Higher Transcendental Functions

CALIFORNIA INSTITUTE OF TECHNOLOGY
BATEMAN MANUSCRIPT PROJECT
A. Erd^lyi, Editor
W. Magnus, F. Oberhettinger, F. G. Tricomi, Research Associates
Higher Transcendental Functions, 3 volumes.
Tables of Integral Transforms, 2 volumes.

HIGHER TRANSCENDENTAL FUNCTIONS
Volume II
Based, in part, on notes left by
Harry Bateman
Late Professor of Mathematics, Theoretical Physics, and Aeronautics at the
California Institute of Technology
and compiled by the
Staff of the Bateman Manuscript Project
Prepared at the California Institute of Technology under Contract
No. N6onr-244 Task Order XIV with the Office of Naval Research
Project Designation Number: NR 013-015
NEW YORK TORONTO LONDON
McGRAW-HILL BOOK COMPANY, INC.
1953

HIGHER TRANSCENDENTAL FUNCTIONS, vol. II.
COPYRIGHT, 1953, BY THE
McGRAW-HILL BOOK COMPANY, INC.
PRINTED IN THE UNITED STATES OF AMERICA
All rights reserved except those granted to the United States
Government. Otherwise, this book, or parts thereof, may not
be reproduced in any form without permission of the publishers.
Library of Congress Catalog Card Number: 53-5555
7 8 9 10 1112 13 HL 9 8 7 6
Copyright Renewed, 1981, by
California Institute of Technology

This work is dedicated to the
memory of
HARRY BATEMAN
as a tribute to the imagination which
led him to undertake a project of this
magnitude, and the scholarly dedication
which inspired him to carry it so far
toward completion.

STAFF OF THE BATEMAN MANUSCRIPT PROJECT
Director
Arthur Erdelyi
Research Associates
Wilhelm Magnus (1948-50)
Fritz Oberhettinger (1948-51)
Francesco G% Tricomi (1948-51)
Research Assistants
David Bertin (1951-52)
W. B. Fulks (1949-50)
A. R. Harvey (1948-49)
D. L. Thomsen, Jr. (1950-51)
Maria A, Weber (1949-51)
E. L. Whitney (1948-49)
Vari-typist
Rosemarie Stampfel

FOREWORD
The purpose and the history of these volumes were described in the
prefatory material to vol, I. The present second volume contains chapters
on Bessel functions and other particular confluent hyper geometric
functions, on orthogonal polynomials and related matters, and on elliptic
functions and integrals* The method of compilation was similar to that
of the first volume. Of the chapters presented here, Magnus participated
actively in the preparation of Chapters IX and XI, Oberhettinger of
Chapter VII, and Tricomi of Chapters VIII, IX, X, and XIII* Since the
final version of several of the later chapters in this volume was
prepared after the author of the first draft left Pasadena, the editorial work
was much more onerous, and in several cases the revised version differs
considerably from the first draft.
For Bessel functions we drew heavily on Watson's Treatise for a
(comparatively) brief summary of the topics to be found there, while
results obtained since the publication of Watson's book are presented in
more detail. Functions of the parabolic cylinder are described fairly
fully, those of the paraboloid of revolution only very briefly: a recent
book by H. Buchholz {Die konfluente hypergeometrische Funktion,
Springer-Verlag, 1953) gives full information on the latter functions. In
the case of functions defined by integrals (error functions, exponential
integral, and the like) we adopted (by no means unanimously) notations
which are a compromise between the notations which seem the best ones
from the mathematical point of view and those most convenient for the
user of existing mathematical tables. In the chapters on orthogonal
polynomials we summarized briefly some aspects of the general theory,
using extensively Szego's book: mainly we presented the properties of
the classical orthogonal polynomials, although we found it useful to
include some of the less well-known polynomials, polynomials of a
discrete variable, hyperspherical harmonics, and some biorthogonal
systems of polynomials of several variables. The chapter on elliptic
ix

X
SPECIAL FUNCTIONS
functions and integrals is comparatively brief but we hope that it will be
found to contain most of the material frequently required when dealing
with these functions. In particular, we have included more material on
elliptic integrals of the third kind than is often found in presentations
as brief as ours, and attempted to include practically everything that may
be required in dealing with Lame functions or ellipsoidal wave functions.
We hope that the tabular arrangement of many of the formulas of Chapter
XIII will contribute to the usefulness of this chapter.
As in the first volume, a list of references has been given at the end
of each chapter. The length of this list varies with the subject in hand.
In the case of elliptic functions and integrals we listed merely some of
the newer books, and those memoirs or older books to which we explicitly
refer. In cases where bibliographies are available, we give very few
references to work covered in the bibliographies, and more numerous
references to books and papers which have appeared since the publication
of the bibliographies.
At the end of the volume there is an Index of notations and a Subject
index. Notations introduced in vol. I are often used here without further
explanation. Their definition may be located by means of the Index of
notations appended to voL I. The system of references is the same as in
vol. !• In the text, references to literature state the name of the author
followed by the year of publication, more details being given in the list
of references at the end of the chapter. Equations within the same section
are referred to simply by number, equations in other sections by the
number of the equation. Chapters are numbered consecutively, Chapters
I to VI being in voh I, Chapters VII to XIII in the present volume. Thus
3,7(27) refers to equation (27) in section 3,7 and will be found on p. 159
of vol. I, while 9*7(12) is on p. 144 of the present volume.
Since the editor had less assistance in the preparation of this volume
than in the preparation of vol. I,errors and mistakes are more likely to
be prevalent here. Suggestions for improvement and corrections will be
gratefully received.
A. ERDELYI

CONTENTS
FOREWORD ix
CHAPTER VII
BESSEL FUNCTIONS
FIRST PART: THEORY
7.1. Introduction 1
7.2. Bessel's differential equation 3
7.2.1. Bess el functions of general order 3
7.2.2. Modified Bessel functions of general order 5
7.2.3. Kelvin's function and related functions 6
7.2.4. Bessel functions of integer order 6
7.2.5. Modified Bessel functions of integer order 9
7.2.6. Spherical Bessel functions 9
7.2.7. Products of Bessel functions 10
7.2.8. Miscellaneous results 11
7.3. Integral representations 13
7.3.1. Bessel coefficients 13
7.3.2. Integral representations of the Poisson type 14
7.3.3. Representations by loop integrals 15
7.3.4. Schlafli's, Gubler's, Sonine's and related integrals
representations 17
7.3.5. Sommerfeld's integrals 19
7.3.6. Barnes' integrals 21
7.3.7. Airy's integrals 22
7.4. Asymptotic expansions 22
7.4.1. Large variable 23
7.4.2. Large order 24
7.4.3. Transitional regions 28
7.4.4. Uniform asymptotic expansions 30
7.5. Related functions 31
7.5.1. Neumann's and related polynomials 32
7.5.2. Lommel's polynomials 34
xi

Xll
SPECIAL FUNCTIONS
7.5.3. Anger-Weber functions 35
7.5.4. Struves* functions 37
7.5.5. Lommel's functions 40
7.5.6. Some other notations and related functions 42
7.6. Addition theorems ................ 43
7.6.1. Gegenbauer's addition theorem 43
7.6.2. Graf's addition theorem 44
7.7. Integral formulas 45
7.7.1. Indefinite integrals 45
7.7.2. Finite integrals 45
7.7.3. Infinite integrals with exponential functions 48
7.7.4. The discontinuous integral of Weber and
Schafheitlin 51
7.7.5. Sonine and Gegenbauer's integrals and
generalizations 52
7.7.6. Macdonald's and Nicholson's formulas 53
7.7.7. Integrals with respect to order 54
7.8. Relations between Bessel and Legendre functions ... 55
7.9. Zeros of the Bessel functions 57
7.10. Series and integral representations of arbitrary
functions 63
7.10.1. Neumann's series 63
7.10.2. Kapteyn series 66
7.10.3. Schlomilch series 68
7.10.4. Fourier-Bessel and Dini series 70
7.10.5. Integral representations of arbitrary functions 73
SECOND PART: FORMULAS
7.11. Elementary relations and miscellaneous formulas ... 78
7.12. Integral representations 81
7.13. Asymptotic expansions 85
7.13.1. Large variable 85
7.13.2. Large order 86
7.1393. Transitional regions 88
7.13.4. Uniform asymptotic expansions 89
7.14. Integral formulas 89
7.14.1. Finite integrals 89
7.14.2. Infinite integrals 91
7.15. Series of Bessel functions 98
References 106

CONTENTS
xm
CHAPTER VIII
FUNCTIONS OF THE PARABOLIC CYLINDER AND OF THE
PARABOLOID OF REVOLUTION
Introduction 115
PARABOLIC CYLINDER FUNCTIONS
Definitions and elementary properties 116
Integral representations and integrals 119
Asymptotic expansions 122
Representation of functions in terms of the D (x). . . 123
1. Series 123
2. Representation by integrals with respect to the
parameter 124
Zeros and descriptive properties 126
FUNCTIONS OF THE PARABOLOID OF REVOLUTION
The solutions of a particular confluent hypergeometric
equation 126
Integrals and series involving functions of the
paraboloid of revolution .............. 128
References 131
CHAPTER IX
THE INCOMPLETE GAMMA FUNCTIONS AND
RELATED FUNCTIONS
Introduction 133
THE INCOMPLETE GAMMA FUNCTIONS
Definitions and elementary properties 134
1. The case of integer a 136
Integral representations and integral formulas .... 137
Series 138
Asymptotic representations 140
Zeros and descriptive properties 141
SPECIAL INCOMPLETE GAMMA FUNCTIONS
The exponential and logarithmic integral 143

xiv SPECIAL FUNCTIONS
9.8. Sine and cosine integrals ♦ ♦ 145
9.9. The error functions 147
9.10. Fresnel integrals and generalizations 149
References 152
CHAPTER X
ORTHOGONAL POLYNOMIALS
10.1. Systems of orthogonal functions 153
10.2. The approximation problem - 156
10.3. General properties of orthogonal polynomials .... 157
10.4. Mechanical quadrature 160
10.5. Continued fractions 162
10*6* The classical polynomials 163
10.7. General properties of the classical orthogonal
polynomials 166
10.8. Jacobi polynomials 168
10.9. Gegenbauer polynomials 174
10.10. Legendre polynomials 178
10.11. Tchebichef polynomials 183
10.12. Laguerre polynomials 188
10*13* Hermite polynomials 192
10.14. Asymptotic behavior of Jacobi* Gegenbauer and
Legendre polynomials 196
10.15. Asymptotic behavior of Laguerre and Hermite
polynomials 199
10.16. Zeros of Jacobi and related polynomials 202
10.17. Zeros of Laguerre and Hermite polynomials 204
10.18. Inequalities for the classical polynomials 205
10.19. Expansion problems 209
10.20. Examples of expansions 212
10.22. Some classes of orthogonal polynomials 217
10.22. Orthogonal polynomials of a discrete variable ... 221
10*23. Tchebichef's polynomials of a discrete variable
and their generalizations 223
10.24. Krawtchouk's and related polynomials 224
10.25. Charlier's polynomials 226
References 228

CONTENTS
xv
CHAPTER XT
SPHERICAL AND IJYPERSPHERICAL HARMONIC POLYNOMIALS
ILL Preliminaries 232
LL.l.L. Vectors 232
11.1.2. Gegenbauer polynomials 235
11.2. Harmonic polynomials 237
11.3 Surface harmonics 240
11.4. The addition theorem 242
1L5. The case p = 1, h(n9 p)= 2n+ 1 248
11.5.1. A generating function for surface harmonics in the
three-dimensional case 248
11.5.2; MaxweIPs theory of poles 251
11.6. The casep = 2, h(n9 p) = U + l)2 253
11 „7. The transformation formula for spherical harmonics . 256
11.8. The polynomials of Bermite-Kampe, de Feriet .... 259
References 262
CHAPTER XII
ORTHOGONAL POLYNOMIALS IN SEVERAL VARIABLES
12.1. Introduction 264
12.2. General properties of orthogonal polynomials in two
variables 265
12.3. Further properties of orthogonal polynomials in two
variables 268
ORTHOGONAL POLYNOMIALS IN THE TRIANGLE
12.4. AppelPs polynomials 269
ORTHOGONAL POLYNOMIALS IN CIRCLE AND SPHERE
12.5. The polynomials V 273
12.6* The polynomials U 277
12.7. Expansion problems and further investigations . . . 280
HERMITE POLYNOMIALS OF SEVERAL VARIABLES
12.8.
Definition of the Hermite polynomials
283

XVI
SPECIAL FUNCTIONS
12.9. Basic properties of Hermite polynomials 289
12.10. Further investigations 289
References 292
CHAPTER XIII
ELLIPTIC FUNCTIONS AND INTEGRALS
13.1. Introduction 294
PART ONE: ELLIPTIC INTEGRALS
13.2. Elliptic integrals 295
13.3. Reduction of elliptic integrals 296
13.4. Periods and singularities of elliptic integrals . . . 302
13.5. Reduction of G (x) to normal form 304
13.6. Evaluation of Legendre's elliptic integrals .... 308
13.7. Some further properties of Legendre's elliptic
normal integrals 314
13.8. Complete elliptic integrals 317
PART TOO: ELLIPTIC FUNCTIONS
13.9. Inversion of elliptic integrals 322
13.10. Doubly-periodic functions 323
13.11. General properties of elliptic functions 325
13.12. Weierstrass'functions 328
13.13. Further properties of Weierstrass'functions .... 331
13* 14. The expression of elliptic functions and elliptic
integrals in terms of Weierstrass' functions .... 335
13.15. Descriptive properties and degenerate cases of
Weierstrass' functions 338
13.16. Jacobian elliptic functions 340
13.17. Further properties of Jacobian elliptic functions • • 343
13.18. Descriptive properties and degenerate cases of
Jacobi's elliptic functions 349
13.19. Theta functions 354
13.20. The expression of elliptic functions and elliptic
integrals in terms of theta functions. The problem
of inversion • 360
13.21. The transformation theory of elliptic functions . . . 365

CONTENTS xvii
13.22. Transformations of the first order 367
13.23. Transformations of the second order 371
13.24. Elliptic modular functions 374
13.25* Conformal mappings 376
References 382
SUBJECT INDEX 384
INDEX OF NOTATIONS 393

ERRATA
HIGHER TRANSCENDENTAL FUNCTIONS, VOL. II.
P. 21, equation (34): Insert the factor * on the left-hand side.
P. 28, line 2: Read 7.13(34) instead of 7.1(34).
P. 29, line 14: Read 7.13.2 instead of 7.3.2.
P. 41, equation (82): The right-hand side should read
rTy sin(v7r)[s0^(2)-vs_lv(2)].
P. 49, equation (17): Read rTVi instead of rrV\
instead of J on the first line,
o 0
P. 91, equation (21): Insert z~] on the right-hand side, and read (2z) instead
of (z).
P. 95, equation (53): Read (t2 -y2)% instead of (*2-62)^,and (y2-*2)*
instead of (62-*2)1*.
P. 124, equation (2): Insert (cos t)v on the right-hand side.
OO 00
P. 148, equation (18): Read 2 instead of £ .
n= 1 n= 0
r °° r°°
P. 149, equation (2): Read J instead of J
P. 214, equation (11): Read (2m +3/2) instead of (2m + 3/4).
P. 226, equation <6): Read ez instead of e~z.
P. 312, line 11 up: Read Byrd instead of Bird.

CHAPTER VH
BESSEL FUNCTIONS
FIRST PART: THEORY
7.1. Introduction
Bessel functions are probably the most frequently used higher
transcendental functions. Broadly speaking they occur in connection with
partial differential equations, usually when the variables are separated,
or else in connection with certain definite integrals. We shall briefly
describe both types of applications and will start with the latter.
In 1770, Lagrange investigated the elliptic motion of a planet about
the sun. Let a, b be the semi-major and semi-minor axes, of the elliptic
orbit; write € = a"1 (a2 - b2)* for the eccentricity; also let r, M, E, be
respectively, the radius vector, mean anomaly, and eccentric anomaly.
The equations obtained by Lagrange are
(1) Af = £-<rsin£,
(2) t = a(l - e cosE) = adM/dE.
They give rise to the expansions
oo oo
(3) sin£ = 2 Ansin(nM)i cosE = B + 2 Bncos(nM)
71 = t ° 71 = 1
in which Bessel, in 1819, expressed the coefficients in the form of
integrals. For instance
A n = (%77rc)~1 f0 cos£ cos(rc£ - Tie sinE) dE.
By easy manipulations the integral occurring here can be expressed in
terms of Bessel coefficients [compare 7y3(2)and the recurrence relations
of 7.2(56)], and the first expansion (3) becomes
(4) sin£ = (% e)"1 2 sin (nM) J (ne)/n.
n= 1 n
Similarly, the second expansion (3) can be transformed into
oo
(5) cos£ = — Vie + 2 2 cos (nM) J'(n€)/n.
1

2
SPECIAL FUNCTIONS
7.1
Later, in 1824, Bessel made the integral, see 7*3(2), the basis for the
examination of the functions which now bear his name,
Bessel functions occur most frequently in connection with differential
equations. In Watson's monumental Treatise (Watson, 1944), which is the
standard work on Bessel functions, the history of these functions is
traced back to James Bernoulli (about 1700)t Since Euler (1764) and
Poisson (1823) Bessel functions are associated most commonly with the
partial differential equations of the potential, wave motion, or diffusion,
in cylindrical or spherical polar coordinates. However, Bessel functions
occasionally occur in connection with other differential equations or
systems of coordinates.
Let xy y, z be Cartesian coordinates, p9 <£, z, cylindrical coordinates,
and r, 69 <£, spherical polar coordinates, determined by the equations
(6) x = p cos <f>, y = p sin<£, z - z,
(7) x - r sin 6 cos<£, y = r sin 8 sin <f>9 z = r cos #♦
In these coordinates we have
(8) AF = F +F + F =F + p'] F + p'2 F. , + F ,
yvj/ xx yy zz pp ^ P <P<P zz
F F F F
(9) AF-F+2-E-+ —|2- + ctn0—2-+ —-^ ,
r rc r rd sin </>
If solutions of the wave equation AF = k 2F = 0 in the form f(p) g (<£) h (z)
or f(r) g(9) h(<f>) are sought, one obtains, in the respective cases, the
ordinary differential equations for /,
d2f df
(10)—L+ p-*-L+(k2-a2-v2p-2)f=Q,
dp dp
(11) r~*££L + [k* - viv + 1) r"2] / = 0,
dr
in which a and v are separation constants. The general solutions of these
equations are respectively:
(12) f(p)=Zv[p(kz-a2)'A]
(13) f(r) = r-y>Zv+,A(kr),
where Z denotes any Bessel function, or a linear combination with
constant coefficients of Bessel functions of order v%
The wave equation, and its solutions in various systems of
coordinates, can be used to give a physically plausible approach to the theory
of Bessel functions (Weyrich, 1937% Spherical waves of frequency v9

7.2.1
BESSEL FUNCTIONS
3
wave length A, and wave number k - 2 n/ A, originating at a source (£, rj,^)9
may be described by the wave function
^-i e-i27r(vt-R/\) = /£-' e-i&?yt+M
where R is the distance between the points (£, 77, £) and (x, y9 z). If the
z-axis is covered with sources of uniform density and phase, the resulting
wave motion may be obtained by superposition in the form
(14) u = e-**" jrjei + (* -o^"A exP }^fp2 + (* - OzVA\ d£
where p2 = x2 + y29 and by\Huyghens' principle this function represents a
cylindrical wave. With £ = z + p sinh r, equation (14) may be written as
(15) u = e~i2n/t f°° e ^coshr^
thus leading to Sommerfeld's integral representation of the Bessel
functions of the third kind.
Notations: In this chapter we adhere to the notations used in Watson's
Bessel functions. It may be worth while to mention a few notations which
occur in the literature but are not used here.
In Gray-Mathews, (1922, p. 25 and 23, respectively), two functions
F (z) and G (z) are introduced by
(16) Fv(t) - z-*" Jv(2z*),
(17) Gv(z)~%iirHixJ(z).
Jahnke-Emde (1945, p. 128) has
(is) AvU) = r(v+i)(y2z)-vJv(z).
In Whittaker-Watson (1946, p. 373),the modified Hankel function Kv(z) is
defined by
(19) Kv(z) = lAn[I_v(z) - £(*)] ctn(i^).
This differs from our notation sec. 7.2(13).
A function closely related to Neumann's^function Yv(z), 7.2(4),is
denoted by ? v(z) (Watson, 1944, p. 63) or by Yv(z) (Gray-Mathews, 1922,
p. 24):
(20) Yv(z) = Yv{z) ~nYv (z) e ivrr sec (vn).
For other notations of the "related" functions see sec. 7.5j6.
7.2. Bessel's differential equation
7.2.1. Bessel functions of general order
Bessel functions are solutions of Bessel's differential equation

4
SPECIAL FUNCTIONS
7.2.1
/ v i-7 ~ d2w dw ^ d ( dw\
dz dz dz \ dz J
+ (z2-v2)w = 0;
v, z are unrestricted, but for the present we assume that v is not an
integer. (For integer values of v see sec. 7.2 A •) The differential equation (1)
is a limiting case of the hyper geometric differential equation (cf. Klein,
1933, p. 156); it has a singularity of the regular type at z = 0 and an
irregular singularity at z = °o; all other points are ordinary points of the
differential equation. The standard method of obtaining solutions of a
linear differential equation in the neighborhood of a regular singularity
(Whittaker-Watson, 1927, 10.3) leads to the solution
(2) Jv(z) = S (-l)"&2)2"^/[m!r(m + i/+l)]
m. = 0
and tJ_ jU). The first solution, Jv(z), is called the Bessel function of
the first kind; z is the variable, v the order of the Bessel function. It is
easily seen that the series for z v J (z) converges absolutely, and
uniformly in any bounded domain of z and v. Equation (2) may be written
as
(3) Jv{z) = {%z)v 0F,(v+ l;-%z2)/r{v+ 1)
= (%z)v e~iz xFx(v+lA;2v + 1; 2iz)/T{v + 1)
by Kummer's relation, 6.3 (7).
The linear combinations
(4) Yv(z) = (aim*)"1 [</>) cos {vn) - J_v(z)],
(5) H™{z) = Jv{z) + iYv(z) = [i sin Wn)]~' [J_VU) - Jjz) e~iV7T],
(6) #<»(*) = Jv(z) - iYv(z) = (i sin wr)"1 [«?„(*) e ivTT - J_viz)]
are likewise solutions of (1). Yv is called the Bessel function of the
second kind or Neumann's function. H (^J and H i2) are the Bessel
functions of the third kind, also called the first and second Rankel
functions. From (5) and (6) we have
(7) Jv{z)^yt[HtlJ{z)+H^(z)]t
(8) Yv(z) = % [H<»(*) - H'« (*)]/i.
From the definition it is seen that
(9) tfi^-e^tf £»(*), HiJl(z)=e-i^Hlz)(z).

7.2.2
BESSEL FUNCTIONS
5
Also, if z denotes the complex number conjugate to z, and similarly for
other quantities, we have
do) 7/7) = J-G), YJz~) = y^G),
H«Kz) = //12>U), H«Kz)=H«Hz).
In particular, Jv and Yv are real if the order, v, is real and the variable z
is positive. All the four Bessel functions are single-valued in thez-plane
cut along the negative real axis from 0 to -<». For general v, they all
have branch points at z - 0. The Bessel function of the first kind is
clearly an entire function of v9 and later it will be seen that, with a
suitable definition for integer v = n, the Bessel functions of the second
and third kind are also entire functions of v%
7.2.2. Modified Bessel functions of general order
If z is replaced by iz, Bessel's differential equation (1) becomes
„ d2w dw
(11) zz — + z U2 + v2)u> = 0.
dzz d.
z
If v is not an integer (for integer values of v see sec. 7<2«5), J (iz) and
J-V(iz) are two linearly independent solutions of (11), but more often the
function
(12) Iv(z) = e-iXv"Jv{zei'A7T)= f (V2z)2a+v/[m\r(m + u+l)]
m. = 0
(Kz)v VAzYe-*
= 2-2*-*z-* Af0fI/(2z)/r(i/+l)
[compare 6*9(11)] and I_v(z) are used. They are known as the modified
Bessel functions of the first kind and are real when v is real and z is
positive.
The function
(13) Kv(zV- Y2 iKsini/ir)-1 [!_„(«) - !,(*)] = (W*)* ^v(22)
[compare 6„9(l4)] is likewise a solution of (11). It is known as the
modified Bessel function of the third kind or Basset's function (although
the present definition is due to Macdonald).
Clearly we have
(14) K_Jz)=Kv(z),
and from (12), (5) and (6) it follows that
(15) Xv(«)=^£ireiX,w^I,(«ei5Jw)—K*ire-'Xvfffl»>Ue-i}«w),

6
SPECIAL FUNCTIONS
7.2.2
so that
(16) Kv(zeilA") = y2ineiX™Hl}\zei") = -y2ine-iX™Hl2Hz)9
(17) H™(z)— — e-M^K^ze-M").
K (z) is real when v is real and z is positive.
7.2.3. Kelvin's function and related functions
Kelvin's functions ber(*)and bei(x)(x real)are defined by the equation
(18) ber(*) + i bei(x) = JQ(xe i%7r) = IQ(xe VArr).
Extensions of this definition to Bessel functions of any order and complex
z are given by the relations
(19) berv(z)±l-beiv(z) = c7v(ze±i^),
(20) k^v(z)±i\ieiv(z)^eTi%V7TKv(ze±iV^).
Instead of (20) we may use
(21) herv(z) + ;heiv(z) = R^ize™*)
(22) herv(z) - i heiv(z) = ff J? (**"** *)
so that
(23) 2kerv(z)=-7rheiv(z); 2 keiv(z) = tt her^z).
The functions ber^(z), beiv(z), kerv(z), keiv(z), herv(z), heiv(z) are
real when v isrealandzisrealandpositive. (For details see McLachlan,
1934, pp. 119, 168.)
7.2.4. Bessel functions of integer order
Bessel functions of the first kind of integer order are known as Bessel
coefficients. If n is a positive integer, the first n — 1 terms in the
infinite series defining J_ (z) vanish because of the poles of the gamma
function in the denominator. The remaining gamma functions may be
rewritten as factorials, and we have
c/_n(z)= 1 (-lVOSz^-VtmUm-n)!],
m = n
or, with m = n + /, I = 0, 1, 2, • •. ,
(24) J (z) = (-l)nJ (z).
This relation holds for all integers n.

7.2.4
BESSEL FUNCTIONS
7
Bessel coefficients are generated by the expansion of exp[%z (t — £~T)]
in powers of t. To prove this we note that
e%zt e-*z/t= f (%zt)i/u s (-y2zr})a\/m\,
J= 0 ra = 0
and the coefficient of tn in this expansion is exactly J (z). This leads to
the generating function
expP/s z(*-r')]= 2 *nc/n(z),
n = —oo
or, replacing z by az and £ by t/a to the more general expression
(25) exp[1/22U-a2rI)]= S (t/a)nJnUz)
n = — oo
= «70(a*) + 2 Jn(az) [(*/a)n + (-t/a)_rt]
for the Bessel coefficients. With a = 1 and t = e1^ we obtain the formula
of Jacobi-Anger
(26) eizsin^ « 1 eirt* J (z)
n = -oo n
= e/ (z) + 2 2 e/rt (z) COS
and with t = ie l<^
(27) e"cos^= I ^"e^^c/ U)-c70U) + 2 1 in J (2) cos (/* <£).
If v is an integer, the right-hand sides of (4), (5), (6) appear in
indeterminate form. However the limits of these right-hand sides as v -> n
(integer) exist and may be taken as the definition of Bessel functions of
the second and third kinds of integer order. Clearly it will be sufficient
to evaluate
Y (z)~ lim Yv(z) n =0, 1, 2, ..• ,
v-* n
By L'Bospital's rule applied to (4) we obtain
[dJ dJ 1
From (2) and 1.7(1)
dJ oo .„ \b{v + m+\)
(29) —v- = J (z) log 04 z ) - 2 (-1) » (K z) *+2» — = —
dv v « = o m\Y(v + m + V)

8
SPECIAL FUNCTIONS
7.2.4
dJ^
\/)(-u + m + 1)
(30)_^=_c7 (2)l0g(i/22)+ 2 (-D-CKz)^2"^- —,
and from formulas 1.17(11) and 1.17(12) for m < n - 1,
lim xfj{-v + m + l)/r(-v + m + 1) = (-1)""" (n - m - 1) ! ,
V -» n
so that from (30) and (24)
(—^ = (-Dn ~ JJz) log(K22) + "l (V2z)2m~n in-m-1) \/n
+ 2 (-!)■"" %*)
2m —n
*/»(to + 1 - ft)
r(/7l + l-ra)/7l!j
(For special values of 1/ in (29) see Mitra, 1925, Airey, 1935 a, and also
Miiller, 1940.) With a new index of summation / = to — n, the infinite sum
in this expression can be written as
I (-l)ICi2)2l+n^(Z + l)/[Z !(/ + ")!],
1 = 0
and so we obtain
(31) vYn(z) = 2Jn(Z)leg(y2z)-n:2} (H2)to""(n-m-l) \/m !
m = 0
-1 (-i)'(x-«)-«^0,^,;1);/,(<+1) »-i.2,3
1=0 I l(n + I) I
which may be written as
(32) 7TYn{z)=2[y+log(y2z)]Jn{z)-n2 (Hz)to""0»-m-l)|/m!
Bl = 0
■ a0 L 771 ! (/I + 77l) I J
where we have used 1.7 (9) and put
hn = 1~* + 2~* + ... +m-t m = 1, 2, 3, ... , AQ -0.
If v - 0, it follows from (30) that the finite sum in (32) is to be omitted.
Therefore, we have
(33) * Y0(z) = 2 [y + logC/2*)]</„(*)-2 1 (-D-O^z)2" (m\rzhK,
with the same meaning of A as in (32). It is to be noticed that according
to (28)

7.2.6 BESSEL FUNCTIONS
-a- \V±
L
(34) Y_»= lb, [cosM-' I -»-COS^)-J-»(z)
= (-DnynU) ti = 1,2,3,....
With this definition of Y (z) and Y_ (2) and with the corresponding
definition of Bessel functions of the third kind, all Bessel functions
become entire functions of 1/.
7.2.5. Modified Bessel functions of integer order
From (24) and (12) we have
(35) InW = JW /i = l,2, 3, ....
We therefore take In(z) and Kn(z)as a fundamental system of solutions of
(11) where
(36) K (z)= lim KJz)=(-l)
V -
4L—- — J„,
In a similar manner as in sec. 7.2*4 we obtain
(37) K (2) = (-l)rt+' In(z) logOS*)+J< "i1(-l)''(1/22)2'''ft(ra",n'"1)!
m = 0 ml
+ 1/2(-Dn 1 (^z)"*2" [^(n+m + l) + ^(m+l)]/[i»!(B + i»)!]
* = 0
n = 1, 2, 3, ... .
!In case n = 0 we have
(38) KAz) =- I. (z)log(y2z)+ I (K*)2"1 0(m + l)/[(m!)2].
u m = 0
With the definition of Kv(z) completed in this manner, we have an entire
function of v.
7.2.6. Spherical Bessel functions
The Bessel functions and modified Bessel functions reduce to
combinations of elementary functions if and only if v is half of an odd
integer (Watson, 1944, 4.7 to 4.75). We shall express here K +1/ (z) for
7i = 0, 1, 2, ..., in terms of elementary functions. The corresponding
expressions for the other Bessel functions follow from (16), (17), (7),
and (8), and are recorded in sec. 7.11. When7i=0, 1,2, •• • , and v -n + %
we have from 7.3 (16)
-<*>-©" T
(39) Kn+1/(*) = [— 1 "7" J^c-'(l + «/22)»«»A.

