Computation of Special Functions [1 ed.] 0471119636, 9780471119630

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COMPUTATION OF SPECIAL FUNCTIONS SHANJIE ZHANG Department of Electronic Science and Engineering Nanjing University Nanjing, China

JIANMING JIN Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign Urbana, Illinois

A Wiley-Interscience Publication

JOHN WILEY & SONS, INC.

New York I Chichester I Brisbane I Toronto I Singapore

This text is printed on acid-free paper. Although the authors made a great effort to test and validate the computer programs, they make no warranties, express or implied, that these programs are free of error, or are consistent with any particular standard of merchantability, or that they will meet your requirements for any particular application. They should not be relied on for solving problems whose incorrect solution could result in injury to a person or loss of property. If you do use the programs in such a manner, it is at your own risk. The authors and publisher disclaim all liability for direct or consequential damages resulting from your use of the programs. Copyright © 1996 by John WHey & Sons, Inc. All rights reserved. Published simultaneously in Canada. Reproduction or translation of any part of this work beyond that permitted by Section 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further i!lformation should be addressed to the Pennissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012.

Library of Congress Cataloging in Publication Data: Zhang, Shanjie Computation of special functions / Shanjie Zhang, Jianming Jin. p. cm. "A Wiley-Interscience publication." Includes bibliographical references and indexes. ISBN 0-471-11963-6 (cloth : acid-free paper) l. Functions, Special. I. Jin, Jian-Ming, 1962- 11. Title. QA35l. C45 1996 515' .5-dc20 95-49544 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

CONTENTS

PREFACE ACKNOWLEDGMENTS liST OF COMPUTER PROGRAMS 1.

BERNOUlll AND EUlER NUMBERS

xi xvii

xix 1

1.1 Bemoulli Numbers / 1 1.2 Euler Numbers / 6 1.3 Mathematical Table / 10 References / 11 2.

ORTHOGONAl POLYNOMIALS

2. 1 Introduction / 12 2.2 Chebyshev Polynomials / 13 2.3 Laguerre Polynomials / 18 2.4 Hermite Polynomials / 20 2.5 Numerical Computation / 23 2.6 Application in Numerical Integration / 27 References / 43

12

iv

CONTENTS

3. GAMMA, BET A, AND PSI FUNCTIONS

44

3.1 Gamma Function / 44 3.2 Beta Function / 53 3.3 Psi Function / 55 3.4 Incomplete Gamma Function / 61 3.5 Incomplete Beta Function / 64 3.6 Mathematical Tables / 66 References and Further Reading / 76

4. LEGENDRE FUNCTIONS

77

4.1 Introduction / 77 4.2 Legendre Functions of the First Kind / 78 4.3 Legendre Functions of the Second Kind / 83 4.4 Associated Legendre Functions of the First Kind / 89 4.5 Associated Legendre Functions of the Second Kind / 96 4.6 Legendre Functions with an Arbitrary Degree / 104 4.7 Mathematical Tables / 113 Refere:pces and Further Reading / 125

5. BESSEL FUNCTIONS

126

5.1 Introduction / 126 5.2 Computation of Jo(x) , J.(x), Yo(x) , and Y.(x) / 131 5.3 Computation of In(x) and Yn(x) with Real Arguments / 140 5.4 Computation of Yn(z) and Yn(z) with Complex Arguments / 149 5.5 Computation of J,lz) and J,lz) with an Arbitrary Order / 161 5.6 Assessment of Validity and Accuracy of Computation / 175 5.7 Zeros of Bessel Functions / 180 5.8 Lambda Functions / 182 5.9 Mathematical Tables / 184 References and Further Reading / 201

6. MODIFIED BESSEL FUNCTIONS 6.1 Introduction / 202 6.2 Computation of lo(x) , l.(x), Ko(x), and K.(x) / 207

202

CONTENTS

V

6.3 Computation of In(x) and Kn(x) with Real Arguments / 213 6.4 Computation of In(z) and Kn(z) with Complex Arguments / 217 6.5 Computation of I,,(z) and K,,(z) with an Arbitrary Order / 225 6.6 Computation of H~I)(Z) and H~2)(Z) for Complex Arguments / 235 6.7 Mathematical Tables / 239 References and Further Reading / 251 7.

INTEGRALS OF BESSEL FUNCTIONS

252

7.1 Simple Integrals of Bessel Functions / 252 7.2 Simple Integrals of Modified Bessel Functions / 261 7.3 Curves and Tables / 268 References / 272 8.

SPHERICAL BESSEL FUNCTIONS

273

8.1 Spherical Bessel Functions / 273 8.2 Riccati-Bessel Functions / 283 8.3 Modified Spherical Bessel Functions / 286 8.4 Mathematical Tables / 295 References and Further Reading / 306 9.

KELVIN FUNCTIONS

307

9. 1 Introduction / 307 9.2 Mathematical Properties / 311 9.3 Asymptotic Expansions / 312 9.4 Numerical Computation / 315 9.5 Zeros of Kelvin Functions / 321 9.6 Mathematical Tables / 321 Reference / 324 10.

AIRY FUNCTIONS

10.1 Introduction / 325 10.2 Numerical Computation / 329 10.3 Mathematical Tables / 336 References / 340

325

vi

CONTENTS

11.

