xz (1 + z)α =

∞  xz = Lα (x)z n z − 1 n=0 n ∞ 

Lα−n (x)z n n

[|z| < 1]

EH II 189(17), MO 109

[|z| < 1]

MO 110, EH II 189(19)

[α > −1]

EH II 189(18), MO 109

n=0 ∞   √  z − 12 α J α 2 xz e (xz) =

zn Lα n (x) Γ(n + α + 1) n=0

3.

8.976 Other series of Laguerre polynomials: 1.

∞  n=0

n!

  √ 1 α n xyz (xyz)− 2 α x+y Lα n (x) Ln (y)z = exp −z Iα 2 Γ(n + α + 1) 1−z 1−z 1−z [|z| < 1]

∞ 

2.

n=0

3.

6

Lα n (x) n+1

2 Lα n (x)

= ex x−α Γ(α, x) n

Γ(n + α + 1)  = 22n n!

k=0

4.6

α Lα n (x) Ln (y) =

[α > −1, 

2n − 2k n−k



EH II 189(20)

x > 0]

EH II 215(19)

1 (2k)! L2α (2x) k! Γ(α + k + 1) 2k

n α+2k Γ(1 + α + n)  Ln−k (x + y) (xy)k n! Γ(1 + α + k) k!

MO 110

MO 110, EH II 192(42)

k=0

8.977 Summation theorems: 1.12

Lnα1 +α2 +···+αk +k−1 (x1 + x2 + · · · + xk ) =

 i1 +i2 +···+ik =n

2.

y Lα n (x + y) = e

∞  (−1)k k α+k y Ln (x) k! k=0

αk α2 1 Lα i1 (x1 ) Li2 (x2 ) · · · Lik (xk )

MO 110

MO 110

1012

Orthogonal Polynomials

8.978

8.978 Limit relations and asymptotic behavior:  2x (α,β) 1. Lα 1 − (x) = lim P n n β→∞ β   x   √  1 = x− 2 α J α 2 x 2. lim n−α Lα n n→∞ n  1 3  √ 1 απ 1 1 1 1 1 π 2 x x− 2 α− 4 n 2 α− 4 cos 2 + O n 2 α− 4 3. Lα nx − − n (x) = √ e 2 4 π [Im α = 0,

EH II 191(36)

x > 0]

8.979 Laguerre polynomials satisfy the following differential equation: du d2 u + nu = 0 x 2 + (α − x + 1) dx dx 8.98011 Orthogonality relation  ∞ 0, m = n α  n+α  e−x xα Lα n (x) Lm (x) dx = , m=n Γ(1 + α) 0 n 8.98110

EH II 191(35)

EH II 199(1)

EH II 188(10), SM 574(34)

MS 5.5.2

Behavior of relative maxima of |Lα n (x)|

1.

Let α be arbitrary and real. The sequence formed by the relative maxima of |Lα n (x)| and by the value of this function at x = 0, is decreasing for x < α + 12 , and increasing for x > α + 12 . The successive relative maxima of |Lα n (x)| form a decreasing sequence for x ≤ 0, and an increasing sequence for x ≥ 0. SZ 174(7.6.1)

2.12

Let α be an arbitrary real number. The successive relative maxima of 1

−x/2 α/2+ 4 x |Lα e−x/2 x(α+1)/2 |Lα n (x)| and e n (x)|

form an increasing sequence provided x > x0 . In the first case ⎧ ⎨0 if α2 ≤ 1, 2 x0 = ⎩ α −1 if α2 > 1 2n + α + 1 In the second case

x0 =

0  2 1  12 α −4

if α2 ≤ 14 , if α2 >

1 4

SZ 174(7.6.2)

In the first case we take n so large that 2n + α + 1 > 0. 8.98210 1.

Asymptotic and limiting behavior of Lα n (x) Let α be arbitrary and real, c and w fixed positive constants, and let n → ∞. Then  

1 1 if cn−1 ≤ qx ≤ qω x−α/2− 4 O nα/2− 4 α Ln (x) = O (nα ) if 0 ≤ qx ≤ qcn−1 These bounds are precise as regards their orders in n. For α ≥ q − 12 , both bounds hold in both intervals, that is,

8.982

The Laguerre polynomials

Lα n (x) = 2.12

  1 1 x−α/2− 4 O nα/2− 4 , O (nα ) ,

0 < x ≤ qω,

Let α be arbitrary and real. Then for an arbitrary complex z   −α/2 1/2 2z , lim n−α Lα (z) = z J α n n→∞

uniformly if z is bounded.

1013

α≥q−

1 2

SZ 175(7.6.4)

SZ 191(8.1.3)

1014

Hypergeometric Functions

9.100

9.1 Hypergeometric Functions 9.10 Definition 9.100 A hypergeometric series is a series of the form α·β α(α + 1)β(β + 1) 2 α(α + 1)(α + 2)β(β + 1)(β + 2) 3 z+ z + z + ... F (α, β; γ; z) = 1 + γ·1 γ(γ + 1) · 1 · 2 γ(γ + 1)(γ + 2) · 1 · 2 · 3 9.101 A hypergeometric series terminates if α or β is equal to a negative integer or to zero. For γ = −n (n = 0, 1, 2, . . .), the hypergeometric series is indeterminate if neither α nor β is equal to −m (where m < n and m is a natural number). However, 1.

lim

γ→−n

F (α, β; γ; z) α(α + 1) . . . (α + n)β(β + 1) . . . (β + n) = Γ(γ) (n + 1)! ×z n+1 F (α + n + 1, β + n + 1; n + 2; z) EH I 62(16)

9.102 If we exclude these values of the parameters α, β, γ, a hypergeometric series converges in the unit circle |z| Re(α + β − γ) ≥0. point z = 1.

2.

Re(α + β − γ) Re β > 0] B(β, γ − β) 0  2π −n  z  cos nt dt Γ(p)n! 9.1128 F p, n + p; n + 1; z 2 = 2π Γ(p + n) 0 (1 − 2z cos t + z 2 )p [n = 0, 1, 2, . . . ; p = 0, −1, −2, . . . ; |z| < 1]

WH

WH, MO 16

9.121

Elementary functions as hypergeometric function

1015

 ∞i Γ(α + t) Γ(β + t) Γ(−t) 1 Γ(γ) (−z)t dt Γ(α) Γ(β) 2πi −∞i Γ(γ + t) Here, |arg(−z)| < π and the path of integration is chosen in such a way that the poles of the functions Γ(α + t) and Γ(β + t) lie to the left of the path of integration and the poles of the function Γ(−t) lie to the right of  it.   p+m p+m (−2)m (p + m) π 9.114 ;1− ; −1 = cosm ϕ cos pϕ dϕ F −m, − 2 2 sin pπ 0 [m + 1 is a natural number; p = 0, ±1, . . . ] EH I 80(8), MO 16

9.113 F (α, β; γ; z) =

See also 3.194 1, 2, 5, 3.196 1, 3.197 6, 9, 3.259 3, 3.312 3, 3.518 4–6, 3.665 2, 3.671 1, 2, 3.681 1, 3.984 7.

9.12 Representation of elementary functions in terms of a hypergeometric functions 9.121 1.8 2. 3. 4. 5. 6. 7. 8.

n

F (−n, β; β; −z) = (1 + z)   n n − 1 1 z2 (t + z)n + (t − z)n ; ; 2 = F − ,− 2 2 2 t 2tn     z n z = 1+ lim F −n, ω; 2ω; − ω→∞ t 2t   n − 1 n − 2 3 z2 (t + z)n − (t − z)n ,− ; ; 2 = F − 2 2 2 t 2nztn−1  z  (t + z)n − tn = F 1 − n, 1; 2; − t nztn−1 ln(1 + z) F (1, 1; 2; −z) = z   1+z ln 1 3 2 1−z , 1; ; z = F 2 2 2z   z z = 1 + z lim F 1, k; 2; lim F 1, k; 1; k→∞ k→∞ k k  z2 z lim F 1, k; 3; = · · · = ez =1+z+ 2 k→∞ k 

9. 10.

lim F

k→∞ k →∞

lim F k→∞

k →∞

11. 12.

 

lim F

k→∞ k →∞

lim F k→∞

k →∞



1 z k, k  ; ; 2 4kk  2

3 z2 k, k ; ; 2 4kk  

z

z2 1 k, k  ; ; − 2 4kk 

GA 127 IIIa GA 127 IV GA 127 V GA 127 VI GA 127 VII

−z

=

e +e 2

= cosh z

GA 127 IX

=

sinh z ez − e−z = 2z z

GA 127 X



z2 3 k, k ; ; − 2 4kk 

GA 127 II

GA 127 VIII





EH I 101(4), GA 127 Ia



sin z z

GA 127 XI

= cos z

GA 127 XII

= 

1016

Hypergeometric Functions

9.121



13.

F

14.

F

15.

F

16.

F

17.

F

18.

F

19.

F

20.

F

21.

F

22.

F

23.

F

24.

F

25.

F

26.

F

27.

F

28.

F

29.

F

30.

F

 1 1 3 z 2 , ; ; sin z = 2 2 2 sin z   3 z 1, 1; ; sin2 z = 2 sin z cos z   1 3 z 2 , 1; ; − tan z = 2 2 tan z   n+1 n−1 3 sin nz ,− ; ; sin2 z = 2 2 2 n sin z   n+2 n−2 3 sin nz ,− ; ; sin2 z = 2 2 2 n sin z cos z   n−2 n−1 3 sin nz − ,− ; ; − tan2 z = 2 2 2 n sin z cosn−1 z   n+2 n+1 3 sin nz cosn+1 z , ; ; − tan2 z = 2 2 2 n sin z   n n 1 , − ; ; sin2 z = cos nz 2 2 2   n+1 n−1 1 cos nz 2 ,− ; ; sin z = 2 2 2 cos z   n n−1 1 cos nz − ,− ; ; − tan2 z = 2 2 2 cosn z   n+1 n 1 , ; ; − tan2 z = cos nz cosn z 2 2 2   1 1 , 1; 2; 4z(1 − z) = 2 1−z   1 , 1; 1; sin2 z = sec z 2   arcsin z 1 1 3 2 = , ; ;z 2 2 2 z   3 arctan z 1 , 1; ; −z 2 = 2 2 z   1 1 3 arcsinh z , ; ; −z 2 = 2 2 2 z   1+n 1−n 3 2 sin (n arcsin z) , ; ;z = 2 2 2 nz   n n 3 sin (n arcsin z) √ 1 + , 1 − ; ; z2 = 2 2 2 nz 1 − z 2

GA 127 XIII GA 127 XIV GA 127 XV GA 127 XVI GA 127 XVII GA 127 XVIII GA 127 XIX EH I 101(11), GA 127 XX EH I 101(11), GA 127 XXI EH I 101(11), GA 127 XXII GA 127 XXIII

|z| ≤ 12 ;

|z(1 − z)| ≤

(cf. 9.121 13) (cf. 9.121 15) (cf. 9.121 26) (cf. 9.121 16) (cf. 9.121 17)

1 4



9.130

Transformation formulas for hypergeometric series

 31.

F

32.

F



n n 1 2 ,− ; ;z 2 2 2

1017

 = cos (n arcsin z)

1+n 1−n 1 2 , ; ;z 2 2 2

 =

cos (n arcsin z) √ 1 − z2

(cf. 9.121 20) (cf. 9.121 21)

The representation of special functions in terms of a hypergeometric function: • for complete elliptic integrals, see 8.113 1 and 8.114 1; • for integrals of Bessel functions, see 6.574 1, 3, 6.576 2–5, 6.621 1–3; • for Legendre polynomials, see 8.911 and 8.916. (All these hypergeometric series terminate; that is, these series are finite sums); • for Legendre functions, see 8.820 and 8.837; • for associated Legendre functions, see 8.702, 8.703, 8.751, 8.77, 8.852, and 8.853; • for Chebyshev polynomials, see 8.942 1; • for Jacobi’s polynomials, see 8.962; • for Gegenbauer polynomials, see 8.932; • for integrals of parabolic cylinder functions, see 7.725 6. • for Lerch function Φ(z, s, v); see 9.559 1. 9.122 Particular values: 1.

F (α, β; γ; 1) =

Γ(γ) Γ(γ − α − β) Γ(γ − α) Γ(γ − β)

[Re γ > Re(α + β)] GA 147(48), FI II 793

2.

3.

F (α, β; γ; 1) = F (−α, −β; γ − α − β; 1) 1 = F (−α, β; γ − α; 1) 1 = F (α, −β; γ − β; 1)   3 1 π = F 1, 1; ; 2 2 2

[Re γ > Re(α + β)]

GA 148(49)

[Re γ > Re(α + β)]

GA 148(50)

[Re γ > Re(α + β)]

GA 148(51)

(cf. 9.121 14)

9.13 Transformation formulas and the analytic continuation of functions defined by hypergeometric series 9.130 The series F (α, β; γ; z) defines an analytic function that, speaking generally, has singularities at the points z = 0, 1, and ∞ (In the general case, there are branch points). We make a cut in the z-plane along the real axis from z = 1 to z = ∞; that is, we require that |arg(−z)| < π for |z| ≥1. Then, the series f (α, β; γ; z) will, in the cut plane, yield a single-valued analytic continuation which we can obtain by means of the formulas below (provided γ + 1 is not a natural number and α − β and γ − α − β are not integers). These formulas make it possible to calculate the values of F in the given region even in the case in which |z| > 1. There are other closely related transformation formulas that can also be used to get the analytic continuation when the corresponding relationships hold between α, β, γ.

1018

Hypergeometric Functions

9.131

Transformation formulas 9.131 1.

11

  z F (α, β; γ; z) = (1 − z) F α, γ − β; γ; z − 1  z −β = (1 − z) F β, γ − α; γ; z−1 −α

GA 218(91) GA 218(92)

= (1 − z)γ−α−β F (γ − α, γ − β; γ; z) 2.

F (α, β; γ; z) =

Γ(γ) Γ(γ − α − β) F (α, β; α + β − γ + 1; 1 − z) Γ(γ − α) Γ(γ − β) Γ(γ) Γ(α + β − γ) +(1 − z)γ−α−β F (γ − α, γ − β; γ − α − β + 1; 1 − z) Γ(α) Γ(β) EH I 94, MO 13

9.132 1.

2.12

9.133

F (α, β; γ; z) =

  1 (1 − z)−α Γ(γ) Γ(β − α) F α, γ − β; α − β + 1; Γ(β) Γ(γ − α) 1−z   1 −β Γ(γ) Γ(α − β) +(1 − z) F β, γ − α; β − α + 1; Γ(α) Γ(γ − β) 1−z

  1 Γ(γ) Γ(β − α) (−z)−α F α, α + 1 − γ; α + 1 − β; F (α, β; γ; z) = Γ(β) Γ(γ − α) z   1 Γ(γ) Γ(α − β) −β (−z) F β, β + 1 − γ; β + 1 − α; + Γ(α) Γ(γ − β) z [|arg(−z)| < π, α − β = ±m, m = 0, 1, 2, . . .]     F 2α, 2β; α + β + 12 ; z = F α, β; α + β + 12 ; 4z(1 − z) |z| ≤ 12 ,

9.134

1 4



GA 220(93)

WH



 2 α α+1 1 z , ;β + ; 1. 2 2 2 2−z   1 4z −2α 2. F (2α, 2α + 1 − γ; γ; z) = (1 + z) F α, α + ; γ; 2 (1 + z)2     1 4z 1 2 −2α 3. F α, α + − β; β + ; z = (1 + z) F α, β; 2β; 2 2 (1 + z)2     1 1 2 2 ϕ α, β; α + β + 2α, 2β; α + β + ; sin ; sin 9.135 ϕ = F F 2 2 2 z −α F (α, β; 2β; z) = 1 − F 2 

|z(1 − z)| ≤

MO 13

ϕ x = sin real; 2 2

MO 13, EH I 111(4)

GA 225(100) GA 225(101)

 √ 1− 2 1 0) or the lower (Im z 0. However, if α − γ − m ≤0, the right member of this expression is a polynomial taken to the (1 − z)th power. 4.

If α, β, and γ are integers, the hypergeometric differential equation always has a solution that is regular for z = 0 and that is of the form R1 (z) + ln(1 − z)R2 (z), where R1 (z) and R2 (z) are rational functions of z. To get a solution of this form, we need to apply formulas 9.137 1–9.137 3 to the function F (α, β; γ; z). However, if γ = −λ, where λ + 1 is a natural number, formulas 9.137 1 and 9.137 2 should be applied not to F (α, β; γ; z) but to the function z λ+1 F (α + λ + 1, β + λ + 1; λ + 2, z). By successive applications of these formulas, we can reduce the positive values of the parameters to the pair, unity and zero. Furthermore, we can obtain the desired form of the solution from the formulas −1 F (1, 1; 2; z) = −z ln(1 − z), F (0, β; γ; z) = F (α, 0; γ; z) = 1

MO 19–20

9.16 Riemann’s differential equation 9.160 The hypergeometric differential equation is a particular case of Riemann’s differential equation   d2 u 1 − β − β 1 − γ − γ  du 1 − α − α 11 + + 1. + dz 2 z−b z−c dz ⎡z − a ⎤    αα (a − b)(a − c) (b − c)(b − a) (c − a)(c − b) ββ γγ u ⎦ +⎣ + =0 z−a z−b z−c (z − a)(z − b)(z − c) WH 

The coefficients of this equation have poles at the points a, b, and c, and the numbers α, α ; β, β  ; γ, γ  are called the indices corresponding to these poles. The indices α, α ; β, β  ; γ, γ  are related by the following equation: α + α + β + β  + γ + γ  − 1 = 0 2. 3.

WH

The differential equations 9.160 1 are written diagramatically as follows: ⎧ ⎫ ⎨a b c ⎬ u=P α β γ z ⎩  ⎭ α β γ

The singular points of the equation appear in the first row in this scheme, the indices corresponding to them appear beneath them, and the independent variable appears in the fourth column. WH

1024

Hypergeometric Functions

9.161

9.161 The two following transformation formulas are valid for Riemann’s P -equation: ⎧ ⎫ ⎧ ⎫ k  l ⎨ a b c  b c ⎬ ⎨ a ⎬ z−c z−a P α β γ z =P α+k β−k−1 γ+l z 1. ⎩  ⎭ ⎩  ⎭ z−b z−b α β γ α + k β − k − l γ + l ⎧ ⎫ ⎧ ⎫ ⎨a b c ⎬ ⎨a1 b1 c1 ⎬ 2. P α β γ z = P α β γ z1 ⎩  ⎭ ⎩  ⎭ α β γ α β γ The first of these formulas means that if

⎧ ⎨a u=P α ⎩  α

then the function

b β β

c γ γ

z

⎫ ⎬ ⎭

WH

WH

,

k  l z−c z−a u u1 = z−b z−b satisfies a second-order differential equation having the same singular points as equation 9.161 2 and indices equal to α + k, α + k; β − k − l, β  − k − l; γ + l, γ  + l. The second transformation formula converts a differential equation with singularities at the points a,b, and c, indices α, α ; β, β  ; γ, γ  , and an independent variable z into a differential equation with the same indices, singular points a1 , b1 , and c1 , and independent variable z1 . The variable z1 is connected with the variable z by the fractional transformation 

Az1 + B [AD − BC = 0] Cz1 + D The same transformation connects the points a1 , b1 , and c1 with the points a, b, and c. z=

WH, MO 20

9.162 By the successive application of the two transformation formulas 9.161 1 and 9.161 2, we can convert Riemann’s differential equation into the hypergeometric differential equation. Thus, the solution of Riemann’s differential equation can be expressed in terms of a hypergeometric function. For k = −α, l = −γ, and z1 = (z−a)(c−b) (z−b)(c−a) , we have 1.

⎧ ⎧ ⎫ b ⎨a b c ⎬  z − a α  z − c γ ⎨ a 0 β+α+γ P u= P α β γ z = ⎩  ⎩  ⎭ z−b z−b α − α β + α + γ α β γ ⎧ ⎫ α  γ ⎨ 0  ∞ 1 ⎬ z−c z−a (z−a)(c−b) 0 β+α+γ 0 = P (z−b)(c−a) ⎭ ⎩  z−b z−b α − α β + α + γ γ − γ

c 0 γ − γ

z

⎫ ⎬ ⎭

MO 23

2.

Thus, this solution can be expressed as a hypergeometric series as follows: α  γ    z−c z−a   (z − a)(c − b) u= F α + β + γ, α + β + γ; 1 + α − α ; z−b z−b (z − b)(c − a)

If the constants a, b, c; α, α ; β, β  ; γ, γ  are permuted in a suitable manner, Riemann’s equation remains unchanged. Thus, we obtain a set of 24 solutions of differential equations having the following form (provided none of the differences α − α , β − β  , γ − γ  are integers): WH, MO 23

9.166

9.163 1. 2. 3. 4.12 9.164 1.10 2. 3. 4. 9.165 1. 2. 3. 4.

Riemann’s differential equation

γ   z−c   (c − b)(z − a) F α + β + γ, α + β + γ; 1 + α − α ; z−b (c − a)(z − b) α  γ    (c − b)(z − a) z−c z−a     α + β + γ, α + β + γ; 1 + α − α; u2 = F z−b z−b (c − a)(z − b) α  γ     z−c z−a     (c − b)(z − a) u3 = F α + β + γ ,α + β + γ ;1 + α − α ; z−b z−b (c − a)(z − b)         α γ z−c z−a (c − b)(z − a)       u4 = F α + β + γ , α + β + γ ; 1 + α − α; z−b z−b (c − a)(z − b) 

u1 =

z−a z−b

α 

α   z−a   (a − c)(z − b) β + γ + α, β + γ + α; 1 + β − β ; F z−c (a − b)(z − c) β   α    (a − c)(z − b) z−a z−b     u6 = F β + γ + α, β + γ + α; 1 + β − β; z−c z−c (a − b)(z − c)        β α z−a z−b     (a − c)(z − b) u7 = F β + γ + α ,β + γ + α ;1 + β − β ; z−c z−c (a − b)(z − c) β   α    z−a z−b (a − c)(z − b)       β + γ + α , β + α + γ ; 1 + β − β; u8 = F z−c z−c (a − b)(z − c) 

u5 =

z−b z−c

β 

γ  β   z−b z−c   (b − a)(z − c) u9 = F γ + α + β, γ + α + β; 1 + γ − γ ; z−a z−a (b − c)(z − a)        γ β z−b z−c (b − a)(z − c)     u10 = F γ + α + β, γ + α + β; 1 + γ − γ; z−a z−a (b − c)(z − a) γ  β     z−b z−c     (b − a)(z − c) γ + α + β , γ + α + β ; 1 + γ − γ ; u11 = F z−a z−a (b − c)(z − a) γ   β     z−b z−c (b − a)(z − c)       u12 = F γ + α + β , γ + α + β ; 1 + γ − γ; z−a z−a (b − c)(z − a) 

9.166

β   z−b   (b − c)(z − a) F α + γ + β, α + γ + β; 1 + α − α ; z−c (b − a)(z − c) α  β    z−b z−a (b − c)(z − a)     α = + γ + β, α + γ + β; 1 + α − α; F z−c z−c (b − a)(z − c) α  β     z−b z−a     (b − c)(z − a) = F α + γ + β ,α + γ + β ;1 + α − α ; z−c z−c (b − a)(z − c) 

1.

u13 =

2.

u14

3.

u15

z−a z−c

α 

1025

1026

Hypergeometric Functions

 4.

u16 =

9.167

u17 =

2.

u18

3.

u19

4.

u20

u21

2.

u22

3.

u23

4.

u24

z−b z−c

β 

  (b − c)(z − a)       F α + γ + β , α + γ + β ; 1 + α − α; (b − a)(z − c)

z−c z−b

γ 

γ   z−c   (c − a)(z − b) = F β + α + γ, β + α + γ; 1 + β − β ; z−a (c − b)(z − a) β   γ    z−c z−b (c − a)(z − b)     = F β + α + γ, β + α + γ; 1 + β − β; z−a z−a (c − b)(z − a) β  γ     z−c z−b     (c − a)(z − b) = F β + α + γ ,β + α + γ ;1 + β − β ; z−a z−a (c − b)(z − a) β   γ     (c − a)(z − b) z−c z−b       β = + α + γ , β + α + γ ; 1 + β − β; F z−a z−a (c − b)(z − a) 

1.

α 

α   z−a   (a − b)(z − c) F γ + β + α, γ + β + α; 1 + γ − γ ; z−b (a − c)(z − b) γ   α    z−a z−c (a − b)(z − c)     γ = + β + α, γ + β + α; 1 + γ − γ; F z−b z−b (a − c)(z − b) γ  α    z−a z−c     (a − b)(z − c) = F γ + β + α ,γ + β + α ;1 + γ − γ ; z−b z−b (a − c)(z − b) γ   α    z−a z−c (a − b)(z − c)       = F γ + β + α , γ + β + α ; 1 + γ − γ; z−b z−b (a − c)(z − b) 

1.

9.168

z−a z−c

9.167

z−b z−a

β 

WH

9.17 Representing the solutions to certain second-order differential equations using a Riemann scheme 9.171 The hypergeometric equation (see 9.151): ⎫ ⎧ ∞ 1 ⎬ ⎨ 0 0 α 0 z u=P WH ⎭ ⎩ 1−γ β γ−α−β 9.172 The associated Legendre’s equation defining the functions P m n (z) for n and m integers (see 8.700 1): ⎫ ⎧ ∞ 1 ⎪ ⎪ ⎬ ⎨ 0 1−z 1 1 WH 1. u=P m n + 1 m 2 2 ⎪ 2 ⎪ ⎭ ⎩ 1 1 − 2 m −n − 2 m ⎧ ⎫ 0 ∞ 1 ⎪ ⎪ ⎪ ⎨ 1 ⎬ 1 ⎪ 1 0 2m 2. u = P −2n WH 2 1−z ⎪ ⎪ ⎪ ⎪ ⎩n + 1 −1m 1 ⎭ 2 2 2

9.180

Hypergeometric functions of two variables

1027

  z2 1 − 2 satisfies the equation 9.173 The function 2n ⎧ 2 ⎫ ∞ 0 ⎨ 4n ⎬ 1 1 2 m n + 1 m z u=P 2 ⎩ 21 ⎭ − 2 m −n − 12 m The function J m (z) satisfies the limiting form of this equation obtained as n → ∞. Pm n

9.174 The equation defining the Gegenbauer polynomials C λn (z), (see 8.938): ⎧ ⎫ ∞ 1 ⎨ −1 ⎬ u = P 12 − λ n + 2λ 12 − λ z ⎩ ⎭ 0 −n 0 9.175 Bessel’s equation (see 8.401) is the limiting form of the equations: ⎧ ⎫ ∞ c ⎨ 0 ⎬ 1 n ic + ic z 1. u=P 2 ⎩ ⎭ −n −ic 12 − ic ⎧ ⎫ ∞ c ⎨ 0 ⎬ 1 n 2. u = eiz P 0 z 2 ⎩ ⎭ −n 32 − 2ic 2ic − 1 ⎧ ⎫ ∞ c2 ⎨ 0 ⎬ 1 1 2 n (c − n) 0 z 3. u=P 2 ⎩ 21 ⎭ − 2 n − 12 (c + n) n + 1

WH

WH

WH

WH

WH

as c → ∞.

9.18 Hypergeometric functions of two variables 9.180 1.

 F 1 (α, β, β , γ; x, y) =

∞ ∞   (α)m+n (β)m (β  )n m n x y (γ)m+n m!n! m=0 n=0

[|x| < 1,

|y| < 1] EH I 224(6), AK 14(11)

2.

  F 2 (α, β, β , γ, γ ; x, y) =

∞  ∞ 



(α)m+n (β)m (β )n m n x y (γ)m (γ  )n m!n! m=0 n=0 [|x| + |y| < 1]

3.

  F 3 (α, α , β, β , γ; x, y) =

∞ ∞  



EH I 224(7), AK 14(12)



(α)m (α )n (β)m (β )n m n x y (γ)m+n m!n! m=0 n=0 [|x| < 1,

|y| < 1] EH I 224(8), AK 14(13)

4.

 F 4 (α, β, γ, γ ; x, y) =

∞ ∞   (α)m+n (β)m+n m n x y (γ)m (γ  )n m!n! m=0 n=0

!√ ! √

! x! + | y| < 1 EH I 224(9), AK 14(14)

1028

Hypergeometric Functions

9.181

9.181 The functions F 1 , F 2 , F 3 , and F 4 satisfy the following systems of partial differential equations for z: 1.

System of equations for z = F 1 :

2.

∂z ∂z ∂2z ∂2z + [γ − (α + β + 1)x] − βy − αβz = 0, + y(1 − x) ∂x2 ∂x ∂y ∂x ∂y 2 2 ∂ z ∂ z ∂z ∂z + [γ − (α + β  + 1) y] − βx − αβ  z = 0 y(1 − y) 2 + x(1 − y) ∂y ∂x ∂y ∂x ∂x System of equations for z = F 2 :

3.

∂z ∂z ∂2z ∂2z + [γ − (α + β + 1)x] − βy − αβz = 0, − xy ∂x2 ∂x ∂y ∂x ∂y 2 2 ∂ z ∂ z ∂z ∂z + [γ  − (α + β  + 1) y] − βx − αβ  z = 0 y(1 − y) 2 − xy ∂y ∂x ∂y ∂y ∂x System of equations for z = F 3 :

x(1 − x)

x(1 − x)

EH I 233(9)

EH I 234(10)

∂z ∂2z ∂2z + [γ − (α + β + 1)x] − αβz = 0, + y ∂x2 ∂x ∂y ∂x 2 2 ∂ z ∂ z ∂z y(1 − y) 2 + x + [γ − (α + β  + 1) y] − α β  z = 0 ∂y ∂x ∂y ∂y

x(1 − x)

EH I 234(11)

4.

System of equations for z = F 4 : ∂z ∂z ∂2z ∂2z ∂2z + [γ − (α + β + 1)x] − (α + β + 1)y − αβz = 0, x(1 − x) 2 − y 2 2 − 2xy ∂x ∂y ∂x ∂y ∂x ∂y EH I 234(12)

y(1 − y)

∂ z ∂z ∂ z ∂ z ∂z + [γ  − (α + β + 1)y] − (α + β + 1)x − αβz = 0 − x2 2 − 2xy ∂y 2 ∂x ∂x ∂y ∂y ∂x 2

2

2

AK 44

9.182 For certain relationships between the parameters and the argument, hypergeometric functions of two variables can be expressed in terms of hypergeometric functions of a single variable or in terms of elementary functions:     −α  x−y 1. EH I 238(1), AK 24(28) F 1 (α, β, β , β + β ; x, y) = (1 − y) F α, β; β + β ; 1−y   y   −α   2. EH I 238(2), AK 23 F 2 (α, β, β , β, γ ; x, y) = (1 − x) F α, β ; γ ; 1−x   xy  −β −β   3. EH I 238(3) F 2 (α, β, β , α, α; x, y) = (1 − x) (1 − y) F β, β ; α; (1 − x)(1 − y) 4. 5.

α+β−γ

F 3 (α, γ − α, β, γ − β, γ; x, y) = (1 − y) 

F (α, β; γ; x + y − xy)

EH I 238(4), AK 25(35)



F 4 (α, γ + γ − α − 1, γ, γ ; x(1 − y), y(1 − x)) = F (α, γ + γ  − α − 1; γ; x) F (α, γ + γ  − α − 1; γ  ; y) EH I 238(5)

9.183

Hypergeometric functions of two variables

6.

 F 4 α, β, α, β; −

7.

 F 4 α, β, β, β; −

8.

 F 4 α, β, 1 + α − β, β; −

9.

−y x , (1 − x)(1 − y) (1 − x)(1 − y)



y x ,− (1 − x)(1 − y) (1 − x)(1 − y)

= 

1029

(1 − x)β (1 − y)α (1 − xy)

EH I 238(6)

= (1 − x)α (1 − y)α F (α, 1 + α − β; β; xy)

y x ,− (1 − x)(1 − y) (1 − x)(1 − y)

EH I 238(7)



  x(1 − y) = (1 − y)α F α, β; 1 + α − β; − 1−x EH I 238(8)

   1 1 1 √ −2α x 1 F 4 α, α + , γ, ; x, y = (1 + y) F α, α + ; γ;  √ 2 2 2 2 2 1+ y  1 √ −2α x 1 + (1 − y) F α, α + ; γ;  √ 2 2 2 1− y

AK 23

Γ(γ) Γ (γ − α − β  )  F (α, β; γ − β ; x) Γ(γ − α) Γ (γ − β  )

10.

 F 1 (α, β, β , γ; x, 1) =

11.

  F 1 (α, β, β , γ; x, x) = F (α, β + β ; γ; x)

EH I 239(10), AK 22(23) EH I 239(11), AK 23(25)

9.183 Functional relations between hypergeometric functions of two variables:    x y 1.12 F 1 (α, β, β  , γ; x, y) = (1 − x)−β (1 − y)−β F 1 γ − α, β, β  , γ; , x−1 y−1  −α = (1 − x) F 1 α, γ − β − β  , β  , γ;

y−x x , x−1 1−x

EH I 239(1)



EH I 239(2)

  y y−x , = (1 − y)−α F 1 α, β, γ − β − β  , γ; y−1 y−1    x−y = (1 − x)γ−α−β (1 − y)−β F 1 γ − α, γ − β − β  , β  , γ; x, 1−y   x−y −β γ−α−β   ,y = (1 − x) (1 − y) F 1 γ − α, β, γ − β − β , γ; x−1

EH I 239(3)

EH I 240(4)

EH I 240(5), AK 30(5)

1030

2.

8

Hypergeometric Functions





−α

F 2 (α, β, β , γ, γ ; x, y) = (1 − x)

9.184

   F 2 α, γ − β, β , γ, γ ;

y x , x−1 1−x

 = (1 − y)−α F 2 α, β, γ  − β  , γ, γ  ;



y x , 1−y y−1

 −α = (1 − x − y) F 2 α, γ − β, γ  − β  , γ, γ  ;

EH I 240(6)



EH I 240(7)

y x , x+y−1 x+y−1



EH I 240(8), AK 32(6)

3.7

 F 4 (α, β, γ, γ ; x, y) =

  Γ(γ  ) Γ(β − α) x 1 −α  (−y) , α, α + 1 − γ , γ, α + 1 − β; F 4 Γ (γ  − α) Γ(β) y y  x 1 Γ(γ  ) Γ(α − β) β  (−y) F 4 β + 1 − γ , β, γ, β + 1 − α; , + Γ (γ  − β) Γ(α) y y EH I 240(9), AK 26(37)

9.184 Integral representations: Double integrals of the Euler type 1.

 F 1 (α, β, β , γ; x, y) =

Γ(γ)  ) Γ (γ − β − β  ) Γ(β) Γ (β    × uβ−1 v β −1 (1 − u − v)γ−β−β −1 (1 − ux − vy)−α du dv u≥0,v≥0 u+v≤1

Re β  > 0,

[Re β > 0, 2.

  F 2 (α, β, β , γ, γ ; x, y) =

3.

Γ(γ) Γ (γ  ) − β) Γ (γ  − β  )

 1 0

1

uβ−1 v β

0 

Re β > 0,



−1

(1 − u)γ−β−1 (1 − v)γ



−β  −1

(1 − ux − vy)−α du dv

Re (γ  − β  ) > 0]

Re (γ − β) > 0,

EH I 230(2), AK 28(2)

  F 3 (α, α , β, β , γ; x, y)

=

Γ(γ)  ) Γ (γ − β − β  ) Γ(β) Γ (β     × uβ−1 v β −1 (1 − u − v)−γ−β−β −1 (1 − ux)−α (1 − vy)−α du dv u≥0,v≥0 u+v≤1

Re β  > 0,

[Re β > 0,

4.