10
SPECIAL FUNCTIONS
7.2.6
Now the binomial expansion of (1 + xAt/z)n terminates and at once leads
to the representation of K n+% (z) in finite terms in the form
(40)K.+yW-f-^y.-. i (2,)-- TU+m + l) .
n+/$ \2V - = ° mIFGi + 1-m)
Using Hankel's symbol
(„, m) l(4v2-l)(4v2-32) ••• [4^2-(2m-D2]}
m!
= r(l/2 + v + m)/[m ITOi + v- m)],
[compare 1,20(3)], this can be written as
(41) Kb+x (z) = (W*)K e_z 2 U + & m) (22)"".
m ~ 0
Hence, for instance if n = 0, we have
(42) Kx(z) = (W*)x e"z.
From (42) and also 7W11(22) we obtain the representation
For the other types of Bessel functions see formulas 7,11 (l)to7.11 (13).
Bessel functions whose order is half of an odd integer often occur in
connection with spherical waves, and in this context Sommerfeld's
notation,
(44) ^.U) = (W*)X </,+„(*),
(45) <r„("(Z)=(Wz)K#<Vz),
(46) <r(^z) = (W^#<Vz),
is often used. Sometimes ift (z) denotes a slightly different function
(Watson, 1944, 3.41). For a class of polynomials connected with the
spherical Bessel functions compare Krall and Frink(l949) and Burchnall,
(1951).
7.2.7, Products of Bessel functions
In order to obtain an expression for the product J (az) J (/3z) of two
Bessel functions in the form of a series of ascending powers of z we use
(2) and Cauchy's rule for the multiplication of power series. Thus the
coefficient of (-l)m (l/2 azV (l/2j3z)v (l/2az)2m is found to be
2 (B/a^/ln ! r(i/ + n + 1) (m - n)! T(/z + m - n + 1)].
n=0

7.2.8
BESSEL FUNCTIONS
11
This may be expressed as a terminating hypergeometric series by means
of formulas 1 «2 (3), 1 <20 (5), and 2.1 (2) and leads to the expansion
(47) r(v+ 1) J(pz) JAaz) = Maz^MfizV
M
I
(-!)• Maz)2'
« = o/n!r(/z+ /71 + 1)
,Ft(-/w, - fi-m; v+ 1; 02 a"2),
This expansion simplifies when /3 = a, because then the hypergeometric
series may be summed by Gauss's formula 2.1 (14), so that
(48) J (2) J (2)= 7, .
v M „^o /n!r(/x + /w + l)r(v + /n + l)r(v + /x + m + l)
In the notation of generalized hypergeometric series
(49) r(i/+i)r(p + \)Jv{z)j^{z)
« (VzzY^ 2F3Oi + Jiv + fa, 1 + Hv + y2fi; 1 + v, 1 + /x,
1 + v+ fi; -z2).
From (48) we easily deduce the expansion
v n^o » !r(2v + w + l)
7.2.8. Miscellaneous results
Differentiation formulas and recurrence relations follow. From (2) we
find that
(50)4-[^>)] = 2v 2 H)"(y22)2fl+v"VU!ru + v)] = ^J1(2),
(5D^-[z-vj (z)] = z-v f (-i)m (1/22)2m+l,"V[(/n-i) iru + v + i)]
and hence by repeated differentiation
(53)f—\ b"v^W]-(-l)--""-^+.W m-1,2,3
V Z</z J
From (50) and (51) it is obvious that
(54) zJ'v(z) + vJv(z)=zJv_y{z\

12
SPECIAL FUNCTIONS
7.2.8
(55) zJl(z)-vJv{z)=-zJv+l(z)
and hence
(56) c7v.l"U) + c7v+l(2)-2i/2-1 J»,
(57) c/I/.lU)-c7I/+IU) = 2c7;U).
By virtue of (4), (5), (6) the same relations are valid for Bessel functions
of the second and third kind* Relations (12), (13), and the previous
results give similar fonnulas for the modified Bessel functions. For these
see sec, 7.11.
From the recurrence relations the following inequality (Szasz, 1950)
may be derived
Wv(x)]2 - Jy^ (x) c/v+1 (x) >{y + I)"1 Wv{x)Y v > 0, x real.
WRONSKIANS
The Wronskian of two solutions w} and w2 of (1) is a constant multiple
of exp [ - J z~} dz\.
(58) W{w}, w2\ = w^w'z -w2w\ =Cz-\
The constant C can be computed from the first terms of the series
expansions of the solutions involved. If we take wf = Jv(z)9w 2 = J-V(z) we
find from the series (2) that
lim zW = - (2i/)/[T(l - i/) HI + i/)] =- 2ir"! sin (vjt) = C,
z -> 0
and therefore we have
(59) W[Jv, J_V)=-2UZ)-' sin (mi).
If v is an integer, this Wronskian vanishes, thus confirming the result
of sec. 7.2,4 about the linear dependence of Jn and J_n. For other
Wronskians of Bessel functions or modified Bessel functions see sec. 7.11.
From (59) and (54) it follows that
(60) J.y+| (z) Jv(z) + J_v(z) Jv_} (z) = 2(nzri sin M.
For other similar formulas see sec. 7.11.
ANALYTIC CONTINUATION
The Bessel function of the first kind of variable ze m7r where m js any
integer, may be expressed by (2) as
(61) J^ze^^-e^^J^z) m = ±l, ±2, ±3, ... .
For the corresponding relations for the othertypes of Bessel functions see
sec. 7.11.

7.3
BESSEL FUNCTIONS
13
DIFFERENTIAL EQUATIONS
A large class of differential equations whose solutions may be expressed
in terms of Bessel functions has been obtained by Lommel. One of
Lommel's transformations is
where £ is the independent variable and v the new dependent variable.
This tranformations carries (1) into
(62) C j£- + (1 - 2a> C yj + UyO1 + («2 - S y2)] v = o.
If w} (z) and w2 (z) are any two linearly independent solutions of Bessel's
equation, the general solution of (62) is
(63) v^Cw^O and v2=Caw2(p£y).
For other differential equations whose solutions can be expressed in
terms of Bessel functions see Kamke(1948, p« 440).
The general solution of the inhomogeneous Bessel equation
„ d2w dw
(64) z2 r + z + {z2-v2)w=f{z)
dz dz
may be obtained by the method of variation of parameters in the form
(65) w =Aw}(z) + Bw2(z) + u(z)
where w (z) and w (z) are two linearly independent solutions of the
homogeneous equation (1), u(z) is a particular solution of (64) defined by
(66) Cu(z) =- w (z) J* r} w2(t) f(t) dt + w2(z) J* t~] w^{t) f(t)dt
0 0
and C is the constant in the Wronskian of if and w [cf. (58)].
The functions J' (z) and azJ1 (z) + bJ (z) satisfy the following
differential equations respectively,
(67) z2(z2 - v2) ^- + z (z2 - Zv2)^- + [{z2-v2)2 - (z2+ v2)] w = 0,
dz dz
(68) z2 [a2 (z2 - v2) + b2] -J?--z[a2(z2 + v2)-b2]-^-
dz dz
+ [a2(z2 - v2)2 + 2abz2 + b2{z2 - v2)] w = 0.
7.3. Integral representations
7.3.1. Bessel coefficients
If Cauchy's theorem of residues is applied to 7.2(25) we obtain

14
SPECIAL FUNCTIONS
7.3.1
(1) 2m Jn(az) = an j£rn"! exp [V2 z {t - a2 t'*)] dt n = 0, 1, 2, ... .
C is any simple closed contour in the Nplane around the origin. If in
(1) we put a = 1 and choose C to be the unit circle around the origin,
t = e l<^, we have
(2) 2n Jn(z) = Jo27re £Usin *-»*>d0 = 2 J^ cos (2 sin <f> - n <f>) d <f>
n = 0, 1, 2, ... .
This is Bessel's representation.
7,3.2. Integral representations of the Poisson type
For general v we have Poisson's integral representation [for a
generalization of this formula see 7.8(11)]
(3) r(i/+ J5) «/„(*) = 2tT* (%zV J ^cosC* sin <f>) (cos tf)*'^
Re v > - &
This result may be proved by expanding cos(z sin <f>) into a series of
powers of z and integrating term by term. In this process one encounters
the integral
/^(sin^)2* (cos eft2* d <h
o
which is found to be equal to
y2 r(v + i/2) ru + y2)/ru + * + d
by virtue of 1.5(19)« Therefore, we have
r(i/ + x)r(in + X)
r(i/+J4)c7,» = ir"x(JiOv S (-1)"*
2m
« = o (2/n) ! T(v + m + 1)
Using the duplication formula from 1.2(15) of the gamma function for
(2/n)! = r (2/n + 1) and remembering also 7-2(2) the result (3) is
established. Slight modifications of (3) are given in sec. 7.12.
jPoisson's integral, in the form of 7.12(6), may be used to derive an
inequality for Jv(z). Let v be real, v > - % and z = x + iy (x, y real);
then we obtain
r(v+1)|Jiz)]<c «r*0£|i|)w/"V*(costf)2"^
and by virtue of 1.5 (19)
(4) :\Jv{z)\£ \%z\v e^/T{v+V)
[see also 7.10(22)].

7.3.3
BESSEL FUNCTIONS
15
7.3.3. Representations by loop integrals
Bessel functions for unrestricted values of the order v may be
represented as loop integrals. Let a be a complex number with Re a > 0; then
we have the representation
(5) 2in JvU2)=ZvJ"_^+,exp[K2aU-Z2r,)]r1'-1 dt
= (^/2)''/<0+,exp[a(^-^^^r,)]^I'-, it
— oo
Rea>0, |arg£|<7T.
Here the symbol, J_ denotes, as usual, integration along a contour
which starts at infinity on the negative real Naxis, encircles the origin
counter-clockwise, and returns to its starting point• Clearly (5) is an
extension of (1), for the integrand in (5) is one-valued, and the loop may
be deformed into a closed contour around the origin, if v is an integer.
To prove (5), we use the expansion
exp[-az2/(4*)]= 1 (-1)" i}/Aaz2Y t~m/ml
m = 0
in (5) and integrate term by term. From 1.6 (6) we obtain
-*— oo
Therefore, we have
Jl0+) exp[at-V4azzt-']rv-'dt
— oo
= 2m(y2z)-v 1 (-1)" {Xaz)'m+v/\jnirbn + v+l)l
m = 0
and using 7,2(2) this establishes (5).
The corresponding loop integral for the other types of Bessel functions
may be obtained using formulas 7,2(4) to 7.2(6) and formulas 7.2(12) and
7.2(13). For these see McLachlan and Meyers (1937).
When Re v > — 1 and a is real and positive, the contour in (5) may be
deformed into one parallel to the imaginary axis, leading to
(6) 2niJ (az)^zvJc\iPOe%a^-z2r')rv-%dt NO R . -,
7 vv Jc-ioo c,a>0, Ke v > -1,
HANKEL'S REPRESENTATIONS
Generalizations of Poisson's integral (3) were given by Hankel. The
first of these is
(7) 2mjv(z) = Tr-xr(y2-v)(y2Zy /"♦•-'-^"'(t2-!)"-* a,

16
SPECIAL FUNCTIONS
7.3.3
v + Y2 not a negative integer. The path of integration is the figure eight
indicated in the diagram below.
The initial amplitude of (t - 1) and (t + 1) at the point of intersection
with the positive real axis on the right-hand side of t = 1 is zero. To
prove (7) we replace the original contour by the dotted one. If we assume
that Re {v + Yz) > 0 and make the radii of the circlesaround ± 1 tend to zero,
then we obtain
J (t+'-t~) eizt(<t2_ l)V-% dt = 2i cosM^e^a^2)^ it
Re v>-V2.
If the integral on the right-hand side is expressed by 7.12(7), we obtain
(7). By the theory of analytic continuation the restriction Re v*>— % may
be omitted as long as v + % is not a positive integer.
Another representation [for a related expression compare 7#8(13)] is
(8) 2m Jv(z) = 7T~'A T(% + v)ei3™<%z)~v
xJ<",*8,,+J»*<*8-l)-"-**
v + Vi^ 0, - 1, - 2, ... , S<arg/<277 + 8, - 8< argz <tt- 8.
The path of integration is indicated in the figure below,
-1
///
/ / I
/ / /
/ /
/ /
/It f
' 1 /
\
t- plane

7.3.4
BESSEL FUNCTIONS
17
and the initial and final values of arg t are taken to be S and 2n + 5. To
prove (8) we take the contour to lie outside the unit circle. Then we have
r(x+v)(t2-irw-5J= I r(K + i> + m)r2i'-2"-7ra!.
m = 0
We insert this in (8) and integrate term by term. Then from 1.6(6) with
^ze'^^we obtain
J(o+) r2v-2*-i ei*tdtTS2niz2v+2m e"i377(x,+tt)/T(2v+2/Ti + l)
- 5 < arg z < n - 5.
Thus we have
v « = o m!r(2m + 2v+l)
Using the duplication formula from 1,2(15) for the gamma function, (8)
is established.
7.3.4. Schlafli's, Gubler's, Sonine's and related integral representations
From the results of sec. 7«3«3 a number of representations in the
form of definite integrals may be obtained.
SCHLAFLI'S REPRESENTATIONS
In (5) we interchange a and z, put a = 1, and deform the loop into a
path consisting of the real axis from — °o to — 1 (arg t - —n)9 the unit
circle in the positive sense around the origin (— 77 < arg t < n), and the
real axis from —1 to — °° (arg t - 77). The result is Schlafli's representation
(9) n J (*)- J77 cos (z sin<£ - v<f>) d<f> - sin(w) J°°e'^^^^dp
0 0
Re z > 0.
It still holds in case Re 2=0 provided that Re v > 0. Formula (9) reduces
to (2) in case v is an integer. Also 7.2(4) and (9) admit a similar
expression for Neumann's function
(10) u Y(z) = F sin (z sin* - vt) dt - f x(evt+e~vt cos vn) <TIsinh *dt
Re 2 >0.
[For the first integral on the right-hand side of (9) and (10) compare
7.5(32).] Generalizations of (9) and (10) are given in formulas 7J2( 17)
and 7.12(18).
GUBLER'S REPRESENTATIONS
From (8) another representation for J (z)maybe derivedby specializing

18 SPECIAL FUNCTIONS 7.3.4
the contour. i-plane
1
1
t
1
1
( , - + J
\ r
^ _ s
1
1
1
k.
1 A'-
1 1
We choose 5 = Ym and deform the contour into the dotted line. If Re v <V2
and the radii of the circles around ± 1 tend to zero, we obtain the result
(11) r(yi-V)Jv(z)=2n-1A(y2zmS* (l-t2rv-'A cos (zt-vn)dt
~sm(v7T)jaa(l + tz)-v~'Ae-ztdt] Re2>0, Re v < Y2.
0
This formula corresponds to Poisson's integral (3)t If in (12) vis replaced
by — v and this is combined with (3) and also 7.2(4), the corresponding
expression for Neumann's function is
(12) r(v+1/2)yv(2) = 2^(1/22)v[J1 (l-*2)*-Ksin(**)ifc
^S°°e'zta+tz)^y2dt] Rez>0, Re v>-V2.
By introducing Struve's function 7.5(78) in (12) we have
(13) [Hv(2)-yv(2)]r(v+1/2)=2^(1/22)vJo°0e-2Hl + ^2)v^^
Re 2 >0.
Now in (8) we take 5=0 and as a path of integration the dotted line.
Replacing z by ze l^rr and v by -v we obtain, as we suppose Re v > -%
(n order that the radii of the indentations around t = ±1 may tend to zero,
(14) I.v(2)=^"3/2r(1/2-i,)(^2)v[sin(2W)J100e^^2-l)^^^
+ cos(i/ir) f1 ezt(\-t2)v~%dt] Rev>-1/2, Re 2 >0.
Hence and by the aid of formulas 7.2(13), 7.2 (12), and 7.2(14) we obtain
the result
(15) T(v+V2)Kv{z) = n*(y2z)v j~ e-«{t2 -l)v-v> dt
Rei/>-y2, Re z >0.

7.3.5
BESSEL FUNCTIONS
19
Hence, with t — 1 = v/z, we have
(16) Hi/+ «)«(*) = &«/*)* e"1/00 e-"vv-*A (1 +W*)""* dv
v o
|argz|.<77, Re v>~\^
or more generally
piS
(17) r(i/+JflK(z).(W*)* e~*r e"'*v~* (\ + Kt/z)v'* dt
o
Re v > - JJ, |g| < i^ 5 - 77.< arg z < 8 + 77.
7.3.5. Sommerfeld's integrals
If we evaluate Je lz cos r e ^(r"^7r) rfr taken along the rectilinear
contours c t (from - l/2rr + ioo to %/r — ioo) and c2 (from % tt — j.oo to 3y2?r + i.oo)
(cf. figure),
a = 0-7r j6 a = 0
a = 0 + 27r
*- a
a=77-0 a=27r-0

20
SPECIAL FUNCTIONS
7.3.5
we find from (9), (10), 7.2(5) and 7.2(6) that
(18) irff™(*)=£ eizcosreij/(T'^dT9
(19) 77#(2>U)=£ e"COSTe£v(T^77)rfr,
both integrals being convergent for Re z > 0. The contour cy may be
replaced by a contour Cy from -rj + i°° to 77 - i°°, where 77 is a suitable
number between 0 and 77. With the notations
<& = arg2, a = Re r, /3=Imr, r = a + £/3,
it is easy to verify that Re (i'z cos /3) is represented asymptotically by
- \z\ cosh/3 sin($ + a)for large /3. The upper or lower sign is to be taken
according as /3 <0. Therefore the integrand of (18) vanishes exponentially
as r -> 00 in the shaded part of the r-plane. We may replace c } by C as
long as—77 < <& < Y277 or - Y277 <Q < n - 77 according as 0 < 7/ < Y277 or
% 77 < 7/ < 77 • Thus we have
(20) 7r#(t)(z)= T ei2Cosre£v(r-^77)rfr
and similarly
(21) 77H{jKz)= Jc e^osTgivd-^)^
2
C2 being a contour from 7/ — i«> to 2 77 — 77 + J°°. The integrals are
convergent for
(22) -7/<$ = arg2 <tt- 77, 0 < 77 < tt,
and by the theory of analytic continuation this is the range of validity for
(20) and (21).
With these results it follows from 7.2(7) that
(23) 2t7J(z) = J eizcosreiv{r^A7T)dr9
— 77 < arg 2 < 77 — 77, 0 ^< 77 < 7T,
C3 being a contour from -77 + 1*00 to 2 77 - 77 + i«».
Very often the contour integrals
(24) 7rff^H2) = -iX.°l+i7762sinha~varfa,
(25) 77H{2)(z) = i f°°~i7Te*sillha-vada,
"V —00
(26) 2uJ (z)=-iJ°°+i'T e'aiBha-vada,
00-177
valid when |arg z\ < Yin, are used. They simply are deduced from (20),
(21), and (23), respectively, taking 77 = Yi 77 and introducing the substitution
t = Yi 77 + i a.

7.3.6
BESSEL FUNCTIONS
21
SPECIAL CASES
We choose 77 = 0, take the contours Cy and C2 to be rectilinear, and
obtain Heine's expressions
(27) 7rfl")U) = -l-e-^V7rJ°°e"<:osht e~vtdt 0 < arg z < n,
(28) 7rHl2)(z) = 2ieiy>V7T[£ ei2COsht cosh {vt-ivir)dt
- i jj7 e~"cos* cos (PL) dt] 0 < arg z < m
If we take 77-77" and C,, C£ to be rectilinear, we obtain
(29) irfl"}(2)— 2ie^i^^[JO°e-"C0shtcosh(v^ + iV7r)^
v 0
+ ije i2COS * cos (vt) dt] - 77 < arg z < 0,
0
(30) 77/^2)U) = iV*™£V*eo*te-,'*<fc - n< arg z .< 0.
From (27) to (30) we obtain respectively using 7.2(7)
(31) 7rJ7(2) = e^V77[J77e-"cos*cos(v^)c/^sin(w) jVvtf£l cosh *efe]
v 0 0
0 < arg z < 77,
(32) irJvU)-c"£Xw[J^c"°"*cos(i/0*-ain(wr) Jj° e"vt'izco&ht dt]
- 7T < arg z < 0.
In (27) let e *= v/a; then we have
(33) n Hl'Kaz) = -if^^ av J" eilAzi^aZ /v) v'v~' dv
Im 2 > 0, Im (az z) > 0.
7.3.6. Barnes' integrals
A representation of the Bessel function of the first kind as a Mellin-
Barnes integral (see 1.19) is
(34) 4tt1V,U) = Jc+i°° (1/2^pr(1^ + 1/2s)/r(l + 1/2^1/2s)rfs
c~i°o
5C€ erfa4-a' * > 0, -Rev<c<l,
and may be proved by evaluating the integral in terms of the residue of the
integrand or applying Mellin's inversion formula to 7.7(19).
If the restriction - Re v < c < 1 is removed, the integral still makes
sense, but it need not represent a Bessel function. We put
(35) 4irf j Ax) =5 (Xxr*mv + y2s)/ra + y2v-y2s) ds
v'm a-ioo
x>0, o<l, -2m-Re v<a<-(2m- 1) - Re v, m = 1, 2, ....

22
SPECIAL FUNCTIONS
7.3.6
the integral being taken along a line parallel to the imaginary axis. The
evaluation of the integral in terms of the residues of the integrand gives
4irJ,BU) = 4ircM*)-4ir» 2 (-l)n
-_ri . (Kxy*2*
v'm v n-o nir(i/+n+D
We define for arbitrary complex values of z and v
(36) J (*)- 2 (-l)"(KZ)v+2B/[n!r(v+n+l)] m-1,2,3
and call this the cut Bessel function of the first kind. From (33) we have
(38)-^- [*-"^>)]--*-,'«U,.<*).
7.3.7. Aiiy's integrals
Airy's formulas,
(39) J"°° cos (tz + 3j*) <fe = U/3)x X 1/3(2*3/*) * > 0,
(40) /e°coflfr3-3to)&--ir/3*x [J}/3 (2x3/2) +J_}/3 (2*V2)] * > 0,
can be proved as follows. In (39) we substitute t = 2x*sinh yv. Since
4(sinh r/3)3 + 3 sinhte/3) = sinh v
we obtain
J°° cos (*3 + Ztx) dt » 2x^/3 J°° cos (2*3/2sinh v) cosh (v/3) rfwf
0 0
and using 7•12(25) this establishes (39)*
To prove (40) we express the right-hand side of (39) by means of its
power series [see 7,2(12) and 7.2(13)] and obtain
f °°cos(*3 + 3tx) dt
= 1/3 77
„ 3m oo 3m
X
A * i n-1/3 +*+ D x A
=0 ... . A v-*,- . ... ^ ., -5.0 T(l/3 + m + 1) m !
Here we replace * by - x and using 7,2(2) we obtain (40). For
generalizations of the formulas (39) and (40) see Watson (1944, pp. 320-324).
7.4. Asymptotic expansions
The asymptotic behavior of Bessel functions is different according as

7.4.1
BESSEL FUNCTIONS
23
the order v9 the variable z, or both of these quantities increase
indefinitely. The power series expansions of 7.2(2) are asymptotic expansions
when z is fixed and v -> «>. It is comparatively easy to derive asymptotic
expansions for the case that v is fixed and z -> 00; when both v and z are
large, the investigation becomes much more involved.
7.4.1. Large variable
We shall derive here the asymptotic expansion of the modified Bessel
function of the third kind, Kv(z). The corresponding expansions of other
Bessel functions may be obtained by means of formulas 7,2(16), 7,2(17),
and 7.2(8); the results are given in sec. 7.13 .1.
We start with the integral representation 7.3(17),
r(v+y2)Kv(z) = (y27T/zYA e~* fo°°el e^t^m^t/zy^dt,
ReV>-1/2, \8\<Y27T9 8-7T<aigZ<8+7T9
substitute the binomial expansion with remainder term,
^„ m\r(v+y2-m) M
(D — 0
and use 1.1(6) obtaining
(1) *„(,)-«./,)*.- TY r!Vrf+v} 1 {2z)" +*«1
L» = o mir(v+lA-m) "J
-37r/2<argz <3tt/2,
where the remainder is given by
(2) (M-i)\r(v+y2-M)RM
= (22)-" fie-Hv-**"it /o' (l-v)u-m + V2vt/z)v-,A-Mdv.
It is easy to see that for any fixed v with Re v > -%,
RM = 0(\z\~M)9 z-><*>, -37r/2 + e< argz <37r/2-e, e>0.
By a more careful discussion of (2) it may be shown that the modulus of
the remainder in (1) is less than the modulus of the first neglected term
(m = M) if v is real, 'M > v - V2 > -1, and Re z > 0 (MacRobert 1947,
p. 272; Watson, 1944, p. 207) and that the remainder is approximately
equal to half of the first neglected term when v and z are both real and
2z - M + Y2 is small in comparison with z (compare Burnett, 1929)* Airey
( see 1937), modified (1) so as to obtain a much closer approximation
suitable for numerical computation to high accuracy.
Using Hankel's symbol 1.20(3)
(3) („, m)--^ 1(4 v2-l2) - [4,2 - (2m-l)2]} = ™*V*m\ ,
ml mil v/z + v-m)

24
SPECIAL "FUNCTIONS
7.4.2
the asymptotic expansion may conveniently be written as
(4) «„(*)-(W*)*e"*[ "i\v, m)(2Z)-"+0(|Z|-*)]
m. = 0
- 3 n/2 < arg z < 3 tt/2.
Since only i/ appears in the definition of (y, m\ the restriction Re v>-Vi
may be omitted.
7.4.2. Large order
The first reliable investigation of Bessel functions with large variable
and order was carried out by Debye (1909) by means of the method of
steepest descents. This method is based on the following consideration
(Copson, 1935, p. 330; Watson, 1944, p. 235).
Suppose a function F (z) is given in the form
(5) F(z) = Jce-*f^g(a)da
where C is a contour in the complex a-plane joining two zeros of e~zf(a\
In many cases it is possible to choose C so that it passes through a zero
aQ of f' (a) and that the imaginary part of f(a) is constant along C. Thus
we have f' {aQ) = 0 and
(6) Im [f(a)] = constant = Im [/"(«<,)]
along C so that Re [jf (z)] changes as rapidly as possible when a traverses
C For large z, the modulus of the integrand has a sharp maximum at aQ
and only that part of C which is in the immediate neighborhood of a will
give a significant contribution to the contour integral (5).
For the sake of simplicity we assume that both order and variable are
positive and put
(7) z=x>0, V=p>0.
Moreover, we shall assume that the quantity vQ determined by
(8) sinhi;o = p/*, cosh vQ = (1 + p2/x2YA, vQ>0
is fixed as p, x -* <». We shall discuss K (x) only; the corresponding
expansions of other Bessel functions are listed in sec. 7.13 .2.
An integral representation for K (x) of the form (5) is immediately
obtained from 7.2(15) and Sommerfeld's expression 7.3(20) in the form
(9) Kp(x)=y2iJce-XC<*aeiPada=y2ifce-*f^da
where
(10) f(a) = cos a - ip a/x.

