Formulas and Theorems for the Special Functions of Mathematical Physics

516 Pages·1966·15.133 MB·English

by  Dr. Wilhelm Magnus, Dr. Fritz Oberhettinger, Dr. Raj Pal Soni (auth.)

Die Grundlehren der
mathematischen Wissenschaften
in Einzeldarstellungen
mit besonderer Beriicksichtigung
der Anwendungsgebiete
Band 52
H erau.fgegeben von
J.
L. Doob . E. Heinz· F. Hirzebruch . E. Hopf
H. Hopf . W. Maak . S. Mac Lane
W. Magnus· D. Mumford· F. K. Schmidt· K. Stein
(;eschaJftsJfuhrende lSlerausgeber
B. Eckmann und B. L. van der Waerden
Formulas and Theorems for the
Special Functions
of Mathematical Physics
Dr. Wilhelm Magnus
Professor at the New York University
Courant Institute of Mathematical Sciences
Dr. Fritz Oberhettinger
Professor at the Oregon State University
Department of Mathematics
Dr. Raj Pal Soni
Mathematician
International Business Machines Corporation
Third enlarged Edition
Springer-Verlag Berlin Heidelberg GmbH 1966
Geschäftsfahrende Herausgeber :
Prof. DΓ. B. Eckmann
Eidgenössische Technische Hochschule Zürich
Prof. Dr. B. L. van der Waerden
Mathematisches Institut der Universität Zürich
ISBN 978-3-662-11763-7 ISBN 978-3-662-11761-3 (eBook)
DOI 10.1007/978-3-662-11761-3
All rights reserved, especially that of translation into foreign languages. It is also
forbidden to reproduce this book, either whole or in part, by photomechanical
means (photostat, microfilm and/or microcard or any other means) without written
permission from the Publishers
© by Springer-Verlag Berlin Heidelberg 1966
Softcover reprint of the hardcover 1st edition 1966
Library of Congress Catalog Card Number 66-28437
Titel No. 5035
Preface
This is a new and enlarged English edition of the book which, under
the title "Formeln und Satze fur die Speziellen Funktionen der mathe
matischen Physik" appeared in German in 1946. Much of the material
(part of it unpublished) did not appear in the earlier editions.
We hope that these additions will be useful and yet not too numerous
for the purpose of locating .with ease any particular result. Compared to
the first two (German) editions a change has taken place as far as the
list of references is concerned. They are generally restricted to books and
monographs and accomodated at the end of each individual chapter.
Occasional references to papers follow those results to which they apply.
The authors felt a certain justification for this change. At the time of
the appearance of the previous edition nearly twenty years ago much of
the material was scattered over a number of single contributions. Since
then most of it has been included in books and monographs with quite
exhaustive bibliographies. For information about numerical tables the
reader is referred to "Mathematics of Computation", a periodical publis
hed by the American Mathematical Society; "Handbook of Mathe
matical Functions" with formulas, graphs and mathematical tables
National Bureau of Standards Applied Mathematics Series, 55, 1964,
1046 pp., Government Printing Office, Washington, D.C., and FLETCHER,
MILLER, ROSENHEAD, Index of Mathematical Tables, Addison-Wesley,
Reading, Mass.) ..
There is a list of symbols and abbreviations at the end of the book.
The formulas within each section are arranged in order of increa
sing complexity (a concept which, cannot of course, be sharply defined).
We gratefully acknowledge the assistance of many people who con
tributed towards the improvement of the book. Finally, the authors
wish to thank the staff of Springer-Verlag for their patience and splen
did cooperation.