EH I 230(1), AK 28(1)

Γ(β) Γ (β  ) Γ(γ ×

[Re β > 0,

Re (γ − β − β  ) > 0]

Re (γ − β − β  ) > 0]

EH I 230(3), AK 28(3)

 F 4 (α, β, γ, γ ; x(1 − y), y(1 − x))  1 1  Γ(γ) Γ (γ  ) = uα−1 v β−1 (1 − u)γ−α−1 (1 − v)γ −β−1  Γ(α) Γ(β) Γ(γ − α) Γ (γ − β) 0 0

×(1 − ux)α−γ−γ [Re α > 0,



+1

(1 − vy)β−γ−γ

Re β > 0,



+1

(1 − ux − vy)γ+γ

Re (γ − α) > 0,





−α−β−1

Re (γ − β) > 0]

du dv EH I 230(4)

9.202

Introduction

1031

9.185 Integral representations: Integrals of the Mellin–Barnes type The functions F 1 , F 2 , F 3 and F 4 can be represented by means of double integrals of the following form:  i∞  i∞ Γ(γ) Ψ(s, t) Γ(−s) Γ(−t)(−x)s (−y)t ds dt F (x, y) = Γ(α) Γ(β)(2πi)2 −i∞ −i∞ Ψ(s, t)

F (x, y) 

Γ(α + s + t) Γ(β + s) Γ (β + t) Γ (β  ) Γ(γ + s + t)

F1 (α, β, β  , γ; x, y)

Γ(α + s + t) Γ(β + s) Γ (β  + t) Γ (γ  ) Γ (β  ) Γ(γ + s) Γ (γ  + t)

F2 (α, β, β  , γ, γ  ; x, y)

Γ(α + s) Γ (α + t) Γ(β + s) Γ (β  + t) Γ (α ) Γ (β  ) Γ(γ + s + t)

  F 3 (α, α , β, β , γ; x, y)

Γ(α + s + t) Γ(β + s + t) Γ (γ  ) Γ(γ + s) Γ (γ  + t)   [α, α , β, β may not be negative integers]

F4 (α, β, γ, γ  ; x, y) EH I 232(9–13), AK 41(33)

9.19 A hypergeometric function of several variables F A (α; β1 , . . . , βn ; γ1 , . . . , γn ; z1 , . . . , zn ) ∞ ∞  ∞   (α)m1 +···+mn (β1 )m1 · · · (βn )mn m1 m2 z1 z2 · · · znmn ... = (γ ) · · · (γ ) m ! · · · m ! 1 n 1 n m m 1 n m =0m =0 m =0 1

2

n

ET I 385

9.2 Confluent Hypergeometric Functions 9.20 Introduction 9.20110 A confluent hypergeometric function is obtained by taking the limit as c → ∞ in the solution of Riemann’s differential equation ⎧ ⎫ ∞ c ⎨ 0 ⎬ u = P 12 + μ −c c − λ z WH ⎩1 ⎭ − μ 0 λ 2 9.202 The equation obtained by means of this limiting process is of the form  1 2 λ d2 u du 4 −μ + + u=0 + WH 1. dz 2 dz z z2 Equation 9.202 1 has the following two linearly independent solutions:   1 2. z 2 +μ e−z Φ 12 + μ − λ, 2μ + 1; z   1 3. z 2 −μ e−z Φ 12 − μ − λ, −2μ + 1; z which are defined for all values of μ = ± 12 , ± 22 , ± 32 , . . .

MO 111

1032

Confluent Hypergeometric Functions

9.210

9.21 The functions Φ(α, γ; z) and Ψ(α, γ; z) 9.21010 1.

The series α(α + 1) z 2 α(α + 1)(α + 2) z 3 αz + + + ... γ 1! γ(γ + 1) 2! γ(γ + 1)(γ + 2) 3! is also called a confluent hypergeometric function.

Φ(α, γ; z) = 1 +

A second notation: Φ(α, γ; z) =

1F 1

(α; γ; z).

Γ(γ − 1) 1−γ Γ(1 − γ) Φ(α, γ; z) + z Φ(α − γ + 1, 2 − γ; z) Γ(α − γ + 1) Γ(α)

2.

Ψ(α, γ; z) =

3.

Bateman’s function kν (x) is defined by  2 π/2 kν (x) = cos (x tan θ − νθ) dθ π 0

EH I 257(7)

[x, ν real]

EH I 267

[0 < Re α < Re γ]

MO 114

9.211 Integral representation: 1

1.

21−γ e 2 z Φ(α, γ; z) = B(α, γ − α)

Φ(α, γ; z) =

2.

4.8

Ψ(α, γ; z) =

1

−1

1 Γ(α)

1

(1 − t)γ−α−1 (1 + t)α−1 e 2 zt dt

1 z 1−γ B(α, γ − α)

Φ(−ν, α + 1; z) =

3.





z

et tα−1 (z − t)γ−α−1 dt

0

Γ(α + 1) z − α e z 2 Γ(α + ν + 1) 



∞

α

e−t tν+ 2

0

[0 < Re α < Re γ]  √  J α 2 zt dt Re(α + ν + 1) > 0,

MO 114

|arg z|
0,

Re z > 0]

EH I 255(2)

0

Functional relations 9.212 Φ(α, γ; z) = ez Φ(γ − α, γ; −z) z Φ(α + 1, γ + 1; z) = Φ(α + 1, γ; z) − Φ(α, γ; z) γ

1. 2. 3. 4. 5.

∗

6.∗ ∗

7.

MO 112 MO 112

α Φ(α + 1, γ + 1; z) = (α − γ) Φ(α, γ + 1; z) + γ Φ(α, γ; z)

MO 112

α Φ(α + 1, γ; z) = (z + 2a − γ) Φ(α, γ; z) + (γ − α) Φ(α − 1, γ; z)

MO 112

Ψ(α, γ; z) = z

1−γ

Ψ(α − γ + 1, 2 − γ; z)

zΨ(α + 1, γ + 1; z) = (γ − α − 1)Ψ(α + 1, γ; z) + Ψ(α, γ; z) αΨ(α + 1, γ + 1; z) = Ψ(α, γ + 1; z) − Ψ(α, γ; z)

The Whittaker functions Mλ,μ (z) and Wλ,μ (z)

9.220

9.213 1.12

dΦ(α, γ; z) α = Φ(α + 1, γ + 1; z) dz γ

1033

MO 112

dΨ(α, γ; z) = −αΨ(α + 1, γ + 1; z) dz   1 n+1 α + n Φ(α, γ; z) = z 9.214 lim Φ(α + n + 1, n + 2; z) γ→−n Γ(γ) n+1 2.∗

[n = 0, 1, 2, . . .] MO 112

9.21510 Φ(α, α; z) = ez

1.

  π 

 1 1 Φ(α, 2α; 2z) = 2α− 2 exp 4 (1 − 2α)πi Γ α + 12 ez z 2 −α J α− 12 ze 2 i  z −p   Φ p + 12 , 2p + 1; 2iz = Γ(p + 1) eiz J p (z) 2 Ψ(α, α + 1; z) = z −α

2. 3. 4.∗

1

MO 15 MO 112 MO 15

For a representation of special functions in terms of a confluent hypergeometric function Φ(α, γ; z), see: • • • • • •

for for for for for for

the probability integral, 9.236; integrals of Bessel functions, 6.631 1; Hermite polynomials, 8.953 and 8.959; Laguerre polynomials, 8.972 1; parabolic cylinder functions, 9.240; the Whittaker functions M λ,μ (z), 9.220 2 and 9.220 3.

9.216 The function Φ(α, γ; z) is a solution of the differential equation dF d2 F + (γ − z) − αF = 0 dz 2 dz This equation has two linearly independent solutions:

1.

z

2.

Φ(α, γ; z)

3.

z 1−γ Φ(α − γ + 1, 2 − γ; z)

MO 111

MO 112

Or alternatively Φ(α, γ; z) and Ψ(α, γ; z).

9.22–9.23 The Whittaker functions Mλ,μ (z) and Wλ,μ (z) z

9.220 If we make the change of variable u = e− 2 W in equation 9.202 1, we obtain the equation  1 2 d2 W 1 λ 4 −μ 1. W =0 + − + + MO 115 dz 2 4 z z2 Equation 9.220 1 has the following two linearly independent solutions:   1 2. M λ,μ (z) = z μ+ 2 e−z/2 Φ μ − λ + 12 , 2μ + 1; z   1 3.11 M λ,−μ (z) = z −μ+ 2 e−z/2 Φ −μ − λ + 12 , −2μ + 1; z MO 115 To obtain solutions that are also suitable for 2μ = ±1, ±2, . . . , we introduce Whittaker’s function

1034

4.

12

Confluent Hypergeometric Functions

9.221

  1 W λ,μ (z) = z e Ψ μ − λ + , 2μ + 1; z 2 Γ(2μ) Γ(−2μ)  M λ,μ (z) +  1  M λ,−μ (z) = 1 Γ 2 −μ−λ Γ 2 +μ−λ 1 μ+ 2 −z/2

WH

which, for 2μ approaching an integer, is also a solution of equation 9.220 1. For the functions M λ,μ (z) and W λ,μ (z), z = 0 is a branch point and z = ∞ is an essential singular point. Therefore, we shall examine these functions only for |arg z| < π. These functions W λ,μ (z) and W −λ,μ (−z) are linearly independent solutions of equation 9.220 1. Integral representations  1 1 1 1 1 z μ+ 2  9.221 M λ,μ (z) = 2μ  (1 + t)μ−λ− 2 (1 − t)μ+λ− 2 e 2 zt dt, 1 1 2 B μ + λ + 2 , μ − λ + 2 −1 if the integral converges. See also 6.631 1 and 7.623 3. 9.222  ∞ 1 1 1 z μ+ 2 e−z/2 11   1. W λ,μ (z) = e−zt tμ−λ− 2 (1 + t)μ+λ− 2 dt Γ μ − λ + 12 0 Re(μ − λ) > − 21 , |arg z|
− 21 , |arg z| < π     z  i∞ Γ(u − λ) Γ −u − μ + 12 Γ −u + μ + 12 u e− 2     z du 9.223 W λ,μ (z) = 2πi −i∞ Γ −λ + μ + 12 Γ −λ − μ + 12 [the path of integration is chosen in such a way that   the poles of the function Γ(u − λ) are separated from the poles of the functions Γ −u − μ + 12 and Γ −u + μ + 12 ]. See also 7.142. MO 118  ∞  ∞ 1 1 9.224 W μ, 12 +μ (z) = z μ+1 e− 2 z (1 + t)2μ e−zt dt = z −μ e 2 z t2μ e−t dt [Re z > 0] WH 

9.225 1.

z

0

 W λ,μ (x) W −λ,μ (x) = −x

∞

tanh2λ

0

t {J 2μ (x sinh t) sin(μ − λ)π 2

+ Y 2μ (x sinh t) cos(μ − λ)π} dt

(z1 z2 ) exp − 12 (z1 + z2 ) W κ,μ (z1 ) W λ,μ (z2 ) = Γ(1 − κ − λ) 

|Re μ| − Re λ < 12 ;

x>0



MO 119

μ+ 12

2.

∞

× 0

×F Θ=

t (z1 + z2 + t) , (z1 + t) (z2 + t)

− 12 +κ−μ

e−t t−κ−λ (z1 + t)

1 2

− 12 +λ−μ

(z2 + t)

 − κ + μ, 12 − λ + μ; 1 − κ − λ; Θ dt [z1 = 0,

z2 = 0,

|arg z1 | < π,

|arg z2 | < π,

Re(κ + λ) < 1] MO 119

See also 3.334, 3.381 6, 3.382 3, 3.383 4, 8, 3.384 3, 3.471 2.

The Whittaker functions Mλ,μ (z) and Wλ,μ (z)

9.233

1035

9.226 Series representations   ∞ 2k  1 z M 0,μ (z) = z 2 +μ 1 + 24k k!(μ + 1)(μ + 2) . . . (μ + k)

WH

k=1

Asymptotic representations For large values of |z| $ $ ⎛ 2 % $ 2  2 % 2 % ⎞   ∞ μ2 − λ − 12 μ − λ − 32 . . . μ2 − λ − k + 12  ⎠ W λ,μ (z) ∼ e−z/2 z λ ⎝1 + k!z k

9.2277

k=1

[|arg z| ≤ π − α < π]

WH

9.228 For large values of |λ|

  √ 1 1 −μ− 14 1/4 z cos 2 λz − μπ − π M λ,μ (z) ∼ √ Γ(2μ + 1)λ 4 π

9.229

1  √ 4z 4 −λ+λ ln λ π 1. W λ,μ ∼ − e sin 2 λz − λπ − λ 4  z  14 √ 2. W −λ,μ ∼ eλ−λ ln λ−2 λz 4λ Formulas 9.228 and 9.229 are applicable for √ |λ| 1, |λ| |z|, |λ| |μ|, z = 0, |arg z| < 3π 4 and |arg λ|
−1;

for x < 0,

arg xp = pπi] MO 122

2.

D p (z) =

2 − z4

e Γ(−p)



∞ 0

e−xz−

x2 2

x−p−1 dx

[Re p < 0]

(cf. 3.462 1)

MO 122

1038

9.242 1.10 2.

3.

4.

Confluent Hypergeometric Functions

9.242

 Γ(p + 1) − 1 z2 (0+) −zt− 1 t2 2 e 4 e (−t)−p−1 dt [|arg(−t)| ≤ π] 2πi ∞    (−1+) Γ p2 + 1 1 1 2 1 1 (p−1) D p (z) = 2 2 e 4 z t (1 + t)− 2 p−1 (1 − t) 2 (p−1) dt iπ −∞

|arg z| < π4 ; |arg(1 + t)| ≤ π    1 − 1 z2 ∞i Γ 12 t − 12 p Γ(−t) √ t−p−2 t e 4 D p (z) = 2 z dt 2πi Γ(−p) −∞i

|arg z| < 34 π; p is not a positive integer D p (z) = −

1 − 1 z2 e 4 D p (z) = 2πi



(0−)

Γ

1

2t

∞

WH

WH

WH

 − 12 p Γ(−t) √ t−p−2 t 2 z dt Γ(−p)

[for all values of arg z; also,the contours encircle the poles of the function Γ(−t) but they do not encircle  the poles of the function Γ 12 t − 12 p ]. WH 9.243 ⎧ ⎨ ∞  π −1/2 √  √  2 cos  1 2 1 n+1 zt n dt n e4z −2n e−n(t−1) 1. D n (z) = (−1)μ ⎩ −∞ sin 2 ⎫  0  ∞$ ⎬ % cos  √   √ 2 2 2 cos  1 zt n dt e 2 n(1−t ) tn − e−n(t−1) zt n dt − e−n(t−1) + ⎭ sin sin 0 −∞ 

2.

[n is a natural number]

WH

∞

2 cos 1 2 (2zt) dt D n (z) = (−1)μ 2n+2 (2π)−1/2 e 4 z tn e−2t sin *n+ 0 [n is a natural number, μ = , and the cosine or sine is chosen according as n is even or odd] 2

WH

9.244 1.

D −p−1 [(1 + i)z] =

e− 2

p−1 2

p+1

2.

D p [(1 + i)z] =

2 2   Γ − p2

Γ

 1



iz 2 2

 p+1 

∞

2

∞

0

e−ix (1 +

2 2

z

p−1

i

e− 2 z

2

x

xp

1+ p x2 ) 2

(x + 1) 2 p dx (x − 1)1+ 2

dx

Re p > −1, Re p < 0;

  Re iz 2 ≥ 0

  Re iz 2 ≥ 0

MO 122

MO 122

See also 3.383 6, 7, 3.384 2, 6, 3.966 5, 6. 9.245     2  ∞ 1 1 1 t x sinh t + pπ √ dt cothp+ 2 sin 1.10 D p (x) D −p−1 (x) = − √ 2 2 π 0 sinh t [x is real,

Re p < 0]

MO 122

Parabolic cylinder functions Dp (z)

9.248

2.



D p ze

π 4i





D p ze

−π 4i



1 = Γ(−p)



∞

0

  2 z dt coth t exp − sinh 2t 2 sinh t |arg z| < π4 ;

1039

p

Re p < 0



MO 122

See also 6.613. 9.246 Asymptotic expansions. If |z| 1 and |z| |p|, then   p(p − 1) p(p − 1)(p − 2)(p − 3) z2 + − . . . 1. D p (z) ∼ e− 4 z p 1 − 2z 2 2 · 4z 4

MO 121 |arg z| < 34 π   2 p(p − 1) p(p − 1)(p − 2)(p − 3) 2.11 D p (z) ∼ e−z /4 z p 1 − + − ... 2z 2 ⎛ 2 · 4z 4 ⎞ √ 2π pπi z2 /4 −p−1 ⎝ (p + 1)(p + 2)(p + 3)(p + 4) (p + 1)(p + 2) e e z + + . . .⎠ − 1+ Γ(−p) 2z 2 2 · 4z 4 1

5 MO 121 4 π < arg z < 4 π   2 p(p − 1) p(p − 1)(p − 2)(p − 3) 3.11 D p (z) ∼ e−z /4 z p 1 − + − ... 2z 2 2 · 4z 4 ⎞ ⎛ √ 2π −pπi z2 /4 −p−1 ⎝ (p + 1)(p + 2)(p + 3)(p + 4) (p + 1)(p + 2) e e z + + . . .⎠ − 1+ Γ(−p) 2z 2 2 · 4z 4 1

− 4 π > arg z > − 45 π MO 121 Functional relations 9.247 Recursion formulas: 1. 2. 3.

D p+1 (z) − z D p (z) + p D p−1 (z) = 0 d D p (z) + dz d D p (z) − dz

1 z D p (z) − p D p−1 (z) = 0 2 1 z D p (z) + D p+1 (z) = 0 2

WH WH MO 121

9.248 Linear relations: 1.12

% Γ(p + 1) $ πpi/2 √ e D −p−1 (iz) + e−πpi/2 D −p−1 (−iz) 2π √ 2π −π(p+1)i/2 −pπi e D p (−z) + D −p−1 (iz) =e Γ(−p) √ 2π π(p+1)i/2 e D −p−1 (−iz) = epπi D p (−z) + Γ(−p)

D p (z) =

MO 121

1040

Confluent Hypergeometric Functions

9.249

  ∞ cos xt −it2 /4 i 21+p/2 π exp − x2 + p e dt Γ(−p) 2 2 tp+1 0 [x real; −1 < Re p < 0] MO 122  n  n z 2 /4 d −z 2 /2 10 e [n = 0, 1, 2, . . .] 9.251 D n (z) = (−1) e WH dz n    ∞ p , ,   b k  p a (bx − ay)2 √ D p−k 9.252 D p (ax+by) = exp a2 + b 2 x D k a2 + b 2 y 4 a k a2 + b 2 9.24910 D p [(1 + i)x] + D p [−(1 + i)x] =

[a > b > 0,

x > 0,

k=0

y > 0,

Re p ≥ 0]

“summation theorem”

Connections with other functions   z2 z −n − 11 9.253 D n (z) = 2 2 e 4 H n √ 2 9.254 (    z2 π z 4 1−Φ √ 1. D −1 (z) = e 2 2 ( (    z2 π 2 − z2 z 11 e 2 −z 1−Φ √ 2. D −2 (z) = e 4 2 π 2

MO 124

MO 123a

MO 123

MO 123

9.255 Differential equations leading to parabolic cylinder functions:   d2 u 1 z2 u=0 1. + p+ − dz 2 2 4 The solutions are u = D p (z), D p (−z), D −p−1 (iz), and D −p−1 (−iz) (These four solutions are linearly dependent. See 9.248) 2.

 d2 u  2 + z + λ u = 0, 2 dz

3.7

d u du + (p + 1)u = 0, +z 2 dz dz

u = D − 1+iλ [±(1 + i)z] 2

EH II 118(12,13)a, MO 123 2

u = e−

z2 4

D p (z)

MO 123

9.26 Confluent hypergeometric series of two variables 9.261 ∞  (α)m+n (β)n m n x y (γ)m+n m!n! m,n=0

1.12

Φ1 (α, β, γ, x, y) =

2.12

Φ2 (β, β  , γ, x, y) =

3.

Φ3 (β, γ, x, y) =

∞ 

(β)m (β  )n m n x y (γ)m+n m!n! m,n=0

∞ 

(β)m xm y n (γ) m!n! m+n m,n=0

[|x| < 1]

EH I 225(20)

EH I 225(21)a, ET I 385

EH I 225(22)

9.301

Definition

1041

The functions Φ1 , Φ2 , Φ3 satisfy the following systems of partial differential equations: 9.262 1.

z = Φ1 (α, β, γ, x, y)

EH I 235(23)

∂z ∂z ∂2z ∂2z + [γ − (α + β + 1)x] − βy − αβz = 0, + y(1 − x) ∂x2 ∂x ∂y ∂x ∂y ∂ 2z ∂z ∂z ∂2z y 2 +x + (γ − y) −x − αz = 0 ∂y ∂x ∂y ∂y ∂x

x(1 − x)

2.

z = Φ2 (β, β  , γ, x, y)

EH I 235(24)

∂z ∂2z ∂ 2z + (γ − x) − βz = 0, +y 2 ∂x ∂x ∂y ∂x ∂ 2z ∂2z ∂z y 2 +x + (γ − y) − βz = 0 ∂y ∂x ∂y ∂y

x

3.

z = Φ3 (β, γ, x, y)

EH I 235(25)

∂z ∂2z ∂ 2z + (γ − x) − βz = 0, + y ∂x2 ∂x ∂y ∂x ∂ 2z ∂z ∂2z y 2 +x +γ −z =0 ∂y ∂x ∂y ∂y

x

9.3 Meijer’s G-Function 9.30 Definition m -

9.30112

 !   ! a1 , . . . , a p 1 j=1 m,n ! = G p,q x ! q b1 , . . . , bq 2πi L j=m+1

Γ (bj − s)

n -

Γ (1 − aj + s)

j=1

Γ (1 − bj + s)

p j=n+1

xs ds Γ (aj − s)

[0 ≤ m ≤ q, 0 ≤ n ≤ p, and the poles of Γ (bj − s) must not coincide with the poles of Γ (1 − ak + s) for any j and k (where j = 1, . . . , m; k = 1, . . . , n]). Besides 9.301, the following notations are also used:  !  ! ar mn , Gmn G(x) EH I 207(1) G pq x !! pq (x), bs

Meijer’s G-Function

1042

9.302

9.302 Three types of integration paths L in the right member of 9.301 can be exhibited: 1.12

The path L runs from −i∞ to +i∞ along the imaginary axis so that the poles of the functions Γ (1 − ak + s) lie to the left, and the poles of the functions Γ (bj − s) lie to the right of L (for j = 1, 2, . . . , m and k = 1, 2, . . . , n). In this case, the conditions under which the integral 9.301 converges are of the form   p + q < 2(m + n), |arg x| < m + n − 12 p − 12 q π EH I 207(2)

2.

L is a loop, beginning and ending at +∞, that encircles the poles of the functions Γ (bj − s) (for j = 1, 2, . . . , m) once in the negative direction. All the poles of the functions Γ (1 − ak + s) must remain outside this loop. Then, the conditions under which the integral 9.301 converges are: q ≥ 1 and either p < q or p = q and |x| < 1

3.

EH I 207(3)

L is a loop, beginning and ending at −∞, that encircles the poles of the functions Γ (1 − ak + s) (for k = 1, 2, . . . , n) once in the positive direction. All the poles of the functions Γ (bj − s) (for j = 1, 2, . . . , m) must remain outside this loop. The conditions under which the integral in 9.301 converges are p ≥ 1 and either p > q or p = q and |x| > 1 EH I 207(4)  !  ! ar x !! is analytic with respect to x; it is symmetric with respect to the The function G mn pq bs parameters a1 , . . . , an and also with respect to an+1 , . . . , ap ; b1 , . . . , bm ; bm+1 , . . . , bq . EH I 208

9.30312 If no two bj (for j = 1, 2, . . . , m) differ by an integer, then, under the conditions that either p < q or p = q and |x| < 1, m n  Γ (b − b ) Γ (1 + bh − aj ) j h  !   m ! ar j=1 j=1 mn = xbh G pq x !! q p bs h=1 Γ (1 + bh − bj ) Γ (aj − bh ) j=m+1



j=n+1

× p F q−1 1 + bh − a1 , . . . , 1 + bh − ap ; . . . , ∗, . . . , 1 + bh − bq ;

1 + bh − b1 , . . .  (−1)p−m−n x EH I 208(5)

The prime by the product symbol denotes the omission of the product when j = h. The asterisk in the function p F q−1 denotes the omission of the hth parameter.

9.304

Functional relations

1043

9.3047 If no two ak (for k = 1, 2, . . . , n) differ by an integer then, under the conditions that q < p or q = p and |x| > 1, n m  Γ (a − a ) Γ (bj − ah + 1) h j  !   n ! ar j=1 j=1 mn = xah −1 G pq x !! p q bs h=1 Γ (aj − ah + 1) Γ (ah − bj ) j=n+1



j=m+1

× q F p−1 1 + b1 − ah , . . . , 1 + bq − ah ; . . . , ∗, . . . , 1 + ap − ah ;

1 + a1 − ah , . . .  q−m−n −1 (−1) x EH I 208(6)

9.31 Functional relations If one of the parameters aj (for j = 1, 2, . . . , n) coincides with one of the parameters bj (for j = m + 1, m + 2, . . . , q), the order of the G-function decreases. For example,  !   !  ! a1 , . . . , a p ! a2 , . . . , a p mn ! = G m,n−1 x 1. G pq x !! p−1,q−1 ! b1 , . . . , bq−1 b1 , . . . , bq−1 , a1 [n, p, q ≥ 1] An analogous relationship occurs when one of the parameters bj (for j = 1, 2, . . . , m) coincides with one of the aj (for j = n + 1, . . . , p). In this case, it is m and not n that decreases by one unit.

2. 3.12

The G-function with p > q can be transformed into the G-function with p < q by means of the relationships: !    !  ! ! 1 − bs mn −1 ! ar nm ! = G qp x ! EH I 209(9) G pq x ! bs 1 − ar  !   !   !  ! ar ! a1 − 1, a2 , . . . , ap ! ap d mn m,n ! ! = Gm,n x + (a x [n ≥ 1] x − 1)G G pq x !! 1 p,q p,q ! b ! bq dx bs  !q   !  ! a , . . . , ap−1 , ap − 1 ! a = −Gm,n x !! 1 + (ap − 1)Gm,n x !! p [n < p] p,q p,q bq  ! bq   !  ! a ! a = −Gm,n x !! p + b1 Gm,n x !! p [m ≥ 1] p,q p,q b + 1, b , . . . , b bq ! 1 2 q   !   ! a ! a + bq Gm,n = −Gm,n x !! p x !! p [m < q] p,q p,q b1 , b2 , . . . , bq−1 , bq + 1 bq

4.

  !   ! ! ap , 1 − r ! 1 − r, ap m+1,n m,n+1 r ! ! = (−1) G p+1,q+1 z ! G p+1,q+1 z ! 0, bq bq , 1

5.

  !   ! ! ap ! ap + k mn ! ! z = z z k G mn G pq pq ! bq ! bq + k

[r = 0, 1, 2, . . .]

EH I 210(13)

MS2 6 (1.2.2) MS2 7 (1.2.7)

Meijer’s G-Function

1044

6.

∗

7.∗

Gm,n+1 p+1,q+1

  !  !  ! α, ap !ap , α α−β m+1,n ! ! = (−1) z! Gp+1,q+1 z ! β, bq bq , β

where ap = a1 , a2 , . . . , ap and bq = b1 , b2 , . . . , bq .   !   ! k !ap !0, ap m,n+1 k d m,n ! ! x = G x x G p+1,q+1 ! bq ! bq , k dxk p,q

9.304

[q ≥ m,

β − α = 0, 1, 2, . . .]

[n ≥ 1]

9.32 A differential equation for the G-function mn G pq

 !  ! ar x !! satisfies the following linear q th -order differential equation bs ⎡ ⎤   p  q  d d ⎣(−1)p−m−n x − aj + 1 − − bj ⎦ y = 0 x x dx dx j=1 j=1

[p ≤ q]

EH I 210(1)

9.33 Series of G-functions mn G pq

 !   !  ∞  ! a1 , . . . , ap ! a1 , . . . , a p 1 r b1 mn ! ! (1 − λ) G pq x ! λx ! =λ b1 , . . . , bq r! b1 + r, b2 , . . . , bq r=0 [|λ − 1| < 1, m ≥ 1, if m = 1 and p < q, λ may be arbitrary]   ! ∞  ! a1 , . . . , a p 1 ! (λ − 1)r G mn x = λbq pq ! b1 , . . . , bq−1 , bq + r r! r=0 [m < q,

EH I 213(1)

|λ − 1| < 1] EH I 213(2)

 r  !  ∞  ! a1 − r, a2 , . . . , ap 1 1 mn ! λ− x = λa1 −1 G pq ! b1 , . . . , bq r! λ r=0

n ≥ 1, Re λ > 12 , (if n = 1 and p > q, then λ may be arbitrary)  r  !  ∞  ! a1 , . . . , ap−1 , ap − r 1 1 ap −1 mn ! − 1 G pq x ! =λ b1 , . . . , bq r! λ r=0 n < p,

EH I 213(3)

Re γ >

1 2



EH I 213(4)

For integrals of the G-function, see 7.8.

9.34 Connections with other special functions 1. 2. 3.

!  1 2 !! x !1 1 1 1 4 2 ν + 2 μ, 2 μ − 2 ν !  1 2 !! 12 μ − 12 ν − 12 x ! Y ν (x)xμ = 2μ G 20 13 4 ! 12 μ − 12 ν, 12 μ + 12 ν, 12 μ − 12 ν − !   1 2 !! μ μ−1 20 x K ν (x)x = 2 G 02 4 ! 12 μ + 12 ν, 12 μ − 12 ν J ν (x)xμ = 2μ G 10 02



EH I 219(44)

1 2

EH I 219(46)

EH I 219(47)

9.304

4. 5.

6. 7.7 8.

9.

Functional relations

1045

  !1 ! √ 20 2 ! K ν (x) = e π G 12 2x ! ν, −ν !  1 2 !! 12 + 12 ν + 12 μ μ μ 11 x ! Hν (x)x = 2 G 13 4 ! 12 + 12 ν + 12 μ, 12 μ − 12 ν, 12 μ + 12 ν !  1 1 2 !! 12 + 12 μ μ−1 31   1−μ+ν  G 13  x ! S μ,ν (x) = 2 4 ! 12 + 12 μ, 12 ν, − 21 ν Γ Γ 1−μ−ν 2 2   ! ! −a, −b Γ(c)x 12 ! x (a, b; c; −x) = G 2F 1 ! −1, −c Γ(a) Γ(b) 22 .q !   ! 1 − a1 , . . . , 1 − a p j=1 Γ (bj ) 1,p ! . −x (a , . . . , a ; b , . . . , b ; x) = F G p q 1 p 1 q p ! 0, 1 − b1 , . . . , 1 − bq Γ (aj ) p,q+1 .j=1 !   q 1 !! 1, b1 , . . . , bq j=1 Γ (bj ) p,1 − = .p G q+1,p x ! a1 , . . . , ap j=1 Γ (aj ) x

 √ 1 2k xe 2 x 40 x2 W k,m (x) = √ G 24 4 2π

EH I 219(49)

EH I 220(51)

EH I 220(55)

EH I 222(74)a

EH I 215(1)

! ! 1 − 1 k, 3 − 1 k !4 2 4 2 !1 1 ! 2 + 2 m, 12 − 12 m, 12 m, − 21 m

EH I 221(70)

9.4 MacRobert’s E-Function 9.41 Representation by means of multiple integrals E (p; αr : q; s : x) =

Γ (αq+1 ) Γ (1 − α1 ) Γ (2 − α2 ) · · · Γ (q − αq ) p−q−1 q  ∞ -  × λ μμ −αμ −1 (1 − λμ )− μ dλμ ×

μ=1  ∞ 0

0

p −1 e−λp λα p



ν=2

λq+2 λq+3 · · · λp 1+ (1 + λ1 ) · · · (1 + λq ) x

0

∞

α

q+ν e−λq+ν λq+ν

−1

dλq+ν

−αq+1 dλp

[|arg x| < π, p ≥ q + 1, αr and s are bounded by the condition that the integrals on the right be convergent.] EH I 204(3)

9.42 Functional relations 1.

α1 x E (α1 , . . . , αp : 1 , . . . , q : x) = x E (α1 + 1, α2 , . . . , αp : 1 , . . . , q : x) + E (α1 + 1, α2 + 1, . . . , αp + 1 : 1 + 1, . . . , q + 1 : x) EH I 205(7)

2.

(1 − 1) x E (α1 , . . . , αp : 1 , . . . , q : x) = x E (α1 , . . . , αp : 1 − 1, 2 , . . . , q : x) + E (α1 + 1, . . . , αp + 1 : 1 + 1, . . . , q + 1 : x) EH I 205(9)

1046

3.

Riemann’s Zeta Functions ζ(z, q), and ζ(z), and the Functions Φ(z, s, v) and ξ(s)

9.511

d E (α1 , . . . , αp : 1 , . . . , q : x) = x−2 E (α1 + 1, . . . , αp + 1 : 1 + 1, . . . , q + 1 : x) dx EH I 205(8)

9.5 Riemann’s Zeta Functions ζ(z, q), and ζ(z), and the Functions Φ(z, s, v) and ξ(s) 9.51 Definition and integral representations 9.511

ζ(z, q) =

1 Γ(z)



0

∞ z−1 −qt

t e dt; 1 − e−t

1 q 1−z +2 = q −z + 2 z−1



∞



q +t 2

 z 2 −2

0



  dt t sin z arctan q e2πt − 1

WH

[0 < q < 1,

Re z > 1] WH

 Γ(1 − z) (0+) (−θ)z−1 e−qθ dθ 2πi 1 − e−θ ∞ This equation is valid for all values of z except for z = 1, 2, 3, . . . . It is assumed that the path of integration (see drawing below) does not pass through the points 2nπi (where n is a natural number).

9.512 ζ(z, q) = −

See also 4.251 4, 4.271 1, 4, 8, 4.272 9, 12, 4.294 11. 9.513  ∞ z−1 t 1 dt [Re z > 0] 1. ζ(z) = (1 − 21−z ) Γ(z) 0 et + 1  ∞ z−1 t t e 2z dt [Re z > 1] 2. ζ(z) = z (2 − 1) Γ(z) 0 e2t − 1

  ∞ z ∞   1−z 2 π2 z 1 −1 −k πt 11 + t 2 + t2 t 3. ζ(z) =  z  e dt z(z − 1) Γ 2 1 k=1  ∞  − z 2z−1 dt z −2 1 + t2 2 sin (z arctan t) πt 4. ζ(z) = z−1 e +1 0    −z/2 ∞ 2 1 2z−1 z dt + z + t2 5. ζ(z) = z sin (z arctan 2t) 2πt 2 −1z−1 2 −1 0 4 e −1 See also 3.411 1, 3.523 1, 3.527 1, 3, 4.271 8.

WH WH

WH

WH

WH

9.524

Representation as a series or as an infinite product

1047

9.52 Representation as a series or as an infinite product 9.521 1.

2.12

∞ 

1 [Re z > 1, (q + n)z n=0

 ∞ ∞ zπ  cos 2πqn 2 Γ(1 − z) zπ  sin 2πqn sin ζ(z, q) = + cos (2π)1−z 2 n=1 n1−z 2 n=1 n1−z

ζ(z, q) =

[Re z > 0, 3.12

ζ(z, q) =

q = 0, −1, −2, . . .]

WH

0 < q ≤ 1]

WH

N 

∞  1 1 − − F n (z) , (q + n)z (1 − z)(N + q)z−1 n=0 n=N

[Re z > 1, where

N is a natural number]

  1 1 1 1 − − F n (z) = z−1 z−1 1 − z (n + 1 + q) (n + q) (n + 1 + q)z  n+1 (t − n) dt =z (t + q)z+1 n

9.522

WH

1.

ζ(z) =

∞  1 nz n=1

[Re z > 1]

WH

2.

ζ(z) =

∞  1 1 (−1)n+1 z 1 − 21−z n=1 n

[Re z > 0]

WH

9.523 The following product and summation are taken over all primes p: 1.7

ζ(z) =

p

2.

ln ζ(z) =

1 1 − p−z

∞  p k=1

1 kpkz

[Re z > 1]

WH

[Re z > 1]

WH

∞  ζ  (z) Λ(k) =− , [Re z > 1] ζ(z) kz k=1 where Λ(k) = 0 when k is not a power of a prime and Λ(k) = ln p when k is a power of a prime p.