STRUVE FUNCTIONS

341

11.1 Struve Functions / 341 11.2 Modified Struve Functions / 353 11.3 Mathematical Tables / 362 References / 365 12.

HYPERGEOMETRIC AND CONFLUENT HYPERGEOMETRIC FUNCTIONS

366

Definition of Hypergeometric Functions / 366 Properties of Hypergeometric Functions / 368 Linear Transfonnation Fonnulas / 369 Recurrence Relations for Hypergeometric Functions / 372 12.5 Special Functions Expressed as Hypergeometric Functions / 373 12.6 Numerical Computation of Hypergeometric Functions / 374 12.7 Definition of Confluent Hypergeometric Functions / 385 12.8 Properties of Confluent Hypergeometric Functions / 387 12.9 Recurrence Relations for Confluent Hypergeometric Functions / 389 12.10 Special Functions Expressed as Confluent Hypergeometric Functions / 394 12.11 Definition of Whittaker Functions / 395 12.12 Numerical Computation of Confluent Hypergeometric Functions / 398 12.13 Mathematical Tables / 411 References and Further Reading / 424 12.1 12.2 12.3 12.4

13.

PARABOLIC CYLINDER FUNCTIONS

13.1. Introduction / 425 13.2 Definitions of Parabolic Cylinder Functions / 428 13.3 Basic Properties / 432 13.4 Series and Asymptotic Expansions / 437 13.5 Numerical Computation / 438 13.6 Mathematical Tables / 455 References and Further Reading / 474

425

CONTENTS

14.

MATHIEU FUNCTIONS

vii

475

14.1 Definition of Mathieu Functions / 475 14.2 Detennination of Expansion Coefficients and Characteristic Values / 477 14.3 Approximate Calculation of Characteristic Values / 482 14.4 Expansion of Mathieu Functions When Iql < 1 / 485 14.5 Properties of Mathieu Functions / 487 14.6 Definition of Modified Mathieu Functions / 489 14.7 Properties of Modified Mathieu Functions / 496 14.8 Numerical Computation: Algorithms and Computer Programs / 501 14.9 Mathematical Tables / 520 References and Further Reading / 535 15.

SPHEROIDAL WAVE FUNCTIONS

536

15.1 Spheroidal Coordinate Systems / 536 15.2 Wave Equation and Its Solution in Spheroidal Coordinates / 540 15.3 Definitions of Angular and Radial Prolate Spheroidal Wave Functions / 542 15.4 Detennination of Characteristic Values and Expansion Coefficients / 550 15.5 Evaluation of Prolate Radial Wave Functions of the Second Kind for Small c~ / 556 15.6 Definitions of Angular and Radial Oblate Spheroidal Wave Functions / 559 15.7 Evaluation of Oblate Radial Wave Functions of the Second Kind for Small c~ / 567 15.8 Numerical Computation: Algorithms and Computer Programs / 569 15.9 Mathematical Tables / 594 References / 619 16.

ERROR FUNCTION AND FRESNEL INTEGRALS

16.1 16.2 16.3 16.4 16.5

Introduction to Error Function / 620 Numerical Computation of Error Function / 621 Gaussian Probability Integral / 624 Introduction to Fresnel Integrals / 625 Series and Asymptotic Expansions of Fresnel Integrals / 629

620

viii

CONTENTS

16.6 Numerical Computation of Fresnel Integrals / 630 16.7 Zeros of Error Function and Fresnel Integrals / 635 16.8 Mathematical Tables / 636 References and Further Reading / 643

17. COSINE AND SINE INTEGRALS

644

17.1 Introduction / 644 17.2 Series and Asymptotic Expansions / 646 17.3 Numerical Computation / 647 17.4 Mathematical Table / 651 References and Further Readings / 653

18.

ELLIPTIC INTEGRALS AND JACOBIAN ELLIPTIC FUNCTIONS

654

Introduction to Elliptic Integrals / 654 Series Expansion of Elliptic Integrals / 659 Numerical Computation of Elliptic Integrals / 661 Introduction to I acobian Elliptic Functions / 666 -Numerical Computation of lacobian Elliptic Functions / 670 18.6 Mathematical Tables / 672 References and Further Reading / 679 18.1 18.2 18.3 18.4 18.5

19. EXPONENTIAL INTEGRALS

680

19.1 Introduction / 680 19.2 Series, Asymptotic, and Continued Fraction Expressions / 682 19.3 Rational Approximations / 683 19.4 Numerical Computation / 684 19.5 Mathematical Tables / 688 References / 693

20.

SUMMARY OF METHODS FOR COMPUTING SPECIAL FUNCTIONS

APPENDIX A

DERIVATION OF SOME SPECIAL DIFFERENTIAL EQUATIONS

A.l Helmholtz Equation and Separation of Variables / 697 A.2 Circular Cylindrical Coordinates / 698 A.3 Elliptic Cylindrical Coordinates / 700

694 697

CONTENTS

ix

A.4 Parabolic Cylindrical Coordinates / 700 A.5 Spherical Coordinates / 701 A.6 Prolate Spheroidal Coordinates / 701 A.7 Oblate Spheroidal Coordinates / 702 A.8 Parabolic Coordinates / 703 References / 703 APPENDIX B

ROOT-FINDING METHODS

704

B.l Newton's Method / 704 B.2 Modified Newton's Method / 706 B.3 Secant Method / 706 Reference / 706 APPENDIX C

ABOUT THE SOFTWARE

707

INDEX

709

INDEX OF COMPUTER PROGRAMS

715