7.4.2
BESSEL FUNCTIONS
25
According to the results in sec. 7 «3«5, the contour C starts at — 77 + /00, ends
at7/—ioo jwhere 0^ 77 .<, n9 and lies entirely within the strip —77 <, Re a<i 77.
of the complex a-plane. The condition / (a) = 0 leads to
(11) sin a = — ip/x = - i sinh i>0 ,
and this equation has an infinite number of solutions
(12) a =s-£t;n +2irm m = 0, ± 1, ± 2
From these only aQ lies within the strip — 77 < Re a < 77. Hence we have
(13) a^-Hogl*-1 &)+(p2 + *2)*]}=-*i70
and from (10)
(14) f(a0) « cosh vQ - vQ sinh 17 0 .
The condition (6) shows that the path of steepest descent is the imaginary
axis, and from (9) with a = iv we obtain
(15) K (x) = y2J°0 e'xCos]iv+Pvdv = y2J°°e'x^v)dv9
p — 00 —00
where
g(v) - cosh v — v sinh v .
The substitution
(16) r =g(v) - g(v0) = cosh v - cosh vQ — (v — vQ) sinh v
maps the v -plane on the r-plane. The mapping is conformal except at the
points i;=i;+ 2nim where dr/dv has a simple zero. Thus
(17) GM-du/dr-fe'k)]-1
may be represented in a neighborhood of r = 0 in the form
(18) 0>(r)= 2 6 r*»-\
n= 1 n
and this expansion is convergent up to the next singular point r, which
corresponds to v = 17 ± 2 m.
As 17 increases from — <» to 17 , the variable r decreases from ©o to 0;
and as 17 continues to increase from 17 to <*>, the variable t increases from
0 to oo. We shall determine the coefficients b in (18) so that we may take
arg t — 2 77 on the former, and arg r = 0 on the latter part of the path of
integration. Then we have
(19) Kb) = y2e~Xf(Vo) J"~ e~" [*(r) - 4(re i270] dr.

26
SPECIAL FUNCTIONS
7.4.2
Here we use (18) and apply Watson's lemma (Copson, 1935, p. 218) to
obtain the desired asymptotic expansion
(20) K(x)-e'ZflVo) ["£ i^.^rfc+W + OU-*^].
F n= 0
The coefficients in (18) are obtained by Cauchy's theorem
(21) 4iti&b = Jr%n 4>(r) dr = J \g(v) -g{v0)Y%n dv,
the last integral being taken around a small closed contour encircling
v = vQ once in the positive direction.
Since [g(v) — g(*0]""n has a pple of order 2n + 1 at v = v we may
represent (v - ^0)2n + 1 [g(v) — g(^0)]~n as a Taylor series. We then have
(22)
k-"0)2n+' fe(»)-g(»„)]■""*- 2 4 {,n) („-i;0)Z
Z = o
On the other hand, Cauchy's theorem gives
(23) 2i» A <n, = / (v-v0Vn-l[g(v) - g(vo)]-n~'A dv,
taken around a closed contour encircling v = v . A comparison between
(21) and (23) gives for the coefficients in (20),
m l» = *<'-2S)!&b--,!,"kM-e''-rrf}„,
We thus obtain the asymptotic expansion
(25) K (*) = 2"* (p2 + **)"* expt - (p2 + *2)'/2 + p sinh"1 (p/x)]
m = 0
where
x [ 2 2" a r(m + »(p2+^T^+OU-*)] P,x>0,
a = 2*"™ (l+pV*2)X+X" 6, +1 •
01 * C.HI T I
The first few coefficients in (25) are
(26) a0 = l, a,=-i+-!-(l+*7p2r\
o zA

7.4.2
BESSEL FUNCTIONS
27
a = — (1 + *Vp2)"' + (1 + *7p2)'2.
2 128 576 3456 ^
A similar expansion derived by the method of the stationary phase
was given by Jt Bijl (1937, p# 23)t He gives the following result valid
f or p > xl/i > 1.
(27) |£p(x)-2-*(p2+*2)-*exp[-(p2+*2)*+p sinh-'(pA)]
x "l 2" d r(i» + J0(p2 + *2)-*"/(2iii)!|
n = 0
<Cw'2M(p2^x2)^exp[-(p2^x2yA^psmh^(p/x)]9
where w - px~" or (p2 + x2)* p~!/3 according asp<#^or>*^.
For the coefficients in (27) there exists the recurrence relation
/ TO- l\ # .
with ^0= 1, d t =J2 = 0. Here I J is interpreted as zero and
the sum is formed over all I for which to — / is odd and 0 < / < m — 3.
From (28) it follows that
(29) </0=l, J2 = 0, dA = -(p2 + x2YA, J6=10p2-(p2 + *2)*,
J8 = 56p2 + 35(p2+*2)-(p2 + *2)^ and
d1o=-2l00p2(p2 + A;a)+246p2 + 2l0(p2 + A;2)-(p2 + A;2)^.
The corresponding expansions for J (x) and H^\x) are obtained ina
similar manner from Sommerfeld's expressions shown in 7.3(20) and
7.3(23) by the method of steepest descents (compare Debye, 1909;Watson,
1944, p. 235; Weyrich, 1937, p. 49)« (For a discussion of the paths of
steepest descents for various cases see Emde, 1937, 1939, and Emde
and Ruhle, 1934.) Different cases are to be distinguished according as p
is larger, less or in the neighborhood of x% They are listed in formulas
7*13(11) to 7. 13 (16). Formulas for the upper bound of the remainder of
the expansions 7*13(11) and 7.13(14) respectively, and recurrence
relations for the coefficients have been given by Meijer (1933, p. 108), and
Van Veen (1927, p. 27), respectively.
Recently (compare Schobe, 1948) two different asymptotic expansions
for the second Hankel function have been derived from the contour integral
of 7.3 (25) • The terms of Schobe's series are not elementary functions
as in Debye's series shownin 7.13(11) and 7. 13(13) but involve the

28
SPECIAL FUNCTIONS
7.4.2
second Hankel function of the orders 1/3 and -2/3. The first term is just
Nicholson's formula 7.13 (27) and Watson's formula 7.3 (34) respectively.
See erwWl
7.4.3. Transitional regions
The asymptotic expansions 7.13 (11), 7.13 (13)and 7.13 (15)for#(u(*),
valid in case x > p, x < p and x nearly equal to p respectively, do not
cover all possibilities since the restriction x — p = 0(x)/3) has to be
imposed in the last case. In the transitional region, that is when p/x is
nearly equal to 1 while \x — p\ is large, other formulas have to be used.
These have been given by Nicholson (Watson, 1<H4, p. 248); Watson
(1944, p. 249); Schobe (1948); Tricomi (1949).
Nicholson's formulas for integer order n of the Bessel function of the
first kind are
(30) Jn(x) ~ n-< 3-"« (£/*)'/3 K,„(£).
(31) J>) - 3"2/3(#*),/3 [Jf/,(f) + J.V3(Ol
according as x < n or x > n and
[For the Yn (x) see 7.13 (24) and 7.13 (26).] Th ese formulas were derived
by means of the principle of the stationary phase (Watson, 1944, p. 229).
For this purpose we start with the integral representation 7.3(2)
(33) ftJn(#) = f cos(nc/) - x sin<£) dcf>.
o
The phase is stationary where d/dcf>{ncf> — x sin0) = 0 or cos <jy-n/x%
Since n is supposed to be nearly equal to x, <$> is small, and in the
neighborhood of the stationary point we may replace sin 0 by 0 - 03/6. Thus
7rJn(x) ~ J^ cos[x<£3/6 -U - n)<j>\d4>
~ J cos[*<£3A> - ix - n) 0] dcf>.
This is Airy's integral 7.3 (39) and 7.3(40) respectively, according as
x < n or x > n and the desired results (30), (31) are established.
This method of deriving Nicholson's formula is a questionable one;
moreover the range of validity and the order of magnitude of the error
cannot be determined. [A rigorous theory of the method of the stationary
phase has been given by van der Corput (1934, 1936). This method was
applied by J. Bijl (1937) to derive asymptotic expansions for the Bessel
functions.]

7.4.4
BESSEL FUNCTIONS
29
WATSON'S FORMULAS
A more precise fonn of Nicholson's formula was given by Watson
(1944, p. 250)
(34) ei7r/* Hi2)(x) = 3~1A we'ip{w ~w 3/3 -t**"1")
xJ^V>3/3)+0(p-f).
Here the order p is not restricted to be an integer, and we have
(35) w-(*2/p2-l)*,
where arg to = 0 for x > p and arg w = Vitt for x < p. The corresponding
formulas for J (x) and Y (x) are listed in formulas 7,13(28) to 7.13(31)t
In case x is nearly equal to p, w can be replaced by (%p) (x - p)
[arg(x — p)^ = 0 or Vnr for x > p or x < p respectively], and Nicholson's
formulas (30), (31) are obtained*
From his asymptotic expansion, Schobe (1948) derives the result (see
end of sec, 7.3.2), CCt trt^W\
-2/2
(36) ei7T/*H[2pHx) = ru* [— '
(±)"° (2..J-)
t=- Mxi-Hix-p)**,
3
and arg (x — p) V2 equal to 0 or 3 tt/2 according as x > p or x < p.
Another formula was given by Tricomi (1949). The results are
(37) u Jp \p + (p/6),/3 t] = (6/p),/3 A , {t)
- l/(10p) [3t2 A\ {$) + 2tAt (t)] +0{p-5/%
(38) nYp[p+ (p/6),/3 t] = (6/p),/3 A2(t)
+ l/(10p) [3i2 ^ (t) + 2t A2(t)] + 0(p-5/3).
Here, A f (t) and /I 2 (0 denote the functions
(39) At(t) = n/% (t/3)* lc7_t/3 [2(f/3)3/2] + JT/3[2(i/3)3/2]S,
(40) A2(t)- rr/3 tA \J_m [2(t/3)3/2] - J,/3 [2(i/3)3/2]}
[see Airy's integral 7.3(40)]

30
SPECIAL FUNCTIONS
7.4.4
7.4.4. Uniform asymptotic expansions
DIFFERENTIAL EQUATION METHODS
The asymptotic formulas discussed so far have been obtained from
integral representations for the Bessel functions mostly from Sommerfeld's
formulas (see 7.3.5). Another approach uses the differential equation
as its starting point.
For the following we restrict ourselves to positive real values of both
order p and argument x and transform the Bessel equation of 7.2(1) by
the substitution x - pey. The resulting equation is
(41) w"(y) + p2(e2y-l)w(y) = 0.
The asymptotic behavior of solutions of differential equations of the form
(42) w " (y) + [p z O2 (y) - K (y)] w (y) -0
in which p is a large parameter, has been investigated by several authors
(Horn, 1899; Schlesinger, 1907; Birkhoff, 1908; Blumenthal, 1912 ; Jeffreys,
1925; Jordan, 1930% The basic principle is that approximately identical
differential equations will have approximately identical solutions. In the
work of earlier authors the comparison equation has a constant 4>, and
therefore, all these methods fail in a region in which O (y) has a zero. In
the case of the Bessel equation this failure occurs in the neighborhood of
y = 0 or x = p.
Langer (1931, 1932, 1934) used a comparison equation in which O(y)
is essentially a suitable power of y and was thus able to cope with zeros
(of any order) of 4>2(y). The solution of Langer's comparison equation
may be expressed in terms of Bessel functions of order L/3. The
application of Langer's results to (28) leads to the following asymptotic
formula which is valid uniformly in 0 < x < <» (Langer, 1931, pp. 60-61).
(43) eirr/*H{pznx) = w-y*(w-tan-] wYA
x H\%{pw-p tan"1 w)+0{V-**) w = (x2/p2-l)X.
For x > p, arg w and arg (w - tan"1 w) are equal to zero; for x < p, arg w
is equal to %nf and arg(w - tan"1 w) is equal to 3^/2, [The results for
J (x) and Yp (x) are listed in formulas 7.13 (32) to 7.13 (35).] For a
comparison between numerical values of J (x) and those obtained by Langer's
formula (43) see Fock (1934), and for an extension of (43) to complex p
and x, see Langer (1932).
In case of sufficiently small w (x nearly equal to p) w - tan"1 w may
be replaced by w3/3 and Watson's formula (34) is obtained.
The method of the "approximately identical" differential equations
was also used by Cherry (1949, p. 121), to obtain uniform asymptotic

7.5
BESSEL FUNCTIONS
31
expansions for the Bessel functions. The differential equation for
yXJAad-y*)*]
IS
d2w
(44)
+ w |^-p2 + (y-2-l)( ly-<-Iy-2 + a2-P2jj =0,
duz
where
(45) u = tanh"1 J - y.
Near y = 0, the coefficient of w in (44) can be developed in the form
-p2 +— u2+ (a2 -p2 - 1/35) {Su)-*3 + P {a*3)
36
where P stands for a power series. Thus (44) is close to
d2W f * 5 ,\
But according to formulas 7.2(62) and 7.2(63) a solution of (46) is
(47) W = (Pu)XKU3(pu),
and if (44)
d2w
(48) —— ■
du*
with
(49) /(u) =
is written
f t
4- u; 1- p * +
36
as
36
(y-2 -1!
»/(«)
)■■
-l)(±y--_i.y-.+a.-p.)f
then, starting with the expression (47) in place of w on the right-handside
of (48), we find the solution of (48) by an iterative procedure using the
method of the variation of parameters. Further results may be found in
Cherry (1949, 1950).
7.5. Related functions
There are certain polynomials and functions which are either similar,
or in some ways analogous, to Bessel functions or which occur in
investigations connected with Bessel functions. These polynomials and
functions are thoroughly discussed in Watson's book (1944, Chapters 9 and 10),
Here we shall give only a very brief account of the basic properties of
some of these functions. For more detailed information the reader may
refer to Watson's book.

32
SPECIAL FUNCTIONS
7.5.1
7.5.1. Neumann's and related polynomials
Neumann's polynomials 0 (z) are defined by the equation
(i) (*-£)"' = 2 cnJM)on(z)
n— 0
e0 = l, en = 2 if B>1, \£\<\z\,
and are of importance in the theory of the expansion of an arbitrary
analytic function f(z) as a series of the form
f(z) = 2 anJnb)-
n= 0
In order to obtain an explicit expression for 0 (z) we start with the
identity
(2) (*-£)"' = *"' S~e-'e'*''dx Re f/2 .< L
0
In 7.2(25) we put a = 1, replace z by £, £ — J~! by 2x/z, and obtain
n= 0
This we substitute in (2), remark that term by term integration may be
justified if \£/z\ < 1 and compare the results with (1), Thus we obtain
Neumann's integral representation
(3) 0,W = X *-"-' J°°\[x + (*2 + zzf]«+[x - (*2+z2)*]"j e"' dx
0
= Ji j" \[t + U2 + l)*]n + [t - (*2 + l)*]"" \ e~ztdt,
where n > 0 and |S + arg z'| .< % 77.
To exhibit the polynomial nature of 0 (z\ we substitute
[(t2 + l)% ± t\» = 2F, (- K n, Kb; Ji; - *2)
± nt .F/K+Kb, J$-K»;3/2;-t2)
in (3) and integrate term by term with the result that
(5) 0ta + l(2)-Ji(n+Ji) > J-rOS*)"2-"1,
or, after some algebra

7.5.1 BESSEL FUNCTIONS 33
<lAn
(6) 0(z) = % 2 n(n-in-l) \{lAz)2sL'n'\/m ! n > 1.
n m = 0
In particular we have
(7) 0o(z).= z-\ C,(2) = 2-2, 02(2) = 2-'+42-3.
Evidently 0 (z) is a polynomial in z"1 of degree n + 1. From (6) we have
the following inequality
(8) \OJz)\<: 2"-1re!|2rn-1exp(^|2|2) «>L
Hence, and from 7«3 (4) it follows that the series
n= 0
is absolutely convergent whenever the series 2 a (£/z)n is absolutely
convergent.
From the definition we have the relations
(9) 0^z)=-0,(z),
(10) 20'n{z)=0n_,{z)-0n+i(z) n>l,
(11) (n - 1) 0n+, (z) + (n + 1) 0n_, (z) - 2r'(/i2 - 1) 0B(z)
= 2nt~* (sin Ymn)2,
(12) rcz 0n_, (z) - (n2 - 1) 0n{z) = (n - 1) z 0B' (z) + n (sin fen *)2,
(13) kO (2)-(nz-l)0 (z) = -(n+ l)zO'(z) + n(sin W)2.
tit i n n
From these relations it follows that 0 (z) satisfies the differential
n
equation
-» d2v dv
(14) z2 + 3z — + (z2 + 1-rc2) v = z(cos ^tt)2
dz c/z
+ n (sin %rc 7r) .
If C denotes any simple closed contour around the origin, then from
(6) and 7,2(2) it follows that
(15) J 0 (z)0 (z)dz = 0 m = n and m ^ n,
(16) i J (z) 0 (z) dz = 0 m + n,
(17) J Jn (z) On (z) dz = ni m>1

34
SPECIAL FUNCTIONS
7.5.1
For some purposes Schlafli's polynomial,
(18) S0 (*) =0, Sn (z) = 2 (n - m - 1) ! (KzVn+**/m ! n £ 1,
n = 0
may conveniently be used (Watson, 1944, Sections 9.3 -9.34). It is
connected with "Neumann's polynomial by the relation
(19) n S (z) = 2zO (z) - 2(cos Y2n n)2.
n n
The polynomials fi (z) defined by the expansion
(20) (22-£2)-'= 2 eAJn{z)]2.QAz) \€\<:\*\,
n = 0
have also been investigated by Neumann, (cf. Watson, 1944, Sections
9,4 and 9,41).
Both of Neumann's polynomials have been generalized by Gegenbauer
(Watson, 1944, Sections 9.2, 9.5). The defining expansions are
<21)£7(*-0- 2 Anv{z)Jv+n{{) \£\<\z.\,
n^ 0 '
(22) r+"/(«-0- ? Bmm„M J^W Jv^i€).
7.5.2. Lommers polynomials
Through repeated application of the recurrence relation, see7«2(56), it
follows that c7v+a may be expressed in the form
(23) Jv+-W-JvC)Jl.fVU)-Jv.f(«)il..f.v+fU),
where R is a polynomial of degree m in z"1; it is called Lommel's
polynomial. Similarly we have
(24) (-1)- J.v.Jz) = J^W*.>>+J^_,WJl._ffV+f <*).
From (23), (24), and 7.11(33) we find that
(25) R^M-Kmkinvn)-' Uv+K{z) J_v+Az)
+ (-l)"«7.v..WJv.f(«)].
Using the power series of 7.2(48) for the product of two Bessel
functions we find from (25) after some reductions
<K.
V (-D* (m-n) !r(i/ + m-7i) nt x ..

7.5.3
BESSEL FUNCTIONS
35
= w , VAz)-" JAVi-Vim, - lAm; v, -m, l-v-m;-z2).
r(w 3
Hence we can find that
(27)RB(>) = (-l)"fiBi_v_„+I(2).
Since the Bessel functions of the second kind satisfy the same
recurrence relations 7.2(56), we obtain a relation analogous to (25)
(28) yv+.(.)-yvu)R.iV(«)-y^Iu-)R..IiV+I(«).
Hence, from (25) and 7.11(36) we have
(29) «,,„(*)—K« [Yv+Jz)Jv_t(z)-Jv,m(z)Yv_,(z)].
Let n be an integer, m = 2n and v = Vm in (25)t Using (26) and 7.11 (5)
we obtain
(30) [Jn + , (Z)V + [e/_, (Z)V = [c/n+, (Z)]2 + [yn + , U)]2
, , . V (22)2--2n(2/i-:2m)!(2ra-m)!
= 2(772)-' Z, — .
m=o in! (ra - m)\ U -m)\
The recurrence and differentiation formulas satisfied by R may be
obtained from (25)« For these formulas and also for the proof of Hurwitz's
limit
(31) lim [(X*)"+WR v+1(*)/r(i/+m+l)]-«T (z)
see Watson (1944, sections 9.63, 9.65)« For other results see Mcdonald
(1926).
7.5.3. Anger - Weber f unc dons
Anger's function &v(.z) and Weber's function E^fz) are defined by
integrals of the Bessel type
(32) Jv(z)±iEv(z)=7r'i J\±i^'zsin^d<f>.
Hence, from 7.3(9) and 7.3(10), respectively, follow the expressions
(33) J (z)=e/ (z)+ tT1 sm(vn) f°° e-'^'-^dt
v v 0
= c7 U)+ it"1 sinU) r *~ZV [f+U + vWn + v^dv
0 Re z > 0,
(34) E (z) = -Y (z)-Tr-1 J°°(ev*+e-vtcos vn)e'taiAtdt
= - *„(*) -=17-' /" e -" 1 [i> -K1 + v W
+ cos vjt[v + (1 + i;2)*]-* J(l + vz)-'A dv Re z > 0.

36 SPECIAL FUNCTIONS 7.5.3
From (33) is it evident that
(35) JBUWB(*) * = 0, ±1, ±2
The expansion of the integrand of (32) in powers of z and terra by
term integration by the aid of 1.5(29) lead to
x v (-i)n(1/2Z)2n
(36) J W = co.(5iMr) I -pT , t/
n/ , £ (-1)"(K2Z)2"+1
+ BmUw) 2, ru + 3/2 + Mv)r(n + 3/2-y2.)'
(37) E„(z) = s.in(W) 2,
v „=0 ru + i + ^)r(n + i-w
(-l)"^*)2"*1
(n + 3/2 + ^v) r (n + 3/2 - H i/)
CONNECTIONS BETWEEN THE ANGER AND THE WEBER FUNCTIONS AND
RECURRENCE RELATIONS
From (33) and (34) we have
(38) sinM Jv(z) = cos(w) E^U) - E_v(z),
(39) sin (vn) Ev(z) = JLVU) - cos (vtt) J^U).
If we differentiate (32), we obtain
2 Wv(z) + i E;U)] = tT1 Jj {e * C (v-D*-»**Lc< [(v+i»-«in*]j^
and hence using (32) again
(40) 2J;U) = J^1U)^JV + 1U),
(41) 2e;(z) = :ev_1U)-ei,+1(z).
In a similar manner, from (32),we derive
(42) J^, U) + Jv+1 (^) = 2i/z"j Jv(z) - 2(^)"1 sin (i/tt),
(43) Ev_1(2) + Ev+1U) = 2v2~,Ev(2)-2(^2)~1 (1- cos vtt).
From (33) and 7,2(1) we find that
*"v(z) + z-' 3'v(z) + (\-vzz~2)3v(z)
= tT' ^2 sind/ir) I — [(- z cosh t + v) e'z ainh *-"] cfc,
Jo dt

7.5.4
BESSEL FUNCTIONS
37
and thus it is evident that
(44) r(z) + z~] r (z) + (1 - vz z~2) Jv(z) = iT% z~2(z- v) sin (vtt\
From (44) and (39) we find that
(45) E^z) + z~' E;(z) + (1 - v2 z"2) Ev(z)
= - iT* Z~2 [z + V+ (z - v) COS {V7T)]
ASYMPTOTIC EXPANSIONS
The asymptotic expansion of J (z) and E (z) for large z and fixed v
may easily be obtained by Watson's lemma. We substitute
(46) [v +(1 + vzYAY (1 + *;T* = 2F, (H + H* lA -lAv;%;- v2)
+ vv 2F,(l + %v,l-%v; 3/2;-*;2)
in (33) and (34) respectively, use 2.1(2), 1.1(5), and obtain
(47) J (z) = J (z) + Or*)"' Binder) [ *£' (-1)" 22n (fc + lAv\(lA-V2v)nz-
n= 0
+ 0C|zr2*) + v *f (-l)"28»(l + Hi') (1-Hv) z~2n~'
n= 0
+ .I/0(|2|-2*-,)]t
(48) Ev{z) = - yy(2) - (ir*)-' (1 + cos w)
x [ *f '(-l)n 22n (V2 + >/2„) & -%v) z"2" + OCIil"2*)]
-i/(frz-|)(l-cosw)[*2l(-l)"22"(l + Xi/) (1-Xi/) z"2""'
n= 0
+ 0(\z\-2M-%
For the asymptotic expansion of Jv(z) and yv(z) in (47) and (48)
respectively see 7.13(3) and 7.13(4)t
The case of large \v\ and \z.\ is discussed in Watson (1944, p. 316).
7.5.4. Struve's functions
Struve's function is defined by a representation similar to Poisson's
integral 7.3(3)
(49) r(v+ y2)Hv(z)=2^(1/2z)vJo1 (l-t2)*-* sin(zt)dt
= 2iT* (%z)vJ 277sin(z cos <f>) (sin $)lv d$ Rev>^,

38
SPECIAL FUNCTIONS
7.5.4
From this expression it may be shown (Watson, 1944, p. 337) that H (x)
is positive when x is positive and vl> %.
If (49) is transformed into a loop integral, the restriction on v may be
removed and we have
(50) n(z)=-iira't Ti}A - v) (Mz)v J(1+> (t2 - I)*-* sinizt) dt
vtl/2, 3/2, 5/2, ....
A further representation follows from 7.2(12)
(51) r{v + V2)[Hv{€Z)-Yv{€z)]*n-X M&v-'z*
xrel/*e-"a + t2C*)V-*dt
0
)8 - % n < arg f .< jS + % n; ->£ir-/8.<args.<J£ir-/8.
(For other integral representations cf. Meijer, 1935 a, p. 628, 744; 1939;
1940, p. 198, 366; Nielsen, 1904, p. 234).
The modified Struve function is
(52) LvU)=-^i^77Hi/Uei^)
Hence we have from (49)
(53) Lv(2) T(v + H) = 2*-* Oizy j'Ansinh (z cos <f>) (sin $)* dcj>
From (51) we have
(54) Ly(*) = I vU) - ^(^)V Jfd + ^)V^ ™ to) dt
x>0, Rev<y2.
A representation of H (z) as a series of ascending powers of z is
obtained from (49) by expanding sin (z cos <fc) in powers of z
(55) hv(z) = f (- iy (y2 z)"+2«+ V tr U + 3/2) r (* + m + 3/2)]
JB = 0
= In'* (YtzV" , F2 (1; 3/2 + v, 3/2; - V*z2)/T(v + 3/2).
Hence it is evident that (%z)~v Hv(z) is an entire function of v and z.
Furthermore we have
(56) Hv{zeim7T)^ei7T{v^u Ujz) m = 1, 2, 3, ... .
From (52) we obtain
(57) L (*)- I (Kz)v+1"+,/[T(m+3/2)r(»/ + m + a/2)]
m= 0

7.5.4 BESSEL FUNCTIONS 39
= 2/T* (Kz)v" , F2 (1; 3/2 + i/, 3/2; % z 2)/T(i/ + 3/2).
From (55) we easily obtain the differentiation formulas
(58)— [zvBv(z)-] = zvBv_}(z),
(59) — lz-vBv(z)] = 2"vn-*/r{v+S/2) - z~vBv^(z),
Carrying out the differentiation on the left-hand sides of (58) and (59) and
comparing the results we find that
(60) Hv_,(z) + Hv+,(2) = 2.w-' Bv(z) + /T* (Ji*)7T(i/+ 3/2),
(61) Hv_, (2) - Hv+, (z) = 2 H^U) - iT* (Ji *)7T (v + 3/2).
From (58) and (59) it follows that the Strove function satisfies the
differential equation
(62) zzb;u) + z h;u) + u2-v2) hvu) - *-* (K*)v"l/r(i/+ jo.
ASYMPTOTIC REPRESENTATIONS
In (51) we put 2 » 1, expand (1 + t2 £~2)v"^ into a series of ascending
powers of t9 integrate term by term and 'obtain for large £ and fixed v
(63) h m = y m + ff-' "f' cru + XXXfr^+^'/r^ + Ji-m)]
■ = o
+ 0(|^|v-2*f-.) |apgf|.<fr.
For the asymptotic expansion of Y (£) see 7.13(4)« Furthennore it may
be proved that if v is real and £> 0, the remainder after M terms is of the
same sign as, and numerically less than, the first neglected term,
provided m + % - v > o.
For the case of large |-i/| and \£\ see Watson (1944, p. 333).
If v = n + % (n = 0, 1, 2, ... , ) , then (1 + t2 £-2)v~% in (51) is a
polynomial, and we have
(64) nn+K(£) = r„+*(£) + *" £ (K£r,"+"-*r(m + xyrGi + i-m).
^n+X ^ *s given ^y 7.11(2). Furthermore from (51) and (54) we obtain
(65) a-ta+wW-(-l)^.+xW; L_{n+,)(2) = I+,(2)
n - 0, 1, 2, ... •
For n =0 we obtain from (64)
— COS 2),

40
SPECIAL FUNCTIONS
7.5.4
When n is a positive integer we may deduce from (37) and (55) (Watson,
1944, p. 337)
(66) BnU)'-it"1 1 r(m+l/2)(y2z)n-2a-*/T(n + 1/2-m)-En(z),
St es 0
(67) B.n(z)«(-l)',+ lir-t 2 r(^~m-y2)(M^)"n+2m+,/r(/7i + 3/2)
m = 0
-■-».
For further results concerning Struve functions see Baudoux{1946)«
7.5.5. Lommers functions
We consider the inhomogeneous Bessel differential equation
dzw dw
(68) zz-——+z—+(z*-v2)w = z»+\
dz d.
z
H, v being unrestricted constants. A solution of (68) is.
Y (_l)«zM+1+2«
(69) s„ (z)
'M,w «*o Kfi + I)2 - v2] [{fi + 3)2 - v2] • • • [in + 2m + l)2 - v2l
u., y (-1)-(M*)2»+2n^ -x*-t-X)rogM + ^v + %]
(ft - V + 1) (/i + V + 1)
x , F2 (I; H/t - Kv + 3/2; if P + fcv + 3/2; - %z2).
The solution (69) becomes nugatory when one of the numbers ft ± v is an
odd integer.
If the differential equation (68) is integrated by the method ofvariation
of parameters and if that solution is determined which is approximately
[(fi - v + 1) in + v + l)]-1 zM+1 for small z, one finds
(70) s^Jz) = HfKsin™)-' [Jv(z) fo* z»J_v(z) dz-J_v(z) j\»Jv(z)dz]
= X * [Yv(z) J* z» Jv(z) dz - Jy(z) Sj z» Y „(*) dz].
The two expressions in (70) for s v are identical when v is not an
integer. When v is an integer, the former expression is not defined, but
the latter is still valid.
Another particular integral of (68) is
(71) Su (z),, 1/W + [2^-,r(X|£-Xi/ + «)r(X|£+^+H)/aittMl

7.5.5
BESSEL FUNCTIONS
41
XICOS [%(/£- V) 77]c7_v(z)-COs[%(u+ V) 77] Jy(z)\
= s^v{z) + 2»-*r{v2iL-v2v + V2)r{v2iL + v2v + lA)
x taints- v) it] Jy(z) - cos [5<(f£- i/) it] y^U)!.
When either of the numbers ^ i v is an odd positive integer, S may be
represented by the following terminating series in descending powers of z
(cf. Watson, 1944, p. 347):
(72) S^iz) = z*-* 11 - [(^ - I)2 - v2] z"2
+ [(M - 1) * _ „*] [(p - 3)2 - V2] 2 -4 - . . • }.
!In case /i + v or fi ^ v is an odd negative integer, s in undefined,
but 5 v(z) approaches a limit (Watson, 1944, p. 348)*
RECURRENCE RELATIONS
From the definitions we have
(74) S;(V(2) + (V*) s^U) = (M +-v- 1) »M_, (V_t (z),
(75) s'^Jz) - {viz) Sflv{Z) -ifL-v-1) Vt >v+t (z),
(76) (2.vA) «MiV(z) - (M + v- 1) Vt ,v-i (z> ~ tf-v- 1) Vt.v+t <z)'
(77) 2S;v(2) = (M + v-l)VtiV_tU) + (M-v-l)Vt(V+t(Z).
From (71) it follows that the same relations are valid if in (73) to (77)
sfi,v^ is rePlaced *>y S^.y CO-
SPECIAL CASES OF LOMMEL'S FUNCTIONS
Several of the functions associated with Bessel functions can be
expressed in terms of Lommel's functions,
(78) 02n(2) = 2-'St>2n(z), O2n+t(z) = (2n+l)Z-'S0i2n+t(Z),
(79) S2n(z)=4*S_,(2„(z), S2n+,(z) = 2S0(2n+t(z),
(80) AZnv(z)=2vzv-<r{v + n){v+2n)S,_VtV+Zn{Z)/n\,
<81) Ain+* ,„<*> * 2V+' zV"' T(v + « + 1) (v + 2« + 1) S_vv+2n+t (z)/n ! ,
(82) J, (z) = sin(vTr) s. „{z)/n- v sln(w7) s . (z), set e»W«t!