9.52411

WH

Riemann’s Zeta Functions ζ(z, q), and ζ(z), and the Functions Φ(z, s, v) and ξ(s)

1048

9.531

9.53 Functional relations 9.531 1.12

2.∗

ζ(−n, q) = −

B n+2 (q) − B n+1 (q) = (n + 1)(n + 2) n+1 [n is a nonnegative integer]

ζ(s, q) = ζ(s, q + m) +

m−1 

(q + k)−s

see EH I 27 (11)

[m = 1, 2, 3 . . . ,

q = 0, −1, −2, . . .]

[m = 1, 2, 3 . . . ,

q = 0, −1, −2, . . .]

WH

k=0

∗

3.

ζ(s, q) = m

−s

m−1  k=0

9.532 9.533 1. 2. 3.

∞  (−1)k−1 k=2

k

  q+k ζ s, m

∞

qz e−Cz Γ(q) z  z ζ(k, q) = ln − + Γ(z + q) q k(q + k) k

ζ(z, q) = −1 z→1 Γ(1 − z)   1 = − Ψ(q) lim ζ(z, q) − z→1 z−1   d 1 ζ(z, q) = ln Γ(q) − ln 2π dz 2 z=0

2.11

2z Γ(1 − z) ζ(1 − z) sin

3.

21−z Γ(z) ζ(z) cos

5.∗

WH

lim

9.534 ζ(z, 1) = ζ(z) 9.535   1 ζ z, 12 1. ζ(z) = z 2 −1

4.

[|z| < q]

k=1

WH WH WH

[Re z > 1]  zπ  2

WH

= π 1−z ζ(z)

WH

zπ = π z ζ(1 − z) 2   z z−1 1−z − z2 π ζ(z) = Γ π 2 ζ(1 − z) Γ 2 2   n    π 2 Γ(s)  π k m = ζ s ζ 1 − s, cos s − 2km s n (2πn) 2 n n

WH WH

k=1

9.536

 lim ζ(z) −

z→1

1 z−1

[n = 1, 2, 3 . . . ,



m = 1, 2, . . . , n]

=C

  (z − 1) Γ z2 + 1 √ 9.537 Set z = + it. Then, Ξ(t) = ζ(z) = Ξ(−t) is an even function of t with real πz JA coefficients in its expansion in powers of t2 . 1 2

The Lerch function Φ(z, s, v)

9.552

1049

9.54 Singular points and zeros 9.5417 1. z = 1 is the only singular point of the function ζ(z)

WH

2.

The function ζ(z) has simple zeros at the points −2n, where n is a natural number. All other zeros of the function ζ(z) lie in the strip 0 ≤ Re z 0, 0 ≤ Re(v) < 1]

EH I 29(7)

Riemann’s Zeta Functions ζ(z, q), and ζ(z), and the Functions Φ(z, s, v) and ξ(s)

1050

9.553

Series representation ∞ 

Φ(z, s, v) = z −v Γ(1 − s)

9.553

e

[0 < v ≤ 1,

Re s < 0,

n=−∞

 9.55412 Φ(z, m, v) = z −v

∞  

ζ(m − n, v)

n=0

9.555 Φ(z, −m, v) =

1.12

s−1 2πnvi

(− ln z + 2πni)

(ln z)m−1 (ln z)n + [Ψ(m) − Ψ(v) − ln(− ln z)] n! (m − 1)!

[m = 1, 2, 3, . . . ,

|ln z| < 2π,

∞ r m! 1  Bm+r+1 (v) (ln z) −m−1 (− ln z) − zv z v r=0 r!(m + r + 1)

Φ(z, s, v) = z −v Γ(1 − s)(− ln z)s−1 +

2.∗

|arg (− ln z + 2πni)| ≤ π] EH I 28(6) 

∞  n=0

ζ(s − n, v)

(ln z)n n!

v = 0, −1, −2, . . .]

[m = 0, 1, 2, . . . ,

EH I 30(9)

|ln z| < 2π] EH I 30(11)

 [s = 1, 2, 3, . . . ,

| ln z| < 2π]

Integral representation 9.556 Φ(z, s, v) =

1.12

1 Γ(s)



 ∞ s−1 −(v−1)t t e t e dt 1 dt = −t t−z 1 − ze Γ(s) e 0 0 [Re v > 0] or [|arg(1 − z)| < π, Re s > 0] or [z = 1, ∞ s−1 −vt

Re s > 1]

EH I 27(3)

∂ Φ(z, s, v) = Φ(z, s − 1, v) − vΦ(z, s, v) ∂z ∂ Φ(z, s, v) = −sΦ(z, s + 1, v) ∂v

2.∗

z

3.∗

Limit relationships 9.557 9.558

lim (1 − z)1−s Φ(z, s, v) = Γ(1 − s)

z→1

lim

[Re s < 1]

Φ(z, 1, v) =1 − z)

EH I 30(12) EH I 30(13)

z→1 − ln(1

Relations to other functions 9.559 1.12 2.

∗

∗ In

Φ(z, 1, v) = v −1 2 F 1 (1, v; 1 + v; z) Φ(1, s, q) = ζ(s, q)

9.554 the prime on the symbol

/

means that the term corresponding to n = m − 1 is omitted.

EH I 30(10)

9.612

Bernoulli numbers

1051

9.56 The function ξ (s)

  Γ 12 s 1 9.561 ξ(s) = s(s − 1) 1 s ζ(s) 2 π2 9.562 ξ(1 − s) = ξ (s)

EH III 190(10) EH III 190(11)

9.6 Bernoulli Numbers and Polynomials, Euler Numbers, the Functions ν(x), ν(x, α), μ(x, β), μ(x, β, α), λ(x, y) and Euler Polynomials 9.61 Bernoulli numbers 9.610 The numbers Bn , representing the coefficients of ∞  t tn = B n et − 1 n=0 n!

are called Bernoulli numbers. Thus, the function bers.

tn in the expansion of the function n! [0 < |t| < 2π] ,

t is a generating function for the Bernoulli numet − 1

GE 48(57), FI II 520

9.611 Integral representations  ∞ 2n−1 x dx 1. B2n = (−1)n−1 4n 2πx − 1 0 e B2n = (−1)n−1 π 2n

3.

B2n

Bn = lim

n

d

x→0 dxn

 ex

x −1

2n

[n = 1, 2, . . .]



See also 3.523 2, 4.271 3. Properties and functional relations 9.6128

A symbolic notation:

hence by recursion

n    n

Bk αn−k k k=0 n    n Bn = (B + 1)[n] = Bk k k=0 n−1  Bk Bn = −n! k!(n + 1 − k)! (B + α)[n] =

in particular

(cf. 3.411 2, 4) FI II 721a

∞

x dx [n = 1, 2, . . .] 2 0 sinh x   2n(1 − 2n) ∞ 2n−2  = (−1)n−1 x ln 1 − e−2πx dx π 0

2.12

4.



[n = 1, 2, . . .]

k=0

[n ≥ 2] [n ≥ 2] [n ≥ 2]

1052

Bernoulli numbers and polynomials, Euler numbers

9.613

9.613 All the Bernoulli numbers are rational numbers. 9.614 Every number Bn can be represented in the form  1 , Bn = Cn − k+1 where Cn is an integer and the sum is taken over all k > 0 such that k + 1 is a prime and k is a divisor of n. GE 64 1 12 9.615 All the Bernoulli numbers with odd index are equal to zero except that B1 = − 2 ; that is, B2n+1 = 0 for n a natural number. GE 52, FI II 521 n−1  2n(2n − 1) . . . (2n − 2k + 2) 1 1 + − B2k B2n = − [n ≥ 1] 2n + 1 2 (2k)! k=1

(−1)n−1 (2n)! ζ(2n) [n ≥ 0] 9.616 B2n = 22n−1 π 2n 2(2n)! 1 9.6177 B2n = (−1)n−1  ∞  (2π)2n 1 1 − 2n p p=2

(cf. 9.542)

[n ≥ 1]

GE 56(79), FI II 721a

(cf. 9.523)

(where the product is taken over all primes p). • For a connection with Riemann’s zeta function, see 9.542. • For a connection with the Euler numbers, see 9.635. • For a table of values of the Bernoulli numbers, see 9.71

! ! 9.619 An inequality !(B − θ)[n] ! ≤ |Bn |

[0 < θ < 1]

9.62 Bernoulli polynomials 9.620 The Bernoulli polynomials B n (x) are defined by n    n Bk xn−k B n (x) = k or symbolically, B n (x) = (B + x)[n] . 9.621 The generating function ∞  ext tn−1 = B n (x) t e − 1 n=0 n!

GE 51(62)

k=0

GE 52(68)

[0 < |t| < 2π]

(cf. 1.213)

GE 65(89)a

9.622 Series representation 1.7

B n (x) = −2

  ∞ n!  cos 2πkx − 12 πn (2π)n kn k=1

[n > 1,

1 ≥ x ≥ 0;

n = 1,

1 > x > 0]

AS 805(23.1.16)

9.628

Bernoulli polynomials

1053

∞

2.7

B 2n−1 (x) = 2

(−1)n 2(2n − 1)!  sin 2kπx (2π)2n−1 k 2n−1 k=1

[n > 1,

1 ≥ x ≥ 0;

n = 1,

1 > x > 0]

AS 805(23.1.17)

∞

3.10

B 2n (x) =

(−1)n−1 2(2n)!  cos 2kπx (2π)2n k 2n

[0 ≤ x ≤ 1,

n = 1, 2, . . .]

GE 71

k=1

9.623 Functional relations and properties: B m+1 (n) = Bm+1 + (m + 1)

1.

n−1 

km

k=1

(see also 0.121)

[n and m are natural numbers] n−1

2.

B n (x + 1) − B n (x) = nx

3.

B n (x)

GE 65(90)

= n B n−1 (x)

[n = 1, 2, . . .]

GE 66

n

B n (1 − x) = (−1) B n (x)

4. 5.

GE 51(65)

n

9.624

GE 66 n−1

(−1) B n (−x) = B n (x) + nx

10

7

B n (mx) = m

m−1 

n−1

k=0

[n = 0, 1, . . .]

AS 804(23.1.9)

  k Bn x + m [m = 1, 2, . . . n = 0, 1, . . .] ;

“summation theorem”

GE 67

9.625 For n odd, the differences B n (x) − Bn vanish on the interval [0, 1] only at the points 0, 12 , and 1. They change sign at the point x = 12 . For n even, these differences vanish at the end points of the interval [0, 1]. Within this interval, they do not change sign and their greatest absolute value occurs at the point x = 12 . 9.626 The polynomials B 2n (x) − B2n and B 2n+2 (x) − B2n+2 have opposite signs in the interval (0, 1). GE 87 9.627 Special cases: 1.

B 1 (x) = x −

2.

B 2 (x) = x2 − x +

3.

B 3 (x) = x −

4.

B 4 (x) = x4 − 2x + x −

5.

B 5 (x) = x −

1 2

3

5

GE 70 1 6

3 2 2x 3

+

5 4 2x

+

GE 70 1 2x 2

GE 70

1 30 5 3 1 3x − 6x

GE 70 GE 70

9.628 Particular values: 1. 2.

12

B n (0) = Bn B 1 (1) = −B1 = 12 ,

B n (1) = (−1)n Bn

[n = 1]

GE 76

1054

Bernoulli numbers and polynomials, Euler numbers

9.630

9.63 Euler numbers 9.630 The numbers En , representing the coefficients of ∞  1 tn = En cosh t n=0 n!

tn n!

are known as the Euler numbers. Thus, the function

is a generating function for the Euler numbers.

1 cosh t

$

in the expansion of the function π% |t| < 2 CE 330

9.631 A recursion formula (E + 1)[n] + (E − 1)[n] = 0

[n ≥ 1] ,

E0 = 1

CE 329

Properties of the Euler numbers 9.632 The Euler numbers are integers. 9.633 The Euler numbers of odd index are equal to zero; the signs of two adjacent numbers of even indices are opposite; that is, E2n+1 = 0, E4n > 0, E4n+2 < 0 CE 329 9.634 If α, βγ, . . . are the divisors of the number n − m, the difference E2n − E2m is divisible by those of the numbers 2α + 1, 2β + 1, 2γ + 1, . . . , that are primes. 9.635 A connection with the Bernoulli numbers (symbolic notation):

3.6

    (4B − 1)[n] − (4B − 3)[n] + 4(−1)n+1 3n−1 − 1 B1 En−1 + 4(−1)n 3n−1 − 1 B1 = 2n n(E + 1)[n−1] [n ≥ 2] Bn = n n 2 (2 − 1)  [2n+1] B + 14 = −4−2n−1 (2n + 1)E2n [n ≥ 0]

4.

En−1 =

1.11 2.

(4B + 3)[n] − (4B + 1)[n] 2n

CE 330 CE 330 CE 341

[n ≥ 1]

For a table of values of the Euler numbers, see 9.72.

9.64 The functions ν(x), ν(x, α), μ(x, β), μ(x, β, α), λ(x, y) 9.640 1.



∞

ν(x) = 0

2.



xt dt Γ(t + 1) ∞

ν(x, α) = 0

 3.

∞

μ(x, β) = 0

4.

xα+t dt Γ(α + t + 1)

EH III 217(1)

xt tβ dt Γ(β + 1) Γ(t + 1)

EH III 217(2)

μ(x, β, α) = 0

 5.



y

λ(x, y) = 0

EH III 217(1)

∞

xα+t tβ dt Γ(β + 1) Γ(α + t + 1)

Γ(u + 1) du xu

EH III 217(2) MI 9

9.653

Euler polynomials

1055

9.6510 Euler polynomials 9.650 The Euler polynomials are defined by n−k  n    1 n Ek x− E n (x) = 2 k 2k 9.651 The generating function:

9.652 Series representation: 1.

AS 804 (23.1.7)

k=0

∞  2ext tn = E n (x) t e + 1 n=0 n!

AS 804 (23.1.1)

  ∞ n!  sin (2k + 1)πx − 12 πn π n+1 (2k + 1)n+1

E n (x) = 4

k=0

[n > 0,

1 ≥ x ≥ 0,

n = 1,

1 > x > 0]

AS 804 (23.1.16)

∞

2.

10

(−1)n 4(2n − 1)!  cos(2k + 1)πx E 2n−1 (x) = π 2n (2k + 1)2n

[n = 1, 2, . . . ,

1 ≥ x ≥ 0]

k=0

AS 804 (23.1.17) ∞

3.

E 2n (x) =

(−1)n 4(2n)!  sin(2k + 1)πx π 2n+1 (2k + 1)2n+1 k=0

[n > 0,

1 ≥ x ≥ 0,

n = 0,

1 > x > 0]

AS 804 (23.1.18)

9.653 Functional relations and properties: 1.

E m (n + 1) = 2

n 

(−1)n−k k m + (−1)n+1 E m (0),

[m and n are natural numbers]

k=1

AS 804 (23.1.4)

2.

En (x) = nEn−1 (x). n

[n = 1, 2, . . .]

AS 804 (23.1.5) AS 804 (23.1.6)

3.

E n (x + 1) + E n (x) = 2x

[n = 0, 1, . . .]

4.8

  m−1  k n k E n (mx) = m (−1) En x − m

[n = 0, 1, . . . , m = 1, 3, . . .]

k=0

AS 804 (23.1.10)

5.

E n (mx) =

  m−1  k −2 mn (−1)k Bn+1 x + n+1 m

[n = 0, 1, . . . , m = 2, 4, . . .]

k=0

AS 804 (23.1.10)

1056

Constants

9.654

9.654 Special cases: 1.

E 1 (x) = x −

2.

E 2 (x) = x2 − x

3.

E 3 (x) = x3 − 32 x2 +

4.

E 4 (x) = x4 − 2x3 + x

5.

E 5 (x) = x5 − 52 x4 + 52 x2 −

1 2

1 4

1 2

9.655 Particular values: E2n+1 = 0

[n = 0, 1, . . .]   E n (0) = − E n (1) = −2(n + 1)−1 2n+1 − 1 Bn+1 [n = 1, 2, . . .] 1 −n E n 2 = 2 En [n = 0, 1, . . .] 1 2     E 2n−1 3 = − E 2n−1 3 = −(2n)−1 1 − 31−2n 22n − 1 B2n

1. 2. 3. 4.

[n = 1, 2, . . .]

AS 805 (23.1.19) AS 805 (23.1.20) AS 805 (23.1.21)

AS 806 (23.1.22)

9.7 Constants 9.71 Bernoulli numbers • • • • • • • • • •

B0 = 1 B1 = − 1/2 B2 = 1/6 B4 = − 1/30 B6 = 1/42 B8 = − 1/30 B10 = 5/66 B12 = − 691/2730 B14 = 7/6 B16 = − 3617/510

• • • • • • • • •

B18 B20 B22 B24 B26 B28 B30 B32 B34

= 43 867/798 = − 174 611/330 = 854 513/138 = − 236 364 091/2730 = 8 553 103/6 = − 23 749 461 029/870 = 8 615 841 276 005/14 322 = − 7 709 321 041 217/510 = 2 577 687 858 367/6

• • • • •

E12 E14 E16 E18 E20

= 2 702 765 = −199 360 981 = 19 391 512 145 = −2 404 879 675 441 = 370 371 188 237 525

9.72 Euler numbers • • • • • •

E0 = 1 E2 = −1 E4 = 5 E6 = −61 E8 = 1385 E10 = −50 521

The Bernoulli and Euler numbers of odd index (with the exception of B 1 ) are equal to zero.

9.744

Stirling numbers

1057

9.73 Euler’s and Catalan’s constants Euler’s constant C = 0.577 215 664 901 532 860 606 512 . . .

(cf. 8.367)

Catalan’s constant ∞  (−1)k = 0.915 965 594 . . . G= (2k + 1)2 k=0

9.7410 Stirling numbers 9.740 The Stirling number of the first kind Sn is defined by the requirement that (−1)n−m Sn is the number of permutations of n symbols which have exactly m cycles. AS 824 (23.1.3) (m)

(m)

9.741 Generating functions: 1.

x(x − 1) · · · (x − n + 1) =

n 

Sn(m) xm

AS 824 (24.1.3)

m=0

2.

{ln(1 + x)}

m

∞ 

= m!

n=m

Sn(m)

xn n!

[|x| < 1]

AS 824 (24.1.3)

9.742 Recurrence relations: 1.8

2.

(m)

Sn+1 = Sn(m−1) − nSn(m) ;

Sn(0) = δ0n ;

Sn(1) = (−1)n−1 (n − 1)!;

n−r m   n  (r) (m+r) S Sn(m) = S k n−k k r

Sn(n) = 1

[n ≥ m ≥ 1]

AS 824 (24.1.3)

[n ≥ m ≥ r]

AS 824 (24.1.3)

k=m−r

9.743 Functional relations and properties 1.

2.

3.

  m  k  hm Γ hx + 1 x (m)  = hm Sk x(x − h)(x − 2h) · · · (x − mh + h) =  x h Γ h −m+1 k=1 −1   −1  p x+m −1 k (m) [(x + 1)(x + 2) · · · (x + m)] = m! = (x + m) Sk m k=1 −1

  m  k  Γ hx + 1 x (m) −1 m = h + m Sk [(x + h)(x + 2h) · · · (x + mh)] = m  x h h Γ h +m+1 k=1

(m)

9.744 The Stirling number of the second kind Sn elements into m non-empty subsets.

is the number of ways of partitioning a set of n

1058

Constants

9.745

9.745 Generating functions: 1.

xn =

n 

Sn(m) x(x − 1) · · · (x − m + 1)

AS 824 (24.1.4)

m=0

2.

m

(ex − 1)

∞ 

= m!

n=m

3.

S(m) n

xn n!

AS 824 (24.1.4) ∞ 

[(1 − x)(1 − 2x) · · · (1 − mx)]−1 =

n=m



|x| < m−1

n−m S(m) n x

AS 824 (24.1.4)

9.746 Closed form expression: m 1  kn (−1)m−k m! k m

1.

Sn(m) =

AS 824 (24.1.4)

k=0

9.747 Recurrence relations: 1.8

2.

(m)

Sn+1 = mSn(m) + S(m−1) , n

S(0) n = δ0n ,

n−r m   n  (r) (m−r) Sn−k Sk Sn(m) = k r k=m−r

3.

Sn(m) =

n−m 

(−1)k



k=0

9.7487

(n) S(1) n = Sn = 1

n−1+k n−m+k



[n ≥ m ≥ 1]

AS 825(24.1.4)

[n ≥ m ≥ r]

AS 825 (24.1.4)

 2n − m (k) Sn−m+k n−m−k

Particular values:

AS 824 (24.1.3)

(m)

Stirling numbers of the first kind Sn m 1 2 3 4 5 6 7 8 9

(m)

S1

1

(m)

S2

-1 1

(m)

S3

2 -3 1

(m)

S4

-6 11 -6 1

(m)

S5

24 -50 35 -10 1

(m)

S6 -120 274 -225 85 -15 1

(m)

S7 720 -1764 1624 -735 175 -21 1

(m)

S8 -5040 13068 -13132 6769 -1960 332 -28 1

(m)

S9 40320 -109584 118121 -67284 22449 -4536 546 -36 1

9.749

Stirling numbers

1059

(m)

Stirling numbers of the second kind Sn m 1 2 3 4 5 6 7 8 9 9.7498 1.

(m)

S1

(m)

S2

1

1 1

(m)

S3

(m)

S4

1 3 1

1 7 6 1

(m)

S5

1 15 25 10 1

(m)

S6

1 31 90 65 15 1

(m)

S8

S9

1 63 301 350 140 21 1

1 127 966 1701 1050 266 28 1

1 255 3025 7770 6951 2646 462 36 1

S7

(m)

(m)

Relationship between Stirling numbers of the first kind and derivatives of (ln x) dn dxn



1 lnm x

 =

1 lnm x

−m

:

n (k)  (−1)k (m)k Sn k=1

xn lnk x

where (m)k = Γ(m + k)/ Γ(m),

[m, n are positive integers]

Table of Integrals, Series, and Products. http://dx.doi.org/10.1016/B978-0-12-384933-5.00010-2 c 2015 Elsevier Inc. All rights reserved. Copyright 

10 Vector Field Theory 10.1–10.8 Vectors, Vector Operators, and Integral Theorems 10.11 Products of vectors Let a = (a1 , a2 , a3 ), b = (b1 , b2 , b3 ), and c = (c1 , c2 , c2 ) be arbitrary vectors, and i, j, k be the set of orthogonal unit vectors in terms of which the components of a, b, and c are expressed. Two different products involving pairs of vectors are defined, namely, the scalar product, written a · b, and the vector product, written either a × b or a ∧ b. Their properties are as follows: 1. 2.

3. 4.

a · b = a1 b1 + a2 b2 + a3 b3   i j k   a × b = a1 a2 a3   b1 b2 b3     a1 a2 a3    a × b · c =  b1 b2 b3   c1 c2 c3 

(scalar product) (vector product)

(triple scalar product)

a × (b × c) = (a · c) b − (a · b) c

(triple vector product)

10.12 Properties of scalar product 1.

a·b=b·a

2.

a × b · c = b × c · a = c × a · b = −a × c · b = −b × a · c = −c × b · a.

(commutative)

Note: a × b · c is also written [a, b, c]; thus (2) may also be written 3.

[a, b, c] = [b, c, a] = [c, a, b] = − [a, c, b] = − [b, a, c] = − [c, b, a]

10.13 Properties of vector product 1.

a × b = −b × a

2.

a × (b × c) = −a × (c × b) = − (b × c) × a

3.

a × (b × c) + b × (c × a) + c × (a × b) = 0

(anticommutative)

1061

1062

Vectors, Vector Operators, and Integral Theorems

10.14 Differentiation of vectors If a(t) = (a1 (t), a2 (t), a3 (t)), b(t) = (b1 (t), b2 (t), b3 (t)), c(t) = (c1 (t), c2 (t), c3 (t)), φ(t) is a scalar and all functions of t are differentiable, then 1. 2. 3. 4. 5. 6. 7.

da da1 da2 da3 = i+ j+ k dt dt dt dt da db d (a + b) = + dt dt dt dφ da d (φa) = a+φ dt dt dt da db d (a · b) = ·b+a· dt dt dt da db d (a × b) = ×b+a× dt dt dt da db dc d (a × b · c) = ×b·c+a× ·c+a×b· dt dt dt dt     da db dc d {a × (b × c)} = × (b × c) + a × ×c +a× b× dt dt dt dt

10.21 Operators grad, div, and curl In cartesian coordinates O {x1 , x2 , x3 }, in which system it is convenient to denote the triad of unit vectors by e1 , e2 , e3 , the vector operator ∇, called either “del” or “nabla”, has the form ∂ ∂ ∂ + e2 + e3 ∂x1 ∂x2 ∂x3 If Φ(x, y, z) is any differentiable scalar function, the gradient of Φ, written grad Φ, is

1.

∇ ≡ e1

2.

grad Φ ≡ ∇ Φ =

3.

4.

∂Φ ∂Φ ∂Φ e1 + e2 + e3 ∂x1 ∂x2 ∂x3 The divergence of the differentiable vector function f = (f1 , f2 , f3 ), written div f , is

∂f1 ∂f2 ∂f3 + + ∂x1 ∂x2 ∂x3 The curl, or rotation, of the differentiable vector function f = (f1 , f2 , f3 ), written either curl f or rot f , is       ∂f3 ∂f2 ∂f1 ∂f3 ∂f2 ∂f1 curl f ≡ rot f ≡ ∇ ×f = e1 + e2 + e3 , − − − ∂x2 ∂x3 ∂x3 ∂x1 ∂x1 ∂x2 or equivalently,    e1 e2 e2   ∂ ∂ ∂  curl f =  ∂x1 ∂x ∂x3  2  f1 f2 f3  div f ≡ ∇ ·f =

Properties of the operator ∇

1063

10.31 Properties of the operator ∇ Let Φ (x1 , x2 , x3 ), Ψ (x1 , x2 , x3 ) be any two differentiable scalar functions, f (x1 , x2 , x3 ), g (x1 , x2 , x3 ) any two differentiable vector functions, and a an arbitrary vector. Define the scalar operator ∇2 , called the Laplacian, by ∂2 ∂2 ∂2 ∇2 ≡ + 2+ 2 2 ∂x1 ∂x2 ∂x3 Then, in terms of the operator ∇, we have the following: MF I 114 1.

∇(Φ + Ψ) = ∇ Φ + ∇ Ψ

2.

∇(ΦΨ) = Φ ∇ Ψ + Ψ ∇ Φ

3.

∇ (f · g) = (f · ∇) g + (g · ∇) f + f × (∇ ×g) + g × (∇ ×f )

4.

∇ · (Φf ) = Φ (∇ ·f ) + f · ∇ Φ

5.

∇ · (f × g) = g · (∇ ×f ) − f · (∇ ×g)

6.

∇ × (Φf ) = Φ (∇ ×f ) + (∇ Φ) × f

7.

∇ × (f × g) = f (∇ ·g) − g (∇ ·f ) + (g · ∇) f − (f · ∇) g

8.

∇ × (∇ ×f ) = ∇ (∇ ·f ) − ∇2 f

9.

∇ × (∇ Φ) ≡ 0

10.

∇ · (∇ ×f ) ≡ 0

11.10 ∇2 (ΦΨ) = Φ ∇2 Ψ + 2 (∇ Φ) · (∇ Ψ) + Ψ ∇2 Φ The equivalent results in terms of grad, div, and curl are as follows: 1.

grad(Φ + Ψ) = grad Φ + grad Ψ

2.

grad(ΦΨ) = Φ grad Ψ + Ψ grad Φ

3.

grad (f · g) = (f · grad) g + (g · grad) f + f × curl g + g × curl f

4.

div (Φf ) = Φ div f + f · grad Φ

5.

div (f × g) = g · curl f − f · curl g

6.

curl (Φf ) = Φ curl f + grad Φ × f

7.

curl (f × g) = f div g − g div f + (g · grad) f − (f · grad) g

8.

curl (curl f ) = grad (div f ) − ∇2 f

9.

curl (grad Φ) ≡ 0

10.

div (curl f ) ≡ 0

11.

∇2 (ΦΨ) = Φ ∇2 Ψ + 2 grad Φ · grad Ψ + Ψ ∇2 Φ

The expression (a · ∇) or, equivalently (a · grad), defined by ∂ ∂ ∂ + a2 + a3 , (a · ∇) ≡ a1 ∂x1 ∂x2 ∂x3 is the directional derivative operator in the direction of vector a.

1064

Vectors, Vector Operators, and Integral Theorems

10.411

10.41 Solenoidal fields A vector field f is said to be solenoidal if div f ≡ 0. We have the following representation. 10.411 Representation theorem for vector Helmholtz equation. If u is a solution of the scalar Helmholtz equation ∇2 u + λ2 u = 0, and m is a constant unit vector, then the vectors 1 X = curl (mu) , Y = curl X λ are independent solutions of the vector Helmholtz equation ∇2 H + λ2 H = 0 involving a solenoidal vector H. The general solution of the equation is 1 H = curl (mu) + curl curl (mu) λ

10.51–10.61 Orthogonal curvilinear coordinates Consider a transformation from the cartesian coordinates O {x1 , x2 , x3 } to the general orthogonal curvilinear coordinates O {u1 , u2 , u3 }: x1 = x1 (u1 , u2 , u3 ) , x2 = x2 (u1 , u2 , u3 ) , x3 = x3 (u1 , u2 , u3 ) Then, 1.

2.

∂xi ∂xi ∂xi du1 + du2 + du3 ∂u1 ∂u2 ∂u3 and the length element dl may be determined from dxi =

(i = 1, 2, 3) ,

dl2 = g11 du21 + g22 du22 + g33 du23 + 2g23 du2 du3 + 2g31 du3 du1 + 2g12 du1 du2 , where

3.3

4.

∂x1 ∂x1 ∂x2 ∂x2 ∂x3 ∂x3 + + = gji , ∂ui ∂uj ∂ui ∂uj ∂ui ∂uj provided the Jacobian of the transformation    ∂x1 ∂x2 ∂x3   ∂u1 ∂u1 ∂u1   1 ∂x2 ∂x3  J =  ∂x ∂u2 ∂u2  2  ∂u ∂x2 ∂x3  1  ∂x ∂u ∂u ∂u

gij =

3

3

gij = 0,

i = j.

3

does not vanish (see 14.313). 5.

Define the metrical coefficients √ √ √ h1 = g11 , h2 = g22 , h3 = g33 ; then the volume element dV in orthogonal curvilinear coordinates is

6.

dV = h1 h2 h3 du1 du2 du3 , and the surface elements of area dsi on the surfaces ui = constant, for i = 1, 2, 3, are

7.

ds1 = h2 h3 du2 du3 ,

ds2 = h1 h3 du1 du3 ,

ds3 = h1 h2 du1 du2

Denote by e1 , e2 , and e3 the triad of orthogonal unit vectors that are tangent to the u1 , u2 , and u3 coordinate lines through any given point P , and choose their sense so that they form a right-handed set in this order. Then in terms of this triad of vectors and the components fu1 , fu2 , and fu3 of f along the coordinate line,

10.613

8.

Orthogonal curvilinear coordinates

f = fu1 e1 + fu2 e2 + fu3 e3

1065

MF I 115

10.611 ∇ Φ, div f , curl f , and ∇2 in general orthogonal curvilinear coordinates. 1. 2.3

3.

4.

e2 ∂Φ e3 ∂Φ e1 ∂Φ + + h1 ∂u1 h2 ∂u2 h3 ∂u3   ∂ 1 ∂ ∂ div f = (h2 h3 fu1 ) + (h3 h1 fu2 ) + (h1 h2 fu3 ) h1 h2 h3 ∂u1 ∂u2 ∂u3    h1 e 1 h2 e 2 h3 e 3   ∂  1 ∂ ∂   curl f = ∂u1 ∂u2 ∂u3   h1 h2 h3  h1 fu1 h2 fu2 h3 fu3         ∂ h2 h3 ∂ ∂ h3 h1 ∂ ∂ h1 h2 ∂ 1 2 ∇ ≡ + + h1 h2 h3 ∂u1 h1 ∂u1 ∂u2 h2 ∂u2 ∂u3 h3 ∂u3 grad Φ =

MF I 21-31

10.612 Cylindrical polar coordinates. In terms of the coordinates O{r, φ, z}, that is, u1 = r, u2 = φ, u3 = z, where x1 = r cos φ, x2 = r sin φ, x3 = z for −π < φ ≤ π, it follows that 1.

h1 = 1,

h2 = r,

h3 = 1,

and 2. 3.

4.

5.

∂Φ 1 ∂Φ ∂Φ er + eφ + ez , ∂r r ∂φ ∂z 1 ∂fφ ∂fz 1 ∂ (rfr ) + + , div f = r ∂r r ∂φ ∂z   e reφ ez  1  ∂r ∂ ∂  curl f =  ∂r ∂φ ∂z  , r fr rfφ fz    ∂ 1 ∂2 1 ∂ ∂2 2 ∇ ≡ r + 2 2+ 2 r ∂r ∂r r ∂φ ∂z grad Φ =

MF I 116

10.613 Spherical polar coordinates. In terms of the coordinates O{r, θ, φ}, that is, u1 = r, u2 = θ, u3 = φ, where x1 = r sin θ cos φ, x2 = r sin θ sin φ, x3 = r cos θ, for 0 ≤ θ ≤ π, −π < φ ≤ π, we have 1.

h1 = 1,

h2 = r,

h3 = r sin θ,

and 2.10 3.

4.

5.