New York, N.Y. WILHELM MAGNUS
Corvallis, Oregon FRITZ OBERHETTINGER
Endicott, N.Y., July 1¢6 RAJ PAL SONI
Contents
Chapter I. The gamma function and related functions 1
1.1 The gamma function ...... . 1
1.2 The function 'P (z) • • • • • ; • • • 13
1.3. The Riemann zeta function C(z) • •• 19
1.4 The generalized zeta function C(z, "') 22
1.6 Bernoulli and Euler polynomials 26
1.6 Lerch's transcendent «P(z, 5, "') 32
1.7 Miscellaneous results 36
Literature ........•••. 36
Chapter II. The hypergeometric function 37
2.1 Definitions and elementary relations .. 37
2.2 The hypergeometric differential equation. 42
2.3 Gauss' contiguous relations. . . • • . . 46
2.4 Linear and higher order transformations . 47
2.5 Integral representations ....... . 54
2.6 Asymptotic expansions . . . . . . . . 66
2.7 The Riemann differential equation . . . 67
2.8 Transformation formulas for Riemann's P-function 68
2.9 The generalized hypergeometric series 62
2.10 Miscellaneous results 64
Literature ........•. 65
Chapter III. Bessel functions 66
3.1 Solutions of the Bessel and the modified Bessel differential
equation .......... 65
3.2 "Bessel functions of integer order. . . . . . . . . . • . " 69
3.3 Half odd integer order. . . . . . . . ..... ". . . . . 72
3.4 The Airy functions and related functions ... ". . . . ... 76
3.6 Differential equations and a power series expansion for the pro-
duct of two Bessel functions . . . . . . . . . . . . . . . 77
3.6 Integral representatio~s for Bessel, Neumann and Hankel func-
tions ................... . 79
3.7 Integral representations for the modified Bessel functions 84
3.8 Integrals involving Bessel functions. . 86
3.9 Addition theorems . . . . . . . . . 106
3.10 Functions related to Bessel functions . 108
3.11 Polynomials related to Bessel functions • 120
3.12 Series of arbitrary functions in terms of Bessel functions. 123
3.13 A list of series involving Bessel functions. . . . . . . . 129
Contents VII
3.14 Asymptotic expansions 138
3.16 Zeros . . . . 146
3.16 Wscellaneous. 148
Literature. . . . . 161
Chapter IV. Legendre functions. 161
4.1 Legendre's differential equation 151
4.2 Relations between Legendre functions. 164
4.3 The functions P~(x) and Q:(x). (Legendre functions on the cut) 166
4.4 Special values for the parameters . . . 172
4.6 Series involving Legendre functions . . 178
4.6 Integral representations . . . . . . . 184
4.7 Integrals involving Legendre functions. 191
4.8 Asymptotic behavior . . . . . . . . 195
4.9 Associated Legendre functions and surface spherical harmonics 198
4.10 Gegenbauer functions, toroidal functions and conical functions 199
Literature. . . • . • . . • . . . . . 203
Chapter V. Orthogonal polynomials. 204
6.1 Orthogonal systems. . . . . . . 204
6.2 Jacobi polynomials . . . . . . . 209
6.3 Gegenbauer or ultraspherical polynomials 218
6.4 Legendre Polynomials. . . . . . . 227
5.6 Generalized Laguerre polynomials. . 239
6.6 Hermite polynomials . . . . . . . 249
6.7 Chebychev (Tchebichef) polynomials 256
Literature. . • . . . . . • • . 262
Chapter VI.' Kummer's function 262
6.1 Definitions'and some elementary results 262
6.2 Recurrence relations. . . .. . . . 267
6.3 The differential equation . . . . . 268
6.4 Addition and multiplication theorems 271
6.6 Integral representations . . . . . . 274
6.6 Integral transforms associated with tFt(a; c; z), U(a, c, z) 278
6.7 Special cases and its relation to other functions . 283
6.8 Asymptotic expansions . . . . 288
6.9 Products of Kummer's functions 293
Literature. . • . . • . . . . . . 295
Chapter VII. Whittaker function 295
7.1 Whittaker's differential equation. 295
7.2 Some elementary results. . . . . 301
7.3 Addition and multiplication theorems 306
7.4 Integral representations . . . . 311
7.5 Integral transforms . . . . . . 314
7.6 Asymptotic expansions . . . . 317
7.7 Products of Whittaker functions 321
Literature. . . . . . . . . . . . . 323
VIII Contents
Chapter VIII. Parabolic cylinder functions and parabolic
functions . 323
8.1 Parabolic cylinder functions 323
8.2 Parabolic functions . 333
Literature. . . . . . . . 335
Appendix to Chapter VIII. 336
Chapter IX. The incomplete gamma function and special
cases. 337
9.1 The incomplete gamma function 337
9.2 Special cases 342
Literature. . . . . . . . . . . . . 357
Chapter X. Elliptic integrals, theta functions and elliptic
functions 357
10.1 Elliptic integrals . . . . . . . . . . . . . 358
10.2 The theta functions. . . . . . . . . . . . 371
10.3 Definition of the Jacobian elliptic functions by the theta func-
tions ............. . 377
10.4 The Jacobian zeta function . . . . . . . . . . . . 386
10.5 The elliptic functions of Weierstrass ...... .. 387
10.6 Connections between the parameters and special cases. 392
Literature 395.