42
SPECIAL FUNCTIONS
7.5.5
(83) Ev(z) = -(1 + cosi/7t) s0fV(z)/it-v(l - cosvri) s_, v(z)/n,
(84) Hv(z)=2,-v^svv(z)/r(v+y2)
= Yv(z) + 2f-v'ir-* Svv(z)/r(^+ K).
where the notations introduced in sec, 7,5 have been used.
Young's function (1912) is
(85) cv(t)= f (-i)^v+2vr^+2m + i) = z^^v.3/2>i/2(z)/r(^i).
ASYMPTOTIC EXPANSION
The series (72)diverges in general, but it can be shown (Watson, 1944,
p. 351) to bean asymptotic expansion of S v(z) when \z\ is large and
|argz|<77.
INTEGRAL REPRESENTATIONS
The integral representation
(86) s^v(z) = 2^2z)^(1 ♦"♦^ros + y2/z-y2u)
xCn J^d+A-v)(2 sin^ (sin^)^(f+v-^)(cos0)l,+^^
Re(v+/x+ 1)>0
may be verified by expansion in ascending powers of z. For further
integral representations see formulas 7,12(48) to 7.12(52)andSzymanski
(1935); Meijer (1935 a, 1938, 1939 a, 1940, pp. 198,366).
Lommel has also investigated functions of two variables defined as
(87) £/>,*) = I (-1)' W2)"+2« J (2),
«= 0
(88) Vv (w, z) = cos (Y2w + 1/2Z2/w + 1/2V7t)+ U_v+2 (w, z).
For the theory of these see Watson (1944, sections 16,5 to 16,59); see
also Shastri (1938).
7.5.6. Some other notations and related functions
In Nielsen's Handbuch der Theorie der ZyUnderfunktionen, some
notations (for a list of those see Nielsen's book, p. 406) different from
those introduced in sec, 7,5 are used. These notations are
Zv(Z)=Hv(z), Vv(z)~Jv{z), nv(z) = -EvU),

7.-6.1 BESSEL FUNCTIONS 43
Furthermore the following functions are investigated there:
IFU) = Y2[Jv(z) + JLV(*)1 XVU) « «[J„(*) - J^U)],
ir*^)-*1' f"eizcOB<1>cos(v<f>)d<f>,
7TAv(z)=i'-vf"e "cos<* sin(v<f>)d<f>.
The last two functions are generalizations of Hansen's integral see
7.12(2) for the Bessel coefficients. [See also fonnulas 7.12(40) to
7,12(45).]
7.6. Addition theorems
There are two types of expansions of Bessel functions which are
known as addition theorems. Roughly speaking, Gegenbauer's type is
connected with the theory of spherical wave functions (in 2 v + 2
dimensions), while Graf's type is more nearly related to the theory of
cylindrical waves. This description is not quite accurate, and the two types
coincide when v = 0. As a matter of fact these two types are developed
as two different generalizations of Neumann's addition theorem for J •
7.6*1. Gegenbauer's addition theorem
Gegenbauer's addition theorem will be established for the modified
Bessel function of the third kind, Kp(z)9 We put
(1) u> = (z2 + Z2-2zZ cos^-KZ-w-^MZ-w*)]*
and assume at first that z, Z, <£, are real and 0 < z < Z. With 2=1 and
a * w in 7.12 (23) we have
(2) w^Kv(w) = y2 f0°°exV[-t -(z2 + Z2-2zZ cos <f>)/t] Cv'x dt.
If v £ 0, we use Sonine's expansion 7.10(5)
expk"1 zZ cos<£]
= [2t/(zZ)YT(v) I (v + n)Cvn(cos<f>)I.n(zZ/t),
n~ 0
substitute in (2), and integrate term by term using here 7,7(37) in the
process. Thus we obtain the addition theorem (for the Cv see sec. 3.15)»
(3) w-vKv(w)^(y2zzrvr(v)
x f {v+n)Cv (cost) I +n(2)£ (Z)
n= 0
v£ 0,-1,-2, ... , z <Z.

44
SPECIAL FUNCTIONS
7.6.1
If we make v tend to zero, we obtain,using 3.15(14),
(4) K0(w) = IQ(z)K0(z) + 2 2 In{z)K(Z)cosn$ z < Z.
n — \
It follows from 7.2(12) and 7.2(13) that the series (3) converges like
2 C* (cos <£) (z/Z)n and therefore from 3.15(1) that (3) and (4) hold,
provided that |ze±i<^|.< \Z\.
The addition theorems for the other Bessel functions follow from (3)
by means of 7.2(16), 7.2(17), 7.2(7), and 7.2(8).'Also see 7.15(28) to
7.15(32),
7.6.2. Graf's addition theorem
Graf's addition formula
(5) ^ I 7 zei* ) ' 3 J^ (Z) Jn(Z) ' "*.
\ Li — ZC ^ J n = —oo
where we have
\ze±i(^\ <|Z|, it; = (z2 + Z2 - 2zZ cos <£)* = [(Z - ze**)(Z-ze **>)]*
may be proved as follows. From 7,3 (5) we obtain
(2.ri)«7 .(Z) J (*)***_ J""**' eKZ(*-*",)rw-,(ei*/*y,«7(2)A.
— ooexp(-iy6)
From 7,2 (25) we have
(2irf) 2 Jv+n(Z) Jn(2)e^
n —— oo
— ooexp(-iy6)
Now we put (Z - ze~z<?) t = wv, (Z - ze lc?)/t = w/v and take thatvalue
of the square root (1) which makes w -* + Z when z -* 0. We then may
take the contour to start from and end at — °o exp {—id) where a = arg to.
Thus we have
(2wi) t Jv+n{Z)Jn(z)ein*~w-v(Z-ze-i'f>)v
x / exp[lAw(v - t>~T)] f"17"1 rft>.
— ooexp (— id)
Using 7,3 (5) again we obtain (5).
Formula (5) may be written in a slightly different manner,
introducing an angle if* by means of the equations
Z — z cos <£ = uj cos ^4, z sin cf> = iv sin ^i,

7.7.2
BESSEL FUNCTIONS
45
so that in case of real <£ and positive z, Z and w, if/ is the angle opposite
to z in the triangle with the sides z, Z, w* We then have
(6) eiv+Jv(w)- 1 c7v+n(Z) </„(*) e^
|^c±£*| <|Z| incase v£>0, ±lf ±2
For the other Bessel functions see formulas 7.15(33) to 7.15(36).
A duplication formula for the Bessel function of the first kind and for
the modified Hankel function in case the order is half an odd integer has
been derived by Cooke (I930)t The results are
(7) Jm+%(2z) = (-l)» n*m\zm+*
x i (-iy(2m-2n+l)c7-.n+xU)c7.(B-n+5{j[z)/[ji!(2iii-ji + l)!].
K +1,(2z) = 7r %mlzn+% > / n+^
n+A n?0 n!(2m-n + l)!
(8)
For other similar formulas see Cooke (1930).
7.7. Integral formulas
7.7.1. Indefinite integrals
From formulas 7.2(52) and 7.2(53), respectively, we have
(1) Jzv+lJv(z)dz = zv+<Jv„(z),
(2) Jz-v"Jv(z)dz—z-v+'Jv_i(z).
From sec. 7.2(57) we obtain
(3) fJv{z)dz = 2 **' Jv+2n+,(z) + J Jv+lm(z)dz m-1,2,3
•n — 0
Equations 7.2(4) to 7.2(6) show that (1) to (3) are valid for Y (z) and
H]])(z), H{JHz). For similar formulas see 7.14(1) to 7.14(13). *
1.1.2. Finite integrals
Many definite integrals involving Bessel functions are of the
convolution type
F *G(t)= ftF(v)G(t-v)dv
Jo
and may be evaluated by means of the convolution formula of the Laplace
transform (Doetsch, 1937, p. 161; Widder, 1941, p. 84). According to

46 SPECIAL FUNCTIONS 7.7.2
this method if
f(s)^J°° e'stF(t)dt^L\F\
o
and
g(s) = L IG}, then f(s) g(s) = L\F * Gl
The formula is valid for example if L{F] and L \G\ are absolutely
convergent.
As an example we shall prove Sonine's second integral in this
manner. With
we have from (24) for Re pc > — 1
f^a,s) = L\F^a,t)}=2->J's-^ ex?(-%a2s^)
'Now we have
y«. *) fv(P. «) = 2^+v+1 Ka2 + 0 *)*, *]
and this leads to Sonine's integral
= 2 a^^(«I + i8Ir«<1'+^l,Jv+/l+| U(a2 + 02)*]
Re v > - 1, Re ii > - 1.
Putting t = 1 and substituting r = (sin 0) 2 we obtain
(4) J ^Sa s^n $) ^(i^ cos $) (s*n ^^+1 (cos
0 M ^
= a^^(a2 + /S2)-^^+,)Jv+M+1[(a2 + iS2)^
Rev>-1, Re/z>-l.
A limiting case of (4) may be mentioned separately. If we divide
both sides of (4) by fiv and let $ -> Q we obtain Sonine's first integral
(5) S**Jp(a sin 0) (sin 6)^y (cos 0)^+1 <*0
- 2PT{p + 1) a^-' c7p+/i+1(a) Re p >- 1, Re p >- 1.
Other formulas of the convolution type are
(6) J"oVcyr)(i-r)"Jva-r)dr
= <2»)-« -r(v + X)r(M+ M)*"^* Jv+M+K (*)/T(v + p. + 1)
ReM'>-H, Rev>-%,

7.7.2
BESSEL FUNCTIONS
47
(see Hardy, 1921, p. 169) and
(7) Jo'r-X J2>rK)(*-rr*cos[/3(*-r)*]rfr
= nJv\Xt* [(a2 + /32)* + /3)l Jv\%tHU*+ /32)* *- 0]}
Rei/>-K,
which may be written as
(8) J?"JZv&kO* sin 0] cos [(* - £) cos 0]rf0 = % n JJz) </,(£)
Re i/>-&
Formulas (6) and (7) result from the convolution theorem in connection
with (17) and (23), (25) respectively.
An integral formula involving Struve*s function, corresponding to
Sonine's first integral (5), is
(9) f*wB„ (z sin0)(sin0)M+1 (cos0)2^+1 dS
0 M
= r(p + l)2"2-/'-,H/0+M+I(2) Rep>-1, ReM>-3/2,
and may be established as follows. We expand the Strove function under
the integral sign according to 7,5(55) and integrate term by term using
1.5(19).
In many cases the representations 7,2(47) to 7,2(49) of a product of
two Bessel functions as a power series may be used for the evaluation
of integrals involving Bessel functions. For instance we have from 7.2(2)
and 1.5(19)
f*"jv{2z sin0)(sin0)v(cos0)2l'</0
1 y i-l)nzv*T(v+m + %)riv+%)
2 S.0 m!r(v + m+l)r(2v+»+l)
= 2
and by 7.2(49) this leads to the result
(10) J*" Jj2z am6)(Bin6)v(coad)tvde
= MZ-v^r(I/+K)[c7v(2)]2 Rev>-X.
Similarly we prove Neumann's formula
(ID S*"Jv+a(2z coa6)coaitii-v)0\de-%nJJz)JU)
Re(v+/z)>-l
in the proof of which we use formulas 7.2(2), 1.5(19) and 7.2(49)t
A generalization of Neumann's formula

48 SPECIAL FUNCTIONS 7.7.2
(12) TT&azrv&pz)-" JjLaz) Jv(fiz)
Re(v + il) > -1, A = [2 cos 6(a2 e^+p2 a"*)]*
may be proved as follows. We expand the Bessel function under the
integral sign according to 7,2(2) and obtain
ir(a*)"'*(/8*)-v«7M(aa)Jy(/8«)
= 2
(-ir 2-" 2
-m _2«
; .T On + v + u + 1)
x fXne *(»-v)kos6)m*v*'l("t ew+l32 e"*)" dd.
But the integral is expressible as a hypergeometric function Fy
compare 2.4(11)] and by 7.2(47) the truth of (12) is obvious. [For a
related representation see 7.14 (60).]
Another class of integral formulas may be derived from the addition
theorem in sections 7.6 and 7.15. From 7.15(31) we have
(13) ir[JB(x)]* = J Jo(2*sin0) cos(2n<£) d<f) n = 0, 1, 2, ... ,
or, more generally, if Z v denotes any Bessel function of the first,
second, or third kind, we obtain from formulas 7.15(28), 7.15(29), and
3.15(17)
(14) ^w-vZy(w) C* (cos<f>) (sin<f>)2vd<f>
= 2^TU+2v) (2zy)-vZv^ (y) Jy^ (z)/[ml Hi/)]
w = (z2 +yz-2zy cos<£)K, Re v>-% m = 0, 1,2, ... .
For other formulas of a similar type see formulas 7.14(14) to 7.14(23)
and Watson (1944, p. 373); Copson (1932); Rutgers (1941); B. N. Bose
(1948); MacRobert (1947, p. 383).
7.73. Infinite integrals with exponential functions
The formula
(15) 2v+^a^j8"vy^+I/r^+l) f~JJLat>) Jv<$t)e'ytth** it
Y ru + n + ^ + 2"0
"« = o wirk + m + i)
x /, (-m, -p-m; i/+l; jS2 a"2) (-if a2 y"2)"
Re(A + jx+y)>0, Re(y±ia±ijS)>0

7.7.3 BESSEL FUNCTIONS 49
may be proved by replacing the Bessel function product by its power
series expansion 7,2(47), then integrating term by term, and also using
1,1 (5)t In some special cases the right-hand side of (15) reduces to
simpler expressions. If,for example, we put A + v = p and let /3 tend to
zero, we obtain Hankel's integral
(16) (2y/aWr(/z + 1) J^° e-WJjiat) t^ it
= r(M + P) 2ft(y2P + y2tl9 y2P + y2fl + y2; M+i; -a2 y~2)
= r(M + p)(l + a2y'2rW^
x 2F} [KP + Km 1/2+1/2tL-l/2p;tL+l; az/Uz + y2)]
Re(p + /z)>0, Re(y ±ia)>0.
The second expression (16) is derived from the first one using the
transformation formula of 2,10(6) for the hypergeometric function.
From the second formula in (16) we see that if p = \l + 1
(17) {"e-TtJUt) t»dt = ** (2«)" H/i + X) (y2 + a1)-*""
0 M
5tree«e*U> Re(2/»+l)>0, Re(y±ia)>0.
If in (16) p = 1, we obtain from 2.8(4)
(18) J"e-y* J (at) dt = a-Hy2 + «T* [(y2 + a2)* - yP
0 ^
Re fi>-l, Re(y ±ia)>0.
Furthermore from the second expression in (16) with y = 0, using
2,1(14) we have
(19) S"JAat) tp-y dt = 2P-< a-trMvi + Kp)/T(l + H#£-^p)
0 A1
-Re /z < Re p < 3/2, a > 0.
In the same manner a number of similar integral formulas containing
the square of the integration variable in the exponential function may be
established. For example the relation
(20) 2v+^+1 a-^jS-'y'+^rtiz+l) r J (at) J (fit) e^^ tK" dt
-i
'o /x
fi0 m\r(m + [jL + l)
x zFy (-w, -p - m; i/+ 1; /32 a"*)(-X a2y~2)»
Re(p + i/+ A) >0, Re y2>0
may be derived using the expression of 7,2(47) and integrating term by

50 SPECIAL FUNCTIONS 7.7.3
tenn* We shall now investigate some special cases in which (20)
reduces to simpler expressions.
Let j8 = a; then we obtain using 2.1(14)
(21) J^J^iat) Jv(at) e-?*'2 l*"' it
I>+l)r(i/ + l)
Re(v+X+/z)>0, Rey2'>0.
Let /3 tend to zero in (20). Then the expression on the right-hand
side of (20) reduces to a confluent hypergeometric function, and we
obtain with <v + A = 'p
(22) T(ii + 1) J^JJiat) e^ ** *'"' it
= Ky-0T(y1^ + %p)(1Aa/.y)ll}Fl(1At + y1p;t+\;-y4a*y-*)
= %y-PT(Xtz + Kp) (H «/V)M exp(- Xa2 .y"2)
x.^^p-Jip+lju+l^o'-y"8)
Re y2 > 0, Re Qt + p) p> 0.
Furthermore we have
(23) J~J<at) e-?*'2 it-Un* y'% exp(-2~3 a2 y"2)X (2~9 a2 y"2)
Rey2»0, Rett'>-1,
(24) J0"JM(a*) e"yV ^+1 A = a^y2)^"1 exP(-^a2 y"2)
Re/t->-l, Rey2>0,
(25) J"" Jv(at) c7v(/30 e^2'2 «it
= &y"2 expt-^y"2 («2 + /S2)] I^Acfiy-*)
Rev>-1, Rey2>0.
Formulas (23) and (24) originate from (22), and (25) from (20).
A formula similar to (16)
(26) r0$ + y) v'* (2/3)"v(a + /3)"+* J^V^t/Si) t»~% it
= T(/i + v) r(/z-.v) 2F, [v+n,'V + y2; fi + X; (a- /3)/(a + /3)]
=(2ara'-^(a+^)a'+^r(^ + v)r(^-I/)
x 2F, (v +/z,/z; 2,v + 2^; 1 -/32/a2)
Re(zt±.v)'>0, Re(a+/3)'>0

7.7.4 BESSEL FUNCTIONS 51
may be proved by inserting here 7.3(15) for K^ifii), interchanging the
order of integration, and then using 2.12(5)* From (26) and 2,8(47) for
a = 0 we have
(27) J^Kv(pt)t^f dt = 2»-2p'»r{y2ti + Y2v)r(y2n-y2v)
Re(/z±i/)>0, Re 0 > 0.
Furthermore from (23) and 7.2(13) we obtain
(28) f^ KJiat) e-V*** dt-Xir* y~* sec (Hfur) exp(2"3 a2/y2)
x K^(2~* a2/y2) - 1< Re p < L
Furthermore one may consult Shabde (1935); Mohan (1942, p. 171); Sinha
(1942).
7.7.4. The discontinuous integral of Weber and Schafheitlin
We shall now investigate the integral J JAat) Jvibt) t"p dt in which
a, i, are positive real. It turns out that even when the integral converges
for all positive a and b, its analytic expression is different, according as
a is smaller, equal to, or larger than i. The results are
(29) 2^i^^,r(M+i)r(1^+^v+y2p~K2/z)
x /o°° J^(at) Jv{bt) t-f>dt = a^T (X + fcv+Xp-Xp)
x AM + Xv+%iL-%p, 1/2 + 1/2fi-1/2v-1/2p; /z+1; a2/b2)
Re (y + ii - A + 1) > 0, Re p > -1, 0 < a < £,
with a corresponding expression for 0 < b < a [interchange a and b in
(29)] and
(y2q)^! r(p)r(Ky+%*+x-Xp)
"2r(^+K2v-V+^p)r^+1^+^+Mr(K2+y2/x-y2v+^p)
Re (i/ + /x + 1) > Re p > 0, a > 0.
The proof of these results follows. We use (12) with a = a, jS = b, z = t in
the integrand of (29),interchange the order of integration, evaluate the
integral with respect to t by (19), and obtain
J°°Jn(at)JJbt)rPdt
rvAv+VivL-Vtp + Vi)
= tT' aH bV tfv+Kn-Kp-*
r«M + Hi/ + Hp + K)
^77
x (a2 e #+ £2 e^)^-1^-1^.
But the integrand on the right-hand side is expressible as a hypergeo-

52
SPECIAL FUNCTIONS
7.7.4
metric function 2F} , compare 2,4(11), and we immediately obtain the
expressions (29) and (30) according as b > a or b = a. In some special
cases the hypergeometric function reduces to a simpler function. For
instance the formulas 7.14(28) to 7.14(31) are derived from (29) and (30)
by putting p = v = Yi%
An integral related to the Weber-Schafheitlin integral but with one
Bessel function replaced by a modified Bessel function of the third kind
can likewise be expressed in terms of hypergeometric functions, but it
has no discontinuity at a - 6. We obtain
(31) 2^ av-?+*r(v+l) f~K^(at) Jv(pt) I"?dt
Re(a±ij8)>0f Re(v-p+ 1 ±/*) > 0,
by expanding JJ<f5t) in a power series of 7.2(2) and integrating term by
term using (27)« Further integrals of a similar type are given in formulas
7.14(35) to 7.14(39)% Here formulas 7.14(35) and 7.14(36) are
consequences of (31)t The other formulas were given by Dixon and Ferrar(1930)t
7.7.5. Sonine and Gegenbauer's integrals and generalizations
Discontinuous integrals of a more general type than (29) to (30) have
been investigated by Sonine and Gegenbauer. The integral
(32) /°VU(tOcaa^ + zW^ + z2)-**^' it
0 p v
= 0 a<6, Rei/>Re/£>-l,
a>b, Re v>Re/*>-l,
may be established by replacing the second Bessel function under the
integral sign by using 7.3 (6), interchanging the order of integration, and
using (24) and again 7.3 (6).
Generalizations of (32) have been given by Bailey (1935 a), and by
Gupta (1943% For instance according to Bailey, we have
(33) ^JJbt)t^x fl Jv[an(t2 + zl)X]UUz2fV2Vndt = 0
6 >o, + a2 + •» +aB , Re(i^ + ••• + v^ + %m - V2) > Re /*>-l,
(34) J°°JJbt)t^ 5 Jy [a^ + z^V + z^^dt
n= 1 n
= 2M-.^r(M) fl,-". Jv (Znan)
b>a +a +••• + a , Re {v. + v + — + v + Vim + 3/2) > Re a > 0.
1 2 m I 2 * n

7.7.6 BESSEL FUNCTIONS 53
Another generalization of (32) is due to Sonine. To obtain this let us
consider for a positive integer m and Re a > 0 the integral
fc zP-< JJhiz* + <T2)*] (z* + <TT^(z2 -■:«*■)-•-' ff™ (az)dz,
where C is a contour consisting of the upper semicircle \z\ = R and its
diameter, with an indentation at z =0, When R approaches *>, and the
indentation shrinks to a point, the contribution of the circular arcs in C
vanishes if a > b9 Re (±v) < Re p < (2m + 4) + Re fi9 Expanding the
integrand in ascending powers of (z 2-a2) we find that the residue at the pole
z1 = a2 is
-^ (-4-)" U^VJ*(«« + ^«.)K](a8+ ^^^ff^taa)!.
m! \ada/ M
2"
mi
From Cauchy's residue theorem and 7,2(16) we find that
(35) f"t*>-% JJbb1* £W*2+ £*)-X»(tz-a2r*-*
x [H "Hat)-he i7r b-^H™(at)] dt
mZl2-m(—)m \aP-*J t&(a2 + ^2)K](a2 + ^2)-^ffcl,(aa)I
ml \adaj M v
a >b, Re(±v) <Re p < 2m + 4 + Re p, Re(ta)<0, m = 0, 1,2,....
Similar formulas and special cases are listed in 7.14(46) to 7.14(59).
7.7.6. Macdonald's and Nicholson's formulas
Representations of a product of Bessel functions as an infinite
integral have been given by Macdonald and Nicholson. Macdonald's formula
06) f°° exP[-y2* - xr1 (z2 + z2)] Kv(zZ/t) r1 it = 2Kv(z) kv(d
|argz.| < 77, |argZ|<77, |argU + Z)\ < % n
is an immediate consequence of
f2Iv(x)Kv(X)
(37) /"exp[-at -**-• (*2 + x*)] ijxx/t)r1 dt = ^
[2Kv(x)Iv(X)
according as X > x or X < x, We prove (37) for positive real x and ^ and
obtain (36) for positive z, Z, by 7,2(13); the extension to complex z, Z,
follows from the theory of analytic continuation. Putting a = x, /3 = X9
y2 = %£, in (25) we have
(38) Iv(xX/t) = t exp[(*2 + *2)/(2*)] /o°° c^Cw) c^tft;) e~*'v* v dv.

54
SPECIAL FUNCTIONS
7.7.6
Inserting this into (37) we obtain
r exP[-'/2* -'Xr1 (*2 + x2)] Uxx/t) r1 it
= J"~ «T(xt;) JJXv) v dv j" e ~V* (1+v 2) it
= 2j°°a + vz)-,J.(xv)J{Xv)vdv
and using 7.14(57) this proves (37).
Nicholson's formulas
(39) KpL(z)Kv(z)=2 f^K^&z cosht)cosh[(ii-v)t]dt
= 2 J X (2z cosh*) cosh[(/£+ v)t\dt Re z > 0
may be proved as follows. From 7.12(21) we have
K(z)K(z)=%r J °° e-2{cosht+c08hv) cosh (//*) cosh (vv)dtdv.
Now we make the transfonnation t + v = 2£, £ - i; = 277, and after some
reductions we obtain
Kv(z)K^U) = H SZ SZe'HecehCcosh" coshfo*+'">£]
x cosh [(/z - 1/) r/ ] rf£ rfr/.
With 7.12(21) tliis proves (39).
Another formula due to Nicholson is (Watson, 1944, p. 444)
(40) [Jv(z)]2 + [Yv(z)]2 = an~2 J~K0(2z sinht) cosh (2vt) dt
Re 2 >0.
For similar formulas, especially integrals for the product of two
Bessel functions, see Watson (1944, p. 445); Chaundy (1931); Dixon and
Ferrar (1930, 1933); Meijer (1935, p. 241, 1935 b, 1936, 1936 a, 1940,
p. 366). For the sum or difference of a product of two Bessel functions,
see Buchholz (1939, 1947).
7.7.7. Integrals with respect to the order
A formula due to Ramanujan (Watson, 1944, p. 449) valid for real y
and a, b > 0, Re (v + p) > 1,
(41) lZa'^X JM+x(a) b'V+X Jv-x{b) eiXydx
-(2 cos y2y)X(v+») (a2 e_i^+ b2 e *Ky)-X&>+/*>c ^^-/x)
* Jv+J[2cosy2y(a2 e-™y + b2 e^y)]*] \y\ < n,
= 0 \y\>n
may be proved by applying Fourier's inversion formula to 7.7(12).