∂Φ 1 ∂Φ 1 ∂Φ er + eθ + eφ , ∂r r ∂θ r sin θ ∂φ 1 ∂  2  ∂ 1 1 ∂fφ div f = 2 r fr + (fθ sin θ) + , r ∂r r sin θ ∂θ r sin θ ∂φ    er reθ r sin θeφ    1 ∂ ∂ , ∂ curl f = 2 ∂r ∂θ ∂φ   r sin θ  fr rfθ r sin θfφ      ∂ 1 ∂ ∂ 1 ∂2 1 ∂ ∇2 ≡ 2 r2 + 2 sin θ + 2 2 r ∂r ∂r r sin θ ∂θ ∂θ r sin θ ∂φ2 grad Φ =

MF I 116

1066

Vectors, Vector Operators, and Integral Theorems

10.614

Special Orthogonal Curvilinear Coordinates and their Metrical Coefficients h1 , h2 , h3 10.614 Elliptic cylinder coordinates O{u1 , u2 , u3 }.  x2 = (u21 − c2 ) (1 − u22 ), 1. x1 = u1 u2 ,   u21 − c2 u22 u21 − c2 u22 2. h1 = , h2 = , 2 2 u1 − c 1 − u22

x3 = u3 h3 = 1

MF I 657

10.615 Parabolic cylinder coordinates O{u1 , u2 , u3 }. 1. 2.

 1 2 u1 − u22 , 2  h1 = u21 + u22 , x1 =

x2 = u1 u2 , x3 = u3  h2 = u21 + u22 , h3 = 1

10.616 Conical coordinates O{u1 , u2 , u3 }.   u1 u1 (a2 − u22 ) (a2 + u23 ), x2 = (b2 + u22 ) (b2 − u23 ), 1. x1 = a b  2.

h1 = 1,

h2 = u 1

(a2

u22 + u23 , − u22 ) (b2 + u22 )

MF I 658

x3 =

u1 u2 u3 ab

with a2 + b2 = 1  u22 + u23 h3 = u 1 2 (a + u23 ) (b2 − u23 )

10.617 Rotational parabolic coordinates O{u1 , u2 , u3 }.   1 2 u1 − u22 1. x1 = u1 u2 u3 , x2 = u1 u2 1 − u23 , x3 = 2   u1 u2 h2 = u21 + u22 , h3 = 2. h1 = u21 + u22 , 1 − u23 10.618 Rotational prolate spheroidal coordinates O{u1 , u2 , u3 }.   x2 = (u21 − a2 ) (1 − u22 ) (1 − u23 ), x3 = u1 u2 1. x1 = (u21 − a2 ) (1 − u22 ),    u21 − a2 u22 u21 − a2 u22 (u21 − a2 ) (1 − u22 ) 2. h1 = , h = , h = 2 3 u21 − a2 1 − u22 1 − u23 10.619 Rotational oblate spheroidal coordinates O{u1 , u2 , u3 }.   x2 = (u21 + a2 ) (1 − u22 ) (1 − u23 ), x3 = u1 u2 1. x1 = u3 (u21 + a2 ) (1 − u22 ),    u21 + a2 u22 u21 + a2 u22 (u21 + a2 ) (1 − u22 ) 2. h1 = , h2 = , h3 = 2 2 2 u1 + a 1 − u2 1 − u23

MF I 659

MF I 660

MF I 661

MF I 662

10.713

Vector integral theorems

1067

10.620 Ellipsoidal coordinates O{u1 , u2 , u3 }.   (u21 − a2 ) (u22 − a2 ) (u23 − a2 ) (u21 − b2 ) (u22 − b2 ) (u23 − b2 ) u1 u2 u3 , x , x3 = 1. x1 = = 2 a2 (a2 − b2 ) b2 (b2 − a2 ) ab    (u21 − u22 ) (u21 − u23 ) (u22 − u21 ) (u22 − u23 ) (u23 − u21 ) (u23 − u22 ) , h , h = = 2. h1 = 2 3 (u21 − a2 ) (u21 − b2 ) (u22 − a2 ) (u22 − b2 ) (u23 − a2 ) (u23 − b2 ) MF I 663

10.621 Paraboloidal coordinates O{u1 , u2 , u3 }.   (u21 − a2 ) (u22 − a2 ) (u23 − a2 ) (u21 − b2 ) (u22 − b2 ) (u23 − b2 ) , x2 = , 1. x1 = 2 2 a −b b 2 − a2   x3 = 12 u21 + u22 + u23 − a2 − b2    (u21 − u22 ) (u21 − u23 ) (u23 − u21 ) (u23 − u22 ) (u23 − u21 ) (u23 − u22 ) , h , h = u = u 2. h1 = 2 2 3 3 2 2 2 2 (u1 − a2 ) (u1 − b2 ) (u2 − a2 ) (u2 − b2 ) (u23 − a2 ) (u23 − b2 ) MF I 664

10.622 Bispherical coordinates O{u1 , u2 , u3 }. 1 − u22 (1 − u22 ) (1 − u23 ) 1. x1 = au3 , x2 = a , u1 − u2 u1 − u2 a 2. h1 = , (u1 − u2 ) u21 − 1

x3 =

a , h2 = (u1 − u2 ) 1 − u22

u21 − 1 u1 − u2 

h3 =

a u1 − u2



1 − u22 1 − u23

MF I 665

10.71–10.72 Vector integral theorems 10.711 Gauss’s divergence theorem. Let V be a volume bounded by a simple closed surface S and let f be a continuously differentiable vector field defined in V and on S. Then, if dS is the outward drawn vector element of area,



f · dS = div f dV KE 39 S

V

10.712 Green’s theorems. Let Φ and Ψ be scalar fields which, together with ∇2 Φ and ∇2 Ψ, are defined both in a volume V and on its surface S, which we assume to be simple and closed. Then, if ∂/∂n denotes differentiation along the outward drawn normal to S, we have 10.713 Green’s first theorem



  ∂Ψ dS = Φ ∇2 Ψ + grad Φ · grad Ψ dV Φ S ∂n V

KE 212

1068

Vectors, Vector Operators, and Integral Theorems

10.714 Green’s second  

theorem   ∂Φ ∂Ψ −Ψ dS = Φ Φ ∇2 Ψ − Ψ ∇2 Φ dV ∂n ∂n S V 10.715 Special cases



2 Φ ∇2 Φ + (grad Φ) dV (Φ grad Φ) · dS = 1. S V



∂Φ 2. dS = ∇2 Φ dV S ∂n V 10.716 Green’s reciprocal theorem. If Φ and Ψ are harmonic, so that ∇2 Φ = ∇2 Ψ = 0, then



∂Ψ ∂Φ dS = Ψ dS Φ 3. ∂n S ∂n S

10.714

KE 215

MV 81

MM 105

10.717 Green’s representation theorem. If Φ and ∇2 Φ are defined within a volume V bounded by a simple closed surface S, and P is an interior point of V , then in three dimensions  



1 1 2 1 ∂Φ 1 1 1 ∂ 4. Φ(P ) = − ∇ Φ dV + dS − dS Φ KE 219 4π V r 4π S r ∂n 4π S ∂n r

5.

6.

If Φ is harmonic within V , so that ∇2 Φ = 0, then the previous result becomes  



1 1 ∂ 1 ∂Φ 1 dS − dS Φ(P ) = Φ 4π S r ∂n 4π S ∂n r In the case of two dimensions, result (4) takes the form

1 Φ(p) = ∇2 Φ(q) ln |p − q| dS 2π S

∂ ∂ 1 1 ln |p − q| Φ(q) ln |p − q| dq − Φ(q) dq + 2π C ∂nq 2π ∂nq MM 116

7.

where C is the boundary of the planar region S, and result (5) takes the form



1 ∂ 1 ∂ Φ(p) = Φ(q) ln |p − q| dq − ln |p − q| Φ(q) dq 2π C ∂nq 2π C ∂nq

VL 280

10.718 Green’s representation theorem in Rn . If Φ is twice differentiable within a region Ω in Rn bounded by the surface Σ with outward drawn unit normal n, then for p ∈ Σ and n > 3 

∇2 Φ(q) ∂ Φ(q) 1 1 1 ∂ −1 dΣq , dΩq + − Φ(q) Φ(p) = (n − 2)σn Ω |p − q|n−2 (n − 2)σn Σ |p − q|n−2 ∂nq ∂nq |p − q|n−2 where 2π n/2 VL 279 σn = Γ(n/2) is the area of the unit sphere in Rn . 10.719 Green’s theorem of the arithmetic mean. If Φ is harmonic in a sphere, then the value of Φ at the center of the sphere is the arithmetic mean of its value on the surface. KE 223 10.720 Poisson’s integral in three dimensions. If Φ is harmonic in the interior of a spherical volume V of radius R and is continuous on the surface of the sphere on which, in terms of the spherical polar coordinates (r, θ, φ), it satisfies the boundary condition Φ (R, θ, φ) = f (θ, φ), then

10.811

Integral rate of change theorems

1069

  π π R R2 − r 2 f (θ , φ ) sin θ dθ dφ Φ(r, θ, φ) = , 3/2 4π 0 −π (r2 + R2 − 2rR cos γ) where cos γ = cos θ cos θ + sin θ sin θ cos (φ − φ ) KE 241 10.721 Poisson’s integral in two dimensions. If Φ is harmonic in the interior of a circular disk S of radius R and is continuous on the boundary of the disk on which, in terms of the polar coordinates (r, θ), it satisfies the boundary condition Φ(R, θ) = f (θ), then  2  π R − r2 f (φ) dφ Φ(r, θ) = 2 2 2π −π r + R − 2rR cos(θ − φ) 10.722 Stokes’ theorem. Let a simple closed curve C be spanned by a surface S. Define the positive normal n to S, and the positive sense of description of the curve C with line element dr, such that the positive sense of the contour C is clockwise when we look through the surface S in the direction of the normal. Then, if f is continuously differentiable vector field defined on S and C with vector element S = n dS, 

f · dr = curl f · dS, MM 143 C

S

where the line integral around C is taken in the positive sense. 10.723 Planar case of Stokes’ theorem. If a region R in the (x, y)-plane is bounded by a simple closed curve C, and f1 (x, y), f2 (x, y) are any two functions having continuous first derivatives in R and on C, then  

 ∂f1 ∂f2 − dx dy, (f1 dx + f2 dy) = MM 143 ∂x ∂y C R where the line integral is taken in the anticlockwise sense.

10.81 Integral rate of change theorems 10.811 Rate of change of volume integral bounded by a moving closed surface. Let f be a continuous scalar function of position and time t defined throughout the volume V (t), which is itself bounded by a simple closed surface S (t) moving with velocity v. Then the rate of change of the volume integral of f is given by



D ∂f dV + f dV = f v · dS, Dt V (t) V (t) ∂t S(t) where dS is the outward drawn vector element of area, and ∂ D ≡ + v · ∇. Dt ∂t By virtue of Gauss’s theorem this also takes the form  



D Df + f div v dV f dV = MV 88 Dt V (t) Dt V (t)

1070

Vectors, Vector Operators, and Integral Theorems

10.812

10.812 Rate of change of flux through a surface. Let q be a vector function that may also depend on the time t, and n be the unit outward drawn normal to the surface S that moves with velocity v. Defining the flux of q through S as

m = q · n dS, then

S

 Dm ∂q = + v div q + curl (q × v) · n dS MV 90 Dt ∂t S 10.813 Rate of change of the circulation around a given moving curve. Let C be a closed curve, moving with velocity v, on which is defined a vector field q. Defining the circulation ζ of q around C by

ζ= q · dr,



C

then Dζ = Dt

 C

 ∂q + (curl q) × v · dr ∂t

MV 94

Table of Integrals, Series, and Products. http://dx.doi.org/10.1016/B978-0-12-384933-5.00011-4 c 2015 Elsevier Inc. All rights reserved. Copyright 

11 Integral Inequalities 11.11 Mean Value Theorems 11.111 First mean value theorem Let f (x) and g(x) be two bounded functions integrable in [a, b] and let g(x) be of one sign in this interval. Then  b  b f (x)g(x) dx = f (ξ) g(x) dx, CA 105 a

with a ≤ ξ ≤ b.

a

11.112 Second mean value theorem (i)

Let f (x) be a bounded, monotonic decreasing, and nonnegative function in [a, b], and let g(x) be a bounded integrable function. Then,  b  ξ f (x)g(x) dx = f (a) g(x) dx, a

with a ≤ ξ ≤ b. (ii)

a

Let f (x) be a bounded, monotonic increasing, and nonnegative function in [a, b], and let g(x) be a bounded integrable function. Then,  b  b f (x)g(x) dx = f (b) g(x) dx, a

η

with a ≤ η ≤ b. (iii)

Let f (x) be bounded and monotonic in [a, b], and let g(x) be a bounded integrable function which experiences only a finite number of sign changes in [a, b]. Then,  b  ξ  b f (x)g(x) dx = f (a + 0) g(x) dx + f (b − 0) g(x) dx, CA 107 a

a

ξ

with a ≤ ξ ≤ b.

11.113 First mean value theorem for infinite integrals Let f (x) be bounded for x ≥ a, and integrable in the arbitrary interval [a, b], and let g(x) be of one sign ∞ in x ≥ a and such that a g(x) dx is finite. Then, 1071

1072

Differentiation of Definite Integral Containing a Parameter



∞ a

 f (x)g(x) dx = μ

∞ a

g(x) dx,

CA 123

where m ≤ μ ≤ M and m, M are, respectively, the lower and upper bounds of f (x) for x ≥ a.

11.114 Second mean value theorem for infinite integrals Let f (x) be bounded and monotonic when x ≥ a, and g(x) be bounded and integrable in the arbitrary ∞ interval [a, b] in which it experiences only a finite number of changes of sign. Then, provided a g(x) dx is finite,  ∞  ξ  ∞ f (x)g(x) dx = f (a + 0) g(x) dx + f (∞) g(x) dx, CA 123 with a ≤ ξ ≤ ∞.

a

a

ξ

11.21 Differentiation of Definite Integral Containing a Parameter 11.211 Differentiation when limits are finite Let φ(α) and ψ(α) be twice differentiable functions in some interval c ≤ α ≤ d, and let f (x, α) be both integrable with respect to x over the interval φ(α) ≤ x ≤ ψ(α) and differentiable with respect to α. Then,       ψ(α) d ψ(α) ∂f dφ dψ f (ψ(α), α) − f (φ(α), α) + dx f (x, α) dx = FI II 680 dα φ(α) dα dα φ(α) ∂α

11.212 Differentiation when a limit is infinite Let f (x, α) and ∂f /∂α both be integrable with respect to x over the semi-infinite region x ≥ a, b ≤ α < c. Then, if the integral  ∞ f (x, α) dx f (α) = a  ∞ ∂f exists for all b ≤ α ≤ c, and if a ∂α dx is uniformly convergent for α in [b, c], it follows that   ∞ ∂f d ∞ dx f (x, α) dx = dα a ∂α a

11.31 Integral Inequalities 11.311 Cauchy–Schwarz–Buniakowsky inequality for integrals Let f (x) and g(x) be any two real integrable functions on [a, b]. Then,   2     b b b 2 2 f (x)g(x) dx ≤ f (x) dx g (x) dx , a

a

a

and the equality will hold if, and only if, f (x) = kg(x), with k real.

BB 21

11.312 H¨ older’s inequality for integrals p

q

Let f (x) and g(x) be any two real functions for which |f (x)| and |g(x)| are integrable on [a, b] with p > 1 and p1 + 1q = 1; then

Gram’s inequality for integrals

 a

b

 f (x)g(x) dx ≤

b a

p

p

1073

1/p 

|f (x)| dx

a

q

b

1/q q

|g(x)| dx

.

The equality holds if, and only if, α|f (x)| = β|g(x)| , where α and β are positive constants.

BB 21

11.313 Minkowski’s inequality for integrals p

p

Let f (x) and g(x)be any two real functions for which |f (x)| and |g(x)| are integrable on [a, b] for p > 0; then  1/p  1/p  1/p b b b p p p |f (x) + g(x)| dx ≤ |f (x)| dx + |g(x)| dx . a

a

The equality holds if, and only if, f (x) = kg(x) for some real k ≥ 0. n

a

BB 21

11.314 Chebyshev’s inequality for integrals Let f1 , f2 , . . . , fn be nonnegative integrable functions on [a, b] which are all either monotonic increasing or monotonic decreasing; then  b  b  b  b n−1 f1 (x) dx f2 (x) dx . . . fn (x) dx ≤ (b − a) f1 (x)f2 (x) . . . fn (x) dx MT 39 a

a

a

a

11.315 Young’s inequality for integrals Let f (x) be a real-valued continuous strictly monotonic increasing function on the interval [0, a], with f (0) = 0 and b ≤ f (a). Then  a  b ab ≤ f (x) dx + f −1 (y) dy, 0

0

where f −1 (y) denotes the function inverse to f (x). The equality holds if, and only if, b = f (a).

BB 15

11.316 Steffensen’s inequality for integrals Let f (x) be nonnegative and monotonic decreasing in [a, b] and g(x) be such that 0 ≤ g(x) ≤ 1 in [a, b]. Then  b  b  a+k f (x) dx ≤ f (x)g(x) dx ≤ f (x) dx, b−k a a b where k = a g(x) dx. MT 107

11.317 Gram’s inequality for integrals Let f1 (x), f2 (x), . . . , fn (x) be real square integrable functions on [a, b]; then   b b b   2 f (x)f (x) dx · · · f (x)f (x) dx 1 2 1 n    a f1 (x) dx a ab   b b 2   a f2 (x)f1 (x) dx f (x) dx · · · f (x)f (x) dx n a 2 a 2  ≥ 0.  .. .. .. ..   .   . . .   b b b 2   fn (x)f1 (x) dx a a fn (x)f2 (x) dx · · · a fn (x) dx

MT 47

1074

Fourier Series and Related Inequalities

11.318 Ostrowski’s inequality for integrals Let f (x) be a monotonic function integrable on [a, b], and let f (a)f (b) ≥ 0, |f (a)| ≥ |f (b)|. Then, if g is a real function integrable on [a, b],      b  ξ        f (x)g(x) dx ≤ |f (a)| max  g(x) dx.  a   a≤ξ≤b a

11.41 Convexity and Jensen’s Inequality A function f (x) is said to be convex on an interval [a, b] if for any two points x1 , x2 in [a, b]   f (x1 ) + f (x2 ) x1 + x2 ≤ . f 2 2 A function f (x) is said to be concave on an interval [a, b] if for any two points x1 , x2 in [a, b] the function −f (x) is convex in that interval. If the function f (x) possesses a second derivative in the interval [a, b], then a necessary and sufficient condition for it to be convex on that interval is that f  (x) ≥ 0 for all x in [a, b]. A function f (x) is said to be logarithmically convex on the interval [a, b] if f > 0 and log f (x) is concave on [a, b]. If f (x) and g(x) are logarithmically convex on the interval [a, b], then the functions f (x) + g(x) and f (x)g(x) are also logarithmically convex on [a, b]. MT 17

11.411 Jensen’s inequality Let f (x), p(x) be two functions defined for a ≤ x ≤ b such that α ≤ f (x) ≤ β and p(x) ≥ 0, with p(x) ≡ 0. Let φ(u) be a convex function defined on the interval α ≤ u ≤ β; then  b  b f (x)p(x) dx φ (f ) p(x) dx a φ ≤ a b HL 151 . b a p(x) dx a p(x) dx

11.412 Carleman’s inequality for integrals If f (x) ≥ 0 and the integrals exist, then   x   ∞  ∞ 1 exp f (t) dt dx ≤ e f (x) dx x 0 0 0

11.51 Fourier Series and Related Inequalities The trigonometric Fourier series representation of the function f (x) integrable on [−π, π] is ∞ a0 + (an cos nx + bn sin nx) , f (x) ∼ 2 n=1 where the Fourier coefficients an and bn of f (x) are given by   1 π 1 π f (x) cos nx dx, bn = f (x) sin nx dx. an = 2π −π 2π −π (See 0.320–0.328 for convergence of Fourier series on (−l, l).)

TF 1

Generalized Fourier series

1075

11.511 Riemann–Lebesgue lemma If f (x) is integrable on [−π, π], then

 lim

π

t→∞ −π

and



π

lim

t→∞ −π

f (x) sin tx dx → 0

f (x) cos tx dx → 0.

TF 11

11.512 Dirichlet lemma



sin n + 12 x π dx = , 1 2 2 sin x 0 2 

in which sin n + 12 x m/2 sin 12 x is called the Dirichlet kernel. 

π

ZY 21

11.513 Parseval’s theorem for trigonometric Fourier series If f (x) is square integrable on [−π, π], then  ∞  a20 2 1 π 2 2 + ar + b r = f (x) dx. 2 π −π r=1

Y 10

11.514 Integral representation of the nth partial sum If f (x) is integrable on [−π, π], then the nth partial sum n a0 + (ar cos rx + br sin rx) sn (x) = 2 r=1 has the following integral representation in terms of the Dirichlet kernel,

  sin n + 12 t 1 π dt. f (x − t) sn (x) = π −π 2 sin 12 t

11.515 Generalized Fourier series ∞

Let the set of functions {φn }n=0 form an orthonormal set over [a, b], so that  b 1 for m = n, φm (x)φn (x) dx = 0 for m = n. a Then the generalized Fourier series representation of an integrable function f (x) on [a, b] is ∞ cn φn (x), f (x) ∼ n=0

where the generalized Fourier coefficients of f (x) are given by  b cn = f (x)φn (x) dx. a

Y 20

1076

Fourier Series and Related Inequalities

11.516 Bessel’s inequality for generalized Fourier series For any square integrable function defined on [a, b],  b ∞ c2n ≤ f 2 (x) dx, n=0

a

where the cn are the generalized Fourier coefficients of f (x).

11.517 Parseval’s theorem for generalized Fourier series ∞

If f (x) is a square integrable function defined on [a, b] and {φn (x)}n=0 is a complete orthonormal set of continuous functions defined on [a, b], then  b ∞ c2n = f 2 (x) dx, n=0

a

where the cn are generalized Fourier coefficients of f (x).

Table of Integrals, Series, and Products. http://dx.doi.org/10.1016/B978-0-12-384933-5.00012-6 c 2015 Elsevier Inc. All rights reserved. Copyright 

12 Fourier, Laplace, and Mellin Transforms 12.1– 12.4 Integral Transforms 12.11 Laplace transform The Laplace transform of the function f (x), denoted by F (s), is defined by the integral  ∞ f (x)e−sx dx, Re s > 0. F (s) = 0

The functions f (x) and F (s) are called a Laplace transform pair, and knowledge of either one enables the other to be recovered. If f is summable over all finite intervals, and there is a constant c for which  ∞ |f (x)|e−c|x| dx, 0

is finite, then the Laplace transform exists when s = σ + iτ is such that σ ≥ c. Setting F (s) = L [f (x); s] , to emphasize the nature of the transform, we have the symbolic inverse result f (x) = L−1 [F (s); x] .   The inversion of the Laplace transform is accomplished for analytic functions F (s) of order O s−k with k > 1 by means of the inversion integral  γ+i∞ 1 f (x) = F (s)esx ds, 2πi γ−i∞ where γ is a real constant that exceeds the real part of all the singularities of F (s). SN 30

12.12 Basic properties of the Laplace transform 1.8

For a and b arbitrary constants, L [af (x) + bg(x)] = aF (s) + bG(s)

2.

(linearity)

If n > 0 is an integer and lim f (x)e−sx = 0, then for x > 0, x→∞   L f (n) (x); s = sn F (s) − sn−1 f (0) − sn−2 f (1) (0) − · · · − f (n−1) (0) (transform of a derivative) SN 32

1077

1078

3.11

Integral Transforms

 x  If lim e−sx 0 f (ζ) dζ = 0, then x→∞



L

0

x



f (ξ) dξ; s =

1 F (s) s

(transform of an integral)

 L e−ax f (x); s = F (s + a)

4.

SN 37

(shift theorem)

SU 143

The Laplace convolution f ∗ g of two functions f (x) and g(x) is defined by the integral  x f ∗ g(x) = f (x − ξ)g(ξ) dξ,

5.

0

and it has the property that f ∗ g = g ∗ f and f ∗ (g ∗ h) = (f ∗ g) ∗ h. In terms of the convolution operation L [f ∗ g(x); s] = F (s)G(s)

(convolution (Faltung) theorem)

SN 30

12.13 Table of Laplace transform pairs f (x)

F (s)

1

1

2

xn ,

n = 0, 1, 2, . . .

3

xν ,

ν > −1

4

xn− 2

5

x−1/2 (x + a)−1 ,

1

 6

1/s

x for 0 < x < 1 1 for x > 1

n! , sn+1 Γ(ν + 1) , sν+1   Γ n + 12 1

sn+ 2

|arg a| < π

,

Re s > 0

ET I 133(3)

Re s > 0

ET I 137(1)

Re s > 0

ET I 135(17)

πa−1/2 eas erfc a1/2 s1/2 ,

1 − e−s , s2

Re s ≥ 0

ET I 136(25)

Re s > 0

ET I 142(14)

7

e−ax

1 , s+a

Re s > − Re a

ET I 143(1)

8

xe−ax

1 , (s + a)2

Re s > − Re a

ET I 144(2)

9a

e−ax − e−bx b−a

(s + a)−1 (s + b)−1 , Re s > {− Re a, − Re b}

AS 1022(29.3.12) continued on next page

Table of Laplace transform pairs

1079

continued from previous page

f (x) 9b11

F (s)

αe−ax + βe−bx + γe−cx (a − b)(b − c)(c − a)

(s + a)−1 (s + b)−1 (s + c)−1 , Re s > {− Re a, − Re b, − Re c}

a, b, c distinct , α = c − b, β = a − c, γ = b − a 1011

ae−ax − be−bx b−a

s(s + a)−1 (s + b)−1 , Re s > {− Re a, − Re b}

11 12 13

eax − 1 a eax − ax − 1 a2  ax 1 2 2  e − 2 a x − ax − 1 a3

AS 1022(29.3.13)

s−1 (s − a)−1 ,

Re s > Re a

s−2 (s − a)−1 ,

Re s > Re a

s−3 (s − a)−1 ,

Re s > Re a

14

(1 + ax)eax

s , (s − a)2

Re s > Re a

15

1 + (ax − 1)eax a2

s−1 (s − a)−2 ,

Re s > Re a

16

2 + ax + (ax − 2)eax a3

s−2 (s − a)−2 ,

Re s > Re a

17

xn eax ,

n!(s − a)−(n+1) ,

Re s > Re a

18 19 20 21 22

n = 0, 1, 2, . . .



 x + 12 ax2 eax

s , (s − a)3

Re s > Re a



 1 + 2ax + 12 a2 x2 eax

s2 , (s − a)3

Re s > Re a

(s − a)−4 ,

Re s > Re a

s , (s − a)4

Re s > Re a

s2 (s − a)−4 ,

Re s > Re a

1 3 ax 6x e

1

2 2x



 + 16 ax3 eax

 x + ax2 + 16 a2 x3 eax

continued on next page

1080

Integral Transforms

continued from previous page

f (x) 23

24 25



F (s)

 1 + 3ax + 32 a2 x2 + 16 a3 x3 eax

aeax − bebx a−b  1 ax 1 bx − be + ae a−b

26

xν−1 e−ax ,

27

xe−x

2

/(4a)

,

1 b

−

1 a

s3 (s − a)−4 ,

Re s > Re a

s(s − a)−1 (s − b)−1 ,

Re s > {Re a, Re b}

s−1 (s − a)−1 (s − b)−1 ,

Re s > {Re a, Re b}



Re ν > 0

Γ(ν)(s + a)−ν ,

Re a > 0

2 2a − 2π 1/2 a3/2 seas erfc sa1/2

Re s > − Re a

ET I 144(3)

ET I 146(22)

28

exp (−aex ) ,

Re a > 0

as Γ (−s, a)

298

x1/2 e−a/(4x) ,

Re a ≥ 0

1 1/2 −3/2 s 2π

ET I 147(37)



  1 + a1/2 s1/2 exp (−as)1/2 , Re s > 0

3012

x−1/2 e−a/(4x) ,

Re a ≥ 0

  π 1/2 s−1/2 exp −(as)1/2 , Re s > 0

3112

x−3/2 e−a/(4x) ,

Re a > 0

ET I 146(26)

ET I 146(27)

  2π 1/2 a−1/2 exp −(as)1/2 , Re s ≥ 0

ET I 146(28)

32

sin(ax)

−1  a s2 + a 2 ,

Re s > |Im a|

ET I 150(1)

33

cos(ax)

−1  s s2 + a2 ,

Re s > |Im a|

ET I 154(3)

34

|sin(ax)|,

πs −1  , a s2 + a 2 coth 2a Re s > 0

ET I 150(2)

a>0

continued on next page

Table of Laplace transform pairs

1081

continued from previous page

f (x) 3511

36

|cos(ax)|,

a>0

1 − cos(ax) a2

F (s) πs   2 −1  , s + a cosech s + a2 2a Re s > 0  −1 s−1 s2 + a2 , Re s > |Im a|

37

38

ax − sin(ax) a3 sin(ax) − ax cos(ax) 2a3

AS 1022(29.3.19)

 −1 s−2 s2 + a2 ,



s2 + a2

−2

Re s > |Im a|

AS 1022(29.3.20)

Re s > |Im a|

AS 1022(29.3.21)

,

39

x sin(ax) 2a

−2  s s2 + a2 ,

40

sin(ax) + ax cos(ax) 2a

 −2 s2 s2 + a2 ,

Re s > |Im a|

Re s > |Im a| 41

x cos(ax)



s2 − a2

43 44 45

4611

cos(ax) − cos(bx) b 2 − a2

1

 2 2 2 a x − 1 + cos(ax) a4

 1 − cos(ax) − 12 ax sin(ax) a4

1  1 b sin(bx) − a sin(ax) a2 − b 2

 1 − cos(ax) + 12 ax sin(ax) a2

ET I 152(14)

AS 1023(29.3.23)

 2 −2 s + a2 , Re s > |Im a|

42

ET I 155(44)

ET I 157(57)

−1  2 −1  s + b2 s s2 + a2 , Re s > {|Im a|, |Im b|}

AS 1023(29.3.25)

 −1 s−3 s2 + a2 ,

Re s > |Im a|

 −2 s−1 s2 + a2 ,

Re s > |Im a|



s2 + a2

−1 

s2 + b 2

−1

, Re s > {|Im a|, |Im b|}

 −2  2  2s + a2 , s−1 s2 + a2

Re s > |Im a| continued on next page

1082

Integral Transforms

continued from previous page

f (x) 47

a sin(ax) − b sin(bx) a2 − b 2

F (s)  −1  2 −1 s + b2 s2 s2 + a2 , Re s > {|Im a|, |Im b|}

48

sin(a + bx)

−1  (s sin a + b cos a) s2 + b2 ,

Re s > |Im b|

49

cos(a + bx)

−1  (s cos a − b sin a) s2 + b2 ,

Re s > |Im b|

1



50

a

 sinh(ax) − 1b sin(bx) a2 + b 2

s2 − a2

−1 

s2 + b 2

−1

, Re s > {|Re a|, |Im b|}

51

cosh(ax) − cos(bx) a2 + b 2

−1  2 −1  s + b2 s s2 − a2 , Re s > {|Re a|, |Im b|}

52

a sinh(ax) + b sin(bx) a2 + b 2

 −1  2 −1 s + b2 s2 s2 − a2 , Re s > {|Re a|, |Im b|}

53

sin(ax) sin(bx)

−1 2 −1

s + (a + b)2 2abs s2 + (a − b)2 , Re s > {|Im a|, |Im b|}

54

cos(ax) cos(bx)



−1 2 −1  s + (a + b)2 s s2 + a2 + b2 s2 + (a − b)2 , Re s > {|Im a|, |Im b|}

55

sin(ax) cos(bx)



−1 2 −1  s + (a + b)2 a s2 + a2 − b2 s2 + (a − b)2 , Re s > {|Im a|, |Im b|}  −1 2a2 s−1 s2 + 4a2 ,

56

sin2 (ax)

57

cos2 (ax)

58

sin(ax) cos(ax)

−1  a s2 + 4a2 ,

e−ax sin(bx)

−1

b (s + a)2 + b2 ,

59



Re s > |Im a|

  −1 s2 + 2a2 s−1 s2 + 4a2 ,

Re s > |Im a| Re s > |Im a|

Re s > {− Re a,

|Im b|}

continued on next page

Table of Laplace transform pairs

1083

continued from previous page

f (x) 60

F (s) −1

(s + a) (s + a)2 + b2 ,

e−ax cos(bx)

Re s > {− Re a, 61

x−1 sin(ax)

arctan(a/s),

62

x−1 [1 − cos(ax)]

1 2

|Im b|}

Re s > |Im a|

ET I 152(16)

Re s > |Im a|

ET I 157(59)

  ln 1 + a2 /s2 ,

63

sinh(ax)

−1  a s2 − a2 ,

Re s > |Re a|

ET I 162(1)

64

cosh(ax)

−1  s s2 − a2 ,

Re s > |Re a|

ET I 162(2)

65

xν−1 sinh(ax),

Re ν > −1

1 2



Γ(ν) (s − a)−ν − (s + a)−ν , Re s > |Re a|

66

xν−1 cosh(ax),

Re ν > 0

1 2



Γ(ν) (s − a)−ν + (s + a)−ν , Re s > |Re a|

67

x sinh(ax)

68

x cosh(ax)

69

sinh(ax) − sin(ax)

−2  2as s2 − a2 , 

s2 + a2

 2 −2 s − a2 ,

cosh(ax) − cos(ax)

ET I 164(19)

Re s > |Re a| Re s > |Re a|

 −1 2a3 s4 − a4 , Re s > {|Re a|, |Im a|}

70

ET I 164(18)

AS 1023(29.3.31)

−1  2a2 s s4 − a4 , Re s > {|Re a|, |Im a|}

AS 1023(29.3.32)

71

sinh(ax) + ax cosh(ax)

 −2 2as2 a2 − s2 ,

Re s > |Re a|

72

ax cosh(ax) − sinh(ax)

 −2 2a3 a2 − s2 ,

Re s > |Re a| continued on next page

1084

Integral Transforms

continued from previous page

f (x) 73 74

F (s)

x sinh(ax) − cosh(ax)

 −2  s a2 + 2a − s2 a2 − s2 ,

1



a

 sinh(ax) − 1b sinh(bx) a2 − b 2

a2 − s2

−1 

b 2 − s2

−1

Re s > |Re a|

, Re s > {|Re a|, |Re b|}

75

cosh(ax) − cosh(bx) a2 − b 2

−1  2 −1  s − b2 s s2 − a2 , Re s > {|Re a|, |Re b|}

76

a sinh(ax) − b sinh(bx) a2 − b 2

 −1  2 −1 s − b2 s2 s2 − a2 , Re s > {|Re a|, |Re b|}

77

sinh(a + bx)

−1  (b cosh a + s sinh a) s2 − b2 ,

Re s > |Re b|

78

cosh(a + bx)

−1  (s cosh a + b sinh a) s2 − b2 ,

Re s > |Re b|

79

sinh(ax) sinh(bx)

−1 2 −1

s − (a − b)2 2abs s2 − (a + b)2 , Re s > {|Re a|, |Re b|}

808

cosh(ax) cosh(bx)



−1 2 −1  s − (a − b)2 s s2 − a2 − b2 s2 − (a + b)2 , Re s > {|Re a|, |Re b|}

81

sinh(ax) cosh(bx)



−1 2 −1  s − (a − b)2 a s2 − a2 + b2 s2 − (a + b)2 , Re s > {|Re a|, |Re b|}  −1 2a2 s−1 s2 − 4a2 ,

82

sinh2 (ax)

83

cosh2 (ax)

84

sinh(ax) cosh(ax)

−1  a s2 − 4a2 ,

Re s > |Re a|

85

cosh(ax) − 1 a2

 −1 s−1 s2 − a2 ,

Re s > |Re a|



Re s > |Re a|

  −1 s2 − 2a2 s−1 s2 − 4a2 ,

Re s > |Re a|

continued on next page

Table of Laplace transform pairs

1085

continued from previous page

f (x) 86 87 88 89

F (s)

sinh(ax) − ax a3

 cosh(ax) − 12 a2 x2 − 1 a4

 1 − cosh(ax) + 12 ax sinh(ax) a4

 −1 s−2 s2 − a2 ,

Re s > |Re a|

 −1 s−3 s2 − a2 ,

Re s > |Re a|

 −2 s−1 s2 − a2 ,

Re s > |Re a|



x1/2 sinh(ax)

π 1/2 /4



 (s − a)3/2 − (s + a)3/2 , Re s > |Re a|

90

ln x

91

ln(1 + ax),

92

x−1/2 ln x

|arg a| < π

Re s > 0

ET I 148(1)

s−1 es/a Ei(−s/a),

Re s > 0

ET I 148(4)

1/2

Re s > 0

ET I 148(9)

− (π/s) 

93

−s−1 ln (Cs) ,

H(x − a) =

0 1

for x < a for x > a

ln (4Cs) ,

s−1 e−as ,

a≥0

(Heaviside step function) 94

δ(x)

95

δ(x − a)

e−as ,

a≥0

96

δ  (x − a)

se−as ,

a≥0

(Dirac delta function)



x

sin ξ 1 dξ ≡ π + si(x) ξ 2

97

Si(x) ≡

98

Ci(x) ≡ ci(x) ≡ −

998

erf

0

 x 2a

∞ x

cos ξ dξ ξ

1

s−1 arccot s,

Re s > 0

ET I 177(17)

  1 − s−1 ln 1 + s2 , 2

Re s > 0

ET I 178(19)

s−1 ea

2 2

s

erfc(as), Re s > 0, |arg a| < π/4

ET I 176(2)

continued on next page

1086

Integral Transforms

continued from previous page

f (x) 100

F (s)

 √  erf a x

 −1/2 as−1 s + a2 ,

  Re s > 0, − Re a2

101

ET I 176(4)

  − 1  1/2 s−1 s + a2 2 s + a2 −a ,

 √  erfc a x

Re s > 0  1028

erfc

a √ x

 s−1 e−2a

√ s

, Re s > 0,

1038

104

105

J ν (ax),

x J ν (ax),

Re ν > −1

Re ν > −2

a−ν



ET I 177(9)

s2 + a2 − s

ν 

Re a > 0 s2 + a2

−1/2

ET I 177(11)

,

ET I 182(1) Re s > |Im a|     1/2 1/2 −ν s + s2 + a 2 a ν s + ν s2 + a 2 −3/2  × s2 + a 2 ,



Re s > |Im a|   1/2 −ν aν ν −1 s + s2 + a2 ,

J ν (ax) x

ET I 182(2)

Re s ≥ |Im a| 106

 −(n+ 12 ) , 1 · 3 · 5 · · · (2n − 1)an s2 + a2

xn J n (ax)

Re s > |Im a| 107

xν J ν (ax),

Re ν > − 12

xν+1 J ν (ax),

Re ν > −1

I ν (ax),

Re ν > −1

ET I 182(7)

 −(ν+ 32 )   , 2ν+1 π −1/2 Γ ν + 32 aν s s2 + a2 Re s > |Im a|

1098

ET I 182(4)

  −(ν+ 12 )  , 2ν π −1/2 Γ ν + 12 aν s2 + a2 Re s > |Im a|

108

ET I 182(5)

ET I 182(8)

 ν   −1/2 s2 − a2 a−ν s − s2 − a2 , Re s > |Re a|

ET I 195(1)

continued on next page

Fourier transform

1087

continued from previous page

f (x) 110

xν I ν (ax),

F (s) Re ν > − 12

  −(ν+ 12 )  , 2ν π −1/2 Γ ν + 12 aν s2 − a2 Re s > |Re a|

111

xν+1 I ν (ax),

Re ν > −1

 −(ν+ 32 )   , 2ν+1 π −1/2 Γ ν + 32 aν s s2 − a2 Re s > |Re a|

112

x−1 I ν (ax),

Re ν > 0

ET I 195(6)

ET I 196(7)

  1/2 −ν ν −1 aν s + s2 − a2 , Re s > |Re a|

ET I 195(4)

113

sin 2a1/2 x1/2

(πa)1/2 s−3/2 e−a/s ,

Re s > 0

ET I 153(32)

114

x−1/2 cos 2a1/2 x1/2

π 1/2 s−1/2 e−a/s ,

Re s > 0

ET I 158(67)

115

x−1 e−ax I 1 (ax)

  −1 1/2 (s + 2a)1/2 − s1/2 (s + 2a) + s1/2 , Re s > |Re a|

116

J k (ax) x

k −1 a−k



s2 + a 2

1/2

−s

k

,

Re s > |Im a|, k > −1 117

x k− 12 J k− 12 (ax) 2a

AS 1024(29.3.52)

AS 1025(29.3.58)

 k Γ(k)π −1/2 s2 + a2 , Re s > |Im a|,

k>0

AS 1024(29.3.57)

118

J 0 (ax) − ax J 1 (ax)

 −3/2 s2 s2 + a2 ,

Re s > |Im a|

119

I 0 (ax) + ax I 1 (ax)

 −3/2 s2 s2 − a2 ,

Re s > |Im a|

12.21 Fourier transform The Fourier transform, also called the exponential or complex Fourier transform, of the function f (x), denoted by F (ξ), is defined by the integral  ∞ 1 F (ξ) = √ f (x)eiξx dx. 2π −∞ The functions f (x) and F (ξ) are called a Fourier transform pair, and knowledge of either one enables the other to be recovered. Setting F (ξ) = F [f (x); ξ] , to emphasize the nature of the transform, we have

1088

Integral Transforms

the symbolic inverse result f (x) = F −1 [F (ξ); x] . The inversion of the Fourier transform is accomplished by means of the inversion integral  ∞ 1 f (x) = √ F (ξ)e−iξx dξ. 2π −∞

12.22 Basic properties of the Fourier transform For a and b arbitrary constants,

1.