Chapter XI. Integral transforms 395
Examples for the Fourier cosine transform 396
Examples for the Fourier sine transform. . 397
Examples for the exponential Fourier transform 397
Examples for the Laplace transform 397
Examples for the Mellin transform . . . . . . 397
Examples for the Hankel transform .. . . . 397
Examples for the Lebedev, Mehler and generalised Mehler transform 398
Example for the Gauss. transform . . . . . . . . . . . . . . • 398
11.1 Several examples of solution of integral equations of the first
kind. . . . . . . 465
Literature . . . . . . . .' 467
Appendix to Chapter XI 467
Chapter XII. Transformation of systems of coordinates. 472
12.1 General transformation and special cases. 472
12.2 Examples of separation of variables . 485
Literature . . . . . . . 492
List of special symbols 493
List of functions 495
Index . . . . . . 500
Chapter I
The gamma function and related functions
1.1 The gamma function
The function r(z) is a meromorphic function of z with simple poles at
z = -no (n = O. 1. 2• ... ) with the respective residue (-~)" .
n.
Definitions by an infinite product
1)'(
r(z) = lim + n+I.n O + = Z-1 !0l0 ( 1 + - 1 + -Z)-1 .
fl-+OOz(z 1) (z 2) ... (z n) n n
II (1 + Z) -,..
1 _ ". 00 n •
F(z) - z e .. =1 e •
(i
m)
y = lim i-log = 0.577215 ...
m ..... oo 1=1
I ntegralrepresentations
J00 J1
)],-1
r(z) = e-tt,-1dt = [loge dt. Re z > O.
o 0
J00
[e- + + ... +
r(z) = t - 1 t - ~2! (_1)"+1 ~"!J f-1 dt.
o
- (n + 1) < Re z <: -no
oosit)
r(z) = if J e-taf-1 dt. - ~ 1& - <5 < arg (1 < ~ 1& - <5. Re z > O.
o
(0+) .
r(z) = if (e2ni• - 1)-1 Je-tat,-1 dt
oosi{J
- 21" 1& - <5 < arg (1 < 21'" 1& - <5. <5 < arg t < ~1& + <5.
± ±
O. 1. 2• •..•
(1,+
1 Magnus/Oberhettinger/Soni, Formulas
2 I. The gamma function
1 (ae-tn)l-Z (J0+ )
r()=-2- .- t-· e -1<1 dt
z :n:z ooeia
(0J+ )
The symbol denotes an integral taken over a contour loop com-.
ooel6
ing from the point 00 ilJ, encircling the origin counter-clockwise and
returning to its starting point
Series expressions lor r(z)
j
r(z) = i'+~+ e-1f-Idt,
+
. n=O n. (z n) I
+ +
r(z) = (_~)n{_+l 1J'(n 1)
n. z n
[!
+ ~ (z + n) n2 + 1J'2 (n + 1) - 1J" (n + 1)] + 0 [(z + n)2]}
(series expansion in the neighborhood of a singular point z = -n,
(n = 0, 1, 2, ...)
r(ll+ z) = 1 + yz + (1'2 _ ~2) ~2 + ...
Functional equations
+
r(1 z) = zr(z) ,
+ + +
r(n z) = z(z 1) •.. (z n - 1) r(z) ,
r(z-n)=(-I)nr(z) r(l-z) = (-1)":n:
+ +
r(n 1-z) sin (:n:z) r(n 1 - z) ,
r(z) r(1 - z) = ~() '
SIn :n:z
r(z) r(-z) = . -:n:
zsm(:n:z) ,
r(~ + z)r(-.!. _ z) = :n:
2 2 cos (:n: z) ,
+
r(n z) r(n - z) :n:z inl-ll ( Z2)
[(n - 1) !J2 = sin (:n:z) 1 - m2 ,
n = 2, 3, 4, ... ,
11
+ ~ + + ~ - cos~:n:Z) ~r
r(n z)r(n z)= [(m - -Z2J.
n = 1,2,3, ... ,