7.8
BESSEL FUNCTIONS
55
The cylindrical and the spherical wave function may be expressed,
respectively, as
(42) X0[(a2 + i2-2aicos<?S)^]
= (2/ir) J™Kix(a)Kix(b)cosh[(7T-<f>)x]dx,
(43) (a2 + i2-2a6cos^rKe^^2+^^2^cos^^--1/27r(airM
x J0°° xe77* H™(ka) H™(kb) tanh(7r%) P_%+ix (-cos<£) dx
Im k < 0.
Equation (42) may be obtained from Macdonald's formula 7.7(36) and
(43) from the residue theorem in connection with 7,15(41); (42) is
a special case of a formula given by Crum (1940),
where A9B9 C, are the angles of the triangle whose sides are of lengths
a, b, c.
Another generalization of (42) and (43) is
(45) w-vKv{w)=V2T{v){V2abYv
x J^ sech(7r%)(v- % + ix) K^^^ (a) ^_^+ix(i)
x C-x+^i-coscf)) dx w = (az + b2 - 2ab cos<£)^
(for the definition of C*y + ix see sec. 3„15)„ For the proof of (45) use
7.6 (3) and the residue theorem.
Other fonnulas are
(46) I^Kix(a)co3(xy)dx = y27re'acoshy
(47) J0 K ^ (a) cosh (K nx) cos (xy) dx = % n cos (a sinhy),
(48) J0 K ^ (a) sin\i fynx) sin (xy)dx = l/27T sin (a sinhy)*
They may be derived from fonnulas 7.12(21), 7.12(25), and 7.12(26)
respectively. For other results compare Ramanujan (1920, 1927, pp.
200, 224, 229); Fox (1929); MacRobert (1931, 1937); Crum (1940).
7.8. Relations between Bessel and Legendre functions
The Bessel and the modified Bessel functions may be expressed as
a limiting case of the Legendre functions. In the expressions 3.2(14)
and 3*4(6)for the Legendre functions we replace z by cosh (z/v)and x by

56
SPECIAL FUNCTIONS
7.8
cos (x/v\ respectively, to obtain
= ttanh(K2/i/)F2F|l-.i/il + i/;l + ^;-tsinh(K2/i/)]2l>
P~*(coB*/i/)r(fi+l)
= [tan QAx/vW 2Fy\-v, l + i/; l+/x; [sin(l/2*/i/)]M.
We now let v approach °o and use 7.2(12) and 7.2(3) to obtain
(1) lim J,Mjp-M(Cosh2/l/)=J^l_f(M+1.^22)=j {z)f
(2) lim ^P7(cosx/v) = -J^- /.(p+l;-***)-*/ (*).
A similar relation (see Poole, 1934) may be derived from 3.2(41). It is
(3) lim {Q^fi/dz^e'^/r^)]
r(v+3/2)
0Ft(i/+3/2;-zV4)
= «'*"»(>£**)* «7„+x(z).
Relations analogous to (1) and (2) may be obtained for Legendre
functions of the second kind either from (1) and 3.3(4) or from (2) and
3.4(13). These relations are
f lim v'^e-1^Q^{cos\iz/v) = K (z)
(4) \
lim ^<?7(cos*/v)=-H^ya(x).
We now turn to some integral relationships between Bessel and Legendre
functions. Comparing the hypergeometric series on the right-hand side of
7.7(26) with 3.2(16) we obtain
(5) r(-i/-M)r(i/-M+i)?"£(*)
0
Rez>-lf Re(v-/z+ 1) > 0, Re(v+/z)<0
and similarly from 7.7(16) and 3.2(41)
(6) <?£(*) = (Kit)* U2 - D^e ^ J~ e"<* I+^ (0 t^A it
Re(v+/z)>-l, Rez>l.

7.8
BESSEL FUNCTIONS
57
Applying Whipple's formulas 3.3 (13) and 3.3(14) to (5) and (6),
respectively, we obtain four further integral representations
(7) r(i/-,»+l) (?£(*)-e^U'*-!)-*"-*
x j~ e-tz(zz-\)-% K {t)tvdt Re(v± fjt)>-l,
0 M
(8) r{-v- M) p^(z) = (z2 - d*v J~0°° e-*»(»2-i)"x i_mu) ^at
Re (v + ji) < 0,
(9) r(v + li+i)p;»(z) = (z*-ir'Av-*S~e-tz(>z2-1)~% i^)tvdt
Re (v + n) > -1,
do) r(v + iM + i)p-»(cOSd) = f°°e-tccs0J (tsiad)tvdt
Re(v+ jLt)>-l, 0 <0<%7r.
Equation (9) follows from(8) by means of 3.3 (1) and (10) follows from (9)
by means of 3.4(1).
A simple example of a representation of Bessel's functions of the
first kind by means of an integral involving Legendre functions is Gegen-
bauer's generalization of Poisson's integral,
(11) 2V rr% r(v+Y2)r{n + 2v) in z~v Jv+n(z)/[n\r(2V)]
= J^e«cos^Cv(cos^)(sin(^)2v^
0
Re v>-lA9 n= 0, 1, 2, ... ,
This can be derived from Sonine's formula 7yl0(5). We replace y by
cos <^>, multiply both sides by Cv(cos<^>) (sin<^>)2v, integrate term by term
with respect to <^>, and use 3.15(17).
A similar formula
(12) (2tt/z)X in(sincf>y-X C*(cos0) Jv+n(z)
= Jo77 e iz cos*cos * Jv_,A (z sin 6 sin <f>) C* (cos 6) (sin d)v+* dd
Rev>-M, \ar%z\<7T, n = 0, 1, 2
may be derived from the addition theorem 7,15(17). For further formulas
of these types see Meijer (1934, 1938); MacRobert (1936, 1940); Bailey
(1935 a).
Finally we mention Whittaker's loop integral which is related to
Hankel's integral 7.3(8). It is
(13) 7r3/2JU)=(y22)'/2e-*<^^
oo e
-1Att+ 8<argz < y27r+8, |8| <xAtt.

58
SPECIAL FUNCTIONS
7.8
To prove this formula we assume that the contour lies entirely outside
the circle \t\ = 1; then we expand Qv~%(t) in descending powers of t
according to 3.2(5) and proceed as in sec. 7.3. From (13) we obtain a
corresponding expression for the second Hankel function
(14) *3/2 H™(z) cos M = (%z)* e *<"+X> - J("'_+'' +' e »*Pv_% (t) dt
oo e
-%7r+S<argz KVin + d, \S\ <VnT,
where we have used 7.2(6) and 3.3 (8)t
An expansion of Legendre's function Pv(cos#) in a series of Bessel
functions,
(15) Pv(cos 0) = (0/sin 6)% I an {$) {v + ViT* J [(v+K)0]
R = 0
has been given by Szego (1933)t The a>n(d) are elementary functions,
regular in 0 < Re 6 < tt. In particular, a = 1, aT = 2"3 (ctn0 - 0"1), etc.
(15) is uniformly convergent in 0 < 6 < 6 - € where e > 0 and
60 = 2(2M - 1) n = (0.828 ••• )ir.
This formula may be derived as follows. In 7.10(15) we put s = 1, z = 0,
r2 = 1 - 12/02, and v = -Yi to obtain
2cos^ = (1/2tt)^ f 21-*(02-*2)* 0~*+J* J v(fl)/ffi!f
and hence
2(cost-cos0)=(Hir)* S(fl2-J2)'2'-"^-*J „(0)/ro!.
m = 1
If we use this expansion in Mehler's integral 3.7(27)> integrate term by
term, and use 7.3 (3) we obtain (15%
In the paper by Szego" already referred to, similar expansions are
given for Pv(cosh £), Qv (cos &) and @v(cosh £) on pages 450, 449, and
448, respectively.
7.9. Zeros of Bessel functions
A detailed discussion of this subject is contained in Chapter XV of
Watson's book. Some further results have been obtained since the
original publication of Watson's book in 1922 and are not included in the
1944 edition. Here we shall discuss briefly the more important results.
GENERAL RESULTS
From general theorems on differential equations (Ince, 1944, Chap. X)

7.9
BESSEL FUNCTIONS
59
follow the statements:
a) Any zero of any solution >of 7.2(1) or 7.2(11) is a simple zero,
the only possible exception being the origin.
b) The real zeros of two real linearly independent solutions of
7.2(1) separate one another. Here a real solution is defined by
a ^ifa} + ^ Yv(x) wftb real a9 b9 v, and positive real x%
BESSEL FUNCTIONS OF THE FIRST KIND
For the special case of the function Jv(z) the following theorems may
be proved.
The zeros of Jy(z) and Jv (z) for real is are symmetrical with respect
to the axes of coordinates.
For real v, Jv(z) has an infinite number of real zeros (Watson, 1944%
p. 478; Wilson, 1939).
If yv ,, yv 2««>are the positive zeros of Jv(pc) arranged in ascending
order of magnitude, then
0<>Vi <yv+i.i <Yv.2<y^t2<yv,3<- ">-!.
(Watson, 1944, p. 479).
When v > -1 and A, 6, C, D, are real numbers such that AD - BC ^ 0,
then the positive zeros of A Jv(x) + BxJ'v(x) and CJv(x) + DxJ'v(x)
separate one another and no function of this type can have a repeated
zero other than x = 0 (Watson, 1944, p. 480).
When A and B are real and v > -1, then the function
AJv(x) + BzJ'v(z)
has only real zeros except that it has two purely imaginary zeros when
A/B + v<0 (Watson, 1944. P» 482). For an asymptotic formula for these
positive zeros see Moore (1920).
For v> 1 the function J_v{z) has an infinity of real zeros and also
2[v] conjugate complex zeros, among them twopure imaginary zeros when
[v]is an odd integer (Hurwitz's theorem). (For different proofs see Watson,
1944, p. 483; Obreschkoff, 1929; Polya, 1929; Falkenberg, 1932; Hille
SzegS, 1943).
A generalization of Hurwitz's theorem due to Hilb (1922) states that
the principal branch of the function
AJv(z)+BJ_v(z), 04, B, real, B 4 0, v>Q)
has [v] complex zeros with a positive real part in case [v] is even; when
[v] is odd there exist [v] - 1 or [v] + 1 complex zeros with a positive real
part according as (A/B) < 0.

60
SPECIAL FUNCTIONS
7.9
The number of zeros of z~v Jy(z) between the imaginary axis and the
line on which
Re z = inn + (% Re v + l/A) 77
is equal to m for sufficiently large m, and all the zeros of J (z) lie
inside a strip |Im z\ <A where 'A is bounded when v is bounded.
•Let >yv**y'v and yv' be the smallest positive zeros of J (*),</' Oc) and
^ (x) respectively; then we have (Watson, 1944, p. 485)
[v(v+ 2)VA <yv< [2(v+ 1) (v+ 3)]* ,
Mv+2)VA<>y'v<[Zv(v+\)]%,
when v '> 0, and
[v{v-ltA<yav.<(vz-l)*
when v > 1. For better bounds and for results on the following zeros see
Mayr (1935).
The formula
>V= v+ 1, 855,757v,/3+ 103,315 v~,/3 + OCiT1)
and similar formulas for other zeros of the Bessel functions of the first
and second kind have been given by Trie omij (1948) • For further information
about the zeros oiJv(x) and J1 (x) see Bickley(1943);Bickley and Miller,
(1945);Gatteschi (1950);01ver (1950).
It has been proved by Siegel (1929) that Jy(z) is not an algebraic
number when v is rational and z is an algebraic number other than zero.
This theorem proves Bourget's conjecture that Jv(z) and Jv+m (2)
0»s* 1, 2, 3, ...) have no common zeros other than zero (Watson, 1944,p.
484).
Investigations about the zeros v of Jv (z) regarded as a function of
•v, with fixed z have been carried out by Coulomb(l936)tThey show that
for positive real values of z, the v are real and simple and asymptotically
near to negative integers (cf. also Gray and Mathews, 1922, p. 88).
The graph of c7v(*) for fixed v > - 1 and variable x ^0 resembles the
graph of a damped oscillation. The successive areas of "half-waves"
above and below the axis, form a decreasing sequence (Cooke, 1937).
The factorization theorem for entire functions (Copson, 1935, p. 158)
leads to the representation of z~v J {z) as an infinite product (Watson ,
1944, p. 497). We consider those zeros of z~v Jy(z) for a fixed v 4 - 1,
- 2, - 3, ... , which lie in the half-plane Re z > 0 (those are symmetrical
to the real axis) and arrange them according to non^decreasing real parts
(in case there exist zeros on the imaginary axis only those with a
positive imaginary part are considered). This sequence is denoted by yy n
(n = 1, 2, 3, . • • ). Then we have

7.9
BESSEL FUNCTIONS
61
(i) r(.i/+i)(Xzrv«7v(*)- n (i-z2-yj2n>
n = 1
V. n '
, 2n-1
'A similar expansion is (Buchholz, 1947),
(2) 2r(v) (Kz)'"" J'(z) = 5 [1 - z2(yj „)-*].
n— 1 '
Here the y' is a sequence formed of the zeros of z,-v J' (z) in the
same manner as the sequence a/ was formed of the zeros of z~vJ (z).
Forming the logarithmic derivative of (1) and using 7t2(5l)we obtain
(3) J,+1(zJ/J>) = -22 I U2-yv2B)"'.
n— 1 *
Hence the following power series valid for \z\ <y may be derived
(4) x-^w/^w-f s^.1
n— 1 '
where
(5) S9. = f >y-21
and in particular (Nielsen, 1904, p. 360)
(6) S2 w = 2" V(v + 1), S4 - 2"4/[Cv + 1)2 („ + 2)].
For further similar expansions andrelations see sec, 7.15; Forsyth (1921);
Buchholz (1947).
BESSEL FUNCTIONS OF THE SECOND KIND
The oldest result on zeros of Bessel functions of the second kind is a
theorem by Schafheitlin (Watson, 1944, p. 482) according to which the
principal branch of Y (z) has no zeros with a positive real part other than
real zeros. This result has been extended by Rilb(1922). When [v] is
even, than Y (z) has [v] complex zeros in |arg z\ £ % 77. When [v] is odd,
then Y (2)has [v] — 1 or [v] + 1 complex zeros in the same range,
according as cos {vn) ^ 0. Thus Y ^ {z) and Y 2n+1(z) <ji = 0, 1, 2, • • • ,) have 2n
complex zeros in |arg z\ < %77.
Yn (z) (n an integer) has complex zeros in the left-half-plane on all
branches and in the right-half-plane on all branches but the principal
branch. Furthermore Y (z) has positive real zeros only if v is rational but
not an integer. In the latter case Y (z) has positive real zeros on the
principal branch and other real zeros only if v is rational but not an
integer. In the latter case Y {z) has real zeros only on the branch for which
2771V in 7,11 (41)is an integer, (Hilhnann, 1949).

62
SPECIAL FUNCTIONS
7.9
For the zeros of linear combinations of J (z) and Y (z) see Watson,
(1944, Chap. XV);Hilb(1922);Hillmann(1949)r For a theorem similar to
Bourget's hypothesis see Banerjee(l936).
For a combination of products of the Bessel functions of the first and
second kind we have the theorem (Gray-Mathews, 1922, p. 82): If v is real
and a and b are positive, then
Jv(aX)Yv(bx)-Jv(bx)Yv(ax)
is a single-valued even function of x, whose zeros are all real and simple
(see also Jahnke-Emde, 1945, p. 204; for similar combinations Carslaw
and Jaeger, 1940; Kline, 1948).
BESSEL FUNCTIONS OF THE THIRD KIND
Investigations about the zeros on the principal branch of the first and
second Hankel functions for real non-negative v have been carried out by
Falkenberg and Hilb (1916), and Falkenberg (1932). The results are1:
H (1 )(z\ v > 0, is free of zeros in 0 < arg z .< 77. The zeros, for V> 0, of
H*x) in -7T < arg z .< 0 and those of H *2) in 0 < arg z < n lie symmetrically
with respect to the imaginary axis.
There are no pure imaginary zeros except when v = (2k - I) + %
(k = 1, 2, 3, • • • , ) in which case there is one such zero.
The total number of the zeros oi'H™9 (2) (z) on the principal branch is
equal to
0 if 0£v<3/2,
2k -1 if .i/«(2jfc-l) + H,
2k if {2k-\) + lA<v<2k + lA k = 1, 2, 3, ... .
A theorem analogous to Bourget's hypothesis states that H^* (x)and
[]W*W(x) have no common zeros when ^ is real > -land m « 1, 2, 3, • •. ,
(Banerjee, 1935).
MODIFIED BESSEL FUNCTIONS OF THE THIRD KIND
For v > 0, K (z) has no zeros for which "|arg z\ ."LYiit* The number of
zeros in |arg z \ < n is the even integer nearest to v - lA unless v — % is
an integer, in which case the number is v - M (Watson; 1944, p. 511).
When v + 1 is positive real, and m a positive integer, K^(z) and
K + (z) have no common zero.
If f(z) and g (z) are given analytic functions without common zeros
such that g(z)Jf(z) is meromorphic, and Re [g(z)//(*)] > 0 for Re z > 0,
then the function
F(z)=ttz)K'v(z)-g{z)Kv(z)

7.10.1
BESSEL FUNCTIONS
63
has no zeros in the right-half of the complex plane (Erde*lyi and Kermack,
1945).
The zeros of Kv{z) and I^az) Kv(bz) — Kv(az)'Iv(bz) regarded as a
function of v are all purely imaginary, and these functions have an
infinite number of zeros (Gray-Mathews, 1922, p. 88); compare also Polya,
(1926) and Bruijn (I950)t The function G (z) corresponding to equation (iii)
in Polya's paper is 2K . (A).
7.10. Series and integral representations of arbitrary functions
7.10.1. Neumann series
A Neumann series is a series of the type
n= 0
By the expansion 7.2(2) it is evident that its circle of convergence is
identical with that of the power series 2 a (%z)v+n/T (v + n + 1).
The "Neumann series expansion of a function f(z) which is given by a
power series can easily be obtained. For this purpose we first give the
'Neumann series of a power of z
(2) (K zY =1 U, + 2n) T(v + ii) J+2n {z\/n ! ,
v not a negative integer, whichmaybe verified by inserting for Jlf+n(z) its
power series, see 7.2 (2), and rearranging the right-hand side in powers
of z. All the coefficients except that of zv vanish.
Now let
f(z) =2 b.z1.
1 = 0 *
If each power of z is replaced by its Neumann series (2), we obtain
f(z)»z""S b.21^'! (i/ + Z + 2ro)r(i/ + Z+ro) J ... (z)/m !,
1 = 0 «= 0 V1-I1-"
and hence
n= 0
where
<lAn
(3) an = Zl'+n(v + n) 2 2-2T(v + ft-s)6n_2ys!.

64
SPECIAL FUNCTIONS
7.10.1
Conversely, the bl may be expressed in terms of the a (Nielsen, 1904,
p. 271) as
(4) bir{v+l + l) = 2~l-v 2 (-lW K-2a •
m = o V m J
Some cases in which a simpler expression may be found for the sum in
(3) are of special interest. For example we take
/(z) = e"r= 1 (iy)lzl/l\.
I =0
Then we have from (3) after some algebra
an = inyn 2*+n r (i/ + n + 1) 2Ff (-Kn, X - Xn; 1 - » - i/; y"2)> !,
or, introducing Gegenbauer's polynomial 3.15(8) we obtain Sonine's
formula
(5) «"e»y*-2wr(i/) 2 »»(i/ + n) C*(y) <7+n(z) v M -1,-2
B= 0
The expansion of a Bessel function as a Neumann series
(6) (y2(K)^vy(aZ)r(v+i)
= 2 Ft (-n, n + »; .i/ + 1; a2) r (/i + n) (/i + 2n) J (z)/n !
may easily be established in a similar manner. We expand the left-hand
side of (6) in a power series of z and use (3). In the same manner we
obtain the Neumann series of Lommel's function 7.5(69),
0i+l+2*)r0i+l + ii)
n=0 ,..'[(2n + l + ,i)2-i/2]
Hence, using 7.5(82) to 7.5(84) similar expressions for Anger's, Weber's,
and Struve's functions may be obtained. For further results compare sec.
7.15; Nielsen, 1904, Ch. XX; Watson, 1944, Ch. XVI; Baudoux, 1945, 1946.
The theory of the expansion of a function f(x) of a real variable x in a
Neumann series is based on the integral formulas [cf. 7.14(32)]
"0 m ^ n,
-C° '"' <7v+2n+.(t><7v+2. + .Wrft=<
.(4n + 2^+2)~'
m =n, v '> — 1.
Hence, we derive formally the expansion
(8) /(*)=! (2^+2 + 4»)<7v+2B+1Wrr,/(*)<7v+2n+1(t)^
n = 0 °
V>-1.

7.10.1
BESSEL FUNCTIONS
65
The theory of this expansion has been given by Wilkins (1948, 1950)t
The special case y - 0 has been formerly investigated by Webb, Kapteyn,
Bateman (Watson, 1944, p. 533); Korn (1931) and Titchmarsh (1948,
p. 352)„ (For the term by term integration of a Neumann series see Hardy,
1926.)
A series of the type
n= 0 ^
is called a Neumann series of the second kind. If the product of the two
Bessel functions is replaced by its power series of 7.2(48) we obtain the
relation
where
(id r(v+i+M/l)r(M+i+K»)tr2-J-v-'i s (-i)ffl ( Ho,
and hence (Nielsen, 1904, p« 292)
(12) an=2v^+n(v+/x+n)
<xAn
y r{v+ii+n-s)r{v+i-s+y2n)r(ii+i-s+y2n)
x ^n n'2s s !r(i/+M+B-2s+i)
provided neither fi9 nor v, nor /£ + v is a negative integer. Formula (12)
gives the expansion of a power series in a Neumann series, and it may be
shown that the Neumann series thus obtained converges uniformly within
the interior of the circle of convergence of the power series.
A simple example is the expansion of a power of z% We easily obtain
from (12)
'■"r^Dr^n-j.-^T ( » ) •W-'-W*'-
(For further results see Nielsen, 1904, Chap. XXI; Watson, 1944, p. 525;
and Banerjee, 1939*) For series involving the product of an arbitrary
number of Bessel functions see Stevenson (1928).
A modified form of Neumann's series is the series
(14) 1 -."•U.W.
n= 0
From the loop integral, see 7.3(5), we immediately obtain the following
equation

66
SPECIAL FUNCTIONS
7.10.1
(15) (s2-r2)-X*J[z(s2-r2)X]= f (V2z rz)n s^'n J ,(zs}/n ! .
n= 0
With s = 1 and r2 = 1 — X2 we obtain the multiplication theorem of the
Bessel function
(16) J(\z)-kvl lXz(l-k*)]HJv.(zVnl.
n= 0
Hence, making K approach 0 we deduce that
(17) {V1zY=Y(iV+\) I (lAz)nJ.(zyn\
n— 0
a formula analogous to (2).
Equation (17) is useful for the conversion of a power series into a
series of the type mentioned above. We obtain
(18) 2 b.zil=Z-v I anznJ (z)
1= 0 n= 0
where
(19)an= i r/v+::1} 2—6g,
" ,£0 (n-s)!
and hence
(20) r(x/+ra + l)6 -i (-1)* 2~v-n-* a /s]
(Nielsen, 1904, Ch. XXI).
7.10.2. Kapteyn series
Series of the form
(21) 2 «„ </,,+„ [(* + »)*]
n= 0
are known as Kapteyn series. From the inequality (Watson, 1944, p. 270)
(22) \JAaz)\£ I 1 +
/ l^ina^lX l.eali-.^[1 + (1.2.)Xr.|
it is evident that (21) converges throughout a domain in which
(23) I an[w(z)]n
n= 0
is absolutely convergent where

7.10.2 BESSEL FUNCTIONS 67
(24) w (*) = ze « -^/[\ + (1 - z2)*].
The expansion of a power of z in a Kapteyn series
(25) {lAz)l={lAzyv(v + l)z
x S r(.v+Z + ii)(i/ + Z + 2ii)",;"1"1 J ..., [Ci/+Z + 2ii)*]/i»!
v not a negative integer, may be verified by replacing each Bessel
function on the right-hand side by its power series 7,2 (2). The series (25)
converges throughout the region
(26) \w(z)\ <1.
With (25) we may transform a power series into a Kapteyn series, if
each power of z in
(27) f(z) =2 b.z1
I = o '
is replaced by its Kapteyn series (25), we find after some algebra
(28) f(z) = z~v I an J [(v + n) z\
n= 0
v not a negative integer, where
< fcn
(29) a =% 2 (,v + /l^25)2^(v + 7l-S)(1/2V + 1/27l)2*•'l"v-^
n 5=0
The series in (29) is absolutely convergent when
\w(z)\ < 1 and \w(z)\ < \w{p)\
where p is the radius of convergence of (27)«
A Kapteyn series of the second kind is a series of the type
(30) 2 anJ%y^n[(&1/ + Yip + n) z] c/^+jJOiv + Kp + »)*].
n — 0 "
It may be shown (Nielsen, 1904, p. 307) that
(31) (y2 2)^=(v+p)r(l+>v)r(l + p)
*2, ( \ ) (v + p + 2*)-^-'
x <7,+n[(v+p + 2rc)z] Jo+n[(v + p + 2«) z\
where v, p, >v + p, are not negative integers.
Now, let
(32) /(*)- f bxz\
Z= o

68
SPECIAL FUNCTIONS
7.10.2
then (Nielsen, 1904, p. 308) we have
(33) /(*)=*^^ 2 anJ^npiv+y2p + n)z]J^^Av^A9^Mz\
n— 0
where
(34) (V2V+ Vip + y2n)^v^P+n+1 2-'Af>'+P+'>)an
^ 'An
S= 0 r
///2I/+//2P + R-S-1\
For further results and examples see Nielsen (1904, Chaps. XXII, XXIII);
Watson (1944, Chap. XVII); Bailey, (1932); Budden (1926).
7.1C= 3. Schlomilch series
Series of the form
(35) f(x)=l/2a0+ 1 amJQ(mx)
have been investigated by Schlomilch. There is an expansion theorem for
an arbitrary function of the real variable x over the interval (0, n) (Gray
and Mathews, 1922, p. 40; Watson, 1944, p. 619).
For a function f(x) which possesses, in the interval 0 < x < /r, a
continuous derivative of bounded variation there is an expansion (35) with
(36) an =2/(0) + 2;^-, J% f % "f (v sin<h) d<hdv,
0 00
(37) a =2w~! f vcos(mv)J f (v sind>)d<h dv.
n ** 0 0
A generalized Schlomilch series is
(38) 2 [am Jy(mx) + bm HVU)] (%mx)~\
The theory of such expansions may be found in Watson (1944, Chap. XIX)
and Nielsen (1904, p. 134). In a paper by Cooke (1928), the results
stated in Watson's book are partly simplified and extended. The theory is
based on the formulas
(39) J*nJjz sin0)(sin0)v+,(cos e)~2v d0
0 v
= 2-vn-'Ar{y2-v)zv-t sinz -l<Rev<1/2,
(40) J^H (: siad)(sind)v+y (cos0)-zv d0
0
-2-*,ir-*r(K-ix/)*v-1 (1-cosz) -3/2<Rev<l/2,

7.10.3
BESSEL FUNCTIONS
69
which may easily bederivedfromequations797(5)and 7„7 (9) putting there
\l - v and p - — v — K. Now, let us assume the validity of the expansions
(41) /(*)= 2 [aa Jv(mx)+bn Hv(mx)] (%mx)'v
-Vl<V<Yi, -7T<X<7T.
Here, we replace x by x sin #, multiply both sides by
(sin0)2v+1 (cosflr2*,
integrate with respect to 6 from zero to %n and use (39) and (40). Thus
we obtain formally
JQV27Tf(x sin 0) (sin 6)2v" (cos dT2v dd
= 7t"/z TvA-v) X [am s\n{mx) +bm(\-cosmx)]/(mx\
m = 1
and hence for the coefficients of the expansion (41)
(42) T{V2-v)a =m77"iA
x f t sm(mt)j%7T fit sin0) (sin 0)2v+l (cos d)~iv dddt,
-77" 0
(43) r(K-v)4B=-iBir"*
x J" < cos {mt) j'A'" f{t sin 0)(sin 0)2v+' (cos 6)-2vdddt.
~77 0
The series (41)with the coefficients (42) and (43) is called the Schlomilch
series of f(x)9
In the paper by Cooke, already referred to, it is proved that the class
of functions for which (41) with the coefficients (42) and (43) is valid in
any interval excluding 0, ± n, is the class for which the theory of Fourier
series applies. Furthermore theorems analogous to the Riemann-Lebesgue,
Parseval, Riesz-Fischer theorems on Fourier series are established. In
this connection see also Cooke (1927, 1929, 1930b, 1936); Wilton (1927);
Jesmanowicz,(1938); Wilkins (1950a).
Let us now consider some simple examples of the Schlomilch
expansion (41). We take f(x) = {ax)~v U^iax) (a arbitrary). This is an odd
function of x [compare 7.5(55)] and we have a^ = 0 in (41). On account
of (40) we obtain from (43)
nbn = - (-l)a 2"V+I m sinianVim2 - a2)
and thus
(44) 7T(ax)-vH(ax) = -2-v"sm(a7T) J (-D" " , {V2mx)~vBAmx)
«= I
-n<x<n, Rei/>-3/2

70
SPECIAL FUNCTIONS
7.10.3
Dividing both sides of (44) by sin {an) and making a approach zero we
obtain [see 7.5(55)]
(45) ir"*/r(i/+3/2)+ £ C-1)* {lAmx)-v-yHv{mx) = Q
m = 1
0<x<77, Rev>-3/2.
Now let fix) = (ax)"v Jv{ax). Here /"(x) is an even function of x, and
therefore, b m = 0. From (42) and (39) we obtain
7ran «-(-!)" 2"v+1 sin(a^)/7i7[aU2^a2)],
and therefore,
(46) ^(oxr^J^Ux)-—2"v+1 a"1 sin(a^)
x f (-!)■ m2(mz- a2r}(lAmx)-v JJmx)
0<%<77, Rev>-y2.
Making a approach zero in (46) we obtain
(47) H/r(i/+l)+ f (-l)m(%mx)-vJvlm») = -0
-lA<v<lA and 0 <x <tt or v > M and 0 < x < tt.
From (45) and (47) one sees that there are generalized Schlomilch
series with non-vanishing coefficients which converge and whose sum
vanishes almost everywhere. Such series are known as null series (Nielsen,
1904, Chap. XXV; Fox, 1926; Cooke, 1930)* The existence of null series
indicates that the Schlomilch expansion of a function, if it exists at all,
is not unique.
For other results and examples concerning Schlomilch and related
series, see Pennell (1932); Bennet (1932); Doetsch (1935); Erdelyi (1937);
Kober (1935); Watson (1931); Infield, et. al. (1947); Magnus and Ober-
hettinger (1948, pp. 58-62). Expansions where the Bessel and Struve
functions in (38) are replaced by their squares have been given by
Thielmann (1934).
7.10,4. Fourier-Bessel and Dini series
Let v > -1 and let x = yn and x = yn be two positive zeros of J (x)
(in this case all the zeros of Jv{x) are real; see sec, 7.9)« Using 7.2(56)
we then find from 7.14(9) and 7.14(10), respectively, that
(48) j; tJv(ynt)Jv(ynt)dt^J
K&UM2 * = *•

7.10.4.
BESSEL FUNCTIONS
71
Similarly if Am and An denote two positive zeros (see sec. 7.9) of the
function z Jv(z)+ a Jy(z\ where v> -K and a is any given constant, we
infer from formulas 7.14(9), 7.14(10), 7.2(54) and 7.2(55) that
(49) f* t JJXj) Jv(Xnt) dt = 0 njm,
The integral formula (48) expresses an orthogonal property of Bessel
functions and suggest the expansion of an arbitrary function f(x) of a
real variable x in the form
(50) /■(*)= £ aj(yx)
m = 1
with
(51) J*[Jv+1(y.)]*a. = £ tfU)Jviymt)dt,
where y} , y2, y3, ... are the positive zeros of the function Jv(x)
arranged in ascending order of magnitude. This expansion is called the
Fourier-Bessel expansion of /(#)•
Similarly from (49) we have
(52) /(*)= S hJ(\x)
«=i - v
with
(53) [\\ [J' (X •)]* + (A* - vz) [J(kM)V\ bm
= 2K /J tJv(xnt)f(t)dt9
where v > — Vi and A? , A2, ... are the positive zeros of the function
z Jv(z) + a Jv(z) arranged in ascending order of magnitude. This
expansion is called the Dini expansion of f(x)9
The theory of the Fourier-Bessel and Dini expansion is given in
Watson (1944, Chap. XVIII) and the following theorem may be stated:
Let t^ f(t) be absolutely integrable over (0, 1) and let v > -%; then if
0 < x < 1, the expansions (50) and (52) behave in the same way as an
ordinary Fourier series (see also Moore, 1911; Stone, 1927; MacRobert,
1931; Titchmarsh, 1946, p. 70). For the behavior near x = 1 and x = 0
see Watson (1944, pp. 594, 602, 615) and Young (1941); for the Gibbs
phenomenon Cooke (1927), Wilton (1928), Moore (1930). For series similar
to (50) and (52) but with the square of the Bessel function see Thielmann
(1934).