F [af (x) + bg(x)] = aF (ξ) + bG(ξ)

(linearity)

If n > 0 is an integer, and lim f (r) (x) = 0 for r = 0, 1, . . . , n − 1 with f (0) (x) ≡ f (x), then

2.

|x|→∞

  F f (n) (x); ξ = (−iξ)n F (ξ)

(transform of a derivative)

SN 27

The Fourier convolution f ∗ g of two functions f (x) and g(x) is defined by the integral  ∞ 1 f ∗ g(x) = √ f (x − ξ)g(ξ) dξ, 2π −∞ and it has the property f ∗ g = g ∗ f , and f ∗ (g ∗ h) = (f ∗ g) ∗ h. In terms of the convolution operation.

3.

F [f ∗ g(x); ξ] = F (ξ)G(ξ)

(convolution (Faltung) theorem).

SN 24

12.23 Table of Fourier transform pairs f (x)

F (ξ)

1

1

(2π)1/2 δ(ξ)

27

1 x

(π/2)

3

δ(x)

(2π)−1/2

SU 496

48

δ(ax + b),

a = 0

(2π)−1/2 eibξ/a

SU 517

a>0

(2/π)1/2 ξ −1 sin(aξ)

 1 0

5 6

8

1/2

a, b ∈ R,

|x| < a , |x| > a  0 x0

SU 496

i sign ξ

1 − √ + iξ 2π



π δ(ξ) 2

SU 50

SN 523 continued on next page

Table of Fourier transform pairs

1089

continued from previous page

f (x)

F (ξ)   (2/π)1/2 Γ(1 − a) sin 12 aπ

7

1 a, |x|

0 < Re a < 1

8

eiax ,

a∈R

(2π)1/2 δ(ξ + a)

SU 50

9

e−a|x| ,

a>0

a(2/π)1/2 a2 + ξ 2

SU 50

107

xe−a|x| ,

a>0

11

|x|e−a|x| ,

a>0

12

e−a|x| |x|

1/2 2 2

a>0

,

a>0

2aiξ(2/π)1/2

2 , (a2 + ξ 2 )   (2/π)1/2 a2 − ξ 2 2

14

1 , 2 a + x2

Re a > 0

(π/2)

x , + x2

Re a > 0

i sign ξ (π/2)

SU 50

SU 50

SN 523

1/2

x (a2 + ξ 2 ) √ −1 2 2 a 2 e−ξ /4a

e−a

a2

ξ>0

(a2 + ξ 2 )   1/2 1/2 a + a2 + ξ 2

13

157

x

,

SN 523

1−a

|ξ|

SU 51

1/2 −a|ξ|

e

SU 51

a 1/2 −a|ξ|



e



169

  sin ax2

17

  cos ax2

18

e−a|x| cos(bx),

19

e− 2 ax sin(bx),

209

sinh(ax) , sinh(bx)

|a| < |b|

(π/2)1/2 sin(πa/b) b [cosh (πξ/b) + cos(πa/b)]

SU 123

219

cosh(ax) , sinh(bx)

|a| < |b|

i (π/2)1/2 sinh (πξ/b) b [cosh (πξ/b) + cos(πa/b)]

SU 123

1

2

1 cos (2a)1/2

a > 0,

b>0

a > 0,

b>0

π ξ2 + 4a 4

SN 523

  2 1 π ξ − cos SN 523 4a 4 (2a)1/2  1 1 a(2π)−1/2 2 + a + (b + ξ)2 a2 + (b − ξ)2   1 (ξ − b)2 1 −1/2 ia exp − 2 2 a  1 (ξ + b)2 − exp − 2 a

continued on next page

1090

Integral Transforms

continued from previous page

f (x) 2212

sin(ax) x

2311

x sinh x

247

xn sign x,

257

|x|ν ,

 a>0

F (ξ) 1/2

|ξ| < a, |ξ| > a  1/2 πξ 2Constantpi3 e (π/2) 0

(1 + eπξ ) n = 1, 2, . . .

SN 523

SU 123

2

(2/π)1/2 (−iξ)−(1+n) n!

SU 506

(2/π)1/2 Γ(ν + 1)|ξ|−ν−1 cos [π(ν + 1)/2] −1 < ν < 0, but not integral

267

SU506

i sign ξ(2/π)1/2 sin [(π/2) (ν + 1)] Γ(ν + 1)

|x|ν sign x,

ν+1

|ξ|

−1 < ν < 0, but not integral   e−ax ln 1 − e−x ,

27

SU 506

π 1/2 cot (πa − iξπ) 2 a − iξ

ET I 121(26)

π 1/2 csc (πa − iξπ) 2 a − iξ

ET I 121 (27)

−1 < Re a < 0   e−ax ln 1 + e−x ,

28

−1 < Re a < 0 In deriving results for the preceding table from ET I, account has been taken of the fact that the normalization factor 1/(2π)1/2 employed in our definition of F has not been used in those tables, and that there is a difference of sign between the exponents used in the definitions of the exponential Fourier transform.

12.24 Table of Fourier transform pairs for spherically symmetric functions

2

e−ar

312

e−ar r

 1 f (||r||)e−ik·r dr E(||k||) = (2π)3/2   21 ∞ E(k) = f (r) sin(kr)r dr πk 0  2 2a π (a2 + k 2 )2  2 1 π a2 + k 2

411

1

(2π)3/2 δ(k)

1

 1 E(||k||)eik·r dk f (||r||) = (2π)3/2   21 ∞ f (r) = E(k) sin(kr)k dk πr 0

Basic properties of the Fourier sine and cosine transforms

1091

12.31 Fourier sine and cosine transforms The Fourier sine and cosine transforms of the function f (x), denoted by Fs (ξ) and Fc (ξ), respectively, are defined by the integrals   ∞   ∞ 2 2 Fs (ξ) = f (x) sin(ξx) dx and Fc (ξ) = f (x) cos(ξx) dx π 0 π 0 The functions f (x) and Fs (ξ) are called a Fourier sine transform pair, and the functions f (x) and Fc (ξ) a Fourier cosine transform pair, and knowledge of either Fs (ξ) or Fc (ξ) enables f (x) to be recovered. Setting Fs (ξ) = F s [f (x); ξ] and Fc (ξ) = F c [f (x); ξ] , to emphasize the nature of the transforms, we have the symbolic inverses f (x) = F s −1 [Fs (ξ); x] and f (x) = F c −1 [Fc (ξ); x] The inversion of the Fourier sine transform is accomplished by means of the inversion integral   ∞ 2 f (x) = Fs (ξ) sin(ξx) dξ [x ≥ 0] π 0 and the inversion of the Fourier cosine transform is accomplished by means of the inversion integral   ∞ 2 f (x) = Fc (ξ) cos(ξx) dξ [x ≥ 0] SN 17 π 0

12.32 Basic properties of the Fourier sine and cosine transforms 1.

For a and b arbitrary constants, F s [af (x) + bg(x)] = aFs (ξ) + bGs (ξ) and

2.12

F c [af (x) + bg(x)] = aFc (ξ) + bGc (ξ) (linearity)  If lim f (r−1) (x) = 0 and lim π2 f (r−1) (x) = ar−1 , then denoting the Fourier sine and cosine x→∞

x→0

transforms of f (r) (x) by Fs (r) and Fc (r) , respectively, (i)

Fc (r) (ξ) = −ar−1 + ξFs (r−1) .

(ii)

Fs (r) (ξ) = −ξFc (r−1) (ξ), r−1  (−1)n a2r−2n−1 ξ 2n + (−1)r ξ 2n Fc (ξ), Fc (2r) (ξ) = −

(iii)

n=0 r−1

(iv)

Fc (2r+1) (ξ) = −



(−1)n a2r−2n ξ 2n + (−1)r ξ 2r+1 Fs (ξ),

n=0

(v) (vi)6

(ξ) = ξar−2 − ξ 2 Fs (r−2) (ξ), r  Fs (2r) (ξ) = − (−1)n ξ 2n−1 a2r−2n + (−1)r ξ 2r Fs (ξ),

Fs

(r)

n=1

(vii)

Fs

(2r+1)

(ξ) = −

r  

n=1

(−1)n ξ 2n−1 a2r−2n+1 + (−1)r+1 ξ 2r+1 Fc (ξ).

SN 28

1092

Integral Transforms

 1 ∞ Fs (ξ)Gs (ξ) cos(ξx) dξ = g(s) [f (s + x) + f (s − x)] ds, 2 0  ∞ 0 ∞ 1 Fc (ξ)Gc (ξ) cos(ξx) dξ = g(s) [f (s + x) + f (|x − s|)] ds 2 0 0 (convolution (Faltung) theorem)

 3.

(i) (ii)

4.

∞

SN 24

(i)

If Fs (ξ) is the Fourier sine transform of f (x), then the Fourier sine transform of Fs (x) is f (ξ).

(ii)

If Fc (ξ) is the Fourier cosine transform of f (x), then the Fourier cosine transform of Fc (x) is f (ξ).

(iii)

(v)

If f (x) is an odd function in (−∞, ∞), then the Fourier sine transform of f (x) in (0, ∞) is −iF (ξ). If f (x) is an even function in (−∞, ∞), then the Fourier cosine transform of f (x) in (0, ∞) is F (ξ). The Fourier sine transform of f (x/a) is aFs (aξ).

(vi) (vii)

The Fourier cosine transform of f (x/a) is aFc (aξ). F s [f (x); ξ] = Fs (|ξ|) sign ξ

(iv)

SU 45

12.33 Table of Fourier sine transforms f (x) 1

x−1

2

x−ν ,

Fs (ξ) (π/2)1/2 , 0 < Re ν < 2

(ξ > 0) ξ>0

ET I 64(3)

(2/π)1/2 ξ ν−1 Γ(1 − ν) cos (νπ/2) , ξ>0

ET I 68(1)

3

x−1/2

ξ −1/2 ,

ξ>0

ET I 64(6)

4

x−3/2

2ξ 1/2 ,

ξ>0

ET I 64(9)

(2/π)1/2 ξ −1 [1 − cos(aξ)] ,

ξ>0

ET I 63(1)

(2/π)1/2 Si(aξ),

ξ>0

ET I 64(4)

5 6 7

 1 0a 1 , a−x

a>0



 (2/π)1/2 sin(aξ) Ci(aξ) − cos(aξ) 12 π + Si(aξ) , ξ>0

ET I 64(11)

continued on next page

Table of Fourier sine transforms

1093

continued from previous page

f (x) 87

x2

Fs (ξ)

1 , + a2

a>0

 (2π)−1/2 a−1 e−aξ Ei(aξ) − eaξ Ei(−aξ) ,

9

−3/2  x x2 + a2 ,

Re a > 0

(2/π)1/2 ξ K 0 (aξ),

10

 −1/2 x−1/2 x2 + a2 ,

Re a > 0

ξ 1/2 I 14

117

−ν− 32  x x2 + a2 , Re ν > −1,

12 13

x , 2 a + x2 x (a2 −1

+

2 x2 )



2

Re a > 0

x

15

x−1 e−ax ,

16

xν−1 e−ax ,

1

2 aξ



K 14

1

2 aξ



,

 2 −1

,

Re a > 0 Re a > 0

ξ>0

ET I 65(14)

ξ>0

ET I 66(27)

ξ>0

ET I 66(28)

ξ>0

ET I 65(15)

ξ>0

ET I 67(35)

ξ>0

ET I 65(20)

ξ ν+1 √  K ν (aξ),  2(2a)ν Γ ν + 32 π 1/2 2

e−aξ ,

 π/8a−1 ξe−aξ ,

x +a

14

Re a > 0

(ξ > 0)

  π/2  1 − e−aξ , 2 a   ξ , (2/π)1/2 tan−1 a

ξ>0

ET I 72(2)

   −ν/2  ξ , (2/π)1/2 Γ(ν) a2 + ξ 2 sin ν tan−1 a Re ν > −1,

Re a > 0

17

e−ax ,

Re a > 0

18

xe−ax ,

Re a > 0

19

xe

−ax2

,

|arg a| < π/2

20

sin ax , x

a>0

21

sin ax , x2

a>0

 2/πξ , 2 a + ξ2 (2/π)1/2 2aξ 2

(a2 + ξ 2 )



−ξ 2 ξ exp 4a   ξ + a ,  ln  ξ − a

−3/2

(2a)

,

1 (2π)1/2   1/2 ξ π2  1/2 a π2

ξ>0

ET I 72(7)

ξ>0

ET I 72(1)

ξ>0

ET I 72(3)

 ,

00

a2 x

 ,

a>0

Fs (ξ) (ξ > 0)

π 1/2 1 a ξ −1/2 J 1 2aξ 2 , 2 ξ>0 ET I 83(6) π 1/2

 2 1/2

Y 0 2aξ 1/2 + K 0 2aξ 1/2 2 π ET I 83(7)

24

2510

x−2 sin

27 28

 2

a x

,

coth

 1 ax − 1, 2



1 − x2

−1

2

a>0

Re a > 0

cosech(ax), 

26



Re a > 0

sin(πx) Re a > 0

29

sin2 (ax) , x

a>0

30

  sin ax2 ,

a>0

31

2

cos ax



,

2

a>0



a−1 ξ 1/2 J 1 2aξ 1/2 ,

1/2

(π/2)

a−1 tanh

1

2 πa

−1

ξ>0

ET I 83(8)

ξ>0

ET I 88(2)

 ξ ,

  (2π)1/2 a−1 coth πa−1 ξ − ξ, 

e−ax sin(bx),



π 1/2

ξ>0 (2/π)1/2 sin ξ 0

0≤ξ≤π π0 ⎧ 1/2 −3/2 ⎪ 0 < ξ < 2a ⎨π 2 1/2 −5/2 π 2 ξ = 2a ⎪ ⎩ 0 2a < ξ    2   −1/2 cos ξ /4a C (2πa)−1/2 ξ a     + sin ξ 2 /4a S (2πa)−1/2 ξ , ξ>0      2 −1/2 sin ξ /4a C (2πa) a ξ     − cos ξ 2 /4a S (2πa)−1/2 ξ ,

ET I 78(7)

ET I 78(8)

ET I 82(1)

−1/2

ξ>0 continued on next page

Table of Fourier sine transforms

1095

continued from previous page

32 337 34 35 367 37

f (x) x , arctan a   2a , arctan x

Fs (ξ) a>0 Re a > 0

1/2 −1 −aξ

ξ

(π/2)

e

(ξ > 0)

,

(2π)−1/2 e−aξ sinh(aξ),

ξ>0

ET I 87(3)

ξ>0

ET I 87(8)

ξ>0

ET I 76(2)

ln x x   x + a ,  ln  x − a   ln 1 + a2 x2 , x

a>0

(2π)1/2 ξ −1 sin(aξ),

ξ>0

ET I 77(11)

a>0

−(2π)1/2 Ei (−ξ/a) ,

ξ>0

ET I 77(14)

J 0 (ax),

a>0

1/2

− (π/2)



0 1/2

(2/π)

(C + ln ξ) ,



2

ξ −a

00

−(2/π)1/2 Ci(aξ),

1 

sin(aξ) , ξ

ET I 10(1)

0 00

1 , x2 + a2

7 8

Fc (ξ)

0 0

1

−2−(ν+ 2 ) Γ

1 2

ET I 11(8)

 − ν (ξ/a)ν Y ν (aξ),

−1  (2/π)1/2 a a2 + ξ 2 ,

ξ>0

ET I 11(9)

ξ>0

ET I 14(1)

continued on next page

Table of Fourier cosine transforms

1097

continued from previous page

f (x) 13

147

Fc (ξ)

xe−ax ,

Re a > 0

ET I 15(7) ξ>0      2 ξ 1/2 2 −ν/2 −1 , (2/π) Γ(ν) a + ξ cos ν tan a

xν−1 e−ax , Re a > 0,

15

  −2 (2/π)1/2 a2 − ξ 2 a2 + ξ 2 ,

x−1/2 e−ax ,

Re ν > a Re a > 0

ξ>0 

a2 + ξ 2

−1/2 

a2 + ξ 2

1/2

+a

1/2

ET I 15(7)

,

ξ>0 167 17 18

19

e−a

2

x2

,

−1 −x

x

e



2

sin ax

Re a > 0

−1/2



2 ξ2

ξ>0 ,

ξ>0

  2  2  1 ξ ξ √ cos − sin , 2 a 4a 4a

,

a>0



a>0

a>0

217

sin2 (ax) , x2

a>0

227

e−bx sin(ax),

a > 0,

  1/2  sin b x2 + a2 (x2 x2 + a2

+

a2 )2

−1/2

Re b > 0

tan

1 √ cos 2 a



 2

ξ 4a

 + sin

ξ>0  ξ2 , 4a

  1/2  , sin b x2 + a2 a>0

ET I 19(7)

ET I 23(1)

ξ>0 ET I 24(7) ⎧ 1/2 ⎪ ξa   1/2  a − 12 ξ ξ < 2a (π/2) ET I 19(8) 0 2a < ξ  a+ξ a−ξ , (2π)−1/2 2 + b + (a + ξ)2 b2 + (a − ξ)2 ξ>0

a>0

ET I 15(11)





  cos ax2 ,



−1

,

(2π)

sin(ax) , x

24

e

sin x

20

23

−1 −ξ 2 /4a2

2−1/2 |a|

ET I 14(4)

ET I 19(6)

(b/a) (π/2)1/2 e−aξ , ξ>0 ET I 26(29)    1/2  0 |Im b|

sinh(ax) sinh(bx)

|Re a| < Re b

Fc (ξ)  1/2 (π/2) (a − ξ) ξ < a ET I 20(16) 0 a0 ET I 23(5)  

 2 2 a +b 2 exp −aξ / 4 a + b2    −1 1 − 2 arctan(b/a) , × cos 14 bξ 2 a2 + b2 −1/2



π 1/2 2

 2 −1/4

2

ξ>0 sin(πa/b) , b [cosh (πξ/b) + cos(πa/b)] ξ>0

29

cosh(ax) , cosh(bx)

|Re a| < Re b

Re a > 0

sech(ax),

1/2

a−1 (π/2)

32 337

πx  , x2 + a2 sech 2a   a2 ln 1 + 2 , x   2 a + x2 , ln b2 + x2 

ET I 31(12)

sech (πξ/2a) , ξ>0

31

ET I 31(14)

(2π)1/2 cos(πa/2b) cosh (πξ/2b) , b [cosh (πξ/b) + cos(πa/b)] ξ>0

30

ET I 24(6)

ET I 30(1)

Re a > 0

2(2/π)1/2 a3 sech3 (aξ),

ξ>0

ET I 32(19)

Re a > 0

  (2π)1/2 ξ −1 1 − e−aξ ,

ξ>0

ET I 18(10)

  (2π)1/2 e−bξ − e−aξ ,

ξ>0

ET I 18(12)

a 0 34



x2 + b2

−1

1/2 −1 −bξ

J 0 (ax), a > 0,

(π/2) Re b > 0

b

e

I 0 (ab),

continued on next page

Mellin transform

1099

continued from previous page

f (x) 35

Fc (ξ)

 −1 x x2 + b2 J 0 (ax), a > 0,

(2/π)1/2 cosh(bξ) K 0 (ab), Re b > 0

0 k. Setting

1 2πi



c+i∞

c−i∞

f ∗ (s)x−s ds,

f ∗ (s) = M [f (x); s] to denote the Mellin transform, we have the symbolic expression for the inverse result f (x) = M−1 [f ∗ (s); x] .

MS 397(6)

12.42 Basic properties of the Mellin transform 1.

For a and b arbitrary constants, M [af (x) + bg(x)] = af ∗ (s) + bg ∗ (s)

2.

If lim xs−r−1 f (r) (x) = 0,

(ii)

3.

r = 0, 1, . . . , n − 1,

x→0

(i)

(linearity)

  M f (n) (x); s = (−1)n

Γ(s) f ∗ (s − n) Γ(s − n) (transform of a derivative)

SU 267 (4.2.3)

(transform of a derivative)

SU 267 (4.2.5)

  Γ(s + n) ∗ f (s) M xn f (n) (x); s = (−1)n Γ(s)

Denoting the nth repeated integral of f (x) by In [f (x)], where  x In [f (x)] = In−1 [f (u)] du, 0

(i)

M [In [f (x)] ; s] = (−1)n

Γ(s) f ∗ (s + n) Γ(n + s) (transform of an integral)

(ii)

M [In∞ [f (x)] ; s] =

Γ(s) f ∗ (s + n), Γ(s + n)

where In∞ [f (x)] =

4.

M [f (x)g(x); s] =

1 2πi



SU 269 (4.2.15)



∞ x

c+i∞

c−i∞

∞ In−1 f (u) du

(transform of an integral)

SU 269 (4.2.18)

f ∗ (u)g ∗ (s − u) du (Mellin convolution theorem)

SU 275(4.4.1)

Table of Mellin transforms

1101

12.43 Table of Mellin transforms f ∗ (s)

f (x) 1

e−x

2

e−x

3

cos x

Γ(s) cos

1



4

sin x

Γ(s) sin

1



5

1 1−x

π cot(πs),

0 < Re s < 1

SU 521(M1)

6

1 1+x

π cosec(πs),

0 < Re s < 1

SU 521(M2)

7

(1 + xa )−b

Γ(s/a) Γ(b − s/a) , a Γ(b)

Re s > 0

SU 521(M13)

Re s > 0

SU 521(M14)

,

0 < Re s < 1

SU 521(M15)

,

0 < Re s < 1

SU 521(M16)

Γ(s), 2

1 2

Γ

1  2s , 2 πs

2 πs

0 < Re s < ab SU 521(M3)

8

9

10

11

T n (x) H(1 − x)  (1 − x2 )

Tn

Γ



 x−1 H(1 − x)  (1 − x2 )

P n (x) H(1 − x)

1 2

+

1 2s

2−s π Γ(s)   , + 12 n Γ 12 + 12 s − 12 n

Re s > 0     2s−2 Γ 12 n + 12 s Γ 12 s − 12 n , Γ(s)

SU 521(M4)

Re s > n  1  1 Γ 2 s Γ 2 s + 12    , 2 Γ 12 s − 12 n + 12 Γ 12 s + 12 n + 1

SU 521(M5)

Re s > 0    2s−1 Γ 2 s + 12 n + 12 Γ 12 s − 12 n √ , π Γ(s + 1)

SU 521(M6)

1

  P n x−1 H(1 − x)

Re s > n 12

1 + x cos φ 1 − 2x cos φ + x2

13

x sin φ , 1 − 2x cos φ + x2

−π < φ < π

SU 521(M7)

π cos(sφ) , sin(sπ)

0 < Re s < 1

SU 521(M11)

π sin(sφ) , sin(sπ)

0 < Re s < 1

SU 521(M12)

continued on next page

1102

Integral Transforms

continued from previous page

f ∗ (s)

f (x) 14

e−x cos φ cos (x sin φ) , 1 2π

15

− 12 π < φ < 12 π 16

x

17

Y ν (x),

ν>

J ν (x),

Re s > 0

SU 522(M17)

Γ(s) sin(sφ),

Re s > −1

SU 522(M18)

< φ < 12 π

e−x sin φ sin (x sin pφ) ,

−ν

Γ(s) cos(sφ),

− 21

ν∈R

  2s−ν−1 Γ 12 s   , 0 < Re s < 1 Γ ν − 12 s + 1     −2s−1 π −1 Γ 12 s + 12 ν Γ 12 s − 12 ν   × cos 12 s − 12 ν π, |ν| < Re s
ν > 0     2s−1 tan 12 πs + 12 πν Γ 12 s + 12 ν   , Γ 12 ν − 12 s + 1   −1 − ν < Re s < min 32 , 1 − ν

2s

πn−1 cosec

πs n

24

arctan x

SU 522(M22)

0 < Re s < n

MS 453

h−1 a−s/h B (s/h, ν − (s/h))

,

h > 0, ln(1 + ax),

SU 522(M21)

a(s/n)−1 ,

n = 1, 2, 3, . . . ,

h > 0, |arg a| < π  ν−1 for 0 < x < 1 1 − xh , 0 for x > 1

23

SU 522(M20)

   + 12 ν Γ 12 s − 12 ν ,

ν∈R

0 < Re s < h Re ν

 22

3 2

SU 522(M19)

h−1 B (ν, s/h)

MS 454 MS 454

Re ν > 0 |arg a| < π

πs−1 a−s cosec(πs),

−1 < Re s < 0

MS 454

− 21 πs−1 sec(πs/2),

−1 < Re s < 0

MS 454

continued on next page

Table of Mellin transforms

1103

continued from previous page

f ∗ (s)

f (x) −1 1 2 πs

25

arccot x

26

cosech(ax)

Re a > 0

  a−s 2 1 − 2−s Γ(s) ζ(s),

27

sech2 (ax),

Re a > 0

4a−s (1 − 22−s ) Γ(s)2−s ζ(s − 1),

28 2911

30

31

32

cosech2 (ax), 

x2 + b2

− 12 ν

Re a > 0   1/2  J ν a x2 + b2

  ⎧ 1   2 2 2ν 2 2 1/2 ⎪ a a b − x J − x ⎪ ν ⎨ for 0 < x < a ⎪ ⎪ for x > a ⎩0

Re s > 1

4a−s Γ(s)2−s ζ(s − 1), 1

1

1

2 2 s−1 a− 2 s b 2 s−ν Γ

1

MS 454

Re s > 2

MS 454

Re s > 2

MS 454

1  2 s J ν−s/2 (ab),

0 < Re s < 2 2 s−1 Γ

MS 454

3 2

+ Re ν

ET I 328

 1  − 1 s ν+ 1 s 2 a ,2 J ν+ 1 s (ab), 2s b 2 Re s > 0

MS 455

Re ν > −1   ⎧ 1    − ν 2 2 2 2 1/2 2 ⎪ a b a − x J − x ⎪ ν ⎨ 1 −1 1 for 0 < x < a 21−ν [Γ(ν)] a 2 s−ν b−, 2 ν s ν−1+ 12 s, 12 s−ν (ab), ⎪ ⎪ for x > a ⎩0 MS 455 Re s > 0

α−s 2s−2 Γ

K ν (αx) 

33

0 < Re s < 1

sec(πs/2),

− 1 ν βa2 + x2 2  1/2  × K ν α βa2 + x2 Re (α, β) > 0

1

2s

   − 12 ν Γ 12 s + 12 ν , Re s > |Re ν|

  1 1 1 α− 2 s 2 2 s−1 β 2 s−ν Γ, 12 s K

MS 455

1 (αβ), ν− 2 s

Re s > 0

MS 455

Bibliographic References (See the introduction for an explanation of the letters preceding each bibliographic reference.) AS AD AK BB BE BEA

BI BL BR BS BU

BY CA CE CL CO

Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions, Dover Publications, New York, 1972. Adams, E. P. and Hippisley, R. L., Smithsonian Mathematical Formulae and Tables of Elliptic Functions, Smithsonian Institute, Washington, D.C., 1922. Appell, P. and Kamp´e de F´eriet, Fonctions hyperg´eometriques et hypersph´eriques, polynomes d’Hermite, Gauthier Villars, Paris, 1926. Beckenbach, E. F. and Bellman, R., Inequalities, 3rd printing. Springer–Verlag, Berlin, 1971. Bertrand, J., Traite de calcul diff´erentiel et de calcul int´egral, vol. 2, Calcul int´egral, int´egrales d´efinies et ind´efinies, Gauthier-Villars, Paris, 1870. Beaulieu, N. C., A Useful Integral for Wireless Communication Theory and Its Application to Rectangular Signaling Constellation Error Rates, IEEE Trans. Commun., 54(5), pages 802–805, 2006. Bierens de Haan, D., Nouvelles tables d’int´egrales d´efinies, Amsterdam, 1867. (Reprint) G. E. Stechert & Co., New York, 1939. Bellman, R., Introduction to Matrix Analysis, McGraw Hill, New York, 1960. Bromwich, T. I’A., An Introduction to the Theory of Infinite Series, Macmillan, London, 1908, 2nd edition, 1926.∗ Bellman, R., Stability Theory of Differential Equations, McGraw-Hill, New York, 1953. Buchholz, H., Die konfluente hypergeometrische Funktion mit besonderer Ber¨ ucksichtigung ihrer Anwendungen, Springer–Verlag, Berlin, 1953. Also an English edition: The confluent Hypergeometric Function, Springer–Verlag, Berlin, 1969. Byrd, P. F. and Friedman, M. D., Handbook of Elliptic Integrals for Engineers and Physicists, Springer–Verlag, Berlin, 1954. Carslaw, H. S., Introduction to the Theory of Fourier’s Series and Integrals, Macmillan, London, 1930. Ces` aro, Z., Elementary Class Book of Algebraic Analysis and the Calculation of Infinite Limits, 1st ed. ONTI, Moscow and Leningrad, 1936. Coddington, E. A. and Levinson, N., Theory of Ordinary Differential Equations, McGraw Hill, New York, 1955. Courant, R. and Hilbert, D., Methods of Mathematical Physics, vol. I, Wiley (Interscience), New York, 1953.

∗ The Bibliographic Reference BR* refers to the 1908 edition of Bromwich T. I.’A., An Introduction to the Theory of Infinite Series; BR refers to the 1926 edition.

