A Treatise on the Theory of Bessel Functions [2 ed.] 052106743X, 0521093821

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Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 lRP 32 East 57th Street, New York, NY 10022, USA 296 Beaconsfield Parade, Middle Park, Melbourne 3206, Australia First published 1922 Second edition 1944 Reprinted 1952, 1958, 1962, 1966 First paperback edition 1966 Reprinted 1980 Printed in The United States of America First printed in Great Britain at the University Press, Cambridge Reprinted by Capital City Press, Montpelier, Vermont ISBN 0 521 06743 X hard covers ISBN 0521 09382 1 paperback

PREFACE HIS book has been designed with two objects in view. The first is the development of applications of the fundamental processes of the theory of functions of complex variables. For this purpose Bessel functions are admirably adapted; while they offer at the same time a rather wider scope for the application of parts of-the theory of functions of a real variable than is provided by trigonometrical functions in the theory of Fourier series.

T

The second object is the compilation of a collection of results which would be of value to the increasing number of Mathematicians and Physicists who encounter Bessel functions in the course of their researches. The existence of such a collection seems to be demanded by the greater abstruseness of properties of Bessel functions (especially of functions of large order) which have been required in recent years in various problems of Mathematical Physics. While my endeavour has been to give an account of the theory of Bessel functions which a Pure Mathematician would regard as fairly complete, I have consequently also endeavoured to include all formulae, whether general or special, which, although without theoretical interest, are likely to be required in practical applications; and. such results are given, SQ far as possible, in a form appropriate for these purposes. The breadth of these aims, combined with the necessity for keeping the size of the book within bounds, has made it necessary to be as concise as is compatible with intelligibility. Since the book is, for the most part, a development of the theory of functions as expounded in the Course of Modern Analysis by Professor Whittaker and myself, it has been convenient to regard that treatise as a standard work of reference for general theorems, rather than to refer the reader to original sources. It is desirable to draw attention here to the function which I have regarded as the canonical function of the second kind, namely the function which was defined by Weber and used subsequently by Schlafli, by Oraf and Gubler and by Nielsen. For historical and sentimental reasons it would have been pleasing to have felt justified in using Hankel's function of the second kind; but three considerations prevented this. The first is the necessity for standardizing the

function of the second kind; and, in my opinion, the authority of the group of mathematicians who use Weber's function has greater weight than the a.uthority of the mathematicians who use any other one function of the second kind. The second is the parallelism which the use of Weber's function exhibits between the two kinds of Bessel functions and the two kinds (cosine and sine]

CONTENTS CHAP.

PAGE

1. BESSEL FUNCTIONS BEII'ORE 1826

1

II. 'fHE BESSEL COEFFICIENTS

U.

III. BESSEL FUNCTIONS

38

IV. DIFFERENTIAL EQUATIONS

85

V. MISCELLANEOUS PROPERTIES O}4' HEtiSEL FUNCTIONS

132

VI. INTEGRAL REPRESENTATIONS OF BESSEL FUNCTIONS

160

VII. ASYMPTOTIC EXPANSIONS OF BESSEL FUNCTIONS VIII. BESSEL }'UNCTIONS OF LARGE ORDER

.

194 225

IX. POLYNOMIALS ASSOCIATED WITH BESSEL FUNCTIONS

271

X. FUNCTIONS ASSOCIATED WITH BESSEL FUNCTIONS.

308

XI. ADDITION THEOREMS .

358

XII. DEFINITE INTEGRALS.

373

XIII. INFINITE INTEGRALS .

383

XIV. MULTIPLE INTEGRALS.

450

XV. THE ZEROS OF BESSEL FUNCTIONS . XVI. NEUMANN SERIES AND LOMMEVS FUNCTIONS OF TWO VARIABLES XVII. KAPTEYN SERIES.

522 551

XVIII. SERIES OF FOURIER-BESSEL AND DIN I XIX. SCHLOMILCH SERIES

477

.

XX. THE TABULATION OF BESSEL FUNCTIONS

576 618 654

TABLES OF BESSEL FUNCTIONS

665

BIBLIOGRAPHY

753

INDEX OF SYMBOLS

789

LIST OF AUTHORS QUOTED

791

GENERAL INDEX .

700

658

THEORY OF BESSEL FUNCTIONS

[CHAP. XX

Tables of K.",(x) to five significant figures for n = 0, 1,2, ... ,10 over the range of values of x from x = 0 to x = 6'0 with interval 0'2. Isherwood's Tables were published in the Mem. and Proc., Manchester Lit. and Phil. Soc., 1903-1904, no. 19. Tables of e- X [",,(x) and K.",(x) to seven places of decimals are given in Table IV infra, pp. 736-739. The earliest Tables of Bessel functions of large order were constructed by Meissel, who has calculated .!'lin(n) to twelve significant figures for n = 10, 11, ... ,21, Astr. Nach. CXXIX. (1892), col. 284; Meissel also calculated J.", (1000) to seven significant figures for n = 1000, 999, ... , 981, ibid. CXXVIII. (1891), col. 154-155. The values of J.",(n), J""_l(n), yl""l(n), Y(.",-ll(n), G.",(n), G""_l(n) to six places of decimals for values of n from n = 1 to 'TI = 50 (interval 1), n = 50 to n = 100 (interval 5), n = 100 to n = 200 (interval 10), 'fI. = 200 to n = 400 (interval 20), n = 400 to n = 1000 (interval 50), n = 1000 to n = 2000 (interval 100) and for vanous larger values of n, are given in the British Association Report, un6, pp. 93-96. . Tables of J.",(n), J,,/(n), Y.",(n), Y,,/(n) to seven places of decimals are given in Table VI infra, pp. 746-747. The functions ber (x), bei (x), ker (x) and kei (x) have been extensively tabulated on account of their importance in the theory of alternating currents. A brief Table of ber (x) and bei (x), computed by Maclean, was published by Kelvin, Math. and Phys. Papers, III. (1890), p. 493. Tables of J, (3: .,ji) and ",12. J l (x .,ji) to twenty-one places of decimals have been constructed by Aldis, Proc. Royal Soc. LXVI. (1900), pp. 42-43; their range is from x = 0 to x = 6'0 with interval 0'1. These are extensions of the Table of J; (x .,ji) to nine places of decimals for the range from o: = 0 to o: = 6'0 with interval 0'2 published in the British Association Report, 189a, p. 228, and reprinted by Gray and Mathews in their Treatise, p. 281. Tables of ber (x), bei (x), ker (x) and kei (x) to four significant figures for x = 1, 2, 3, ... ,30, have been published by Savidge, Phil. Ma,g. (6) XIX. (1910), p.53. The functions ber (x), bei (x), ber' (x) and bei' (x) are tabulated to nine places of decimals, from x = 0 to x = 10'0 with interval 0'1 in the British Association Report, 1912, pp. 57-68; and a. Table of ker (x), kei (x), ker' (x) and kei' (x) of the same scope (except that only six or seven significant figures were given) appeared in the Report for 1915, pp. 36-38. Tables of squares and products of the functions to six significant figures from o: = 0 to x = 10'0 with interval 0'2 were given in the Report for 1916, pp. 118-121. The functions J±(n+!)(x) have been tabulated to six places of decimals by Lommel, Munchener Abh, xv. (1886), pp. 644-647, for n = 0, 1, 2, ... ,6 with x = 1, 2, ... ,50, and (in the case of functions of positive order) n = 7,8, ... , 14