72 SPECIAL FUNCTIONS 7.10.4
Let for example f(x) = xv, then we obtain from (50), (53) and 7,7(1)
(») xv = I 2 Jv(ya *)/[y. Jv+] (y,)] 0 < % < 1,
«- i
(55)... I 2x.j,g.«)jyMu.)
t, u«-.^>wv(A.)]'+A:u',,tt.)]'
0^*£l, a+.i>'>0.
If/(x) = Jy(xz), then we obtain from 7.14(9)
(56)
J
v(xz) n y y. «/yty.»)
(57) J,y(xz)
^ J.
^w ^V^a^7 L/v« 'V+l v/v«7 'V*7 * ^V^a7 ^y + 1 '
£i Ui-^UiEJXW' + Cx: -»*)[Jv(km)]* !
0^*^1.
For further examples see sec. 7«15t
An expansion in series of Bessel functions which is suitable for a
positive finite interval has been given by TitchmarsW 1923a, XHI-XVI)
(see also MacRobert, 1931).
Let fix) be defined for a < x < b (a > 0). Then the expansion in
question is
(58) fix) -2 a. [Jv(ya x) Yy(ym b) - Yv(ym x) Jv(yn b)),
where z — y is the m - th positive root of
Jv(az) Yjbz) - Yv{az) JJbz) =0,
and
(59) i[J>„ a)]1-[«/„(?. «]«U.
= K"yIUv(ym «)]2 /' UJy.') rv(y.*) - V*.0 «?>.WW&.
GENERALIZED DIRICHLET SERIES
Series of the form
/(*)- f aAks)*K(ks),
* * n n is n
n- t
5 = cr + i r, A, < A2 < • ■ • < An < " • , lim An = «

7.105
BESSEL FUNCTIONS
73
have been investigated by Greenwood (1941)« For v = %they reduce to the
Dirichlet series
fis) =(%*)* 2 «ne"V.
n= 1
For various theorems concerning these series see Greenwood (1941).
7.10.5. Integral representations of arbitrary functions
The theory of Schlomilch series (compare 7.10.3) gives a method of
expressing an arbitrary function as a series of the functions J and H •
Similar methods may be applied to corresponding expressions of an
arbitrary function as an integral involving Bessel and related functions.
We always suppose for the following that f(t) is a real function of the
real variable t and is of bounded variation in the neighborhood of t = x%
If f(t) is not continuous at t = x, in the following formulas f(x) must
be replaced by %[/(* + 0) + fix - 0)]. The conditions on v in some of
the following expansion formulas have been relaxed by Cherry (1949a).
The simplest type of such a representation is Hankel's integral formula
(60) f(x) = r Jjtx)tdt f°° f(v)J(vt)vdv,
0 v 0 v
valid if v > —% and
0
is convergent, or v > — 1 and
f~tX\f{t)\dt and J*1 tv*' \f(t)\ dt
0 0
are convergent. The theory of the expression (60) has been thoroughly
discussed by Watson (1944, Chap. XIV); Titchmarsh (1948, p.240)and
Tricomi. In case v=±%, (60)reduces to Fourier's sine and cosine integral
respectively.
A generalization q£ Hankel's integral is due to Hardy (1925), who gave
the formula
(61) fix) = J7 Gv(xt)tdt J*0°° Fv(vt)vf(v) dv,
where
(-1)* «*)v+2B+ta
(62) Fv(z)=2
J^Q Tip + m + 1) T(a + m + v + 1)
r(a)T(v + a)

74 SPECIAL FUNCTIONS 7.10.5
(63) G v(z) = cos (a tt) Jv(z) + sin (a tt) Yv (z\
valid under the following conditions (Cooke, 1925):
(i) a>-l, a+.v>>-1, v+2a<3/2, |.v| < 3/2,
(ii) ta fit) integrable over (0, 8), a = min(l +-v+ 2a, %), S'> 0,
(iii) ^ /(*) integrable over (8, «>).
The theory of the expansion formula (61) has been given by Cooke(1925).
SPECIAL CASES OF HARDY'S FORMULA
If a = 0, we obtain Fv(z) = J^U), G^iz) = Jviz)* This case reduces
to Hankel's formula (60). If a = %, we obtain F^U) = H^U), G viz) =
^ (z). This leads to
(64) fix) = J" yv (*t) tdt r flv (t;0 v/M dv.
0 V 0 V
If a = - % we obtain
(65) fix)
/°°r(vt)v~'* tt* "i
[«"!■<,+»"-(,rt.r',)*-
If v= %, we obtain
Fv(z) = (Kir*)"* C2ft+|.(*), Gvfe) = &**)"* sin (z -arf
where C2a+| (2) is Young's function 7,5 (85).
Weber and Orr's formula *
(66) ,U) = f W*M'WW tdt *^^
A [Jv(^)]2 + [yv(^)]2
x Jo [Jv{vt) Yv{at) - Yv{vt) Jv{at)] vf{v) dv,
valid forvreal and J™ t*\f(t)\ dt convergent, reduces to Fourier's sine
integral in case v = ± lA (Titchmarsh, 1923; Watson, 1944, p. 468).
Another formula due to Titchmarsh (1925) is
(67) fix) =* TT J°° Fvixt) t dt J°°id/dt) [t Avivt)] Vfiv) dv
'0 v ' -'o
where
(68) Tv(z) = sin(an)\[JvU)]2 - [y„(z)]2i - 2 cos(a,r) JJz) Yv(z),
(-1)" T(v + m + a + M) iT* z2*+2o+2»
(69)
V (-!)" r(y + m + a + %) n n z^—
v Z _.^0r(o+m+l) r{v + a + m + l) I"(2v + a + m + 1)'

7.10.5
BESSEL FUNCTIONS
75
valid under the following conditions (Cooke, 1925)
(i) a>-l, fl+2v>-l, 1> a+ !/£-& \v\<\
(ii) ta f(t) integrable over (0, 8), a = min (1 + 2v + 2 a, 1),
(iii) £/(0 integrable over (8, *>), 8 '> 0.
The theory of the expansion formula (67) has been given by Cooke (1925)»
Special cases of (68) and (69) are
a = 0, r„(z) = - 2 Jv(z) Yv(z), Av(z) = [Jv(z)]\
a = -v, Av(z) = Jv(z) J_v(z),
a=-2„, Av(z)=[J_v(z)]\
A generalization of'Laplace's integral involving Bessel functions has
been given by Meijer (1940, pp. 599, 702):
(70) /U)-(iri)"1 IC + i0°'Uxt) (xt)* dtf°°K(tv)(tv)Xf(v)dv.
C-ioo v o v
As Kv(z) = K_v(z) 12 (14) we also have
(71) /(%)= {IniY1 J°*£ [Iv{xt) +T v(xt)] ixt)* dt
xj°° Kivt)(vt)X f(v)dv
o v
(cf. also Boas, 1942), In case v- ± %, (71) reduces to 'Laplace's formula.
Other integral representations of arhitrary functions are
(72) fix) = -V2Jio°t Jt{x) dt J°° H?Hv) t>-' f{v) dv
-ioo 0
(Kontorovich and'Lebedev, 1938),
(73) f(x) = n-* £VwU,+t,K<(x+t,(«) *
(Cram, 1940),
(74) f(x) = V2 I Ji«(c^,y-i«(c')t<fc /'" [J. f«T+«/.,(«•)]/(,,)</»
(Titchmarsh, 1946, p. 83),
(75) xf(x) = 2w~2 /J" K ./%) t sinh Grt) c/t f" Ktt(v) f(v) dv
(Lebedev, 1946),
(76) fix) = (it*)"1 Ja+l°° tKXx) dt j°° v~* f(v) liv) dv
a-ioo r o x
(Lebedev, 1947). For further examples see Hardy.(1927), and Hardy and
Titchmarsh (1933%

76
SPECIAL FUNCTIONS
7.10.5
DUAL INTEGRAL EQUATIONS INVOLVING BESSEL FUNCTIONS
In some problems of potential - and electromagnetic or acoustic
radiation theory the unknown function satisfies one integral equation
over part of the range (0, <») and a different equation over the rest of the
range (Nicholson, 1924; King, 1935, 1936; Sommerfeld, 1943). The pair
of equations (Titchir.arsh, 1948, p. 337; Busbridge, 1938)
J0°° yaf(y) Jvby)dy ~ g(%) o < x < \
(77)
J™ f(y)Jv(xy)<ty = o *>i
has the solution
(78) T{% a) fix)
= (2xy-,A°-Jo't'+'AaJv+iia(xt)dt Jlg{vt)vv+i a-v2YAar' dv,
supposed a> 0.
The special case a - 1, v = 1, g(x) = 1 has the solution
%?r/(x) = x~2 sin x -x~] cos x.
The pair (Tranter, 1951)
J°°y $ (y) Jixy) dy - f(x) 0 < x .< 1,
o v
(79)
J^Qiy) Jvixy) dy = Fix) x > 1
has the solution
(80) $(y) = Hiy) + {% ny)% /„' tv+* L U) Jv+% (ty) dt,
where
(81) H(y) = F(l)Jv+i (y) +y J™xF(x) JJxy) dx,
(82) LW = (2/77)r^Jo'*I'+' [f(x)-J"yH(y) Jv{xy) dy] {t* - x2)'* dx.
The solution of
Jo°° $ (y) Jv(xy) dy=G (x) 0 < x < 1,
(83)
J°° y <My) c7vUy) dy = g (x) x > 1
is

7.10.5 BESSEL FUNCTIONS 77
(84). *(y) = K(y) + (Kiry)* /J «v+* f (t) «/„_„Uy) A,
where
(85) A (y) = Jj°°.*g (*) c/v(«y) dx,
(86) Mffi2v^(i) -If (0) + i J0V-**)"* If' U)d*
(87) «(x) =x"CW - *VJ" X(y) Jv(xy) dy.

78 SPECIAL FUNCTIONS 7.11
SECOND PART: FORMULAS
7.11. Elementary relations and miscellaneous formulas
SPHERICAL BESSEL FUNCTIONS
In (l)to (13)/i=0, 1, 2, ... ,
(D Jn+»(z)=(Y2 7TzrlA[sin(z-Y2n7T) 2 (-ir(i» + &2in)(22)-2*
n A « = o
+ cos(z-J4inr) 2 (-lrdi + H, 2m + D(22)"2""1],
m = 0
(2) yn+KU)-(J*ir*r*[sinfe-J4i»ir) 2 (-1)* (» + & 2m + l)(2z)'
0=0
-cos (2 -^/xtt) 2 (-l)"0*+& 2m) (2z)"2"]f
m = 0
(3) #'"(*) -(Kir*)-* f-'e" 2 »"(» + X, »») (2z)-",
n * «=o
(4) H «x(z) = (H,rz)-K i"+l e"** £ (-i)• U + K, m) (2*)"-,
(5) J.n_,(2)=(-ir+'yn+,(z); *-.-*■<*>-<-ir «w*>.
(6) ■fli».xw-.-(-ir».%u); <>_,(*)=-a-ir//'V*>-
(7) ^(,,..^«..)-*.-^.)"±i.
(8) yB+J|(,,..(.V«„,-«,.«(^j!^.
(9) ^jy *>--*• <-ir (***>-* *"+l£iJ -il,
(10) ^(2) = ,(-1)" (Xir*)"* 2n+1 f ~J ^-,
/ </ \" sin z
(11) ^B(z) ».(W*)* Jm+H(z) = (-Dn W— J —,
(12) £n(,)(z) = (Xit/*)* fl^ (*) = - i(-1)" *n QLJ ll,
•2jn -1

7.11 BESSEL FUNCTIONS 79
(13) (™(z) = (H n/z) * H*\% (z) = i (- IT z »
(14) Jk(2) = y_^(2) = 0^2r* sin 2,
(15) * „ (*) = -J_x (2) = - (X IT*)"* COS 2,
(16) '^U)-(J£jr«)"*sinh*,
(17) # »>(*) = - » fl »} (2) = - i (Hff *)"X e ",
(18) #,{»(*) = * H™(z) = iOJff*)"K e~".
RECURRENCE RELATIONS AND DIFFERENTIATION FORMULAS FOR
MODIFIED BESSEL FUNCTIONS
(20) (-^J [2--^(2)]=2--»I+ii(2),
(2l)(^)^Kv(z)U(^:
(22) (-^-J b"wAvU)]-(-l)"
v-
2_V_"X . (.2).
(23) I.1(2)-jr/+1U) = 2v2-,^(2),
(24) v.w+'UW-2^^
(25) K^to-K^to--^*"' Kv(z),
(26) *„_,(*)+*:,,+,(*)«-2K'„(*).
WRONSKIANS AND RELATED FORMULAS
W{wy, wz) =wy vo'z -w'f wz
(27) V (Jv, J_J =- 2(ir«)-' sinter),
(28) IP (<7„, V )-2(m)-',

80 SPECIAL FUNCTIONS 7.11
(29) W(Jv, Hl">ui) = ±2i(nz)-\
(30) W{R™,R™)=-U{ttz)-\
(31) WO^, I_v) =-2(772)"' sin(n7),
(32) W (I, Xv)=-2-'.
(33) «/„(*) «7_v+t (*) + «/_„(*) Jv_,(2) = 2(772)-' sinW),
(34) #'"(2) flv« (2) -flj,", (2) fl ««(z) = - 4i(772)-',
(35) JAz) Yv^(z) - YAz) Jv_, (2) = 2(772)"',
(36) Jv_, (2) H »>(*) - J>) "fl"2, (2) = 2(7712)"',
(37) JAz) Hlv% (2) - Jv_, (2) fl ^(2) = 2(7712)-' ,
(38) I (z)I_v+I(z)-I_v(z)^_f(z) =-2(772)"' sin(„77),
(39) K^zYW + KAzY^M-z-*.
FUNCTIONS OF VARIABLE ze'77*, (m integer)
(40) JAze^^^e^^JAz),
(41) YAzei«iT)=e-i"™Y(z) + 2i 81°(""r,*/ co»(it.i/)-J (z),
*^ sin (v 77)
,,., . sin[(m — 1) nv] ,,., , sin(m77j/) . .
(42) H™(ze «") = • //<' »(z) - e"'™ , , ■ #<2,(z),
^ sii^nv) sin (771/)
_, v sin[(/n+ 1) ;7v] ,.,*, v . sinimnv) „m ,
(43) fl «2)(ze »w) !— H1%) + e ™ - //"''(z),
SUHTTv) Sin(7Tv)
(44) T(zei-7r) = ei»7rvJv(z),
sin (m nv)
(45) X (ze ™77) = e~l»™ K.(z) - in . , v 7 (z).
Vv~ f^
s
in WO v
In case v is an integer equal to n9 then
V-+ n sin(77v)
where I is equal to m — 1, m or m + 1 respectively.

7.12 BESSEL FUNCTIONS 81
7.12. Integral representations
BESSEL COEFFICIENTS
(1) 7T J (z) - J cos (z sin <fi — ncfi) d<f>,
-7T
0
rTT
(2) n Jn (z) = Tn JoW e "C08<*> cos fc <£) </<£,
(3) n <J2n(z) = 2 J cos (2 sin <£) cos (2rc <£) */<£,
(4) 77 </2n + 1 (2) = 2 J sin (z sin <£) sin [(2n + 1) <£] c/<£.
In (1) to (4) n equals 0, 1, 2, ... .
POISSON'S INTEGRAL
(5) r(i/+ K) '«7VU) « 2iT* (^y J"*" cos (2 sin <£) (cos <f>)2v d<f>,
(6) = iT* i}AzY J*" e izain<t> (cos cf)^ d<f>,
(7) = n~'A (Az)v J' e «« (1 - t2)I'-« «fo,
(8) = 2rT* (1/22)vJ"' (l-f2)v_* cos (zt)dt,
0
(9) = *"* (Hz)v /%•■"»* (sin 0)*(ty,
(10) r(i/ + K) I (2) = n~* QizVj^ e-zt(l -tz)v-* dt,
in (5) to (10) Re .i/>-&
HEINE'S FORMULA
(11) 77Yv(2) = e<*I/77!;Jo7Vi"»tcosU)</,
- f° e i"osht [cosh (i/t - iw) + e~ivn cosh (i/t)] <fei
U .< arg 2 < 77.
MEHLER - SONINE FORMULAS
(12) r(H -i/) «/„(*) = 2rT* («x)-w J"~ (*2 - irv-y> sinte) dt,
(13) r(H-i/) Yv(x) =-%,-* tiix)-vS~ (t2 - IT*'-* cosOct)rft,
in both formulas x > 0, —XA< Re v < %.

82 SPECIAL FUNCTIONS 7.12
(14) n Jv(x) = 2 J sin (x cosht - 1Avtt) cosh (vt) dt,
(15) nYv{x) = -2] cos (a; coshfc - % vtt) cosh(w) dt9
in formulas (14) and (15) x > 0, -1 < Re v < 1.
(16) nJ,(x)= f e~vt sin(x cosht-V2V7t) dt + J cos (x sint - vt) dt
x >0, Re v>0.
Generalizations of Schlafli's integrals (Lambe, 1931)
/x + v \ ^
(17) 77^—_LJ </J(x2-y2)^]= J^ e y"* cos (x sint- vt) dt
- sin(vn) j'°° e-vte-ycosli1r**^tdt Re(* + y)>0.
o
/x + v \ ^ v
(18) 7rr_Lj yv[(x2-y2)^]- /JV^'sinfrsin^w)^
- J°° (evt + ycosht+e-vt-yc06htcosvn)e-xsil,htdt
° Re x > Re y > 0.
MODIFIED HANKEL FUNCTIONS
(19) r(%-v)Kv(z) = nH(%Zrvf00e-*t(t2-l)-v-*dt
Rez>0, Rev<V2,
(20) rOi + i/) Ky(*) = ** (VizY r e~zcosht(sinh *)2v rft
° Rez>0, Rev>-1/2,
(21) #v(*) = J°V2COshtcosh(i4)^ Rez>0,
o
"0
(22) T{v+ lA)Kv(2)=(yin)'A zve-* f^e-'U"-* (1 + %t)v~* *
Rez>0, Rei/>-%,
(23) Kv(az)=1/2av_f0OOe-^(t+a2r,)ri/-' Jt
Rez>0, Re(a2z)>0,
(24) ^ (az) = 1/2eiKl/7ral/ re^2(t-a2r1)rv-'Jt
lmz>0, lm(a2z)>0,
(25) Kv(x) cos (¥2 vrr) = J cos(x sinhj) cosh(^t) dt,
(26) Kix) sin(V2vn) = } sin(x sinhj) sinh(w) dt,
0
in formulas (25) and (26) x > 0, -1 < Re v < 1.

7.12 BESSEL FUNCTIONS 83
(27) Kv(z) = iT* (2z)vr(i/ + Vi) fQ (t2 + **yv~XA cos t dt
Hei/>>- % !argz| .< %tt.
HANKEL FUNCTIONS
(28) i Y(lA - i/) -fl^fe) = 2*"* (H*)"" J^° e i2t(t2 - 1)-^ ^
lmz>0, Rev<K,
(29) - » Hfc - v) flJ2,U) = 2*"* (>£*)-* J^e-^U2 - l)*1^ ^
Im z < 0, Re v < l/2,
(30) »r(}f-i/)fli!,U) = (Jizr,/2^ J"o°°r2v(i + ^)-^6"(,+t2^^
lmz>0, Rev<^,
(3D -»ros-1/)#<2,u) = 2*-* («o"v J-V^q + *2r* e-*»+«2>* a
V 0
Im z < 0, Re i/ < %,
(32) r(l/ + K)^"(Z) = (^7rZ)-'<ei(*-^'T-«'T)
x J"0"6'8 e-'t"^ (1 + Kite-')"-* A
» 3
Rei/>-K, |fi|.<H»r, 5-Kn-<argz.<5+—*,
(33) r(l/+^)fl^'(Z)=(H7T2)-!<e-i(J-^^-«w)
x J"006^-'*"-* (i- y2itz-Y~* dt
0
Re.v>-H. ]8\<%u, -3n/2 + 8.<aTgz<y2„ + 8>
BARNES' REPRESENTATION
(34) 2n2 H « >(z) = - e-'^7r J-c + io° r(_ „_ s) r(TS) (_fcfa)v+* ds
|arg(-*'z)| < y27T,
(35) 2TT*H{2)(z) = eilAv7TJ~C*i0°r{-.v-s)r(-s) GHz)"*2* ds
\avg(iz).\ <%n9
C is any positive number exceeding Re v.
(36) 2ir» J (*) = Sie°r(-s) IT(v + s +.!)]-' (K*)v+I,(fc
*>>0, Rev'>0,
(37) tt^ Hllr(z) = - e iiz'V7r) cos fcir) (2*)"
x Jioor(-s) r(-2.i/-5) r(v + s + y2) (-2**)* <b
|arg(-jz)| .< 3tt/2, %v not an odd integer,

84 SPECIAL FUNCTIONS 7.12
(38) 7T5/i Hll\z) = e-il'-V7r)cos(vn) (2z)v
xjt°°r(-s)r(-2v-s)r(.v + s + 1A)(2izyds
- ioo
|arg(iz)| < Zjt/2, 2v not an odd integer,
(39) 27r2iKv(Z)=(Wz)He-*cosW)
x Jie°T{s) T(K - s -,„) TiVi - s + v) (2z)s ds
|arg z\ < 3 77/2, 2v not an odd integer.
INTEGRALS EXPRESSED IN TERMS OF RELATED FUNCTIONS
(40) J " cos (z cos 0) cos v$ d<£ = 77[4 cos (54w)]""1 [J {z) + J (z)]
o v
= -,vsin(W) s.1|VU) = 77 [4 sin(K2w)]""1 [E^U) - E_vU)]f
(41) J^ sin (z cos <£) cos v<£ d$ = tt[4 sin (fr/ir)]"1 [J (2) - J (2)]
0 v 1/
= cos(M^) s0fV(z) = - tt[4 cos (Hi/it)]"1 [EVU) + E_v(z)]
(42) J" cos (z sin 0) cos v<p dcfa = - v sin(v7r) s_t VU),
(43) ] cos (z sin <£) sin vcp d<fj = — v(l — cos> 1/77) s _f (z)
(44) J77 sin (z sin 0) sin vcj) dcf> = sin(1/77) s 0 v (z),
0
(45) J77 sin (z sin <£) cos v$ dcj> = (1 + cos 1/77) sQ ^(z),
(46) f"e'«-"i*tdt = %[JSn(z)-frEn(z)-nYn(z)]
rc = 0, 1, 2, ... , Re z>0,
(47) ;o°° e-nt-zsinhtdt = JJ(-iyi+l [5Jz) + ;rEn(z) + n Yn(z)]
n = 0, 1, 2, ... , Re z >0,
(48) S^Jz) = 2" J~ e'u 2Fx(K-%fi + %v,%- Kfi- lA>v; 'Al-t2) dt,
Re z>0,
(49) Sfi VU) = ^+I /"aT" 2Ff(«-HM+>U K-y2ii-Kv;A/2;-t2)dt9
Re z '> 0,
(50) S0 (z) = /" e-"inh * cosh (w) dt,

7.13.1 BESSEL FUNCTIONS 85
(51) vSQtV(z) = z £° e"23inh*sinhCvi) cosh t dt,
(52) S {z)~z /°°e~2sinh*cosh(w) cosh t dt,
in (50) to (52) Re z > 0.
7.13. Asymptotic expansions
7.13.1. Large variable
(1) H™{z) =(V2ttz)-X e H*-Xvrr-Xrr)[ *£* ^ m) ^2iz)'n + 0 (\z\ ~M)]
ffl= 0
— 7T < arg z < 2 77,
(2) !?»>(*) = O^z)-* e-<(*^^-^)["f' U m) (2j2)-« + 0(:|Zr*)]
« = 0
- 2t7.< arg z < 77,
For an appraisal of the remainder after the M -th term for complex v and
- Vnr < arg 2 < 3t7/2 and for - 3 77/2 < arg z < Y2 n see Watson (1944,
p.219). These results have been extended to the range — 77 < arg z .< 277
and - 277 < arg z < tt by Meijer (1932, pp. 656, 852, 948, 1079). For the
asymptotic behavior of a function expressed as an infinite Hankel
function series see Meixner (1949).
(3) JV(Z) = (V27TZ)-IA [COS (z-VtW-YtTT)
xtf? (-1)* U 2m) (2z)~2» + 0(|z.r^)]
»i=0
-s-Mz-Vivn-Vtn)^ {-\Y(v, 2m + l)(22)-2"-,+ OCM-2*-')]!
— n < arg z < tt,
(4) Yv(Z) = (}/i7TZ)-ii{8m(z-y2V7T-y4n)
x ["£' (-1)" („, 2m) (22)-2» + OXIzr^)]
m = 0
+ cos(z-1^V7^-^7^)[,'£l(-l)•,U2m + l)(22)-2»-,+0(|^^2*'-,)]}
m = 0
- 77 < arg 2 < 77.
For formulas for the remainder after the M-th term see Watson (1944,
pp. 206, 209) and in case of complex v Meijer, (1932, ref. above)• For
further formulas see Burnett (1929).