1105

1106

Bibliographic References

DLMF Digital Library of Mathematical Functions, 2011-08-29, National Institute of Standards and Technology, http://dlmf.nist.gov/ Dwight, H. B., Tables of Integrals and Other Mathematical Data, Macmillan, New York, 1934. DW DW61 Dwight, H. B., Tables of Integrals and Other Mathematical Data, Macmillan, New York, 1961. Efros, A. M. and Danilevskiy, A. M., Operatsionnoye ischisleniye i konturnyye integraly EF (Operational calculus and contour integrals). GNTIU, Khar’kov, 1937. Erd´elyi, A., et al., Higher Transcendental Functions, vols. I, II, and III. McGraw Hill, EH New York, 1953–1955. Erd´elyi, A. et al., Tables of Integral Transforms, vols. I and II. McGraw Hill, New York, 1954. ET Euler, L., Introductio in Analysin Infinitorum, Bousquet, Lausanne, 1748. EU Fikhtengol’ts, G. M., Kurs differentsial’nogo i integral’nogo ischisleniya (Course in differential FI and integral calculus), vols. I, II, and III. Gostekhizdat, Moscow and Leningrad, 1947–1949. Also a German edition: Differential-und Integralrechnung I–III, VEB Deutscher Verlag der Wissenschaften, Berlin, 1986–1987. Gauss, K. F., Werke, Bd. III. G¨ottingen, 1876. GA Gonczarek and Czerwonko, Bulletin of the Polish Academy of Sciences, vol. 48, no. 4, 2000. GC Gel’fond, A. O., Ischisleniye konechnykh raznostey (Calculus of finite differences), part I. ONTI, GE Moscow and Leningrad, 1936. obner, W. and Hofreiter, N., Integraltafel, vol. 2, Bestimmte Integrale, Springer, Wien, 1961. GH2 Gr¨ Giunter, N. M. and Kuz’min, R. O. (eds.), Sbornik zadach po vysshey matematike (Collection of GI problems in higher mathematics), vols. I, II, and III. Gostekhizdat, Moscow and Leningrad, 1947. Gantmacher, F. R., Applications of the Theory of Matrices, translation by J. L. Brenner. Wiley GM (Interscience), New York, 1959. Goursat, E. J. B., Cours d’Analyse, vol. I, Gauthier–Villars, Paris, 1923. GO Guillera, J. and Sondow, J., Double Integrals and Infinite Products for some Classical Constants GS via Analytic Continuation of Lerch’s Transcendent, arXiv:math/0506319 v2, 2005. Gr¨ obner, W. et al., Integraltafel, Teil I, Unbestimmte Integrale, Akad. Verlag, Braunschweig, GU 1944. Gr¨ obner, W. and Hofreiter, N., Integraltafel, Teil II, Bestimmte Integrale, Springer–Verlag, Wien GW and Innsbruck, 1958. Hille, E., Lectures on Ordinary Differential Equations, Addison- Wesley, Reading, HI Massachusetts, 1969. Hardy, G. H., Littlewood, J. E., and Polya, G., Inequalities, Cambridge University Press, HL London, 2nd ed., 1952. Hobson, E. W., The Theory of Spherical and Ellipsoidal Harmonics, Cambridge University HO Press, London, 1931. Hurewicz, W., Lectures on Ordinary Differential Equations, MIT Press, Cambridge, HU Massachusetts, 1958. Ince, E. L., Ordinary Differential Equations, Dover, New York, 1944. IN Jahnke, E. and Emde, F., Tables of Functions with Formulas and Curves, Dover, New York, JA 1943. Jackson, J. D., Classical Electrodynamics, Wiley, New York, 1975. JAC James, H. M. et al. (eds.), Theory of Servomechanisms, McGraw Hill, New York, 1947. JE Jolley, L., Summation of Series, Chapman and Hall, London, 1925. JO

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Kellogg, O. D., Foundations of Potential Theory, Dover, New York, 1958. Krajcik, R. A. and McLenitham K. D., Integrals and Series Related to the Surface Area of Arbitrary Ellipsoids, Los Alamos LA-UR-04-4398, http://arxiv.org/abs/math/0605216, 2004. Krechmar, V. A., Zadachnik po algebre (Problem book in algebra), 2nd ed. Gostekhizdat, KR Moscow and Leningrad, 1950. Kuzmin, R. O., Besselevy funktsii (Bessel functions). ONTI, Moscow and Leningrad, 1935. KU Laska, W., Sammlung von Formeln der reinen und angewandten Mathematik, Friedrich Viewig LA und Sohn, Braunschweig, 1888–1894. Legendre, A. M., Exercises calcul int´egral, Paris, 1811. LE Lei, X., Fan, P. and Chen, Q., Exact Symbol Error Probability of General Order Rectangular LEI QAM with MRC Diversity Reception over Nakagami-m Fading Channels, IEEE Commun. Lett., 11(12), pages 958–960, 2007. Lindeman, C. E., Examen des nouvelles tables d’int´egrales d´efinies de M. Bierens de Haan, LI Amsterdam, 1867, Norstedt, Stockholm, 1891. Lobachevskiy, N. I., Poloye sobraniye sochineniy (Complete works), vols. I, III, and LO V. Gostekhizdat, Moscow and Leningrad, 1946–1951. LUKE Luke, Y. L., Mathematical Functions and their Approximations, Academic Press, New York, 1975. Lawden, D. F., Elliptic Functions and Applications, Springer–Verlag, Berlin, 1989. LW McLachlan, N. W., Theory and Application of Mathieu Functions, Oxford University Press, MA London, 1947. Computation by Mathematica. MC McLachlan, N. W. and Humbert, P., Formulaire pour le calcul symbolique, L’Acad. des Sciences ME de Paris, Fasc. 100, 1950. Morse, M. P. and Feshbach, H., Methods of Theoretical Physics, vol. I, McGraw Hill, New York, MF 1953. Marden, M., Geometry of Polynomials, American Mathematical Society, Mathematical Survey MG 3, Providence, Rhode Island, 1966. McLachlan, N. W. et al., Suppl´ement au formulaire pour le calcul symbolique, L’Acad. des MI Sciences de Paris, Fasc. 113, 1950. Mirsky L., An Introduction to Linear Algebra, Oxford University Press, London, 1963. ML MacMillan, W. D., The Theory of the Potential, Dover, New York, 1958. MM Magnus, W. and Oberhettinger, F., Formeln und S¨ atze f¨ ur die speziellen Funktionen der MO mathematischen Physik, Springer–Verlag, Berlin, 1948. Magnus, W., Oberhettinger, F. and Soni, R. P., Formulas and Theorems for the Special MS Functions of Mathematical Physics, 3rd ed. Springer–Verlag, Berlin, 1966. Mathai, A. M. and Saxens, R. K., Generalized Hypergeometrics Functions With Applications in MS2 Statistics and Physical Science, Springer–Verlag, Berlin, 1973. Mitrinovi´c, D. S., Analytic Inequalities, Springer–Verlag, Berlin, 1970. MT Milne, E. A., Vectorial Mechanics, Methuen, London, 1948. MV Meyer Zur Capellen, W., Integraltafeln, Sammlung unbestimmer Integrale elementarer MZ Funktionen, Springer–Verlag, Berlin, 1950. Natanson, I. P., Konstruktivnaya teoriya funktsiy (Constructive theory of functions). NA Gostekhizdat, Moscow and Leningrad, 1949. Nielsen, N., Handbuch der Theorie der Gammafunktion, Teubner, Leipzig, 1906. NH KE KM

1108

NO NT NV

OB PBM PE SA SI

SN SM ST STR SU SZ TF TI VA VL WA WH ZH ZY

Bibliographic References

Noble, B., Applied Linear Algebra, Prentice Hall, Englewood Cliffs, New Jersey, 1969. Nielsen, N., Theorie des Integrallogarithmus und verwandter Transcendenten, Teubner, Leipzig, 1906. Novoselov, S. I., Obratnyye trigonometricheskiye funktsii, posobive dlya uchiteley (Inverse trigonometric functions, textbook for students), 3rd ed. Uchpedgiz, Moscow and Leningrad, 1950. Oberhettinger, F., Tables of Bessel Transforms, Springer–Verlag, New York: 1972. Prudnikov, A. P., Brychkov, Yu. A., and Marichev, O. I., Integrals and Series, Gordan and Breach, New York, vols. I (1986), II (1986), III (1990). Peirce, B. O., A Short Table of Integrals, 3rd ed. Ginn, Boston, 1929. Sansone, G., Orthogonal Functions (Revised English Edition), Interscience, New York, 1959. Sikorskiy, Yu. S., Elementy teorii ellipticheskikh funktsiy s prilozheniyama k mekhanike (Elements of theory of elliptic functions with applications to mechanics). ONTI, Moscow and Leningrad, 1936. Sneddon, I. N., Fourier Transforms, 1st ed. McGraw Hill, New York, 1951. Smirnov, V. I., Kurs vysshey matematiki (A course of higher mathematics), vol. III, Part 2, 4th ed. Gostekhizdat, Moscow and Leningrad, 1949. Strutt, M. J. O., Lam´esche, Mathieusche und verwandte Funktionen in Physik and Technik, Springer–Verlag, Berlin, 1932. Stratton, J. C., Phys. Rev A, 43(3), pages 1381–1388, 1991. Sneddon, I. N., The Use of Integral Transforms, McGraw Hill, New York, 1972. Szeg¨ o, G., Orthogonal Polynomials, Revised Edition, Colloquium Publications XXIII, American Mathematical Society, New York, 1959. Titchmarsh, E. C., Introduction to the Theory of Fourier Integrals, 2nd ed. Oxford University Press, London, 1948. Timofeyev, A. F. Integrirovaniye funktsiy (Integration of functions), part I. GTTI, Moscow and Leningrad, 1933. Varga, R. S., Matrix Iterative Analysis, Prentice Hall, Englewood Cliffs, New Jersey, 1963. Vladimirov, V. S., Equations of Mathematical Physics, Dekker, New York, 1971. Watson, G. N., A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge University Press, London, 1966. Whittaker, E. T. and Watson, G. N., Modern Analysis, 4th ed. Cambridge University Press, London, 1927, part II, 1934. Zhuravskiy, A. M., Spravochnik po ellipticheskim funktsiyam (Reference book on elliptic functions). Izd. Akad. Nauk. U.S.S.R., Moscow and Leningrad, 1941. Zygmund, A., Trigonometrical Series, 2nd ed. Chelsea, New York, 1952.

Supplementary References (Prepared by Alan Jeffrey for the English language edition.)

General reference books 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

Bromwich, T. I’A., An Introduction to the Theory of Infinite Series, 2nd ed., Macmillan, London, 1926 (Reprinted 1942). Carlitz, L., “Generating Functions”, 1969, Fibonacci Quarterly, 7 (4): 359–393. Copson, E. T., An Introduction to the Theory of Functions of a Complex Variable, Oxford University Press, London, 1935. Courant, R. and Hilbert, D., Methods of Mathematical Physics, vol. I, Interscience Publishers, New York, 1953. Davis, H. T., Summation of Series, Trinity University Press, San Antonio, Texas, 1962. Erd´elyi, A. et al. Higher Transcendental Functions, vols. I to III, McGraw Hill, New York 1953–1955. Erd´elyi, A. et al., Tables of Integral Transforms, vols. I and II. McGraw Hill, New York, 1954. Fletcher, A., Miller, J. C. P., and Rosenhead, L., An Index of Mathematical Tables, 2nd ed., Scientific Computing Service, London, 1962. Gr¨ obner, W. and Hofreiter, N., Integraltafel, I, II. Springer–Verlag, Wien and Innsbruck, 1949. Hardy, G. H., Littlewood, J. E., and P´ olya, G., Inequalities, 2nd ed., Cambridge University Press, London, 1952. Hartley, H. O. and Greenwood, J. A., Guide to Tables in Mathematical Statistics, Princeton University Press, Princeton, New Jersey, 1962. Jeffreys, H. and Jeffreys, B. S., Methods of Mathematical Physics, Cambridge University Press, London, 1956. Jolley, L. B. W., Summation of Series, Dover Publications, New York, 1962. Knopp, K., Theory and Application of Infinite Series, Blackie, London, 1946, Hafner, New York, 1948. Lebedev, N. N., Special Functions and their Applications, Prentice Hall, Englewood Cliffs, New Jersey, 1965. Magnus, W. and Oberhettinger, F., Formulas and Theorems for the Special Functions of Mathematical Physics, Chelsea, New York, 1949. McBride, E. B., Obtaining Generating Functions, Springer–Verlag, Berlin, 1971. National Bureau of Standards, Handbook of Mathematical Functions, U.S. Government Printing Office, Washington, D.C., 1964. Prudnikov, A. P., Brychkov, Yu. A., and Marichev, O. I., Integrals and Series, Vols. 1–5, Gordon and Breach, New York, 1986–1992. Truesdell, C. A Unified Theory of Special Functions, Princeton University Press, Princeton, New Jersey, 1948.

1109

1110

21. 22.

Supplementary References

Vein, R. and Dale, P., Determinants and Their Applications in Mathematical Physics, Springer–Verlag, New York, 1999. Whittaker, E. T. and Watson, G. N., A Course of Modern Analysis, 4th ed., Cambridge University Press, London, 1940.

Asymptotic expansions 1. 2. 3. 4. 5. 6. 7.

De Bruijn, N. G., Asymptotic Methods in Analysis, North-Holland Publishing Co., Amsterdam, 1958. Cesari, L., Asymptotic Behavior and Stability Problems in Ordinary Differential Equations, 3rd ed., Springer, New York, 1971. Copson, E. T., Asymptotic Expansions, Cambridge University Press, London, 1965. Erd´elyi, A., Asymptotic Expansions, Dover Publications, New York, 1956. Ford, W. B., Studies on Divergent Series and Summability, Macmillan, New York, 1916. Hardy, G. H., Divergent Series, Clarendon Press, Oxford, 1949. Watson, G. N., A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, London, 1958.

Bessel functions 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Bickley, W. G., Bessel Functions and Formulae, Cambridge University Press, London, 1953. Erd´elyi, A. et al., Higher Transcendental Functions, vols. I and II. McGraw Hill, New York, 1954. Erd´elyi, A. et al., Tables of Integral Transforms, vols. I and II. McGraw Hill, New York, 1954. Gray, A., Mathews, G. B. and MacRobert, T. M., A Treatise on Bessel Functions and Their Applications to Physics, 2nd ed., Macmillan, 1922. McLachlan, N. W., Bessel Functions for Engineers, 2nd ed., Oxford University Press, London, 1955. Luke, Y. L., Integrals of Bessel Functions, McGraw Hill, New York, 1962. Petiau, G., La th´eorie des fonctions de Bessel, Centre National de la Recherche Scientifique, Paris, 1955. Relton, F. E., Applied Bessel Functions, Blackie, London, 1946. Watson, G. N., A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, London, 1958. Wheelon, A. D., Tables of Summable Series and Integrals Involving Bessel Functions, Holden-Day, San Francisco, 1968.

Complex analysis 1. 2. 3. 4. 5. 6. 7.

Ahlfors, L. V., Complex Analysis, 3rd ed., McGraw Hill, New York, 1979. Ahlfors, L. V. and Sario, L., Riemann Surfaces, Princeton University Press, Princeton, New Jersey, 1971. Bieberbach, L., Conformal Mapping, Chelsea, New York, 1964. Henrici, P., Applied and Computational Complex Analysis, 3 vols, Wiley, New York, 1988, 1991, 1977. Hille, E., Analytic Function Theory, 2 vols. 2nd ed., Chelsea, New York, 1990, 1987. Kober, H., Dictionary of Conformal Representations, Dover Publications, New York, 1952. Titchmarsh, E. C., The Theory of Functions, 2nd ed., Oxford University Press, London, 1939. (Reprinted 1975).

Supplementary References

1111

Error function and Fresnel integrals 1. 2. 3. 4. 5.

Erd´elyi, A. et al., Higher Transcendental Functions, vol. II, McGraw Hill, New York, 1953. Erd´elyi, A. et al., Tables of Integral Transforms, vol. I, McGraw Hill, New York, 1954. Slater, L. J., Confluent Hypergeometric Functions, Cambridge University Press, London, 1960. Tricomi, F. G., Funzioni ipergeometriche confluenti, Edizioni Cremonese, Turan, Italy, 1954. Watson, G. N., A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, London, 1958.

Exponential integrals, gamma function and related functions 1. 2. 3. 4. 5. 6. 7. 8. 9.

Artin, E., The Gamma Function, Holt, Rinehart, and Winston, New York, 1964. Busbridge, I. W., The Mathematics of Radiative Transfer, Cambridge University Press, London, 1960. Erd´elyi, A. et al., Higher Transcendental Functions, vol. II, McGraw Hill, New York, 1953. Erd´elyi, A. et al., Tables of Integral Transforms, vols. I and II, McGraw Hill, New York, 1954. Hastings, Jr., C., Approximations for Digital Computers, Princeton University Press, Princeton, New Jersey, 1955. Kourganoff, V., Basic Methods in Transfer Problems, Oxford University Press, London, 1952. L¨ osch, F. and Schoblik, F., Die Fakult¨ at (Gammafunktion) und verwandte Funktionen, Teubner, Leipzig, 1951. Nielsen, N., Handbuch der Theorie der Gammafunktion, Teubner, Leipzig, 1906. Oberhettinger, F., Tabellen zur Fourier Transformation, Springer–Verlag, Berlin, 1957.

Hypergeometric and confluent hypergeometric functions 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

Appell, P., Sur les Fonctions Hyperg´eometriques de Plusieures Variables, Gauthier-Villars, Paris, 1926. Bailey, W. N., Generalized Hypergeometric Functions, Cambridge University Press, London, 1935. Buchholz, H., Die konfluente hypergeometrische Funktion, Springer–Verlag, Berlin, 1953. Erd´elyi, A. et al., Higher Transcendental Functions, vol. I, McGraw Hill, New York, 1953. Jeffreys, H. and Jeffreys, B. S., Methods of Mathematical Physics, Cambridge University Press, London, 1956. Klein, F., Vorlesungen u ¨ber die hypergeometrische Funktion, Springer–Verlag, Berlin, 1933. N¨ orlund, N. E., Sur les Fonctions Hyperg´eometriques d’Ordre Superior, North–Holland, Copenhagen, 1956. Slater, L. J., Confluent Hypergeometric Functions, Cambridge University Press, London, 1960. Slater, L. J. Generalized Hypergeometric Functions, Cambridge University Press, London, 1966. Snow, C., The Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory, 2nd ed., National Bureau of Standards, Washington, D.C., 1952. Swanson, C. A. and Erd´elyi, A., Asymptotic Forms of Confluent Hypergeometric Functions, Memoir 25, American Mathematical Society, Providence, Rhode Island, 1957. Tricomi, F. G., Lezioni sulla funzioni ipergeometriche confluenti, Gheroni, Torino, 1952.

Integral transforms 1.

Bochner, S., Vorlesungen u ¨ber Fouriersche Integrale, Akad. Verlag, Leipzig, 1932. Reprint Chelsea, New York, 1948.

1112

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

Supplementary References

Bochner, S. and Chandrasekharan, K., Fourier Transforms, Princeton University Press, Princeton, New Jersey, 1949. Campbell, G. and Foster, R., Fourier Integrals for Practical Applications, Van Nostrand, New York, 1948. Carslaw, H. S. and Jaeger, J. C., Conduction of Heat in Solids, Oxford University Press, London, 1948. Doetsch, G., Theorie und Anwendung der Laplace-Transformation, Springer–Verlag, Berlin, 1937. (Reprinted by Dover Publications, New York, 1943) Doetsch, G., Theory and Application of the Laplace-Transform, Chelsea, New York, 1965. Doetsch, G., Handbuch der Physik, Mathematische Methoden II, 1st ed., Springer–Verlag, Berlin, 1955. Doetsch, G., Guide to the Applications of the Laplace and Z-Transforms, 2nd ed., Van Nostrand-Reinhold, London, 1971. Doetsch, G., Handbuch der Laplace-Transformation, Vols. I–IV, Birkh¨ auser Verlag, Basel, 1950–56. Doetsch, G., Kniess, H., and Voelker, D., Tabellen zur Laplace- Transformation, Springer–Verlag, Berlin, 1947. Erd´elyi, A., Operational Calculus and Generalized Functions, Holt, Rinehart and Winston, New York, 1962. Exton, H., Multiple Hypergeometric Functions and Applications, Horwood, Chichester, 1976. Exton, H., Handbook of Hypergeometric Integrals: Theory, Applications, Tables, Computer Programs, Horwood, Chichester, 1978. Hirschmann, J. J. and Widder, D. V., The Convolution Transformation, Princeton University Press, Princeton, New Jersey, 1955. Marichev, O. I., Handbook of Integral Transforms of Higher Transcendental Functions, Theory and Algorithmic Tables, Ellis Horwood Ltd., Chichester (1982). Oberhettinger, F., Tabellen zur Fourier Transformation, Springer–Verlag, Berlin (1957). Oberhettinger, F., Tables of Bessel Transforms, Springer–Verlag, New York (1972). Oberhettinger, F., Fourier Expansions: A Collection of Formulas, Academic Press, New York, 1973. Oberhettinger, F., Fourier Transforms of Distributions and Their Inverses, Academic Press, New York, 1973. Oberhettinger, F., Tables of Mellin Transforms, Springer–Verlag, Berlin, 1974. Oberhettinger, F. and Badii, L., Tables of Laplace Transforms, Springer–Verlag, Berlin, 1973. Oberhettinger, F. and Higgins, T. P., Tables of Lebedev, Mehler and Generalized Mehler Transforms, Math. Note No. 246, Boeing Scientific Research Laboratories, Seattle, Wash., 1961. Roberts, G. E. and Kaufman, H., Table of Laplace Transforms, McAinsh, Toronto, 1966. Sneddon, I. N., Fourier Transforms, McGraw Hill, New York, 1951. Titchmarsh, E. C., Introduction to the Theory of Fourier Integrals, Oxford University Press, London, 1937. Van der Pol, B. and Bremmer, H., Operational Calculus Based on the Two Sided Laplace Transformation, Cambridge University Press, London, 1950. Widder, D. V., The Laplace Transform, Princeton University Press, Princeton, New Jersey, 1941. Wiener, N., The Fourier Integral and Certain of its Applications, Dover Publications, New York, 1951.

Jacobian and Weierstrass elliptic functions and related functions 1. 2. 3. 4.

Erd´elyi, A. et al., Higher Transcendental Functions, vol. II, McGraw Hill, New York, 1953. Byrd, P. F. and Friedman, M. D., Handbook of Elliptic Integrals for Engineers and Physicists, Springer– Verlag, Berlin, 1954. Graeser, E., Einf¨ uhrung in die Theorie der Elliptischen Funktionen und deren Anwendungen, Oldenbourg, Munich, 1950. Hancock, H., Lectures on the Theory of Elliptic Functions, vol. I, Dover Publications, New York, 1958.

Supplementary References

5. 6. 7. 8. 9.

1113

Neville, E. H., Jacobian Elliptic Functions, Oxford University Press, London, 1944 (2nd ed. 1951). Oberhettinger, F. and Magnus, W., Anwendungen der Elliptischen Funktionen in Physik und Technik, Springer–Verlag, Berlin, 1949. Roberts, W. R. W., Elliptic and Hyperelliptic Integrals and Allied Theory, Cambridge University Press, London, 1938. Tannery, J. and Molk, J., El´ements de la Th´eorie des Fonctions Elliptiques, 4 volumes. Gauthier-Villars, Paris, 1893–1902. Tricomi, F. G., Elliptische Funktionen, Akad. Verlag, Leipzig, 1948.

Legendre and related functions 1. 2. 3. 4. 5. 6. 7.

Erd´elyi, A. et al., Higher Transcendental Functions, vol. I, McGraw Hill, New York, 1953. Helfenstein, H., Ueber eine Spezielle Lam´esche Differentialgleichung, Brunner and Bodmer, Zurich, 1950 (Bibliography). Hobson, E. W., The Theory of Spherical and Ellipsoidal Harmonics, Cambridge University Press, London, 1931. Reprinted by Chelsea, New York, 1955. Lense, J., Kugelfunktionen, Geest and Portig, Leipzig, 1950. MacRobert, T. M., Spherical Harmonics: An Elementary Treatise on Harmonic Functions with Applications, Methuen, England, 1927. (Revised ed. 1947; reprinted Dover Publications, New York, 1948). Snow, C., The Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory, 2nd ed., National Bureau of Standards, Washington, D.C., 1952. Stratton, J. A., Morse, P. M., Chu, L. J. and Hunter, R. A., Elliptic Cylinder and Spheroidal Wave Functions Including Tables of Separation Constants and Coefficients, Wiley, New York, 1941.

Mathieu functions 1. 2. 3. 4.

Erd´elyi, A., Higher Transcendental Functions, vol. III, McGraw Hill, New York, 1955. McLachlan, N. W., Theory and Application of Mathieu Functions, Oxford University Press, London, 1947. Meixner, J. and Sch¨ afke, F. W., Mathieusche Funktionen und Sph¨ aroidfunktionen mit Anwendungen auf Physikalische und Technische Probleme, Springer–Verlag, Heidelberg, 1954. Strutt, M. J. O., Lam´esche, Mathieusche und verwandte Funktionen in Physik und Technik, Ergeb. Math. Grenzgeb. 1, 199–323 (1932). Reprint Edwards Bros., Ann Arbor, Michigan, 1944.

Orthogonal polynomials and functions 1. 2. 3. 4. 5. 6. 7.

Bibliography on Orthogonal Polynomials, Bulletin of National Research Council No. 103, Washington, D.C., 1940. Courant, R. and Hilbert, D., Methods of Mathematical Physics, vol. I, Interscience, New York, 1953. Erd´elyi, A. et al., Higher Transcendental Functions, vol. II, McGraw Hill, New York, 1954. Kaczmarz, St. and Steinhaus, H., Theorie der Orthogonalreihen, Chelsea, New York, 1951. Lorentz, G. G., Bernstein Polynomials, University of Toronto Press, Toronto, 1953. Sansone, G., Orthogonal Functions, Interscience, New York, 1959. Shohat, J. A. and Tamarkin, J. D., The Problem of Moments, American Mathematical Society, Providence, Rhode Island, 1943.

1114

8. 9. 10.

Supplementary References

Szeg¨ o, G., Orthogonal Polynomials, American Mathematical Society Colloquim Pub. No. 23, Providence, Rhode Island, 1959. Titchmarsh, E. C., Eigenfunction Expansions Associated with Second Order Differential Equations, Oxford University Press, London, part I (1946), part II (1958). Tricomi, F. G., Vorlesungen u ¨ber Orthogonalreihen, Springer–Verlag, Berlin, 1955.

Parabolic cylinder functions 1. 2.

Buchholz, H., Die konfluente hypergeometrische Funktion, Springer–Verlag, Berlin, 1953. Erd´elyi, A. et al., Higher Transcendental Functions, vol. II, McGraw Hill, New York, 1954.

Probability function 1. 2. 3.

Cramer, H., Mathematical Methods of Statistics, Princeton University Press, Princeton, New Jersey, 1951. Erd´elyi, A. et al., Higher Transcendental Functions, vols. I, II, and III. McGraw Hill, New York, 1953–1955. Kendall, M. G. and Stuart, A., The Advanced Theory of Statistics, vol. I: Distribution Theory, Griffin, London, 1958.

Riemann zeta function 1. 2.

Titchmarsh, E. C., The Zeta Function of Riemann, Cambridge University Press, London, 1930. Titchmarsh, E. C., The Theory of the Riemann Zeta Function, Oxford University Press, London, 1951.

Struve functions 1. 2. 3.

Erd´elyi, A. et al., Higher Transcendental Functions, vol. II, McGraw Hill, New York, 1954. Gray, A., Mathews, G. B. and MacRobert, T. M., A Treatise on Bessel Functions and Their Applications to Physics, 2nd ed., Macmillan, London, 1922. Watson, G. N., A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, London, 1958.

Index of Functions and Constants This index shows the occurrence of functions and constants used in the expressions within the text The numbers refer to pages on which the function or constant appears.

Symbols

arccot function . . . . xxix, xxx, 51, 55–57, 63, 64, 244, 246, 265, 282, 328, 503, 560, 565, 603, 605–609, 611, 629, 774, 901, 1085, 1103 arccoth function . . . . . . . . . . xxix, xxx, 56, 60, 61, 75, 131–135, 172, 177, 178, 242, 653, 655 arcosech function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 arcsec function . . . . . . . . . . . . 61, 66, 99, 243, 245, 246 arcsech function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 arcsin function . . . . . . . . . . . . xxix, xxx, 27, 55–61, 64, 66, 94, 97, 99, 116, 124–126, 133, 135–138, 173, 179–182, 186–193, 195, 196, 198, 202–213, 225, 242–247, 256, 265, 267, 275, 278, 282, 299, 310, 384, 561, 562, 567, 570, 571, 586, 591, 592, 594, 603–605, 607–610, 625, 626, 628, 629, 637, 638, 646, 670, 676, 707–709, 720, 724–726, 735, 736, 750, 751, 755, 762, 774, 775, 801, 823, 868, 923, 997, 1016, 1017 arcsinh function . . . . . . . . . . . . . . . . . . xxix, xxx, 53, 56, 60, 61, 64, 94, 97, 126, 133, 136, 139, 242, 373, 384, 388, 452, 591, 628, 646, 1016 arctan function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxix, xxx, 27, 30, 48, 49, 51, 52, 54–61, 63–67, 71–77, 79, 83–85, 87, 90, 97, 103, 104, 106, 114, 116, 117, 119, 126, 128–133, 148, 149, 171–175, 177, 178, 190, 205, 239–241, 243–247, 256, 265, 274, 277, 282, 296, 297, 310, 320, 326, 331, 348, 365, 374, 375, 383, 395, 411, 457, 497–505, 511, 513, 520–525, 527, 560, 564, 566, 568, 569, 597, 603, 605–611, 616, 626–628, 637, 638, 645, 647–649, 652, 654–657, 754, 770, 868, 893, 894, 899–902, 907, 932, 1016, 1046, 1083, 1095, 1098, 1102 arctanh function . . . . . . . . . . . . . . xxix, xxx, 56, 60, 61, 64, 75, 79, 97, 125–128, 131, 132, 134, 135, 172, 177, 178, 242, 625–627, 986

! and !! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . see factorials (m) Sn . . . . . . . . . . . . see Stirling numbers, second kind (m) Zn . . . . . . . . . . . . . . . . . . . . . . . . see Bessel functions, Z ∇ . . . . . . . . . . . . . . . . . . xlii, 774, 1062–1065, 1067–1069 β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . see beta function δ(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . see delta function δij . . . . . . . . . . . . . . . . . . . . . . . . . . . . . see Kronecker delta γ and Γ . . . . . . . . . . . . . . . . . . . . . . see gamma functions λ function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvii, 1054 μ function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvii, 1054 ν function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvii, 1054 Φ . . . . . . . . . . . see Lerch function and hypergeometric functions, confluent Ψ . . . . . . . . . . . see Euler function and hypergeometric functions, confluent Θ function . . . . . . . . . . . . . . . see Jacobi theta function ℘(x) . . . . . . . . . . . . . . . . . . . . . . see Weierstrass function ξ function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvii, 1051 || · || . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1090

A Airy function (Ai) . . . . . . . . . . . . . . . . . . . . xxxvi, xxxvii am function . . . . . . . . . . . . . . . . . . . xxxvii, 629, 875, 876 Anger function (J) . . xxxix, 341, 354, 373, 386, 425, 427, 448–450, 678, 679, 955, 957, 958, 1001 arccos function. . . . xxix, xxx, 24, 55–60, 64, 99, 135, 136, 139, 173, 179–183, 210, 211, 243–246, 265, 275, 282, 295–297, 299, 310, 314–316, 321, 395, 458, 515, 562, 565, 592, 603–605, 611, 613, 628, 703, 737, 749, 774, 775, 870, 899, 945, 1002 arccosec function . . . . . . . . . . . . . . . . . . . . . 243, 246, 736 arccosh function . . . . . . . . . . . . . . . . . . . . . . . . . xxix, xxx, 56, 60, 61, 64, 97, 126, 133, 135, 137, 138, 242, 384, 388, 515, 535, 625, 628, 736, 775

1115

1116

Index of Functions and Constants

associated Legendre functions first kind (P ) . . . . . xxxix, 329, 335, 336, 338, 408, 409, 490, 669, 670, 673, 674, 693, 694, 710, 712, 713, 735, 762, 767, 768, 774–796, 800, 804, 814–816, 818, 831, 839, 847, 848, 968–981, 983, 984, 989–992, 1001 second kind (Q) . . . xxxix, 335, 338, 376, 385, 515, 669, 670, 674, 693, 694, 707, 709, 710, 713, 735, 776–788, 790–792, 798, 802, 803, 839, 847, 848, 968–982, 990

376, 707, 799, 856, 409, 712, 831,

B Bn (x) . . . . . . . . . . . . . . . . . . . see Bernoulli polynomials B(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . see Beta function Bateman function (k) . . . . . . . . . . . . . xxxix, 351, 1032 bei(z) . . . . . . . . . . . . . . . . . . . . . . see Thomson functions ber(z) . . . . . . . . . . . . . . . . . . . . . . see Thomson functions Bernouli number (Bn ) . . . . . . . . . . . . . . . . . . . xxx, xxxi, xxxvii, 1–3, 9, 26, 42, 46, 55, 145–148, 221–224, 355, 358, 378, 381–384, 389, 475, 554, 556, 557, 563, 570, 578, 584, 590, 592, 595, 771–773, 908, 915, 945, 1049–1056 Bernouli number (Bn∗ ) . . . . . . . . . . . . . . . . . . . . . xxxi, 62 Bernoulli polynomial (Bn (x)). . . . .xxx, xxxi, xxxvii, 46, 1048, 1052, 1053, 1056 Bessel functions Hankel . . . . . . . . . . . . . . . . . . . . . . see Hankel function Zn (x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xl, 633, 634 In (x) . . . . . . . . xxxvi, xxxix, 14, 322, 341, 342, 347, 349, 352, 353, 370, 384, 387, 423, 439, 445, 448, 449, 474, 481, 483, 496, 498, 500, 511, 517–519, 527, 528, 599, 608, 621, 622, 668, 669, 671–690, 692–695, 697, 699, 700, 703–706, 709–722, 724, 726, 727, 729–732, 735–737, 742, 743, 746, 748, 750, 752–754, 758–767, 769, 786–788, 790–794, 796, 802, 805, 807, 828, 840, 841, 846, 854, 855, 910, 920, 925, 926, 928, 929, 934–942, 952, 963, 1011, 1036, 1037, 1086, 1087, 1093, 1095, 1098 Jn (x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxv, xxxvi, xxxix, 14, 341, 352, 354, 373, 386, 387, 421–425, 427, 439, 444–447, 449, 450, 481–484, 486, 487, 495, 496, 511, 518, 519, 525, 528, 581, 633, 651, 661, 662, 667–760, 763–766, 768–770, 774, 775, 784, 786, 787, 789–794, 800–802, 805– 807, 810, 816, 819, 820, 826–828, 838–846, 849, 853–856, 862, 863, 909, 919–923, 925, 927–940, 942–947, 949–959, 963–966, 972, 973, 981, 1001,

Bessel functions (continued) 1009, 1011–1013, 1027, 1032–1034, 1037, 1044, 1086, 1087, 1094–1099, 1102, 1103 Kn (x) . . . . . . . . . . . . . . . . . . . . . . xxxvi, xxxix, 3, 339, 341, 347, 349, 350, 352–354, 367, 369–373, 386, 387, 421, 423, 439, 445, 446, 448, 449, 481–484, 486, 487, 494, 495, 508, 509, 511, 518, 519, 533, 576, 578, 579, 581, 599, 647, 653, 656, 662, 665, 668–703, 705–707, 710–723, 725–727, 729–739, 742, 743, 745, 747, 749, 752–754, 756–760, 763– 766, 768, 775, 783–794, 796, 801, 807, 810, 819, 822, 825–828, 836, 840–842, 845, 846, 849, 853, 854, 856, 862, 863, 909, 920, 926–929, 932, 934– 942, 948, 951, 954, 964, 966, 1036, 1037, 1044, 1045, 1093–1096, 1099, 1102, 1103 Nn (x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxv, 919 Yn (x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxv, xl, 341, 347, 348, 352, 354, 355, 373, 386, 387, 423, 439, 444, 446, 447, 449, 450, 481–484, 486, 487, 496, 511, 518, 519, 576, 581, 655, 656, 662, 667–672, 674–690, 692–703, 705–707, 712–716, 718, 721–727, 729–731, 733–735, 737, 739–741, 743–745, 747–749, 752, 754–759, 763–766, 768, 774, 775, 784, 786, 789, 790, 792, 794, 801, 807, 825–827, 840, 841, 843–845, 855, 856, 862, 863, 919, 920, 923, 927–929, 931, 932, 934–940, 942, 946–948, 950–952, 954, 955, 958, 963–966, 984, 1034, 1044, 1094, 1096, 1102 Zn (x) . . xl, 633, 634, 774, 919, 920, 935, 940–942, 946, 949, 950, 984 beta function (β) . . . . . . 321, 324, 327, 336, 374, 377, 385, 397, 398, 405, 475, 557, 562, 566, 576, 590, 606, 913, 915, 916 Beta function (B) . . . . . . . . . . . . . . . xxxvii, 6, 129, 175, 318–323, 325–332, 335–337, 340, 349–351, 353, 361, 363, 366, 370, 372, 374, 376, 377, 384, 385, 397–399, 401–404, 409, 410, 415, 417, 444, 446, 448, 463, 473, 475, 489, 490, 494, 516, 543–545, 547, 551, 556, 562, 563, 588, 712, 756, 761, 767, 809, 818, 822, 824, 825, 829, 903, 904, 917–919, 1000, 1014, 1032, 1034 Bi function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvi binomial coefficients . . . . . . . . . . . . . . . . . . . . xxxi, xli, 1, 2, 4–6, 12, 13, 16, 23–25, 31, 33, 46, 84, 86, 88, 89, 100–103, 106, 110, 111, 114, 115, 119, 120, 140, 143, 144, 148, 153, 157, 161, 173, 215, 220, 221, 223, 228, 232–238, 242, 243, 318, 319, 328,

Index of Functions and Constants

binomial coefficients (continued) 331, 356, 359, 360, 363, 365, 388, 395, 396, 399– 404, 420, 435, 441, 463–465, 469, 470, 472–474, 482, 492, 502, 503, 508, 509, 549, 552, 553, 556, 616, 816, 919, 943, 1002, 1005, 1007, 1012, 1033, 1040, 1051, 1052, 1055, 1057, 1058

C Cn (x) . . . . . . . . . . . . . . . . . see Gegenbauer polynomials C(x) . . see Fresnel sine integral and Young function Catalan constant (G) . . . . xxx, xxxviii, 10, 377, 382, 437, 438, 452, 453, 456, 457, 474, 475, 534–541, 559, 561, 564, 567, 568, 583, 603, 605, 606, 639, 640, 1057 cd function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi Ce function . . . . . . . xxxvi, xxxviii, 770–773, 962, 963 ce function . . . . . . . . xxxvi, xxxviii, 770–774, 960–966 Chebyshev polynomials first kind (Tn (x)) . . . . . . . . . . xl, 452, 675, 725, 798, 807–810, 992, 997, 1002–1005, 1008, 1101 second kind (Un (x)) . . . . . . . . . . . xxxv, xl, 807–810, 1002–1005, 1008 chi function . . . . . . . xxxiv, xxxviii, 143, 144, 653, 895 ci function . . . . . . . . xxxiii, xxxviii, 219–221, 342–346, 427–430, 440, 441, 451, 499, 509, 510, 531, 532, 575, 576, 581, 584, 585, 597, 598, 603, 608, 632, 633, 647–653, 655, 665, 666, 755, 769, 895, 896, 1085 Cin function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiv cn function . . . . . . . xxxi, xxxii, xxxviii, 627–630, 721, 875–882, 884, 888, 889 complex conjugate . . . . . . xli, 295, 296, 343, 344, 425, 426, 428–430, 515, 531, 532, 936, 940, 942 confluent hypergeometric functions . . . . . . . . . . . . . see hypergeometric functions, confluent constants Catalan . . . . . . . . . . . . . . . . . . . . see Catalan constant Euler . . . . . . . . . . . . . . . . . . . . . . . . . see Euler constant cos function . . . . . . . . . . . . . . . . . xxvii, xxxiv, xxxvi, 4, 14, 20, 21, 26–53, 55, 56, 64, 74–76, 78, 79, 148, 151–237, 251, 252, 255, 256, 320–322, 325, 326, 329–335, 341–347, 355, 360, 374, 375, 379–385, 387, 390–412, 415–532, 534–538, 540, 544–546, 548, 549, 553–555, 564, 566, 568, 569, 571–576, 579, 582–604, 607–610, 612–616, 621, 625–627, 632, 633, 637–639, 643–645, 647–652, 655–658, 660, 663–667, 670–672, 675, 677, 679, 681–683, 685, 687–689, 693, 694, 696, 697, 699, 700, 702, 703, 710, 712–714, 716, 718, 720, 722, 724–755,

1117

cos function (continued) 757–759, 761, 763–777, 786–789, 791, 792, 794– 797, 800–802, 804–807, 810, 814, 816, 818–820, 823, 825, 826, 831, 833, 837–839, 841, 844, 845, 852, 853, 855, 857, 862, 871–878, 886–889, 891– 895, 897–903, 905, 907–909, 913, 915–919, 921– 927, 929, 931, 933, 934, 937, 939, 942–949, 951– 960, 962, 963, 968–973, 975–990, 993–996, 998– 1002, 1006, 1007, 1009, 1012, 1014–1017, 1032, 1034, 1035, 1038, 1040, 1047, 1048, 1052, 1053, 1055, 1065, 1069, 1074, 1075, 1080–1083, 1085, 1087, 1089–1092, 1094–1098, 1101, 1102 cosec function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv, xxvii, 36–39, 42–44, 49, 50, 64, 126, 155–157, 160, 225, 256, 318, 320–334, 336, 337, 341, 351, 353, 354, 356, 357, 360, 375, 383–385, 389–391, 402, 403, 405–409, 412, 415–418, 425, 426, 438, 441, 442, 449, 457, 458, 475, 483, 489, 500, 512, 513, 532, 544, 551, 554, 555, 569, 589, 591, 592, 597, 604, 606, 607, 666, 667, 671, 672, 677, 678, 685, 687, 688, 690, 697–700, 702, 703, 705, 707, 720, 722, 724, 725, 731, 732, 762, 767, 768, 787, 794, 795, 852, 861, 878, 909, 930, 936, 938, 939, 941, 973, 976, 1101, 1102 cosech function . . . . . . 27, 42, 113–115, 126, 389, 505, 517, 642, 722, 758, 1081, 1094, 1103 cosh function . . . . . . . . . . . . xxvi, xxxiv, 27–33, 35, 36, 38, 41–45, 47, 48, 50–52, 64, 110–151, 231–237, 253, 326, 340, 341, 374–392, 409, 423, 429, 436, 437, 442, 443, 452, 455, 456, 458, 459, 472, 485, 488, 489, 506, 508, 513–530, 572, 573, 576, 581– 584, 599, 609, 614, 615, 625, 626, 642, 643, 651, 653, 656, 693, 709, 712, 713, 717, 718, 720–724, 730, 731, 736, 742, 753, 754, 757–759, 762, 767, 770–774, 785, 788, 794–796, 798, 799, 810, 813, 851, 895, 897, 905, 916–918, 921–926, 930, 931, 961–967, 969–972, 976–978, 982, 986, 989–991, 1007, 1015, 1054, 1082–1085, 1089, 1098, 1099 cosine integral (Ci) . . . . . xxxiii, xxxiv, 219, 895, 939, 1085, 1092, 1096 cot function . . . . xxv, xxvi, 28, 36, 37, 39, 42, 44, 46, 49, 51, 56, 64, 148, 158–161, 168, 174, 176–178, 185, 188, 189, 194, 195, 204–207, 213, 222–225, 230, 277, 321–326, 333, 334, 336, 337, 356, 357, 360, 374, 381, 383–386, 390, 397, 398, 402, 403, 405–408, 415–418, 426, 438, 452, 458, 459, 475, 488, 489, 496, 497, 500, 510, 513, 515, 536–538, 543–546, 550, 562, 563, 569, 571, 572, 591, 592, 594, 598, 608, 610, 615, 627, 637, 638, 645, 671,

1118

Index of Functions and Constants

cot function (continued) 672, 677, 678, 684, 687, 688, 690, 698–700, 705, 707, 724, 725, 731, 760, 761, 857, 876, 877, 885, 891, 912–914, 921, 923, 924, 931, 936, 938, 939, 941, 963, 976–978, 980, 988, 996, 998, 1022, 1090 coth function . . . . . . . 28, 39, 40, 42, 44, 64, 110, 116, 118–120, 124, 130–134, 139, 146–148, 383, 386, 388–390, 489, 494, 506, 513–517, 523, 524, 526, 583, 585, 598, 626, 722–724, 885, 930, 931, 965, 966, 976, 990, 1038, 1039, 1080, 1094 cs function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi curl . . . . . . . . . . . . . . . . . . . . . . . . . 1062–1065, 1069, 1070 cylinder function . . . . see parabolic cylinder function

D dc function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265–267 delta function (δ(x)). . . . . . . . . . . . . . . . . . . . . 669, 1085, 1088–1090 dilogarithm (L2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650 div . . . . . . . . . . . . . . . . . . . . . . . . . 1062, 1063, 1065, 1067, 1069, 1070 dn function . . . . . . xxxi, xxxii, xxxviii, 627–630, 721, 875–882, 884, 888, 889 double factorials . . . . . . . . . . . . . . . see factorial, double ds function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi

E En (x) . . . . . . . . . . . . . . . . . . . . . . . see Euler polynomials E(x) . . . . . . . . . . . . . . . . . . . . . . see MacRobert function elliptic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 867 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxviii, 868, 869 complete . . . . . . . . . . . . . . . . . . . . . xxxviii, 869, 870 E . . . . . . . . . . . . xxix, xxxviii, 59, 135–140, 179–183, 185–195, 197, 198, 200, 202–214, 225, 257–264, 266–276, 283–286, 288–299, 302–317, 412, 608, 610, 625–627, 629, 638, 639, 784, 822, 860, 861, 863, 865, 868, 869, 871–873, 889 complete . . . . . . . . . . . . . . . . . . . . . . . . xxxii, xxxviii, 315, 396, 410–414, 476–478, 566, 596, 600–602, 604, 623–626, 638, 640, 641, 679, 704, 712, 721, 869–874, 878, 889, 999 F . . . . . . . xxix, 60, 61, 135–139, 179–183, 186–195, 197–200, 202–214, 225, 252, 253, 256–317, 409, 410, 412, 415, 473, 490, 567, 571, 606, 608, 610, 625–627, 637, 639, 662, 668, 669, 691, 692, 695, 707, 711, 714, 738, 740, 741, 765, 812, 817–828, 830, 831, 851, 857, 864, 868–873, 898, 927, 968,

elliptic functions (continued) 972, 973, 977, 979, 980, 983–985, 988, 990, 991, 993, 995, 1000, 1003 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680 complete . . . . . . . . . . . . . . . xxxii, xxxix, 276, 277, 315, 316, 396, 410–414, 476–478, 542, 566, 571, 588, 592, 595, 596, 600–604, 615, 616, 623, 624, 626, 637–641, 703, 704, 712, 721, 727, 763, 795, 869–879, 884, 888–890, 999 Π . . . . . . xxix, xxxii, xxxvii, 51, 136–138, 180, 181, 183, 197, 198, 200–202, 204, 205, 210, 212–214, 257, 264, 265, 267, 279–282, 286, 287, 296–302, 609, 624, 625, 627, 639, 868, 889 erf . . . . . . xxxiv, xxxviii, 107–109, 338, 368, 643, 654, 896–898, 1085, 1086 erfc . . xxxiv, xxxviii, 896, 897, 899, 900, 1078, 1080, 1085, 1086 error functions . . . . . . . . . . . . . . . . . . . . . see erf and erfc Euler constant (C ). . . . . . . . . . . . . . . . . . . . . . . . . . . . xxx, xxxiii, xxxiv, xxxviii, 3, 16, 324, 325, 333, 335, 336, 338, 361, 364, 366, 369, 370, 372, 373, 415, 416, 426, 451, 479, 482, 487, 488, 505, 538, 539, 544, 556–558, 562, 573–578, 581–584, 589, 590, 592, 596–598, 603, 609, 632, 648, 652, 664, 666, 754, 755, 892–895, 903–905, 907, 912–915, 920, 928, 946–948, 953, 1048, 1057, 1095 Euler function (ψ) . . . . . . . xxxvii, 321, 324, 325, 333, 335–338, 357–359, 361–363, 366, 371, 384, 389, 390, 402, 405, 407, 415–418, 470, 490, 494, 505, 513, 514, 526, 538–541, 543–545, 547, 556–558, 562, 563, 566, 573–578, 580–583, 588–590, 592, 598, 611, 621, 666, 667, 754, 776, 777, 828, 850, 911–916, 920, 928, 938, 978, 1021, 1022 Euler number (En ) . . . . . . . . xxx, xxxi, xxxviii, 9, 42, 145–147, 221, 223, 378, 381, 382, 536, 554, 584, 1054–1056 Euler polynomial (En (x)). . . . .xxx, xxxi, 1055, 1056 exponential function (exp) . . . . . . . . . . . . 27, 108, 109, 144, 215, 337, 339–342, 347–351, 353, 367–373, 385–387, 392, 430, 432–434, 444, 484, 485, 488– 490, 492, 493, 496–502, 505–513, 517, 520, 524, 527, 530, 578–580, 585, 622, 647, 653–657, 659, 660, 663–666, 701, 705, 706, 713–717, 719, 720, 729, 730, 755–760, 766, 767, 775, 785, 788, 792, 798, 812, 818, 819, 823, 834, 836, 837, 839, 842, 845, 849–851, 856–858, 863, 885, 886, 888, 889, 895–897, 899–902, 904, 905, 922, 924–926, 929– 932, 937, 942–944, 976, 1006, 1011, 1033–1035, 1039, 1040, 1074, 1080, 1089, 1093, 1094, 1098

Index of Functions and Constants

exponential integral (En (x)) . . . . . . . . . . . . . . . . . xxxiii exponential integral (Ei(x)) . . . . . . . . . xxxiii, xxxviii, 107, 109, 144, 150, 151, 238, 340, 342–346, 364, 372, 377, 388, 425, 426, 428–430, 436, 465, 471, 487, 488, 496, 499, 531, 532, 534, 538, 557, 559, 574–577, 580, 581, 597–599, 609, 610, 631, 632, 646–649, 651–655, 657, 664, 666, 755, 892–896, 909, 911, 940, 1085, 1093, 1095

F F . . . . . . . . . . . . . . . . . . . . . . . . . . . . see Fourier transform F (x) . . . . . . . . . . . . . . . . . . see hypergeometric function factorial ! . . . . . . . . . xxx–xxxii, xxxv, xli, 3–6, 9, 13, 14, 19, 23–27, 33, 34, 42, 43, 46, 49–51, 54, 55, 60–62, 66, 68, 77, 79, 85, 90, 92, 94, 106–109, 114, 115, 127, 128, 140–143, 145–148, 152, 153, 155–157, 163–167, 174, 215–219, 221–224, 228, 230, 242, 243, 318, 319, 323, 326–329, 331–334, 338, 342– 344, 346, 348, 355, 356, 361, 363, 367, 369, 381, 383, 384, 388, 391, 395, 398–400, 402–404, 407, 410, 421, 423, 433–435, 440, 441, 445, 448, 469, 473–475, 491, 492, 499–503, 506–509, 512, 514, 516, 521, 525, 531, 534–536, 539, 554–556, 559, 563, 570, 576–580, 582, 584, 589, 590, 595, 597, 604, 605, 611, 617, 620, 623, 631, 643, 644, 668, 680, 685, 695, 696, 706, 711, 712, 714, 716, 733, 776–778, 797–799, 801–808, 810–820, 848–850, 852, 869–871, 875, 876, 878, 893–895, 898, 901, 902, 904–906, 908–910, 913, 916, 918–920, 922, 927–930, 932–934, 938, 939, 943–945, 949, 950, 953, 958, 959, 970, 971, 977, 982–984, 986, 988, 991–993, 995, 997–1002, 1004, 1006–1011, 1014, 1020–1022, 1027, 1031, 1032, 1035, 1037, 1040, 1044, 1049–1055, 1057, 1058, 1078, 1079, 1090 double (!!) . . . . . . . . . . . xli, 24, 77, 79, 94, 110, 111, 113, 114, 127, 128, 146, 147, 152, 153, 155, 156, 174, 222, 226, 247, 252, 319, 322, 326–329, 338, 347, 365–367, 369, 371, 397–400, 403, 404, 407, 410–412, 424, 434, 439, 463, 464, 470, 471, 473, 475, 482, 492, 535, 541, 543, 547, 555, 577–579, 589, 590, 604, 605, 611, 620, 801, 813, 869–871, 898, 906, 918, 932, 943, 949, 983, 986, 991, 993, 995, 997–999, 1001, 1003, 1006, 1008 Fe function . . . . . . . . . . . . . . . . . . . . . . . . xxxviii, 962–964 fe function . . . . . . . . . . . . . . . . . . . . . . . . xxxviii, 962–964 Fek function . . . . . . . . . . . . . . . . . . . . . . xxxviii, 964, 966

1119

Fey function . . . . . . . . . . . . . xxxviii, 772, 774, 964, 965 Fourier transform . . . . . . xlii, 1087, 1088, 1091, 1092, 1099 cosine . . . . . . . . . . . . . . . . xlii, 1091, 1092, 1096–1099 sine . . . . . . . . . . . . . . . . . . . . . . . . xlii, 1091–1095, 1099 Fresnel integral cosine (C) . . . . . xxxiv, xxxviii, 171, 225, 226, 419, 438, 479, 480, 496, 633, 649, 658, 667, 896–899, 944, 1096 sine (S) . . . . . . . . . xxxiv, xxxix, 170, 171, 225, 226, 419, 438, 479, 480, 496, 633, 649, 657, 658, 667, 896–899, 944, 1069, 1094

G

a1 ,... Gpq nm (x | b1 ,... ) . . . . . . . . . . . . . . . see Meijer G function gamma function incomplete (Γ(x, y)) . . . . . . . . xxxvii, 215, 340, 342, 348–350, 353, 369, 371, 440, 442, 502, 580, 665, 666, 717, 756, 794, 908–911, 1011, 1036, 1080 incomplete (γ(x, y)) . . . . . . . . . xxxvii, 442, 908–911 Γ(x) . . . . . . . . . . . . . . . . . . . . . . . . . xxxii, xxxv–xxxvii, xli, 6, 10, 68, 107–109, 121–123, 163–167, 266, 299, 320, 324, 327, 329–331, 333–335, 338–340, 348–357, 360–363, 367, 368, 370–373, 376, 378, 379, 381–386, 388–392, 397, 398, 400–403, 408, 409, 415, 418, 423, 425, 427, 440–449, 463, 464, 466, 470, 475, 483, 490, 495, 496, 501–503, 507, 510, 513, 515, 516, 519, 525, 526, 532, 538, 539, 542, 549–552, 554–558, 563, 570–572, 574, 575, 577, 578, 580–583, 588, 592, 598, 606, 608, 609, 617–622, 640, 642–649, 653, 654, 657–670, 673– 676, 678, 680–696, 698–702, 704–707, 709–719, 722–724, 732–735, 737–744, 746, 748, 751–756, 759–768, 776–796, 798–800, 802–861, 864, 873, 898, 901–911, 913, 918, 919, 921–930, 932, 938, 949–958, 968–973, 975–984, 987, 988, 990–992, 1000–1002, 1004, 1008, 1011, 1012, 1014, 1015, 1017–1019, 1022, 1029–1043, 1045–1051, 1054, 1057, 1059, 1068, 1078, 1080, 1083, 1086, 1087, 1089, 1090, 1092, 1093, 1096, 1097, 1100–1103 γ(x) . . 215, 337, 340, 342, 348, 349, 444, 496, 500, 647, 665, 685, 714, 908, 911, 1036 gd(x) . . . . . . . . . . . . . . . . . . see Gudermannian function Ge function. . . . . . . . . . . . . . . . . . . . . . . . . xxxix, 962–964 ge function . . . . . . . . . . . . . . . . . . . . . . . . . xxxix, 962, 964 Gegenbauer polynomial (Cn (x)) . . . . . . . . . . . . xxxviii, 329, 408, 802–807, 936, 949, 978, 992, 999–1002, 1004, 1006, 1008, 1027

1120

Index of Functions and Constants

Gek function . . . . . . . . . . . . . . . . . . . . . . . xxxix, 964, 966 Gey function. . . . . . . . . . . . . . . . . . . xxxix, 772, 964–966 grad . . . . . . . . . . . . . . . . . . 1062, 1063, 1065, 1067, 1068 Gudermannian (gd) . . . . . . . . . . . . . . . . . . . . xxxviii, 51, 52, 116

H H function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix, 888, 889 Hn (x) . . . . . . . . . . . . . . . . . . . . see Hermite polynomials H(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . see Struve function H(x) . . . . . . . . . . . . . . . . . . . . . . . . . . see Hankel function H(x) . . . . . . . . . . . . . . . . . . . . . . . . see Heaviside function Hankel function (Hn (x)) . . . . . xxxv, xxxix, 341, 352, 353, 371, 373, 387, 496, 661, 662, 671, 696, 699, 701–703, 705, 709, 716, 730, 757, 759, 760, 775, 785, 796, 858, 919, 920, 923–925, 929, 931, 932, 934–937, 940, 949, 953 Hen (x) . . . . . . . . . . . . . . . . . . . see Hermite polynomials Heaviside Function (H(x)) . . . . . xlii, 757, 1085, 1088, 1101 heiν (z) . . . . . . . . . . . . . . . . . . . . . see Thomson functions herν (z) . . . . . . . . . . . . . . . . . . . . . see Thomson functions Hermite polynomials Hen (x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiv, xxxv Hn (x). . . . . xxxiv, xxxv, xxxix, 367, 507, 810–814, 818–820, 992, 1001, 1005–1007, 1010, 1040 Hermitian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlii hyperbolic cosine integral . . . . . . . . . . . . . . . . . . . see chi function sine integral . . . . . . . . . . . . . . . . . . . . . see shi function hypergeometric functions confluent (Φ) . . . . . xxxvii, 1031–1033, 1036, 1037, 1040, 1041 confluent (Ψ) . . 824, 1032, 1033, 1036, 1037, 1048, 1050 F . . . . . . . . . . . . . . . . . . . . . . . xxxvii, xxxviii, 318–320, 322, 323, 329–332, 337, 349–351, 353, 370, 372, 376, 377, 396, 400, 440, 443, 444, 446, 448, 492, 494, 507, 516, 520, 647, 654, 657, 662, 665, 678, 679, 681, 685, 687–689, 693, 696, 698, 706, 710, 711, 713, 715, 719, 744, 752, 756, 761, 762, 766, 767, 778–783, 786, 787, 791, 799, 802, 804, 808, 809, 811, 813–818, 821–826, 829–832, 834–843, 846, 849, 852, 854, 856, 857, 898, 919, 955, 991, 1008, 1014–1032, 1034, 1042, 1043, 1045, 1047, 1050

I incomplete beta function B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxviii, 919, 1102 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix, 919 incomplete Gamma function . . see gamma function, incomplete inverse functions . . . . . . . . . . . . . . . . . . . . . . . . 1088, 1091

J Jacobi elliptic functions . . . see cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, sn Jacobi polynomial (pn (x)) . . xxxix, 1007–1009, 1012 Jacobi theta function (Θ) . . . . xxxii, xxxvii, 888, 889 Jacobi zeta function (zn) . . . . . . . . . . . . . . . . . . . . . xxxii

K kei(z) . . . . . . . . . . . . . . . . . . . . . . see Thomson functions ker(z) . . . . . . . . . . . . . . . . . . . . . . see Thomson functions K  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876 k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlii, 135, 184–200, 204, 206, 225, 265, 412, 476–478, 566, 571, 588, 592, 595, 596, 600–604, 606, 608–610, 623–630, 637–641, 867–877, 879–882, 884, 888, 890, 1015 Kronecker delta . . . . . . . . . . . . . . . . . . . . xlii, 1057, 1058

L L. . . . . . . . . . . . . . . . . . . . . . . . . . . . see Laplace transform L2 (x) . . . . . . . . . . . . . . . . . . . . . see dilogarithm function Ln (x) or Lα n (x) . . . . . . . . . . . see Laguerre polynomials L(x) . . . . . . . . . . . . . . . . . . . . see Lobachevskiy function L(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . see Struve function Laguerre function (Lα n (x)) . . . . . . . . . 445, 714, 716, 810–814, 816–820, 848, 910, 992, 1009–1013, 1037 polynomial (Ln (x)) . . . . xxxix, 346, 814, 816, 817, 819, 820, 852 Laplace transform . . . . . . . . . . . . xlii, 1077, 1078, 1099 Legendre functions first kind (Pn (x)). . .xxxix, 93, 106, 329, 393, 407, 408, 411, 517, 616, 706, 714, 726, 776–779, 781, 783–789, 792, 793, 795–802, 809, 817, 823, 837, 943, 945, 949, 968–970, 972, 974–978, 981–999, 1001, 1008, 1026, 1027, 1101 second kind (Qn (x)) . . . . . . . . xxxix, 327, 375, 376, 385, 704, 726, 727, 776–778, 780, 784, 787, 792, 795, 797, 798, 968, 969, 974, 975, 977, 981, 982, 984–990, 995 Lerch function (Φ) . . . . . . . . . . xxxvii, 650, 1049, 1050

Index of Functions and Constants

li function . . . . . . . . . xxxiii, xxxix, 238, 342, 530, 556, 644–646, 892, 893, 896, 911, 1036 limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxx, 7, 8, 15, 22, 26, 53, 252–254, 515, 614, 615, 621, 643, 892, 896, 899, 903, 904, 913, 914, 940, 960, 972, 1000, 1001, 1009, 1012–1015, 1033, 1048, 1050, 1051, 1075, 1077, 1078, 1088, 1091, 1100 ln function . . . . . . . . . . . . . xxv–xxvii, xxix, xxx, xxxii– xxxv, 3, 10–12, 24, 26, 27, 43, 45–50, 52–56, 60, 61, 63–67, 69–85, 87, 90, 94, 97, 99, 103, 104, 106, 113–120, 123–130, 133, 143, 145–148, 151, 155–161, 167–172, 174–176, 178, 186–197, 199, 200, 204–207, 220–225, 237–247, 252, 319, 324, 326, 328, 333–337, 340, 342, 355–366, 372–375, 377, 378, 380–383, 385, 388–392, 397, 404, 412, 435, 437, 438, 442, 451–453, 455–461, 466–470, 474–476, 487–489, 499, 501, 503–506, 521–524, 530–611, 626–632, 637–640, 644–650, 652, 653, 655–657, 664–667, 669, 676, 679, 680, 703, 709, 710, 726, 736, 754, 755, 762, 770, 870, 871, 877, 889, 891–896, 900–902, 904, 907, 908, 911–916, 918, 920, 923, 928, 938, 946–948, 953, 972, 978, 981, 986–988, 990, 991, 998, 999, 1015, 1021– 1023, 1035–1037, 1047–1051, 1057, 1059, 1068, 1083, 1085, 1090, 1093, 1095, 1098, 1102, see log function Lobachevskiy function (L) . . . . xxxix, 148, 225, 377, 382, 383, 534, 536, 537, 591–593, 597, 900 log function . . . . . . . . . . . 27, 650, 651, see ln function Lommel function (S) . . . xxxiv, xxxix, 341, 348, 354, 373, 386, 388, 421, 682, 684–686, 688, 689, 763, 765, 767, 768, 786, 789, 790, 792, 794, 795, 801, 823, 824, 827, 836, 954–956, 959, 1045 Lommel function (s) . . . . . . xxxix, 423–425, 443, 447, 688, 700, 732, 767, 768, 954–956, 1103 Lommel function (U). . . . . . . . . . . . . . . . . . . xl, 956, 957 Lommel function (V). . . . . . . . . . . . . . . . . . . xl, 956, 957

M M . . . . . . . . . . . . . . . . . . . . . . . . . . . . see Mellin transform Mλ,μ (z) . . . . . . . . . . . . . . . . . . . see Whittaker functions MacRobert function (E) . . . . . . . . . . . . . . . . 1045, 1046 Mathieu functions Se . . . . . . . . . . . . . . . . . . . xxxvi, xl, 771–773, 962, 963 se . . . . . . . . . . . . . . . . . . . xxxvi, xl, 770–773, 960–966 max . . . . . . . . . . . . . . . . . . . . . . . 859, 862, 864, 996, 1074

1121

Meijer function (G) . . xxxix, 353, 448, 662, 698, 699, 711, 718, 765, 783, 785, 825–827, 833–840, 843, 846, 852, 854, 858–864, 1041–1045 Mellin transform . . . . . . . . . . . . . . . . . . . . . . . . . xlii, 1100 min . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859, 862, 864

N Nν (z) . . . . . . . . . . . . . . . . . . . . . . . see Bessel function, Y nc function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi nd function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi Neumann function . . . . . . . . . . . see Bessel function, Y Neumann polynomial (On (x)) . . . . xxxv, xxxix, 347, 386, 388, 955, 958, 959 norm . . . . . . . . . . . . . . . . . . . . . . . . . . . see || · || and || · ||p ns function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi

O On (x) . . . . . . . . . . . . . . . . . . . see Neumann polynomials orthogonal function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805

P Pn (x) . . . . . . . . see Jacobi polynomials and Legendre polynomials Pn (x) . . . . . . . . . . . see Legendre functions (first kind) Pnm (x) . . . . . see Legendre functions (associated, first kind) parabolic cylinder function (D) . . . . . . xxxvi, xxxviii, 350, 351, 354, 367, 386, 392, 507, 508, 510, 661, 665, 666, 705, 715, 719, 747, 753, 809, 812, 819, 849–858, 1037–1040 Phi function (Φ) . . . . . . . . . . . . . . . . . . . . . xxxiv, xxxvii, xxxviii, 240, 338–340, 346, 347, 355, 356, 360, 366, 367, 369, 373, 378, 381, 383, 386, 392, 493, 507, 508, 529, 578, 608, 633, 649, 653–657, 755, 756, 762, 788, 809, 843, 846, 896–900, 908, 911, 1006, 1007, 1010, 1062, 1063, 1068, 1069 Pochhammer symbol . . . xli, 644, 680, 712, 909, 927, 956, 1020, 1021, 1027, 1031, 1040, 1059 polynomials . . . . . . . . . . . . . . . . . . . . . . see specific name principal value (PV) . . . . xli, 325, 331, 336, 337, 339, 344, 429–431, 433, 436, 437, 458, 465, 531, 538, 567, 576, 892

Q Qn (x) . . . . . . . . see Legendre functions (second kind) Qm n (x) . . . see Legendre function (associated, second kind)

1122

Index of Functions and Constants

R RC (x, y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix, 868 RD (x, y, z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix, 868 RF (x, y, z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix, 868 RJ (x, y, z, p) . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix, 868 root. . . . . . . .16, 84, 104, 333, 543, 545, 556, 579, 580 3 . . . . . . . . . 72, 86–88, 266, 333, 366, 542, 574, 927 4. . . . . . . . .73, 78, 83, 105, 136, 139, 140, 210, 211, 265–267, 274, 297, 299, 315–317, 487, 497–499, 511, 527, 528, 877, 884, 887–890, 1006 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 2k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903 rot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1062

S Sn (x) . . . . . . . . . . . . . . . . . . . . . . see Schlafli polynomials (m) Sn . . . . . . . . . . . . . . . . see Stirling numbers, first kind s(x) . . . . . . . . . . . . . . . . . . . . . . . . . . see Lommel function S(x) . . . . . . . see Lommel function and Fresnel cosine integral sc function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi Schlafli polynomial (Sn (x)) . . . . . . . . . xxxix, 958, 959 sd function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi Se(x) . . . . . . . . . . . . . . . . . . . . . . . . see Mathieu functions se(x) . . . . . . . . . . . . . . . . . . . . . . . . see Mathieu functions sec function . . 36, 39, 42–44, 49, 50, 52, 64, 155, 156, 318, 325, 330, 332, 374, 379–381, 391, 397, 398, 402, 403, 405–407, 412, 415–418, 425, 426, 440, 443, 450, 475, 483, 545, 555, 590, 654, 661, 662, 670–672, 677, 682, 690, 697, 699, 706, 713–717, 720, 723, 725, 726, 733–735, 741, 747, 756, 764, 768, 790, 814, 828, 833, 853, 873, 930, 931, 941, 1016, 1102, 1103 sech function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27, 42, 62, 113–115, 326, 389, 513, 642, 722, 757, 758, 794, 807, 809, 849, 892, 1098, 1103 shi function . . . . . . . xxxiv, xl, 143, 144, 499, 653, 895 Si function . . . . xxxiii, 219, 651, 652, 895, 939, 1085, 1092 si function . . . . . . . . . . . . . xxxiii, xl, 219–221, 342–346, 425, 427–430, 451, 499, 509, 510, 531, 532, 575, 576, 581, 584, 585, 597, 598, 603, 608, 632, 633, 647–652, 655, 657, 658, 665, 666, 755, 769, 895, 896, 1000, 1085 sigma function . . . . . . . . . . . . . . . . . . . . . xxxvii, 885, 886 sign function . . xliii, 46, 177, 243, 244, 253, 324, 325, 351–353, 367, 372, 427, 441, 442, 451, 469, 489, 597, 599, 607, 614, 615, 648, 650, 660, 757, 775, 808, 894, 1088–1090, 1092

sin function . . . . . . . . . . . . . . . . . . xxv, xxvii, xxix, xxx, xxxii–xxxvi, 4, 5, 14, 20, 21, 24, 26–52, 55, 56, 64, 74–76, 79, 148, 151–237, 251, 252, 255, 256, 265–267, 318, 320, 321, 323, 325–327, 329–331, 333–335, 341–347, 356, 357, 360, 362, 373–375, 377, 379–385, 387, 390–412, 415–532, 534–538, 540, 541, 543–546, 548–551, 553–555, 558, 562– 564, 566, 569, 571–576, 579, 582–604, 608–610, 612, 614–616, 621, 625–627, 632, 633, 637–639, 641, 644–652, 655–659, 663–667, 670–673, 675, 677, 679–683, 685, 688, 692–694, 696, 697, 699, 700, 703, 706, 710, 712, 713, 715, 716, 718, 720, 722, 724–755, 758, 759, 761–763, 765–767, 769– 777, 780, 781, 783, 784, 786–789, 791, 796, 800– 802, 804–807, 810, 814, 816, 818–820, 825, 828, 833, 837, 838, 841, 844, 845, 847, 850, 852–854, 856–858, 861, 867, 868, 870–878, 885–888, 891– 902, 905–907, 909, 913, 915–919, 921–927, 929, 931, 933, 934, 936, 937, 939, 942–947, 949, 951– 960, 963, 967, 968, 970–973, 975–990, 993–996, 998–1002, 1006, 1007, 1009, 1015, 1016, 1018, 1022, 1034, 1035, 1038, 1046–1048, 1053, 1055, 1065, 1069, 1074, 1075, 1080–1083, 1085, 1087– 1098, 1101, 1102 sinh function . . . . . . . . . . xxxiv, 27–33, 35, 36, 38, 40, 42–44, 47, 48, 50–52, 64, 110–151, 231–237, 253, 341, 361, 373–392, 409, 423, 429, 436, 437, 443, 452, 455, 456, 458, 459, 464, 470, 471, 481, 488, 489, 493, 495, 500, 506, 508, 512–530, 573, 576, 581–585, 599, 609, 614, 615, 625, 626, 642, 643, 651–653, 672, 693, 705, 706, 709, 711–713, 718, 720–724, 730, 731, 736, 742, 753, 754, 758–760, 762, 767, 770–773, 785, 794–796, 813, 851, 895, 897, 905, 907, 922–924, 926, 951, 962, 964–966, 969–972, 976–978, 989, 990, 1006, 1015, 1034, 1038, 1039, 1051, 1082–1085, 1089, 1090, 1094, 1095, 1098 sn function . . xxxi, xxxii, xl, 627–630, 721, 875–882, 884, 888, 889 special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 867 square root . . . . . . . . . . . . . . . . xxx, xxxii, xxxiv, xxxvi, xlii, 3, 10–12, 15, 16, 24–26, 30, 37, 43, 53, 54, 56–61, 63–67, 71–79, 83–99, 103–109, 125–140, 158, 170–175, 177–184, 197, 199, 200, 202–214, 225, 226, 230, 239, 242–247, 251, 253, 256–317, 320–322, 324, 326–333, 335, 336, 338, 339, 341, 342, 346–356, 361, 365–376, 378, 382, 384–387, 392, 393, 395–398, 402–404, 406–415, 418–423, 425, 429, 430, 432, 433, 438–440, 444–450, 452,

Index of Functions and Constants

square root (continued) 455–458, 460, 461, 463, 464, 476–483, 485–487, 489, 490, 492–501, 503, 505–511, 515, 517–519, 521, 525, 527–530, 532–543, 546, 547, 549, 553, 555–557, 560–562, 565–569, 571, 574–583, 585, 587, 588, 591, 592, 594–597, 599–610, 612–623, 625–627, 633, 637–643, 646, 647, 649, 651, 653– 659, 661, 664–667, 669–676, 678, 681–683, 685, 686, 688–691, 693–730, 732–735, 737–770, 773– 775, 778, 780, 784, 785, 787–789, 792–796, 798, 800, 801, 807–816, 818–820, 822, 823, 827, 836, 837, 845, 849, 851–853, 856, 861, 862, 864, 867, 868, 870–875, 877, 879–885, 888–890, 896–900, 902–911, 914, 917, 918, 922–924, 926, 927, 929– 935, 937, 940–949, 951–955, 959, 960, 967, 969– 983, 985–992, 994, 996, 997, 999–1007, 1009, 1011, 1012, 1016–1019, 1027, 1029, 1032, 1035– 1040, 1045, 1048, 1064, 1066, 1067, 1086–1091, 1093, 1095–1097, 1099, 1101 step function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805 Stirling number first kind (Snm ) . . . . . . . . . . . . . . . . . . xliii, 1057–1059 second kind (Sm n ) . . . . . . . . . . . . . . . xliii, 1057–1059 Struve function H(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix, 347, 352, 425, 439, 446, 447, 576, 655, 656, 668, 671, 672, 677, 683–685, 687, 688, 700, 702, 716, 729, 732, 743, 760–766, 794, 846, 856, 864, 951, 952, 955, 1045, 1102 modified (L(x)) . . . . . . . . . . . . . . . . . xxxix, 347, 352, 353, 439, 445, 519, 599, 608, 671, 672, 677, 679, 684, 686, 687, 700, 729, 743, 760–766, 794, 802, 951, 952

T Tn (x) . . . . . . . . . . . . . . . . . . see Chebyshev polynomials tan function . . . . . . . xxv–xxvii, 27–30, 35, 36, 39, 40, 42–44, 46, 49, 51, 52, 55, 56, 59, 64, 126, 151, 155–161, 167–178, 180, 183–185, 188, 189, 194, 195, 199, 200, 202–207, 209, 212, 214, 222–225, 230, 277, 324, 334, 342, 357, 373, 374, 377, 379– 381, 383, 384, 386, 390, 391, 395, 397, 398, 402, 405–407, 411, 412, 415–418, 425–427, 437, 438, 455–459, 461–463, 475–478, 487–490, 496, 497, 500, 502, 509, 510, 513, 519, 522, 536–538, 541, 545, 549, 550, 571–574, 582, 586, 590–592, 594, 595, 597, 598, 600–603, 608–610, 626, 637–639, 667, 672, 677, 679, 690, 694, 706, 723, 725, 732,