86 SPECIAL FUNCTIONS 7.13.1
(5) Iv(z) = (2«rx \e* f~i{-\Y(v, m) (22)-" + OCIil"*)]
«i= 0
+ ie-' + iv"[''il (u, m) (.2z)-"+0(\z\-u)]\
- VittK argz < Stt/2,
(6) I U) = (2z)"^ t^2 cos(77v) l^f1 [e*-i(-ir e^771^]
«=0
xru + y2-.v)ru+i/2+v)(22p>! + ezo(kr*))
-3 n/2 < arg 2 .< lA n,
(7) X (z) = (W*)* e~z [V (i/, m) {2z)'m +Ot\z\~")]
- 3 n/ 2 Org z.< 3^/2,
Throughout these formulas
(v, m) = 2"2"1 I (4i/* - 1) (4 ^2 - 32) • • • [4 v2 - (2m - l)2]}/ro !
-rW+.v+in)/[m!r(JS+.i/-iii)l
7.13.2. Large order
(8) 2irTp(*)«2* (p2 + *2)-^ exp[(p2 + *2)*-p sinh_,(pA)]
x[W2(-2)- a r(m + K)(p2 + ^2)_^+0(^-w)]
» = 0 m
P, X '> 0,
(9) a0 = l, a.^-i+^d+^VpT1,
a2=4-^(1+^2r,+^(1 + ^2)'2'--
(For other expansions of I (x) see Lehmer, 1944; Montroll, 1946.)
(10) Kp(X) = 2-lHp2+x*)-% exp[-(p2+%2)^+psinh-,(pA)]
x [*£' 2" a rOn + M) (p2 + **)-*■ + 0(x"")]
p, x >0, aB as in (9),
(11) 77 fl»»(x) = 2y' (*2 - p2)"« exp[i(*2 -p2)* + ip sin-' (p/x)]
xe-'^W'l'i1 2-6 r(m+'/2)(-i)"U2-P2)-^ + 0U-*f)]
x>p>0,

7.13.2 BESSEL FUNCTIONS 87
(12) 6, = 1, ^-—JLu-xVpV',
b ._L-iI(i_xVp2)-,+—a-**/?2)-2
2 128 576 V 3456 F
(13) n ■//«»(*) - - i 2* (p2 - x2)"* exp[-(p2 -x2)* + p cosh"1 (p/x)]
p
x[ 1 {-l)'2'bKr(m+1A)(p2-x2rii' +0{x-M)]
p > x > 0, bu as in (12),.
(14) 2n Jp(x) = 2*(p2-x2)-* exp[(p2 - x2)* - p sinh"1 (p/x)]
x [*£' 2" 6 rU+W (p2-x2r*» + 0(x"*)]
«=o "
pi>x>0, 6B as in (12),
(15) »#"'(*) ~ - 2/3 f e2("+1)77i/3 B (ex) sin [On + 1) n/3]
p n = 0 "
xr(m + l/3)(x/6)-(n+1)/3
P£x, p, x>0, e-l-p/x, f = o(x_2/3),
(16) B0Ux) = l, B,(fx) = €x, B,(fx)-H(«)2-i.,
B,(fx)oi(fx)3-icx, B.(ex)=i;(ex)4--L(fx)2+ —,
3 6 15 A 24 24 280
5. («) —=r<«)5 ~-(«)3 + — «•
5 120 60 8400
(ForB6, B7, B9, see Airey, 1916, p. 520.)
PURE IMAGINARY ORDER
(17) 2n Jip(x) = 2*(p2 + xT* exp[»(p2 +x2)^ - ip sinh"1 (p/x) - % in]
xe^["i' (2i)"a r(m + K2)(p2+x2)-^+0(x-")]
«=o *
p, xi>0, ob as in (9),
(18) K. :(*)-2_x (x2-p2)"^ exp[-(x2-p2)*-p sin"'(p/x)]
Af-l
x[2 (-l)-2-6 r(m+X) (x*-pTX" +0(x"w)]
—.is n
1^ f~2_p2)~^
*>>p<>0, 6m as in (12)

88 SPECIAL FUNCTIONS 7.13.2
(19) K ,p(x) = 2'A (p2 -x2)-* e^^
x|"f' 2-5 r(m + K2)(p2-x2)-K»
«=o "
x sin [ !4 * m + p cosh"' (p/x) - (p 2 - x 2) *+!£ d + 0 (*"")!
pOxOO, bn as in (12),
(20) K Ax) ~ 1/3n- e-'^ I (-1)"1 C (<rx) sin[U + 1) a/3]
« = o m
xr(H« + ]/3)(V6)",,+,,/'
p ~ x, p, x'>0, e = l-p/x, e = o(x~2/3),
(21)C(«)-L C, («)=«, C,(f*) = H(<r*)2+ —,
20
c.uri -*«,,).4„, c4u-> ^,r+±M- ♦ JL,
(22) ^,<^)=2^(p2+^2)-^ expb'^ + ^-ipsinh"1^)]
xe^^[^' (-0"2-6 r(w + Jd(p2+*2)-x- + 0(x"lf)]
p, * »> 0, bn as in (12).
7.13,3. Transitional regions
NICHOLSON'S FORMULAS
{x ^ n, n integer > 0)
(23) jAx) ~z-v* (^/x),/3 tJ1/3(a + j.uaoi
(24) yn(x)~3-,/6(^x),/3[J|/3(^)-J_J/3(^)],
*>«, £=-(>/2x)-*(»-x)3/2,
3
(25) J Ax) ~ a"' 3-,/6(£A)1/3 £,/3(£),
(26) Yn(x) ~ -3~,/6 (&x),/3 [i,^(^) + XT/3(£)],
re>>x, £=|-(^x)-*(x-re)3/2,
(27) e i7r/6 #» >(*) ~ 3~,/6 (#*),/3 ««>(£)
3
arg (x — re) = 0; for x .< re, arg (* — n) — n.

7.14.1 BESSEL FUNCTIONS 89
WATSON'S FORMULAS
(28) Jp(x) = 3~* w [J)/3 ipw*/3) cos 8 - Y 1/3(pw>73) sinS] +0 (p-1),
(29) Yp(x) = 3-« «;[J1/3(p«;3/3) sin8 + yi/3(pM,3/3) cosS] +0(p_,X
x > p, 8 =pw - piv3/3 - p tan-' w + rr/6, w = (x2/p2 - 1)%,
(30) Jp(x) =3"x tT' we'a«t/s(pu;s/3) +0(p-'),
(31) yp(x) =-3"x «;ePa[TI/3(p«;3/3) + I_I/3(p«;3/3)] +C,(p-')
x.<p, a = p(u> + u>3/3 - tanh-' w), w = (1 -xVp2)5*,
7.13.4. Uniform asymptotic expressions
LANGER'S FORMULAS
(32) Jp(x) = w-'A {w - tan-' u>)* [Jt/3(0 cosU/6) - y|/s(z) sin(jr/6)]
+ o(P-^),
(33) yp(x) = w~y'(w - tan-' «;)* [Jt/S(z) sinU/6) + Yx/a (z) cos (jr/6)]
+ 0(p-4^),
x > p, m; = (x2/p2 - D^, z = p(u> - tan-1 u;),
(34) Jj,(*)-it-ih;-X (tanh"' tv -«;)* K%/a(z) + 0(p_4/3),
(35) yp(x) = -w~x (tanh'' a, - w)5^ (z) + £,„(*)! + 0(p-4/3)
x <p, w = (1 -x2/p2) » z = p(tanh_I «; - w).
7.14. Integral formulas
7.14.1. Finite integrals
(1) JzI'+,T(z)rfz = zv+,^+I(z),
(2) Jz"I'+,T(z)(iz=z-v+,^_I(z),
(3) Jzv+<Kv(z)dz=-zv+xKv„(z),
(4) JV1"" Xv(z) dz = - z-"+' £„_, (z),
(5) Jzv Jv(z) dz = 2""' ct* r(v + K) z [Jy(2) Hv_, (z) - Hv(z) Jv_, (z)],
(6) J"zI'£v(zWz=2''-1 n'A r(u+1A)z[Kv(z)I.v^(z) +-Lv(z) Kv_x(z)],

90 SPECIAL FUNCTIONS 7.14.1
(7) Jz^JJz) dz-di + .v-Dz Jv(z) Sm_1fV_, (z)-zJv_, (z) S^ „(*),
(5) and (7) are also valid, when the Bessel function of the first kind is
replaced by the Bessel function of the second or third kind.
'Let w (z) and W (z) be any Bessel function of the first, second, or
third kind and the order v and fi respectively; then
(8) J"[(£2 - a2) z + (v2- f)/z] wv(az) WJ.fiz) dz
- z [a WJfiz) w'Jaz) - fiwjaz) WJfiz)]
= az W Jfiz) w.v_Jaz) - fi z W M_, (/3 z) wjaz)
+ {fi-.v)WJfiz)wJaz),
(9) jzwJaz)WJfiz)dz = z{fi2-a2)-'
x [01?v+l (0*) wjaz) - a WJfiz) w^ (a*)],
(10) jzwjaz) Wjaz) dz
= y4Z2[2wJaz)WJaz)-wv+}{az)Wv_i{az)-u;v_i{az)Wv+t(az)],
(11) Jz"1 wjaz) Wjaz) dz = (2v)~% wjaz) Wjaz)
/o n-. T / ,3WJaz) 3WV+Aaz)l
+ (2„) ' az ^v+, («) ^ - wjaz) —^ -J .
'Let f (z) and V (z) be any modified Bessel function of the first or
second kind and the order v and //respectively, then,
(12) J[(fi2 -a2)z + {fiz-,v2)/z] vjaz) V Jfiz) dz
= z[-aVJfiz)vJaz)+fivJaz)VJfiz)l
(13) Jz [vjaz)]2 dz
= -V*z2 {[vjaz)]2 - [vjaz)]2 (1 * <T2 z'2 u2)\.
For other indefinite integrals see Watson (1944, pp. 163-138); Thielrnann
(1929); McLachlan (1934, p. 115); McLachlan and Meyers (1936, p. 437);
Straubel (1941, 1942); Picht (1949); Horton (1950); Luke (1950).
(14) J*" J [z (sin 6)2] J[z (cos 6)z] (sin 6 cos 0)""1 d$
0 M v
= %(i/~' +/i"') Jv+M(z)/z Rev>0, Re^t>0,
(15) J^J [z(sine)2] J [z(cos0)2]ctn0rf0=K2J +„(z)/^
Rev>-1, Re/x>0,

7.14.2 BESSEL FUNCTIONS 91
(16) J%n J [z(sin0)2]J [z(cos<?)2] sin 6 cos Odd
= z"' f (-1)» J +,(z) Rev>-1, ReM»-l.
(17) J"*77 J [z(sin0)2] J^(costO'ltsine)2*--' (cose)28-' d6,
0 M *'
(Bailey, 1930, p. 419, 1930c, p. 203; Rutgers, 1931.)
(18) J*" Jk{z sin0)Jv{z sin 6) (sin 0)2S+l (cos 0)2"+1 rffl
(19) /""'Jxfe sin 60 J (* costf) (sine)2S+1 (cos6l)2^+1 dQ,
o ^ v
(Bailey, 1938, p. 145.)
(20) J'An[J (z sin0)]2 (sin0)zS+1 (cos0)2"+1 dd
0 V
(Bailey, 1938, p. 141.)
(21) Jf* [Jv(z sin 0)]2 sin BdQ = I J2v+2b +, (z) Re v > -1,
(22) J* tksin{z-t) J-(t)dt,
o
(23) J"ZAcos(z-t)Jv(t)^,
(24) sinjr(v+/i) f^K (2z cos6!) cos[(n-v) 0]d6
(Bailey, 1930c, p. 204, 205.)
f*nK + (
= ^ [7^(z) T.MU) - Tv(z) 7M(z)]. |Re (M + ,„):| .< 1.
7.14.2. Infinite integrals
INTEGRALS WITH EXPONENTIAL FUNCTIONS
= -y2^y-'exP^ia2/y2)
|Re v| .< Ji.
(26) /^e"tr|-fl^"(2*V«)dt-2«v(2*)-flJ,I)(2*),
(Hardy, 1927.)

(i °w) t*(t oVrZ)
92 SPECIAL FUNCTIONS 7.14,2
(27) S~'Iv(at) e~yZ ** dt = H** y~! exp (
Rev>~l, Rey2>0.
[see also 7.14 (60) to 7.14 (79)].
SPECIAL CASES OF THE WEBER-SCHAFHEITLIN INTEGRAL
(28) J°° t"x J'(at) sin (bt) dt = iT* sin[tt sin"1 (b/a)] b < a,
0 r*"
= a%-' sini'Anti) [b + (62 - a2) ^ ft > a,
Re m >-l.
(29) J°° J*1 J (a*) cos (60 d« = jT! cos [fi sin"1 (6/a)] £ < a
= p-» a^ cos «^) [6 + (62 - a2)^ 5 ,> a9
Re /z>0.
(30) /°° J (at) cos (60 dt = (a2- b2)^ cos [p sin"1 (6/a)] b < a.
o ^
= -a^sin(1/2^)(62-a2)"J/2[6 + (i2^a2)1/2]^ 6 <> a,
Re /i > - 1.
(31) J°° J (at) sin (60 <b =(a2 - 62)~* sin^shT1 (b/a)] b <a
o ^
= a^cos(y2^)(62~a2)^ [6 + (62-a2)^ 6 > a,
Re p > - 2.
For the corresponding integrals for the Neumann function, see Nielsen,
(1904, p. 195).
(32) \{tt(<v2 - fi2) J~ J (at)'Jv(at) *~f dt = smlA(v - p) 77,
Re(i/+p)>0.
(33) r J MJ Mt-^dt= '"W^j^
Re(i/+ /i)>0.
(34) r (»/ - p) J°° J„ («) J (bt) t^v+y dt
= 0 A .< O,
Re v>Re n>- 1,

7.14.2
BESSEL FUNCTIONS
93
INTEGRALS RELATED WITH THE WEBER -SCHAFHEITLIN INTEGRAL
(35) 2^+1 T(v+ 1) av+*-P f K^(atyiv(bt)t-Pdt
= bvr(lA-lAp + lAli + Vw) Y(y2-lAP- Vzy. + Vw)
x Jxi}A-lAp+lAii+V2'V, 1A-yrP-y2fi+1Av;v+ 1; b2/az),
.Re(i/-p+1 ±A0>O, a>b,
(36) 2p+zr(i-P) JT/y<*«) xv(/s») r^rft
= cTv-xI3vzFx i}A+xA,v+%ii-%p, Vi+Vw-ViiL-Vrp; 1-p; l-jSVa2)
xT(K + lAv+ lAy.-lAp) T(XA + 1Av-iAp.-1Ap)
xr{1A-1Av + 1Ali-1Ap)r(1A-1A.v-1Afi-1Ap)
Re(a+/3)'>0, Re(p±^±i/+1)>0,
(37) K2ff S°°Y(at)J (bt)rPdt = &my2iT(v-(i-p)
x J00 K(at)I(bt)t-Pdt a>>b, ReU-p+li^'X),
(38) J""? (bt)J (at)t-Pdt
0 v M
= - J"00 {^(at) Jv(k) + (4/7r2)cos[^ff(p+v+M)]/<:M(aO/^(^)ir''rf«
a'>b, Re(p+v-fi)>-l, Rep>-1,
(39) S^JJpDK^aritvWdt
= (2jS)v (2a)^r(v + ^ + 1) (a2 +/S2)-"-^-'
Ke(v+ 1)> |Re /*|, Re a>|Im£|.
For further combinations see Dixon and Ferrar (L930).
INTEGRALS INVOLVING PRODUCTS OF THREE AND MORE BESSEL
FUNCTIONS
(40) J~ tP-' J^atYJ^U) Jk(ct) dt,
fJk(ct)'
J0 M
TCatson,(1934).
(41) f tP~x J (at)Jv(bt)\
[K&tl
dt,

94
SPECIAL FUNCTIONS
7J4.2
(42) J tP~] I (at) Kx(ct) 1 \dt,
(43) /000^"' K^ariK^btiK^ctidt,
(Bailey, 1935 a, 1936.)
(44) J"[Jv(ax)]2 [Jv{bxy\*x*-ivdx
--2„bZ~ZrL + y2) SM*-«WW
0 < Re v,
(45) f~Jv(ax) Yv(ax)Jv(bx) Yv(bx) x2v+1 dx
a2v6-2-4vr(3v+1)
-1/3 < Re v < 1/2.
(For other formulas see Nicholson, 1920, 1927; Titchmarsh, 1927; Mitra,
1933; Mayr, 1933; Sinha, 1943.)
INTEGRALS OF THE SONINE GEGENBAUER TYPE
(46) J~ J(bt) K\a(t2 + z*)*](t2 + z*riAv I**' dt
0 M v
= b»a-vz»-v"(a* + b*)%v-iAtJ--1A Kv_fi_J[z(a2 + b2YA]
Refi>-1, Rez>0.
(47) {"j/btiK^ok'-yW-W-y'r^t^dt
= % ire ~ HT(v-n-y*> b * a ~v y ' +^v (a 2 + b 2) Vl v~% ^
^H^_][y(a2 + b2Y/'}
Rev<l, Refi>-1, arp,(t2-y2)'A =0 if t > y,
argU2 - y2)a = 77CT if t < y, where o = M and - '4^, respectively.
(48) So°°^(bt) ll™[a(t2 + x2)'A] (t2 + x2)"^ ^+l rft
= a~v b^x ' +M-^(a2 - fc*)X v-Xm-X #U>^ ^ (fl2 _ ^ 2)!4]
a > 6,
= 2*ir-' b»a~vx ,+M"^(i2 - a*)X"-X/*-X A'^_m_, [* (i2 - a2)'^]
a < b,
Rev>Re/z>-l, % > 0.

7.14.2 BESSEL FUNCTIONS 95
(49) J"00#(2,[aa2+*2)*](t2+*T*l't^'i+, dt
0 V
Re (% v - %) > Re y. > -1.
(50) /0°°KJ/[a(*2+z2)*]a2 + z2^5<l'«2'i+, dt
a >0, Re/i>-l.
(51) r",J.(6«)(«* + 22)-»'«'i+IA-«6)v-IzI-+'*-''K ..(izVTM
Re(2v-M)>Re/*>-l, Re 2 > 0.
(52) /"V^t)^'*-^* (t2-y*-rx trft-e-''r(o2+&2)X (a2 + 62)"*
arg(i2—y2)x = %77 if t <y,
(53) ire^^2 + fe2)* (a2 + &2)-* = 2 /" cos(bt) K0[a(t2 -b*)*] dt
= -ffi/~cos(&*)//<2>[a(&2-*2)*]<fc, & e»*U[
(For similar formulas see Watson, 1944, p. 417-418; Mayr, 1932; Gupta,
1943 b.)
(54) eiXir(p-v>j*V-t J [6(t2 + y2.)^]a2+:y2.)-^M(f2_a2)-»-<
0 M
x{cos[/^n-(p- v-)] Ji/(a«)+ sin[%;r(p - v)] Yv(at)}dt
=11 2--- C-4-Y U"-2 Ju[6(a2+y2)^] (ol+ yT^ff«I)(aa)|
a>6, Re(±v) <Rep < 2m + 4 + Re/i, Re(ia).<0f ro = 0, 1, 2,.*.,
(55) J°°^-? c7k[6U2-fy2)^]a2-fy2)^^a2-fig2)-""?
o /*
x }cos[%77(p - v)] Jy(at) + sin[M7r(p - v)] lrl/(a^)} cfc
=(-1)"+,^r (i^f)" }£p~2JMC6(y2-£2)*](y2-£2r*MVa^
a>b, Re(±i/)<Rep<2m + 4+Re/i, Re/3 > 0, ro = 0, 1, 2,... ,
(56) J"°V+I J [6(«2+.y2)^]a2+y2)-^a2 + /32)-, J(at)dt
= )SvJM[6(y2-^2)^](y2-^2)-^/i:v(a/3)
a>b, Re/3>0, -1 <Re v<2 + Re/z,

96 SPECIAL FUNCTIONS 7.14.2
(57) Jj tv'^ J^bt) JJat) (t2 + 02)"1 dt = jS^T/ijS) Kv(a&
a>b, Re.i/>-l, Re(v-^)<2, Re j8 >0.
(58) J^ ^+l J, (a*) (*2 + j8T' & - PvKv(a(3)
a>>0, Rej8>0, - l.<Re.i/.<3/2.
(59) Jo°° tv" Jv(at) (t2 + /82)-/*-1 ^ = a^£^ 2-»Kv_/JL(ap)/T(ii + 1)
-KRe v.<2Re p + 3/2.
For similar integrals see Watson (1944 p. 434-435.)*
PRODUCTS OF BESSEL FUNCTIONS
, ,.( Ze*+ ze'* V
XZe-t+ze* /
(60) KAZ)K (*)« fe'
xAv+JXZ2+z2+2Zz cosh2t)*]<fe Re(Z)<>0, Re * >0.
(61) 2, JmU) J>) - J^e "* ^-se" J [cos lMr J^u,)
(x+xe* V(v+/x)
- sin vn Y , (w)]dd - 2 sin w J°°e~vt ( )
v M o \X+xe x J
x [cos ^77 J +„($) - sin .wr y^+v (0)] dt
X>x>> 0, Re (M - ,1/) .< K, " = tf 2+ % 2 - 2xX cos 60,K
* = (*2 + %2 + 2Z%cosh0?
(Dixon and Ferrar, 1933, p. 193, 194).
(62) J^z)Jv(z)+Yj<z)Yv(z)
= 4;T2 Jo°° K (2z sinh 0 [c^^cosw + e^^'cos ^] <fe
Re 2 > 0, "|ReCi/+^:|.< 1.
(63) JMU)JvU) + yMU)yvU)
= 4^"2 J°°A _ (22sinhO[e(/x+v)t+e-(^)tcos(M -.,/)«]*
Rez>0, |Re(v-^)| < 1-
(64) «7m(k)'JvU) - yMW y„(*) = 4ff-1 Jo°° yM+v(2*cosh*)cosh[(/t-j/)t]A
z>>0.

7.14.2 BESSEL FUNCTIONS 97
(65) JJx) Yv(x) + Jv(x) Y^Ge)--4*'' /0°° ^+v(2* cosht) cosh[(^ v)t ]dt
x>0.
(66) J^Z)Yv{z)~Jv(z)Y^z)
=4tT2 Jo°° Kv^(2z sinht) [e ^^'sinO^) - e k-^'sindV)]^
Rez>0, |Re(v+p)|<l.
(67) JJ,z)Yv(.z)-Jv(z)Yj,z)
=4tt-2 sin[(p- v)tt] J"0~Kv_m(2z sinht)e-(l'+'i)t^
Rez>0, |Re(i/-/i)| < 1,
(68) Kv(x) I>) = Jo°° Jv+I1{2x sinht) e (v_'i)tdt
Re(v- fi) < 3/2, Re(v+/i)>-l, * > 0.
(69) [K„(x)]2 sin(wr)= n-J~°° J0 (2x sinht) sinh(2w)cfe
\Rev\<%, x>0.
(70) [#v(x)]2 cos(i/7r) = -7r.fo°C'yo(2xsinht) cosh (2 vt) dt
|Rei/|<J4, *>0.
(71) I (x) /^(x) + IM(*) K„(*) = 2 f° Jv+M(2x sinht) cosh [(ft - v)t]dt
Re{v+fi)>-l, |Re(ft- v)| < 3/2, x > 0.
(72) I (x) /^(x) - IM(x) «v(x) = 2 J"" Jv+M(2x sinhi) sinh[(^ - v)t]dt
Re(v+fi)>-l, \Re{fi-v)\<3/2, x > 0.
(73) !>) Kv(x) - cosKi/ - p) it] Iv(x) Xm(x)
= sin [An -v)]. £° ^V-M(2* sinhOe"(v+M)t^
*>0, |Re(v- //)| < 1, Re(v+ii)>-V2,
(74) Jv(z)£^ -y U)!^? --4fr"' S~KA2z sinht)e-2vtdt
ov ov
Re z > 0,
(75) I (*) ^ * -£>) ^-1 - n J"~y0(2* sinhO sinh(2i/0A
dv Ov °
+ cos(w)Kv(x)]2 %>0, |Rei/| <#.

98 SPECIAL FUNCTIONS 7.14.2
For most of these formulas see Dixon and Ferrar(l930) and Meijer (1936,
p. 519).
(76) H <2,(*) H <2)(y) = (%n)-%if e~<"'»>* )
x Hl2lfi[(x2+y2 + 2xy cosh2t)y']dt |Re(i/-/c)| < 3/2,
(77) 2*K/>)^>-Iw '""* (^T^J
x Kv+fj[{x2 + yz- 2xy cos0)'x] d<f> - 2sin(w)
/*+ve_£ V(v+/i>
xj"eyt y , ) K . [(*l + y* + 2«ycosht),<]rft
0 \x+ye J M
* >y,
Dixon and Ferrar (1933).
(78) tyz) Kv(0 = ,r°V2V[2(z£)* sinht] e-<f-')«-fc 'rft
Rei/>-K, Re(£-z)>0,
(79) ^v(2)KI/(f)=2cosU)/o0°K2v[2(2^^ sinht]e-e+*>e-htA
-H<Rev<H, Re(**+£*)2>0.
[see also 7.14(25) to 7.14(27)].
INTEGRALS INVOLVING STRUVE'S FUNCTIONS
roc „ , /x rO^"-"-1 tan (H far)
-KBe(i<l, Re p> Re p-3/2,
(81, jr-.M^M'-^^-r^^r^lor^M
Re(f£+v)>0,
(82) Sj°Bv{2Zt)($2-irv^rvdt^%^r{%^v)2v[Jv{z)]2
z>0, |Rei>|<&.
For further integrals involving Struve's functions see Mohan (1942);
Horton(1950).
7.15. Series of Bessel functions
SERIES OF THE NEUMANN TYPE
(1) zver* ~2vr(v) 2 (v+n) &(y) Iv+n(z),

7.15 BESSEL FUNCTIONS 99
« ,x V r(li + n)r(tV+l-li)(^ + 2n)
(2) ew«7,>) = y /i /i/\ +1^ <W*>-
^ n^0 rc I F vv + 1 - \i - rc) 1 vv + n + 1) ^
(If v-fi is a non-negative integer this expression reduces to a finite sum.)
(3) Jv (z sin ft = & /r*)"* (sin 6)v > — T {v + K)
n=0 * ^ + 'v + ^
xC^(cos0)Jv+, + 2n(2),
,, V (v + 1 + 2n) r(i/ + 1 + n) T
(4) Hv(2)r(v + M)=4^2, ,(o „9 ,lW2B,I) Jw.+2n(z)'
(5) J„M ,- .in-Mr [*-' J0U) ^1 (_1_+ tlLj J. (,) J .
(6) (lAz)V-v~» Jiaz) J (/3z) «[a" jSV/T(„ + 1)] 2 (y + 2m) J +2m (*)
^ m= 0
|~V (-l)nr(y+m + n)a2n lrt2, zxl
|_,= o n •' (m - n) l\T (re + /x + lj z J
(7) {%zy-m-v j (az) -j (|82) = a^ j8vcr(m + D r(v + D]
LL V
V (v + 2/7z) r(y + 77l) , 0 -x T , x
x 2, - 1J-L F4(-m,.y+m;p+l,.i/+l;o2,/3*)Jy + 2.(*)f
m = 0
(Bailey, 1935.)
(8) Viz'J iz cos<£ cos$) J (z sin<£ sin<&)
= (cos<£ cos*)"- (sin<£sin$)v I (-1)" (/x +.|/+ 2n + l/<7 +v+2n+J(z
n = 0 ^
x—— —— —rj- 2F [-W, p+.i/+fi+l;i/+l;(sin<0) ]
i» ir(^ + i» + D trcv + i)]2 2 !
x 2FT (-rc, p + i/ + fl + l;i/+l; (sin$)2]
/z, v not a negative integer,
(Watson, 1944, p- 370; Bailey, 1929).
(9) 2v-2vr(l+ «■!/) 1 <&z)*v+nJxv+n(zyn\
n— 0
do) r(v-p)Jvw-r(p+i) I J'({v~,l + "\ MzY-^J^jz),
v n= o 1 (v+n + l)/i! ^
v £ p, \l not a negative integer

100 SPECIAL FUNCTIONS 7.15
(11) Jv(z cos*) Jv(z sin0 - lo nlTiv + n+1) Jv«n(z)
(12) (z + h)^ J [(z + h)'A] = y{^LzMv-**J (**),
m = 0
v not a negative integer*
(±%KT_
ml v+m.
\h\<\z\,
(is) (s + w^y„[<*+«*]- Y lz^!2±^-^y ^
|A|<|*|,
(u) jyy'fcd-a)*]- 1 (^a2)--v//jI.,l/(2)/rU-^ + i),
(15) //'J' [«(l-a)x] = (l-a)x _2 (K2a2)--v//^v+I(2)/r(m-l.+ l),
(16) (M;72)-Kcos[(22-2Zf)'i]= 1 t"-"-J , .,(2)/r(m-^+l),
/7l = — OO
(17) (W)-* sin[(22+22f)*] = 2 *'~v«7_( X)(2)/T(m-i/+D,
0l = — OO
(18) (S2-r2)-^fl^[2(S2-r2)^]= 1 (1^2r2)"'S-l'-"'fl^m(zs)/m!.
(19) (s2- r2)-*" £„[*(*2 - r2)*] = 2 (W)m s~v~" £ . (2S)/m ! ,
(20) W)_I sin(vir) = JJz) J_y(z) + 22 «7„+y(z) «/„-„(* X
n= t
(21) Jv(2z cos 0) = [J^U)]2 +22 «7Xv_„(z) ^V+B(*) cos (2» 0)
Rei/>0, -H»r.$ 6»^ '^77,
(22) [ J (z)]2 = 2 I (-I)""' J (2) J Jz) Re z > 0,
n— I
(23) J2V(2*)-K«X 2 (-l)n[n!r(3/2-«)]"■ Jv+n(z) J^+Jz),
n — 0
., oo
(24) j0[x(t-r})*] = j0(x)io(x)+ 2 K-tr + t""] jn(*rino*),
n= 1

7.15
BESSEL FUNCTIONS
101
(25) 'JAx{t + rx)]**[jn(x)i2+ s (-iru2n + r2n)[j u)]2,
n= I
(26) ber (2*x) = J0 (*) IQ (x) + 22 (-1)" J2n(*) 'J2n (x),
(27) bei(2*%) = 2 2 (-l)nJ2n+IU)I2n+I(^).
n= 0
(For further examples see Bailey, 1935, p. 235; Wise, 1935; Banerjee,
1939; Bateman and Rice, 1935; Fox, 1927; Rice, 1944; Rutgers, 1942;
Nielsen, 1904, Ch. XIX to XXL)
ADDITION THEOREMS AND RELATED SERIES
w - (z2 + Z2 — 2zZ cos<£)^ and C^(z) is Gegenbauer's polynomial (see
section 3.15).
(28) ^"^"^W^^ZPrW 2 {v+n)CvJcos<t)JV+Bfe)ff"+\W(Z)
n = 0
v^ 0,-1,-2, ... , \ze±i*\<\Z\,
(29) /^"•(2,M = J0(*)#(!,)'(2)(Z) + 2 2 Jn(Z)#',M2)(Z)cos(n0)
Ue±i*|.<|Z|,
(30) u>_w«7 M-O^TCi/) 2 (v+riCHcos&J (z)J (Z)
n— 0
v^O, -1,-2, ... ,
(31) c/0 M - J0 (z) J0 (Z) + 2 I Jn (z) Jn (Z) cos (/*<£),
n = I
(32) w~vJ_v(w) =(yiZZ)-vr(v) 2_(-Dn{v+n)Cvn(cos$)J_v_n(z)Jv+n{Z)
1/^0,-1,-2,..., .|2e±'*|.<|Zf,
Let e ^ - (Z - ze "#)/«; and |ze ±#| .<.|Z|.
(33) y^We'"^ 2 yv+n(Z)Jn(2)e£n*,
(34) # ^'(2,M e iv* = 1 fl "MO (Z) c7n (z) e in* ,

102 SPECIAL FUNCTIONS 7.15
(35) K (w) e ** = 2 K. (Z) I(z)e in*,
n = —oo
(36) I>)e**« 2 (-irJv+n(Z)Tn(Z)ein*,
n =— oc
(37) (2z sin^)"v Jv(2z sinK<£)
= 2TW 2 (,„ + «) [2"vJv+n(2)]2C* (cos,£)
n= 0
<vtO, -1, -2, ....
(38) J. (2z sin^) = [J0(z))z + 22 [JJz)]2 cos(re<£),
or
(39) tvJ [z(t + r')]= 2 t2nJv_n(z)Jn(z),
n=—oo
|t | < 1 in case v £ 0, ± 1, ± 2, .. . ,
(40) t"T [2(rf-t)]= I (-l)nt2nJp_n(z)Jn(z)
n = —oo
"|t | < 1 in case v £ 0, ± 1, ± 2, ... ,
(41) (z2 + Z2-2zZ cos<£)^ exp [±i[z2 + Z2 - 2zZ cos<£)^]
= ±,VUZ)-^ 2 (n + y2)Jn+lA(z)H{W)(Z)Pn(coscf>)
n— 0
|ze±i*|.<|Z|,
(42) (%z)*T(2.p) = r{v)ra + v) 2 Cl/ + re)r(21v + re)[Jv+n(z)]2A!,
(43) (sin a sin/3)*~v J„_^ (z sin a sin/?) e <"««*/*
= 2^-^ Uz)-* [rw]2 2 r("'vTI) Jv+n(z)c"(cosa)C"(cos^
(44) cos (2 cos <£) = 2V T (v) I (-l)n (v + 2n) z~v Jv+2n (z) C£ (cos <£),
n — 0
(45) sin (2 coscf) = 2vT(v) I i-lTiv + 2n + Dz"v^+2n+t(z)C2vn+I(cos<^).