1123

tan function (continued) 735, 751, 761, 773, 857, 871–874, 876–878, 891, 895, 899, 903, 914, 918, 931, 941, 963, 978, 980, 988, 995, 998, 1016, 1032, 1093, 1097, 1102 tanh function. . . . . . . . . . . . . . . . . . . . . . . . . . . . 13, 27–29, 31, 39, 40, 42, 43, 51, 52, 62, 64, 110, 113–120, 123–126, 128–132, 134–139, 146–148, 340, 382– 385, 389, 392, 475, 488, 489, 494, 506, 513, 516, 517, 520, 522–524, 573, 609, 625, 626, 643, 724, 758, 760, 773, 795, 796, 849, 930, 931, 965, 966, 1034, 1094 theta function (θ) . . . . . . . . . . . . . xxxii, 525, 642, 643, 886–892 Thomson functions bei(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxviii, 768–770, 953 ber(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxviii, 768–770, 953 hei(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix, 953 her(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix, 953 kei(x) . . . xxxix, 649, 671, 680, 682, 754, 769, 770, 953, 954 ker(x) . . . xxxix, 649, 671, 680, 682, 754, 769, 770, 953, 954 toroidal function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 990 tr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . see trace transpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliii

U Un (x) . . . . . . . . . . . . . . . . . . see Chebyshev polynomials

W Wλ,μ (z) . . . . . . . . . . . . . . . . . . . see Whittaker functions Weber function (E) . . . . 341, 348, 355, 373, 386, 425, 427, 678, 679, 758, 952, 955, 957, 958 Weierstrass function (℘) . . . . . . . xxxvii, xxxviii, 630, 882–886, 889 Whittaker functions M . . xxxix, 340, 350, 449, 662, 690, 705, 711–713, 716, 718, 722–724, 743, 755, 756, 791, 792, 794, 827–849, 1033–1036 W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xl, 340, 348–351, 370, 386, 427, 449, 644, 661, 662, 690, 705, 711, 713, 714, 716, 717, 719, 722–724, 733, 734, 743, 752, 756, 763, 766, 768, 783–785, 788, 789, 791–795, 810, 823–825, 827–849, 851, 854, 855, 865, 988, 1034–1037, 1045

1124

Index of Functions and Constants

X

Z

Xb . . . . . . . . . . . . . . . . . . . . . . . . see bilateral z transform Xu . . . . . . . . . . . . . . . . . . . . . . see unilateral z transform

zeta function (ζ) . . xxxvii, 9, 340, 355, 356, 360, 361, 378, 379, 381–383, 388–392, 437, 438, 453, 474, 513, 543, 546, 547, 554, 555, 563, 570, 572, 573, 580, 584, 590, 595, 597, 611, 630, 642, 643, 666, 667, 809, 810, 885, 886, 889, 903, 907, 912–914, 916, 918, 1046–1052, 1103 zn(x) . . . . . . . . . . . . . . . . . . . . . . see Jacobi zeta function

Y Y . . . . . . . . . . . . . . . . . . . . . . . . . . . see Bessel function, Y Young function (C) . . . . . . . . . . . . . . . . . xxxiv, xxxviii, 421, 443

Index of Concepts A

arctangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 and algebraic functions . . . . . . . . . . . . . . . . . . . . . . 246 and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 754 argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875 of a complex number. . . . . . . . . . . . . . . . . . . . . . . . .xlii arithmetic mean theorem . . . . . . . . . . . . . . . . . . . . . 1068 arithmetic progression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 arithmetic-geometric progression . . . . . . . . . . . . . . . . . 1 associated Legendre functions . . . 776, 795, 967, 981, 983 and algebraic functions . . . . . . . . . . . . . . . . . . . . . . 796 and Bessel functions . . . . . . . . . . . . . . . . . . . . 789, 794 and exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783 and hyperbolic functions . . . . . . . . . . . . . . . . . . . . . 785 and powers . . . . . . . . . . . . . . . . . . . . . . . . 777, 783, 786 and probability integral . . . . . . . . . . . . . . . . . . . . . . 788 and rational functions . . . . . . . . . . . . . . . . . . . . . . . 796 and trigonometric functions. . . . . . . . . . . . . . . . . . 786 associated Mathieu functions . . . . . . . . . . . . . . . . . . . 961 asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . 1110 asymptotic result . . . . . . . . . . . . . . . . . . . . . 22, 358, 904, 1035, 1039 asymptotic series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21, 22

absolute convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 absolute values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 addition theorems . . . . . . . . . . . . . . . . . . . . . . . . . 982, 984 algebraic functions . . . . . . . . . . . . . . . . . . . . . . . . . 82, 255 and arccosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 and arccotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 and arcsine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 and arctangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 and associated Legendre functions . . . . . . . . . . . 796 and Bessel functions . . . . . . . . . . . . . . . . . . . . 682, 722 and exponentials . . . . . . . . . . . . . . . . . . . . . . . . 346, 365 and hyperbolic functions . . . . . . . . . . . . . . . . . . . . 132, 377, 722 and logarithmic functions . . . . . . . . . . . . . . . . . . . . 541 and logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 and rational functions . . . . . . . . . . . . . . . . . . . . . . . 796 and trigonometric functions. . . . . . . . . . . . . . . . . . 438 alternating series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875 analytic continuation . . . . . . . . . . . . . . . . . . . . . 979, 1022 Anger functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 957 angle of parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 anticommutative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061 approximation by tangents . . . . . . . . . . . . . . . . . . . . . 930 arccosecant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 arccosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 and algebraic functions . . . . . . . . . . . . . . . . . . . . . . 244 arccotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 and algebraic functions . . . . . . . . . . . . . . . . . . . . . . 246 arcsecant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 arcsine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 and algebraic functions . . . . . . . . . . . . . . . . . . . . . . 244

B

Bateman’s function . . . . . . . . . . . . . . . . . . . . . . . . . . . 1032 Bernoulli numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1051, 1056 polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 1051, 1052 Bessel functions. . . 633, 667, 755, 756, 760, 919, 921, 923, 925–929, 933, 934, 937, 940, 942–946, 948, 950, 963, 1110, see Constant/Function index and algebraic functions . . . . . . . . . . . . . . . . . 682, 722 and arctangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754 and associated Legendre functions . . . . . . 789, 794 and Chebyshev polynomials . . . . . . . . . . . . . . . . . 810

1125

1126

Index of Concepts

Bessel functions (continued) and exponentials . . . . 702, 706, 716, 718, 720, 723, 750, 842 and Gegenbauer functions . . . . . . . . . . . . . . . . . . . 806 and hyperbolic functions . . . . . . . . . . 720, 722, 723, 753 and hypergeometric functions . . . . . . . . . . . . . . . . 825 confluent . . . . . . . . . . . . . . . . . . . . . . . . 838, 839, 842 and Legendre polynomials . . . . . . . . . . . . . . . . . . . 801 and logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754 and MacRobert functions . . . . . . . . . . . . . . . . . . . . 862 and Mathieu functions . . . . . . . . . . . . . . . . . . . . . . . 774 and Meijer functions . . . . . . . . . . . . . . . . . . . . . . . . . 862 and parabolic cylinder functions . . . . . . . . . . . . . 853 and powers . . . . . 672, 683, 697, 706, 716, 718, 734, 750, 839, 842 and rational functions . . . . . . . . . . . . . . . . . . . . . . . 678 and Struve functions . . . . . . . . . . . . . . . . . . . . . . . . 763 and trigonometric functions . . . . . . . . . . . . 724, 734, 750, 753 generating functions . . . . . . . . . . . . . . . . . . . . . . . . . 942 imaginary arguments . . . . . . . . . . . . . . . . . . . . . . . . 920 Bessel inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076 binomial coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xli binomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318, 324 Bonnet–Heine formula . . . . . . . . . . . . . . . . . . . . . . . . . 996 bounded variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 branch points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875, 1034 Buniakowsky inequality . . . . . . . . . . . . . . . . . . . . . . . 1072

C

Carleman inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 1074 Catalan constant . . . . . . . . . . . . . . . . . . . . xxx, 1057, see Constant/Function index Cauchy principal value . . . . . . . . . . . . . . . . . . . . . . . . . 531 Cauchy–Schwarz–Buniakowsky inequality . . . . . 1072 change of variables . . . . . . . . . . . . . . . . . . . 250, 611, 612 Chebyshev inequality . . . . . . . . . . . . . . . . . . . . . . . . . 1073 Chebyshev polynomials . . . . . . . . . . . . . . . . . . 997, 1002 and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 810 and elementary functions . . . . . . . . . . . . . . . . . . . . 809 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807 Christoffel formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 992 Christoffel summation formula . . . . . . . . . . . . . . . . . 995 circle of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

classification system . . . . . . . . . . . . . . . . . . . . . . . . . . . xxix classified references . . . . . . . . . . . . . . . . . . . . . . . . . . . 1109 complementary error function. . . see error functions, complementary complementary modulus . . . . . . . . . . . . . . . . . . . . . . . 867 complete elliptic integrals . . . . . . . . . . . . . 623, 639, 867 complex analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1110 complex conjugate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xli conditional convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 7 conditions, Dirichlet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 confluent hypergeometric function . . . . . . . . . . . . . . see hypergeometric function, confluent conical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 989 constant of integration . . . . . . . . . . . . . . . . . . . . . . . . . . 63 constants . . . . . . . . . . . . . see Constant/Function index Catalan . . . . . . . . . . . . . . . . . . . . see Catalan constant Euler . . . . . . . . . . . . . . . . . . . . . . . . . see Euler constant continued fraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 911 converge absolutely . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 conditionally . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 uniformly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 convergence circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7, 20 convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074 convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088 theorem . . . . . . . . . . . . . . . . . . 1078, 1088, 1092, 1100 coordinates, curvilinear . . . . . . . . . . . . . . . . . . . . . . . 1064 cosine and rational functions . . . . . . . . . . . . . . . . . . 171, 392 and square roots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632, 648, 895 hyperbolic . . . . . . . . . . . . . . . . . . . . . . . . . . . 653, 895 multiple angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 cosine-amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875 cube roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1062 curvilinear coordinates . . . . . . . . . . . . . . . . . . . . . . . . 1064 cycles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1057 cylinder function . . . . see parabolic cylinder function

D

Darboux–Christoffel formula . . . . . . . . . . . . . . . . . . . 992 definite integrals . . . . . . . . . 249, see integrals, definite delta amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875 derivative of a composite function . . . . . . . . . . . . . . . 23

Index of Concepts

1127

differential equations . . . . . . . . . . . . 882, 883, 919, 940, 952, 956–959, 961, 967, 983, 984, 989, 990, 992, 1001, 1004, 1007, 1009, 1012, 1020, 1023, 1024, 1033, 1040, 1044 hypergeometric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1020 partial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1028, 1041 Riemann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023, 1031 second-order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026 differentiation of integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22, 1072 of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1062 dilogarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650 directional derivative. . . . . . . . . . . . . . . . . . . . . . . . . .1063 Dirichlet conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Dirichlet lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075 div . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1062 divergence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067 double factorial symbol . . . . . . . . . . . . . . . . . . . . . . . . . xli double integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 614, 1030 doubling formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905 doubly-periodic function . . . . . . . . . . . . . . . . . . 875, 883

E

eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 960 elementary functions . . . . . . . . . . . . . . . . . 25, 249, 1015 and Chebyshev polynomials . . . . . . . . . . . . . . . . . 809 and Gegenbauer polynomials . . . . . . . . . . . . . . . . 805 and Legendre polynomials . . . . . . . . . . . . . . . . . . . 799 and MacRobert functions . . . . . . . . . . . . . . . . . . . . 858 and Meijer functions . . . . . . . . . . . . . . . . . . . . . . . . . 858 indefinite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 elliptic functions . . . . . . . . . . . . . . . . 623, 637, 874, 1112 Jacobian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .875, 879 order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874 Weierstrass . . . . . . . . . . 630, see Weierstrass elliptic functions elliptic integrals . . . 104, 184, 623, 625, 637–639, 867 complete . . . . . . . . . . . . . . . . . 396, 476–478, 639, 867 derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 396, 872, 874 functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . 872 generalized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643 Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627 kinds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 867 equations differential . . . . . . . . . . . . . see differential equations error functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 896, 1111 complementary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896 essential singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034

Euler constant . . . . . . xxx, see Constant/Function index dilogarithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650 integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 901, 917 numbers . . . . . . . . . . . . . . . . . . . . . . . . 1051, 1054, 1056 polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055 substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 expansions, asymptotic . . . . . . . . . . . . . . . . . . . . . . . 1110 expansions, Weierstrass . . . . . . . . . . . . . . . . . . . . . . . . 878 exponential integrals . . . . . . . . . . . . 631, 644, 646, 892, 894, 1111 and exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631 exponentials . . . . . . . . . . . . . . . . . . . . . . 26, 106, 336, 340 and algebraic functions . . . . . . . . . . . . . . . . . 346, 365 and associated Legendre functions . . . . . . . . . . . 783 and Bessel functions . . . . . 702, 706, 716, 718, 720, 723, 750, 842 and complicated arguments . . . . . . . . . . . . . . . . . . 338 and exponential integrals . . . . . . . . . . . . . . . . . . . . 632 and gamma functions . . . . . . . . . . . . . . . . . . . . . . . . 660 and hyperbolic functions . . . . . . 149, 341, 384, 388, 526, 529, 720, 723 and hypergeometric functions . . . . . . . . . . . . . . . . 822 confluent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 830, 842 and inverse trigonometric functions . . . . . . . . . . 608 and logarithmic functions . . . . . 341, 575, 577, 603 and parabolic cylinder functions . . . . . . . . . . . . . 850 and powers . . . . . 149, 348, 355, 365, 366, 388, 501, 529, 577, 706, 716, 718, 750, 761, 783, 842, 850 and rational functions . . . . . . . . . . . . . . 106, 342, 355 and Struve functions . . . . . . . . . . . . . . . . . . . . . . . . 761 and trigonometric functions . . . 227, 341, 489, 497, 499, 501, 526, 529, 603, 750 series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

F

factorial symbol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xli field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061 figures. . . . . . . . . . . . 612–614, 901, 922, 924, 925, 1046 finite sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 first mean value theorem . . . . . . . . . . . . . . . . . . . . . . 1071 footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvii, xxix, 82, 132, 249, 250, 276, 399, 551, 664, 867, 876, 917, 929, 940, 990, 1000, 1050, 1105 Fourier series . . . . . . . . . . . . . . . . . . . . 20, 45, 1074, 1075 generalized . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075, 1076

1128

Index of Concepts

Fourier transform . . . . . . . . . . . . . . . . . . . . . . . 1077, 1087 basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088 cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1091, 1099 properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1091 table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096 exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099 sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1091, 1099 properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1091 table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1092 tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088, 1090 fourth roots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 fractional transformation . . . . . . . . . . . . . . . . . . . . . . 1024 Fresnel integrals . . . . . . . . . . . . . . . . 633, 657, 896, 1111 functional series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 functions . . . . . . . . . . . . . see Constant/Function index inner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvi ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvi orthogonal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805 outer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv

G

gamma functions . . . . . . . . . . 659, 901, 903, 904, 1111 and exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 660 and logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 660 and trigonometric functions. . . . . . . . . . . . . . . . . . 663 incomplete. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .665, 908 Gauss divergence theorem. . . . . . . . . . . . . . . . . . . . . 1067 Gegenbauer functions and Bessel functions . . . . . 806 Gegenbauer polynomials . . . . . . . . . . . . . . . . . . . . . . . 999 and elementary functions . . . . . . . . . . . . . . . . . . . . 805 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802 general formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . 65, 251 generalized elliptic integrals . . . . . . . . . . . . . . . . . . . . 643 generalized Fourier series . . . . . . . . . . . . . . . . 1075, 1076 generalized Legendre polynomials . . . . . . . . . . . . . . 999 generating functions Bernoulli numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 1051 Bernoulli polynomials . . . . . . . . . . . . . . . . . xxx, 1052 Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 942 Chebyshev polynomials . . . . . . . . . . . . . . . . . . . . . 1004 Euler numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054 Euler polynomials . . . . . . . . . . . . . . . . . . . . . xxx, 1055 Hermite polynomials . . . . . . . . . . . . . . . . . . . . . . . 1006 Jacobi polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 1009 Legendre polynomials . . . . . . . . . . . . . . . . . . . . . . . 997 Neumann polynomials . . . . . . . . . . . . . . . . . xxxv, 959 Stirling numbers. . . . . . . . . . . . . . . . . . . . . . 1057, 1058

geometric progression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 grad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1062 gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1062 Gram inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073 Green theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1067, 1068 Gudermannian (gd) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

H

Hankel functions . . . . . . . . . . . . . . . . . . . . . . . . . . 919, 934 Heaviside step function . . . . . . . . . . . . . . . . . . . . . . . . . xlii Heine formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996 Helmholtz equation . . . . . . . . . . . . . . . . . . . . . . 774, 1064 Hermite method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Hermite polynomials . . . . . . . . . . . . . . . 810, 1005, 1006 H¨ older inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072 homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884 hyperbolic amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 cosine integral . . . . . . . . . . . . . . . . . . . . . . . . . . 653, 895 sine integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653, 895 hyperbolic functions . . . . . . . . . . . . . . . . . . . 28, 110, 374 and algebraic functions . . . . . . . . . . . . . 132, 377, 722 and associated Legendre functions . . . . . . . . . . . 785 and Bessel functions . . . . . . . . . . . 720, 722, 723, 753 and exponentials . . . . 149, 341, 384, 388, 526, 529, 720, 723 and inverse trigonometric functions . . . . . . . . . . 609 and linear functions . . . . . . . . . . . . . . . . . . . . . . . . . 120 and logarithmic functions . . . . . . . . . . . . . . . . . . . . 582 and Mathieu functions . . . . . . . . . . . . . . . . . . . . . . . 770 and parabolic cylinder functions . . . . . . . . . . . . . 851 and powers . . . . . . . . . . . . . . 110, 120, 140, 149, 388, 520, 529 and rational functions . . . . . . . . . . . . . . . . . . . . . . . 125 and trigonometric functions . . . . . . . 231, 513, 520, 526, 529, 753, 770 inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55, 242 and logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 hypergeometric differential equation . . . . . . . . . . . . . . . . . . . . . . . . 1020 series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014, 1017 confluent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1040 generalized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1020 hypergeometric functions. . . . . . . 820, 849, 955, 1014, 1015, 1050, 1111 and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 825 and exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 820

Index of Concepts

1129

hypergeometric functions (continued) and trigonometric functions. . . . . . . . . . . . . . . . . . 825 confluent . . . . . . . . . . . . . . 828, 849, 1031, 1032, 1111 and Bessel functions . . . . . . . . . . . . . 838, 839, 842 and exponentials . . . . . . . . . . . . . . . . . . . . . 830, 842 and Legendre functions . . . . . . . . . . . . . . . . . . . 847 and parabolic cylinder functions . . . . . . . . . . . 857 and polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 848 and powers . . . . . . . . . . . . . . . . . . . . . . 828, 839, 842 and special functions . . . . . . . . . . . . . . . . . . . . . . 847 and Struve functions . . . . . . . . . . . . . . . . . . . . . . 846 and trigonometric functions . . . . . . . . . . . . . . . 837 several variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1031 two variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1027

I

improper integrals. . . . . . . . . . . . . . . . . . . . . . . . . 253, 254 incomplete beta functions . . . . . . . . . . . . . . . . . . . . . . 919 incomplete gamma function . . . . . . . . . . . . . . . . . . . . 665 indefinite integrals elementary functions. . . . . . . . . . . . . . . . . . . . . . . . . . 63 special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 inequalities . . . . . . . . . . 959, 972, 988, 995, 1006, 1052 Carleman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074 integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071–1074 infinite products . . . . . . . . . . . . . . . . . . . . . . . . . 7, 15, 871 inner function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxix integral differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . 22, 1072 formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994 inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071–1074 inversion . . . . . . . . . . . . . . . . . 1077, 1088, 1091, 1099 part (symbol) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xli representations . . . . . . . . . . . . . . . 896, 897, 901, 907, 909, 911, 915, 917, 921, 923, 925, 951, 955, 959, 969, 983, 985, 989, 990, 994, 1000, 1005, 1014, 1030–1032, 1034, 1037, 1045, 1046, 1050, 1051, 1075 theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061 transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077, 1111 relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099 integrals definite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 special functions . . . . . . . . . . . . . . . . . . . . . . . . . . 637 double. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614, 1030 elliptic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .104, 867 Fresnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1111 improper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253, 254 indefinite . . . . . . . . . . . . . . . . . see indefinite integrals

integrals (continued) Mellin–Barnes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1031 multiple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611, 617 pseudo-elliptic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 triple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 integration constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 termwise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883 inverse hyperbolic functions . . . . . . . . . see hyperbolic functions inverse trigonometric functions . . . . . . . . . . . . . 603, see trigonometric functions and exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608 and hyperbolic functions . . . . . . . . . . . . . . . . . . . . . 609 and logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610 and powers . . . . . . . . . . . . . . . . . . . . . . . . 603, 605, 610 and trigonometric functions . . . . . . . . . . . . . 609, 610 inversion integral . . . . . . . . . . . . 1077, 1088, 1091, 1099

J

Jacobi polynomials . . . . . . . . . . . . . . . . . . . . . . . 814, 1007 Jacobian elliptic functions . . . . . . 875, 879, 888, 1112 Jacobian elliptic integrals . . . . . . . . . . . . . . . . . . . . . . 627 Jensen inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074

L

Laguerre polynomials . . . . . . . . . . . . . . . . . . . . 816, 1009 Laplace formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996 Laplace integral formula . . . . . . . . . . . . . . . . . . . . . . . 994 Laplace transform . . . . . . . . . . . . . . . . . . . . . . 1077, 1099 basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077 table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078 Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774, 1063 least common factor . . . . . . . . . . . . . . . . . . . . . . . . . . . 805 least common multiple . . . . . . . . . . . . . . . . . . . . . . . . . 805 Lebesgue lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075 Legendre functions . . . . . . . . . . . . . . . . . . . . . . . 984, 1113 and hypergeometric functions confluent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847 associated . . . . . see associated Legendre functions special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 978 Legendre normal form . . . . . . . . . . . . . . . . . . . . . . . . . 867 Legendre polynomials . . . . . . . . . . . . . . . . . . . . . 992, 997 and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 801 and elementary functions . . . . . . . . . . . . . . . . . . . . 799 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798

1130

Index of Concepts

lemmas Dirichlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075 Riemann–Lebesgue . . . . . . . . . . . . . . . . . . . . . . . . . 1075 letters, conventions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Lobachevskiy’s “angle of parallelism” . . . . . . . . . . . . 51 Lobachevskiy’s function . . . . . . . . . . . . . . . . . . . . . . . . 900 logarithm integrals . . . . . . . . . . . . . . . . . . . . . . . . 644, 896 logarithms . . . . . . . . . . . . . . . . . . . . . . . . 52, 237, 530, 533 and algebraic functions . . . . . . . . . . . . . . . . . 238, 541 and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 754 and exponentials . . . . . . . . . . . . . . 341, 575, 577, 603 and gamma functions . . . . . . . . . . . . . . . . . . . 664, 907 and hyperbolic functions . . . . . . . . . . . . . . . . . . . . . 582 inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 and inverse trigonometric functions . . . . . . . . . . 610 and powers . . . . . . . . . . . . . . . . . . . 543, 546, 556, 559, 577, 597 and rational functions . . . . . . . . . . . . . . . . . . 538, 556 and trigonometric functions . . . 341, 584, 597, 603 Lommel functions . . . . . . . . . . . . . . . . . . . . . . . . . 767, 954 two variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 956

M

MacRobert functions . . . . . . . . . . . . . . . . . . . . . 858, 1045 and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 862 and elementary functions . . . . . . . . . . . . . . . . . . . . 858 and special functions . . . . . . . . . . . . . . . . . . . . . . . . 864 Mathieu functions . . . . . . . . . . . . . . . 770, 959, 960, 962, 963, 1113 and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 774 and hyperbolic functions . . . . . . . . . . . . . . . . . . . . . 770 and trigonometric functions. . . . . . . . . . . . . . . . . . 770 imaginary argument . . . . . . . . . . . . . . . . . . . . . . . . . 961 matrix determinants . . . . . . . . . . . . . . . . . . . see determinants mean value theorems . . . . . . . . . . . . . . . 249, 1071, 1072 Meijer functions. . . . . . . . . . . . . . . . . . . . . . . . . .858, 1041 and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 862 and elementary functions . . . . . . . . . . . . . . . . . . . . 858 and special functions . . . . . . . . . . . . . . . . . . . . . . . . 864 Mellin transform . . . . . . . . . . . . . . . . . . . . . . . . 1077, 1099 basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1100 table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101 Mellin–Barnes integrals . . . . . . . . . . . . . . . . . . . . . . . 1031 metrical coefficients . . . . . . . . . . . . . . . . . . . . . 1064, 1066 Minkowski inequality . . . . . . . . . . . . . . . . . . . . . . . . . 1073 modulus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .639, 867, 869 multiple angle expansion . . . . . . . . . . . . . . . . . . . . . . . . 31 multiple integrals . . . . . . . . . . . . . . . . . . . . . . . . . 611, 617

N

natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlii Neumann functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 919 Neumann polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 958 nome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886 normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 867 notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xli

O

order of Presentation . . . . . . . . . . . . . . . . . . . . . . . . . . xxv orthogonal curvilinear coordinates. . . . . . . . . . . . . 1064 orthogonal polynomials . . . . . . . . . . . . . . 802, 991, 1113 Ostrogradskiy–Hermite method . . . . . . . . . . . . . . . . . 67 Ostrowski inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 1074 outer function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxix

P

parabolic cylinder functions. . . .849, 857, 1037, 1114 and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 853 and exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 850 and hyperbolic functions . . . . . . . . . . . . . . . . . . . . . 851 and hypergeometric functions . . . . . . . . . . . . . . . . 857 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 850 and Struve functions . . . . . . . . . . . . . . . . . . . . . . . . 856 and trigonometric functions. . . . . . . . . . . . . . . . . . 852 parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886 parameter of the integral . . . . . . . . . . . . . . . . . . . . . . . 867 Parseval theorem . . . . . . . . . . . . . . . . . . . . . . . 1075, 1076 partial fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 partial sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075 periodic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Mathieu functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 960 periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874, 879 permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057 Pochhammer symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . xli points, singular . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 967 Poisson integral . . . . . . . . . . . . . . . . . . . . . . . . . 1068, 1069 poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874, 879, 883, 901 polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 256, 315, 324 and hypergeometric functions confluent . . . . . . 848 Chebyshev . . . . . . . . . . . see Chebyshev polynomials degree 3 or 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 867 Gegenbauer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 999 Hermite . . . . . . . . . . . . . . . . see Hermite polynomials Jacobi . . . . . . . . . . . . . . . . . . . see Jacobi polynomials Laguerre . . . . . . . . . . . . . . . see Laguerre polynomials Legendre . . . . . . . . . . . . . . see Legendre polynomials orthogonal . . . . . . . . . . . . . . . . . . . . . . . . 802, 991, 1113

Index of Concepts

power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–19, 25 expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 and algebraic functions . . . . . . . . . . . . . . . . . . . . . . 365 and arccosecant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 and arcsecant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 and associated Legendre functions . . . . . . . . . . . 777, 783, 786 and Bessel functions . . . . . 672, 683, 697, 706, 716, 718, 734, 750, 839, 842 and binomials . . . . . . . . . . . . . . . . . . . . . . . 24, 318, 324 and Chebyshev polynomials . . . . . . . . . . . . . . . . . 807 and exponential integrals . . . . . . . . . . . . . . . . . . . . 631 and exponentials . . . . 149, 348, 355, 365, 366, 388, 501, 529, 577, 706, 716, 718, 750, 761, 783, 842, 850 and gamma functions . . . . . . . . . . . . . . . . . . . . . . . . 660 and Gegenbauer polynomials . . . . . . . . . . . . . . . . 802 and hyperbolic functions . . . . . . 110, 120, 140, 149, 388, 520, 529 and hypergeometric functions . . . . . . . . . . . . . . . . 820 confluent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 828, 839, 842 and inverse trigonometric functions . . . . . 603, 605, 610 and Legendre polynomials . . . . . . . . . . . . . . . . . . . 798 and logarithmic functions . . . . . 543, 546, 556, 559, 577, 597 and parabolic cylinder functions . . . . . . . . . . . . . 850 and rational functions . . . . . . . . . . . . . . 355, 403, 556 and square roots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 and Struve functions . . . . . . . . . . . . . . . . . . . . . . . . 761 and trigonometric functions . . . 152, 214, 397, 399, 403, 407, 415, 440, 463, 479, 501, 520, 529, 597, 610, 734, 750, 786 principal function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxix values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55, 254, 531 probability function . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114 probability integrals . . . . . . . . . . . . . . . . . . 633, 653, 896 and associated Legendre functions . . . . . . . . . . . 788 product finite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 infinite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7, 15, 44 of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061 theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905 progressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1, 9 pseudo-elliptic integrals . . . . . . . . . . . . . . . . . . . 105, 184

1131

Q

q-series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 889 quasiperiodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887

R

radius of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 rate of change theorems . . . . . . . . . . . . . . . . . . . . . . . 1069 rational functions . . . . . . . . . . . . . . . . . . . . . . 66, 255, 256 and algebraic functions . . . . . . . . . . . . . . . . . . . . . . 796 and associated Legendre functions . . . . . . . . . . . 796 and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 678 and cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171, 392 and exponentials . . . . . . . . . . . . . . . . . . . 106, 342, 355 and hyperbolic functions . . . . . . . . . . . . . . . . . . . . . 125 and logarithmic functions . . . . . . . . . . . . . . . 538, 556 and powers . . . . . . . . . . . . . . . . . . . . . . . . 355, 403, 556 and sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171, 392 and trigonometric functions . . . . . . . . 403, 427, 451 real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliii reciprocal theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1068 reciprocals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3, 13 references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105 supplementary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1109 remainder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 representation theorem. . . . . . . . . . . . . . . . . . . . . . . .1068 residues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 879 Riemann differential equation . . . . . . . . . . . . . . . . . 1023 Riemann hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . 1049 Riemann zeta functions . . . . . . . . . . . . . . . . . 1046, 1114 Riemann–Lebesgue lemma . . . . . . . . . . . . . . . . . . . . 1075 Rodrigues’ formula . . . . . . . . . . 1002, 1004, 1007, 1009 roots . . . . . . see square roots and Constant/Function index fourth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

S

saltus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 scalar product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061 Schl¨ afli integral formula . . . . . . . . . . . . . . . . . . . . . . . . 994 Schl¨ afli polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 958 Schwarz inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072 second mean value theorem . . . . . . . . . . . . . 1071, 1072 second-order equations . . . . . . . . . . . . . . . . . . . . . . . . 1026 semiconvergent series. . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 series . . . . . . . . . . . . . . . . . . . . . . . . . 869, see specific type alternating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 asymptotic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21, 22 convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1132

series (continued) diverge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Fourier . . . . . . . . . . . . . . . . . . . . . . . . 20, 45, 1074–1076 generalized . . . . . . . . . . . . . . . . . . . . . . . . . 1075, 1076 functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . 51 hypergeometric . . . . . . . . . . . . . . . . . . . . . . . 1014, 1017 generalized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1020 of exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 of logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–19 rational fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 remainder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 semiconvergent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Taylor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 trigonometric . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45, 871 sign function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliii signum function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliii sine and rational functions . . . . . . . . . . . . . . . . . . 171, 392 and square roots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632, 648, 895 hyperbolic . . . . . . . . . . . . . . . . . . . . . . . . . . . 653, 895 multiple angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 sine-amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875 singular points . . . . . . . . . . . . . . . . . . . . . . . . . . . 967, 1049 solenoidal fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064 special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvii and hypergeometric functions confluent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847 and MacRobert functions . . . . . . . . . . . . . . . . . . . . 864 and Meijer functions . . . . . . . . . . . . . . . . . . . . . . . . . 864 indefinite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 spherical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983 square roots . . . . . . . . 84, 88, 92, 94, 99, 103, 179, 256 and cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 and sin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 trigonometric functions . . . . . . . . . . . . . . . . . . . . . . 410 Steffensen inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 1073 step function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlii Stieltjes’ theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996 Stirling numbers . . . . . . . . . . . . . . . . . . . . . . . . 1057, 1059 table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1058, 1059 Stokes phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 929 Stokes theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069 Struve functions . . . . . . . . . . . . . . . . . . . . . 760, 951, 1114 and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 763

Index of Concepts

Struve functions (continued) and exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761 and hypergeometric functions confluent . . . . . . 846 and parabolic cylinder functions . . . . . . . . . . . . . 856 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761 and trigonometric functions. . . . . . . . . . . . . . . . . . 762 substitutions, Euler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 summation formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995 summation theorems . . . . . 948, 949, 995, 1000, 1007, 1011, 1040, 1053 sums binomial coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 partial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 powers of trigonometric functions . . . . . . . . . . . . . 37 products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 products of trigonometric functions . . . . . . . . . . . 38 reciprocals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3, 13 tangents of multiple angles. . . . . . . . . . . . . . . . . . . . 39 trigonometric and hyperbolic functions . . . . . . . . 36 supplementary references . . . . . . . . . . . . . . . . . . . . . 1109 symbol binomial coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . xli factorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xli double . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xli integral part. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xli Pochhammer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xli synonyms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv

T

table usage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxix tangent approximation . . . . . . . . . . . . . . . . . . . . . . . . . 930 Taylor series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 termwise integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 tests, convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7, 20 theorems addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 982, 984 arithmetic mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1068 convolution . . . . . . . . . . . . . . . 1078, 1088, 1092, 1100 divergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067 Gauss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067 general nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Green . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067, 1068 integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061 mean value . . . . . . . . . . . . . . . . . . . . . . 249, 1071, 1072 Parseval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075, 1076 product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905 rate of change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069

Index of Concepts

theorems (continued) reciprocal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1068 representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1068 Stieltjes’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996 Stokes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1069 summation . . . . . . 948, 949, 995, 1000, 1007, 1011, 1040, 1053 vector integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067 theta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642, 886 Thomson functions . . . . . . . . . . . . . . . . . . . . . . . . 768, 953 total variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 transformation formulas . . . . . . . . . . . . . . . . . 1017, 1018 transforms Fourier . . . . . . . . . . . . . . . . . . . . see Fourier transform fractional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024 integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1111 Laplace . . . . . . . . . . . . . . . . . . . see Laplace transform Mellin . . . . . . . . . . . . . . . . . . . . . . see Mellin transform of a derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088 trigonometric functions . . . . . . . . . . . 28, 151, 392, 419 and algebraic functions . . . . . . . . . . . . . . . . . . . . . . 438 and associated Legendre functions . . . . . . . . . . . 786 and Bessel functions . . . . . . . . . . . 724, 734, 750, 753 and exponentials . . . . 227, 341, 489, 497, 499, 501, 526, 529, 603, 750 and gamma functions . . . . . . . . . . . . . . . . . . . . . . . . 663 and hyperbolic functions . . . . . . 231, 513, 520, 526, 529, 753, 770 and hypergeometric functions . . . . . . . . . . . . . . . . 825 confluent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837 and inverse trigonometric functions . . . . . 609, 610 and logarithmic functions . . . . . 341, 584, 597, 603 and Mathieu functions . . . . . . . . . . . . . . . . . . . . . . . 770 and parabolic cylinder functions . . . . . . . . . . . . . 852 and powers . . . . . . . . . 152, 214, 397, 399, 403, 407, 415, 440, 463, 479, 501, 520, 529, 597, 610, 734, 750, 786

1133

trigonometric functions (continued) and rational functions . . . . . . . . . . . . . . 403, 427, 451 and square roots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 and Struve functions . . . . . . . . . . . . . . . . . . . . . . . . 762 inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55, 242 trigonometric series . . . . . . . . . . . . . . . . . . . . . . . . 45, 871 triple integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 triple vector product . . . . . . . . . . . . . . . . . . . . . . . . . . 1061

U

uniform convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Use of the Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxix

V vector differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1062 field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061 integral theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067 operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061 product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061

W

Weber functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 957 Weierstrass elliptic functions . . . . . . . . . . . . . . 630, 882, 889, 1112 Weierstrass expansions . . . . . . . . . . . . . . . . . . . . . . . . . 878 weight function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 991 Whittaker functions . . . . . . . . . . . . . . . . . . . . . . . . . . 1033

Y

Young inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073

Z

zeros . . . . . . . . . . . . . . . . . . . . . . 874, 879, 888, 981, 1049 simple. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1009 zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114