7.15 BESSEL FUNCTIONS 103
SERIES OF THE KAPTEYN TYPE
(46) wrJ(.v2)«8!nMr[l-2v2 I (-l)n Jn(nz)/(nz - v2)],
n= 1
(47) virE Ci/*)-2(sinJS*!i)2-4.!/2 f [sin(W + W)]2J>z)/(n2-^2),
n= 1
(48) (l-**)-*-l + 2 f [</>*)] S
(49) [(1 - 2 2r* - 1] = f J [(« + JO z] <7., [0, + X)*],
n= 0
(50) *~fsin*«=l-* f (4ii*-l)"f[c7 Gu)]2,
n= 1 n
(51) (l-z)-f«l + 2 f J (nz\
SCHLOMILCH AND RELATED SERIES
(52)r(i/+l) I cos(mO(^/7^)"v^ (rox)=-% 0<%<*<t7,
» = i v
= X + 77* x-1 (1 - tz/xz)v-* 0<t<x< n,
Re.i/>-%, (Cooke, 1928),
(53) f UimxyrJim) WimyYv J (my) = ~ [2r.(p + l)T(v+ l)]"1
+ 77* [yr^+Drdz+X)]"1 2Ff(K-i/, X;ji+l;*7y2)
77 »> y •> % > 0, /x, v>-%, (Cooke, 1928),
(54) TCv+3/2) I cos On*) (Mm*)"""1 H (m*) = - (i/+ X) *"*
0 < % < s < 77, Re i/> - 1, (Cooke, 1930, p. 58),
= -77-* + 77*%-I(l-*V*2)v+* 2Ff0i>+H, X; v+3/2;l-*2/*2)
0 <t<x < 77, Revl>- 1, (Cooke, 1930, p. 58),
(55) xv=- 2r(v+ 1) I (-1)* (J477/arvm-vJ {mm/a)
«= i v
0 < x < a, v £ 0,

104
SPECIAL FUNCTIONS
7.15
(56) irJvW-23"v f ml-v{4m2-l)-lnv(2mx)
Bt = I
0 < X < 7T, V £ ~ %
(57) a*"1 tt* - tt r(i/ + Vi) ('AaV^U^ax) + iri Y (v + M% aV'vJv(ax)
0 <X <7T, >V>2 V*.
For (55) to (57) see Fennel (1932).
EXPANSIONS OF THE FOURIER-BESSEL TYPE
In the following formulas v and z are arbitrary, but v^ — 1,-2, — 3,... •
The zeros of z~v J (z) arranged in ascending magnitudes of Re (y n) > 0,
are ±yv^,(n = 1, 2, 3, ... ,). Then (Buchholz, 1947 )
(58) ^ M ^ y } ^^ Uz) y U)]
4J U) v v
- 2 </>yv.n) </vUyv.n) [Jvtf (yv,n)]"2 (*2 -<„>"
71— 1
o^*^*^i.
(59) «7VU«)/JVW - 2 2 yVinJv(*yvJKy2Vin-*2YJvU(yvjr
n — 1
= *"+2*2 2'^.^[y^2,^2)^^,,,)]-'
71 — I
0,£* <],
(60) Jv+I(*«J/JvU)-2« 2 <7v+I(yVjn*)[(y^n-Z2) Jv+1(yV(n)]-',
(61) K log * = - 2 J>yn) J0iXyn) [yn J, (yn)r2
o&xzx&i, yn = y0,n,
(62) U0U)]--i-2 2 Ey0>»-y.^r ♦r^H^r'.
(63) [j0(z)]-z=i + 41 [y2>2-<n)'2 + (z2-<B>"'J[J.(yo.„)r2
n— 1

7.15 BESSEL FUNCTIONS 105
(64) [J,(z)]-' = 2z-' + 2z | [U2-yf/n)"' [c70(y1/n)r'
71 — 1
(65) [J,(z)]-2 = 4z-2 + l
n — I
For (62) to (65) see Forsyth (192!)•

106
SPECIAL FUNCTIONS
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Airey, J. R., 1935: Philos. Mag. 19, 230-235-
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Boas, R. P., 1942a: Bull. Amer. Math. Soc. 48, 286-294-

BESSEL FUNCTIONS 107
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Burnett, B. H., 1929: Ptoc. Cambridge Phil. Soc. 26, 145-151.
Busbridge, I. W., 1938: Ptoc. London Math. Soc (2) 44, 115-129.
Carslaw, H. S., and J. C. Jaeger, 1940: Ptoc. London Math. Soc. 46, 361-388.
Chaundy, T. W., 1931 : Quart. J. Math. Oxford Ser. 2, 144-154.
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108
SPECIAL FUNCTIONS
REFERENCES
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Fox, Cyril, 1926: Proc. London Math. Soc. 24, 479-493.
Fox, Cyril, 1927: Proc. London Math. Soc. 26f 35-87; 201-210.

BESSEL FUNCTIONS
109
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Gatteschi, L., 1950: Revista di MathematicadellaUniversita di Parma, 1, 347-362.
Gray, Andrew, and G. B. Mathew, 1922: A treatise on Bess el functions, Macmillan
& Co., Ltd., London.
Greenwood, R. E., 1941: Ann. Of Math. 42, 778-805.
Gupta, H. C, 1943: Proc. Nat. Acad. ScL India Sect. A, 13, 225-231.
Gupta, H. C, 1943 a: Proc. Benares Math. Soc. 5f 1-16.
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Hardy, G. H., 1925: Proc. London Math. Soc. 23, IX
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CHAPTER Vin
FUNCTIONS OF THE PARABOLIC CYLINDER AND OF THE
PARABOLOID OF REVOLUTION
8.1. Introduction
Let *t, x2, x , be Cartesian coordinates in the three-dimensional
space. We define coordinates of the parabolic cylinder £, 77, £, by
(1) *, = £,, xz = y2e-y2r,\ x3 = c
and coordinates of the paraboloid of revolution f, 77, <fi, by
(2) xf = f77 cos<£, *2 = £77 sin<£, *3 = %f2-%772.
)Let
a2 a2 a2
^ - "7 + 7 + 3-
<?*2 a*2 a*2
be 'Laplace's operator, and let / be any function of x only. The partial
differential equation
(3) ^u+f(x^) a = 0
transformed to the coordinates of the parabolic cylinder is
fdZu dZu\ d2u
and it has particular solutions of the form U (£) V(rf) W {£) where U9 V9
W, satisfy the ordinary differential equations
(5) -— +(a£2 + A)t/=0, —- +(aj,2-\)V=0,
d q a 77
(6) — +[/(£)-a] IP = 0,
with arbitrary Constantsa, A. Again, with a constant k2, the partial differ-
115

116
SPECIAL FUNCTIONS
tf.l
ential equation
A u + kz u = 0
transformed to the coordinates of the paraboloid of revolution is
(7)
*^H('*yM$)]
d2u
♦ (fry)-2 ~— + k2u=0,
and it has particular solutions of the form U (£) V (rj) W (eft), where V
satisfies the ordinary differential equation
d2U dU
(8) -— +T' -TT+U2^-4M2r2 + A)^/ = 0.
V satisfies an equation similar to (8)except that the sign of A is reversed,
and W satisfies
d2W
(9) —— + 4M2 JF=0.
d <fe
For solutions which are one-valued and continuous on the paraboloids
f = constant or rj = constant, 2// must be an integer.
In the case of a more than three-dimensional space several
generalizations of this approach to the investigation of (3)are possible. For some
of them see P. Humbert (1920 a, b, c, d).
The solutions of (5) and (8) can be expressed in terms of confluent
hypergeometric functions. Although (8) contains two essentially
independent constants, and therefore is as general as the confluent
hypergeometric equation 6,1(2) itself, the special cases where 2// is an integer and
where k, A, are real are particularly important for certain boundary value
problems. These cases, and the solutions of (5) will be discussed in
this chapter.
PARABOLIC CYLINDER FUNCTIONS
8.2. Definitions and elementary properties
By a simple change of variable, 8.1 (5) can be transformed into
(1) -^+(v + %-Xz2)y-0.
dz
The solutions of (1) are called parabolic cylinder functions or Weber'
Hermite functions. They can be expressed in terms of confluent
hypergeometric functions. If we define

8.2
PARABOLIC CYLINDER FUNCTIONS
117
(2) Di/(z)=-2*b,-,>e-*z2 z <P(l/2-i//2; 3/2; z2/2)
(3) -2"(*+«.-5<^fc,+JOi.J<(X.«)
(4) = 2^e-V4 [ r<*> $(-^,; X; Kz2)
L 1 \72-72V)
+^ra*(1/2-"/2'3/2i'!/2)]
[see 6,1(1), 6,9(2) and sec. 6,5, for the notations] we find that
(5) Dv(z\ DA-z), D_v_,(iz), D_v_^-iz\
satisfy (1), The values of Dv(z) and of its derivative at z = 0 are seen
from (4). Since a solution of (1) is completely determined by its value
and the value of its derivative at z - 0, and since there are precisely
two linearly independent solutions of (1), we find the following relations:
(6) Dv(z) =-^r- le™*D_v_, {iz) + e~™*D_v_y (-£*)]
(7) .e-v^D (_z)+^i_e-^')-^D (fz)
i \-v)
(8) =e^iDv(-z) + ^L e^WtD^i-iz)
1 \—v)
and those which can be obtained from these by substituting — z for z9
These relations show how any three of the solutions in (5) are connected.
The parabolic cylinder functions are entire functions of z. If v = n is
a non-negative integer, we find from (4) that
(9) ez2/4Dn(z) = 2-%nHn(2-y> z)
is a polynomial; H (as) is called the Hermite polynomial of degree n (see
Chap. 10). If v is not an integer, D^(z) and Dv(-z) are linearly
independent. For all values of v9 Dv(z) and D_v_t(± iz) are linearly independent.
The Wronskian determinants are
(10) Dv(*)4- Dv(-z) - D(-z) —D(z) = (2„)Vr(-i/),
dz dz
(11) D (z) — D_Aiz)-D_v_,(iz) — D Az)--i exp(-W).

118
SPECIAL FUNCTIONS
8.2
If v and z are real, the values of D^(z) are also real. For the differential
equation 8.1(5) we can also give real and linearly independent solutions
in terms of the D if a, A, are real. If we assume a]> 0, we can transform
8.1(5) into
(l2)-^~ + {lAx2-p)y=0
ax
where £= (4a)"^,p=- A(4a) , and we find that the real and imaginary
parts of
(13)
V* (*¥')
satisfy (12). Other sets of solutions of (12) which are real on the real
axis are
r (% - %p) i7T/A (-e^/4x)l- y (x)
_ F^~^tp) \D (e inMx) + D {-e i7rMx)~\ - v (x)
-Im|2^e^/4[(l + e-^)^ + l]^e-,'^'/2+'T/8)Z?. _J*ei7r/4)}=y»
where y' = arg Y(lA + ip). y0 and y behave simply at x = 0:
yo(0) = l, y,(0)=0, y^(0)=0, y,'(0) = l,
and y2 and y behave simply at x = .«>:
y2 = (2/xf e'^mi + e-^i^ + l]54 sinfe(*)] [1 + OU"1)]
y3 = (2/*)* e^Rl^ e-^^-l]54 cos [*(*)] [1 + 0(*~')],
where
gW = ^2-p log* + %7T+ My'.
We also have
y3(-%) = y2U).
J. C. P. Miller has made y2 and y3 the basis for the computation of
numerical tables; y0 and y^ have been discussed by Wells and Spence,
(1945)and by Darwin (1949).
From (2) and 6.6(6), 6.6(7) we find

8.3 PARABOLIC CYLINDER FUNCTIONS 119
(14) DJ/+!U)-2DJ,U)+ i/D„_,(*) = (>,
and from 6.6 (10) we have
(15) ^1 [«*** D(*)] = (-I)" (-1/). «K«2 D (*),
(16) ^1 [e-*»2 Dv<*)]= (-1)" c^*2 Dv+a (*) m = 1, 2, 3
Therefore, we obtain from (15), (16) and from Taylor's theorem
(17)D(,+y)-e*^2) I <-y)» U!)"' D . (*)
m =5 0
oo
« = o\w /
and for i/ = 0 this gives the generating function of the D (z) [i.e., of the
Hermite polynomials, see (9) and Chap. 10L
(18) e-^2+z'"^2= V ^rDJz).
« 71 I
n=0
If i/ is a negative integer, the D^(z) can be expressed in terms of the
error function
(19) D f2)=2K —e^*2 — [e**2Erfc (2:*z)]f
*"1 m! dzm
and if v = -% in terms of a modified Eessel function of the third kind,
(20) D_%(z)=Qizn)*K%(]iz2).
83. Integral representations and integrals
INTEGRAL REPRESENTATIONS OF PARABOLIC CYLINDER FUNCTIONS
(1) --^-e^*2J^e-»"2rI-X"(l + «)Xb'-,,d«
r(_y2l/) Rei/<0, |argz|<Jfir,
(2) "t(Y-Kv "^V/^2^"^' (1 + «)X"A
Re i/ < 1, |argz| <% 77,
e^z2 2
<3> = W , I/"Zt"'/2t r^1 it Rev<0,
1 l-i/) 0

120 SPECIAL FUNCTIONS 8.3
(4) DvW=f-j ey*z* f~ e^ tv cos (zt - Vvvn) dt
Re -vO-l,
(5) D_,„ [(«/*)*]
= «-*•/« ," 2"-' Dr(2v)]-' J~ e"***"-1 exp[-(2ai)X] A
Rci/>0,
(6) Dv(z) = 2^X «»«-«*' (2ff/)-«
'*°r{s)r{%v + y4-s)r(y2v-y4-s)
"J.'
{Viz2)*ds
r«,i/+X)r«.i/-X)
|arg*l|<&;r, 1^1/2,-1/2,-3/2, ...,
(7) D^(r) D^I/.|U)-2j00c7j/+xa2)cos(rt-Ji'Mr)c"Jtift
Rezi>0, Re.i/»>-lf
(® =rn/ x J°° J „_„(Xt2) «"**<** Rev<0, RezfeO,
r(-v) o v *
23/2
<9) - *w x J°X+k <'2)cos <*' - ^ ^rt*
0 .< - Re -i/ < 1,
1 r«
(!0) J (cosh tf (sinh i)*"1 exp (- Y2z2 sinh t) <fc
77 °
Re.i/>0, |argzj£%7r,
The integral representations (1), (2), (3), (4), (5), can be proved by
verifying that the right-hand sides satisfy the differential equation 8.2(1)
and assume the correct initial values at z = 0. Equations (1) and (2) can
also be derived from 6.5(2), 6.5(6) and 8.2(1). In (6), the path of
integration must be chosen in such a way that it separates the poles of
T(s) from those of the two other gamma-factors in the numerator of the
integrand; the formula is a consequence of 6.11(9).
The integral representations (7), (8), (9),were proved by Meijer (1935b,
1937a) and (10) was proved by Bailey(1937). Jv, Ky denote Bessel
functions of order v\ see 7.2(2), 7.2(13).

8.3
PARABOLIC CYLINDER FUNCTIONS
121
There exists a very large number of other integral representations for
the D^ora product of two parabolic cylinder functions. For
representations of D^ see Meijer (1934, 1935 a, 1938 a). An integral representation
of D involving other confluent hypergeometric functions was given by
Meijer (1941). For results related to (7), (8), (9), see also Meijer (1937b).
INTEGRALS INVOLVING PARABOLIC CYLINDER FUNCTIONS
(11) f°° e-'U-WtD^VUktftdt
2x-0-v/2n*ri&)
M),.F(Z,l.,^..L±)
r(y2V+y2p+y2)
Re /3 > 0, Re z/k > 0,
(12) J"o°° e*** D_v{t) *2e"' $(a, c; -'Apt1) dt
|l-p|.<l, Re c >0, Re v>2Re(c-a),
(13) Cey^D_v(t)t2c-^(a, c;-Hp^)A
n-H r{2c-i)r(y2v+y2-c+a) ■
-j^w^ r(y2+y2,)r(a+y2v) ^-^^i-p)
|l-p|<l, Rec>%, Re !/>2Re(c-a)- 1,
(14) J"tve-V<t2D2v{t)Jv_,{tz)dt
= ivr{v+y2)„-* zv~< e-*z2$(-v, y2-, y2z2) Re v>-y2,
(15) (2^)-* jx e-(^)2/(2M) eMyz D ^ dy
-CO
Ml-^'e*^4"4"3 Dv[*(l-p)'x] 0 < Re p < 1,
(16) /^e'*r-(l + ^V4 Dv|y(l-W]rfy
= (2ff)U^e-(1+^2/^Z)v[jU-,-l)^] ReX>0,
(17) J" (*y)* <7>y) y"-* e^2 D_^{y)dy
= xv-Xey*x2D_2v(x) Re !/>-«,
(18) /0OODv(y)e-1^2yvU2 + y2)-,iy=(^^r(v+lHI/-,e^2D_v_l(*)
Re !/>-!,

122
SPECIAL FUNCTIONS
8.3
(19) f°° e-3^/4 tv-\ D (t) dt = 2"Xvr(v) cos(fc.vir) Re v> 0,
k2
(20) J" e**' t^x D_JLt) dt
n* 2-**"ftT(p)
"" r.(X/i+H.v+X) Re^">0,
(21) S'rD^(±t)Dv(t)dt
li-w
r » : i
|_r (H - K n) r (-«.*) + r.(-x id r (K - k*) J
Re /z > Re i/ if lower signs are taken,
<22, riD up *-.»*■« **-*«»-«-*•»
o v r (-i/)
(23) J°° [Dn(0]2 * = (2ir)x ti ! n = 0, 1, 2,
In these formulas, i*1, <$, J, 0, denote the hypergeometrie and the
confluent hypergeometric series, the Bessel function of the first kind and
the logarithmic derivative of the T-function.
Equation (11) follows from 6,11(12) and the inversion formula of the
'Laplace transformation. 'According to 2,1(26), 2,1(2), the right-hand side
in (11) reduces to an elementary function if /S = v + 1 or if i/ = — 2zi,
n = 0, 1, 2, ,,, , For the proof of (12), and (13) see Erdelyi (1936).
Wore general formulas of this type involving an pFq ((see,4,1) instead of
$ have been given by Mitra(1946). A proof of (14) was given by Meijer,
(1938),To prove (15) it suffices to express Dv by (3) and to interchange
the order of integrations; if p tends to 1, the right-hand side in (15)tends
to xv. Formula (16) is essentially the same as (15) and formulas (17),
(18) are due to R. S. Vanna(1936, 1937); Watson(1910) has proved (19),
and formulas (20), (21) were given by Erdelyi (1938); for v - /*, we
obtain (22) and (23) from (21). We also see from (21), that the Dn(t),
n = 0, 1, 2, • • • , form an orthogonal system in (—.«>, «>).
8.4. Asymptotic expansions
From 8,3(6) it can be shown that (see Whittaker-Watson, 1927) for
large values of \z.\ and a fixed value of v
(1) *„(.)-.*e " [ I nlll/22>)n n+0\Z>\ ' 'J
n ° -%n<argz <%n,

8.5.1
PARABOLIC CYLINDER FUNCTIONS
123
■ 2.-W-1
L n= 0
r(-„)e e 2, „!(.k2«)» +°|z ' J
n= o 7r/4 < argz < 5^/4,
L n= 0
z BI<X,«)"—+olz' j
(3) D, J(z).8"e^1
r(-v)
n= 0
- 5 7r/4 < arg z < - tt/4,
where the notation
(4) (a)0 = l, (aTn= a(a+l)'^;(a+7i-l), ti = 1, 2, 3, ••. ,
is used.
The behavior of D^(z) for |v| -» oo and for arbitrary values of z which
satisfy \z\ < \vy* has been completely discussed by Schwid (1935)t His
results are based on Langer's method (1932). As a special case we have
the following result which, in the form given here, has been stated by
Cherry (1949):
If \z\ is bounded and |arg(— v)\ <%7r, then, for \v\ -* °°
(5) Dv(z)=2'% exviYzvlogi^-Yzv-i-v)* *] [1+0 H"*].
8.5. Representation of functions in terms of the D (x)
8.5.1. Series
From 6,12(3) we have as a special case, for positive real values of x:
2*v V (-Dn DZnM
(1> D» = F71Z3 I
T(-lAv) rit0 n\2n(n-V2v)
rW-Xv) nr0 ml 2n(n + X-Xv)
This can be considered as an interpolation formula for the function
Dv(x) of v9 the points of interpolation being the non-negative even or
odd integers. An expansion of D^(x) D (x) in terms of the Dn(2^ x),
(n = 0, 1, 2, ... ,) has been given by Dhar (1935), Shanker (1939) proved
the addition theorem

124 SPECIAL FUNCTIONS 8.5.1
(2) Dv(x cost + y sinO= exp[%{x sint - y cos*)2]
(t*nt)nDv_n(x)Dn(y) ?« e***U)
1(0
which holds for real values of t, x9 y and 0 < t < /r/4, Re v > 0.
Erdelyi (1936) proved the expansion [see 8.4(4)]
in which R denotes a remainder term. If \i is half of an odd integer, the
series terminates. In all other cases the series is divergent in general,
but the remainder term can be estimated, in particular if |argz| <%u and
p is large, showing the asymptotic nature of the expansion
From the expansion 6.12(6), an expansion of Dv(z) in terms of the
Bessel functions can be derived, where the Bessel functions become
elementary functions because their order is half an odd integer. In
particular, we have
D (*)=_— {cos £-2-6K-2a(l-2£73)sin£-£cos£]+»,.}
■lsin£-2-6K-2[(l-£2)sin£-£(l-2£2/3)cos £]+-...}
where
k = Kv+X>0, £-(2k)**,
and the terms indicated by ••» are of the orcler of k"3 provided that £ is
bounded.
The Sturm-Liouville problems connected with 8.2(12) lead to certain
orthogonal sets of functions for a finite interval (0, xQ), These are
essentially parabolic cylinder functions whose order is of the type ip - lA
(p real) and for which the variable has an argument, lAn or -%tt9 (see
sec. 8.2). For an application see Magnus (1941); for Sturm-Liouville
problems in general see Chap. 10 in the book by Ince (1944).
8.5.2. Representation by integrals with respect to the parameter
Cherry's theorem (1949). If f(x) is of bounded variation in any finite
interval of the real variable x and is absolutely integrable in (— <», oc), then
* dv r \Pv(hx)D_v_. (ht)
+ Di/(-hx)D_l/_](-It)]f(t)dt

8.5.2
PARABOLIC CYLINDER FUNCTIONS
125
where
(5) A = e*i77, h = e-*i7T.
The condition that f be absolutely integrable, can be replaced by
(6)
,(,)...*<*' (f^Ji-) [1+0(1x1"]
for x -+ ± oo, where a is real and > l/2 and where c y> c2> are constants
(which may be different for x -+ + oo and x -+ - oo). Condition (6) is needed
in some boundary value problems (see Magnus, 1940). Equation (4) is
analogous to the inversion theorem for Fourier integrals. It can be
simplified if f(x) is an even or an odd function of x%
Cherry (1949) has applied (4) to the function f {x) = D Jhx) for x > 0,
f(x) == 0 for x < 0. In a formal sense [although (4) and (6) are not
satisfied], Erdelyi's formula for the expression of a plane wave in
coordinates of the parabolic cylinder is a special case of Cherry's theorem,
viz.:
(7) -2i(2n)X exvl-Kiig2 - rj2) cos<f> - l/2i fr sin<f>]
/-*+io° dv f"(tan%Q5>)v
sin irr cos/id) v
%-ioo L ^
sin Vi<f> v v J
(cf.Erdelyi, 1941% Here h is given by (5), and (7) holds for all real values
of f, r) • For the diffraction problem of a plane wave incident on a half-
* plane, Cherry (1949), gives the formula
(8) -2» DQ [hU cos %<f> + 77 sin V2cf>)] D_} [h (rj cos Y2<j> -•£sinV2<f>)]
r-X+io° dv (tan1/2«?5))v
J-%-i*o smvn cos%9
for the secondary wave ( "Sommerf eld's wave").
A special case of 6,15(15) is the expression of a "cylindrical
wave " in terms of solutions of 8.2(1), viz.
(9) 2%n*H™[y2k(e+V*)]
= J"e+i~ Dv[k*(i+i) f] z) , [k*a+i)v] r(-y2v)r(lA+1Av)dv
c-ioo
where -1 < c < 0, f, 77, real, Re ik >0, Another expression for the left-
hand side in (9) in terms of an integral taken over the parameter of parabolic
cylinder functions can be obtained from Cherry's theorem; see also
Magnus (1941).

126
SPECIAL FUNCTIONS
8.5.2
Erdelyi (1941) also proved the following formulas which can be
considered as linear and bilinear continuous generating functions of D [see
also 6.2(20)]:
1 p c + ioo
277 c-i«. "*'
(10) _L_ JC * D(z)tvr{-v) dv=e-X* -*'-*'
c <0, |argt| <77/4,
0Sir)X .c + i~ rv~' dv
2«r» J«-<- v ~v-' J v ~v~' J sinC-w)
= (l+t2)-* expfl 2l4u2+y2) + i^lJ]
[4 l+* l+*2J
—1 <c <0, |arg*| <y27T,
8.6. Zeros and descriptive properties
For any fixed value of v the formulas 8.4(1) to 8.4(3) give a
description of Dv(z) for large values of \z\; if v and z are real, then D (z) is
also real in spite of the appearance of 8.4(2) and 8.4 (3)« If v is real,
D v(z) has [v -f 1] real zeros, where [v + 1] denotes the greatest positive
integer less than v + 1 or zero if such a positive integer does not exist.
This result can be derived from a discussion of the differential equation
8.2(1). If v = n = 0, 1, 2, ... 9 Dn(z) has exactly n real zeros (and no
other zeros). For other results about real zeros of the solutions of 8.2(1)
which are real on the real axis see Auluck (1941); for asymptotic formulas
for the real zeros of D^(%) if v is real see Tricomi (1947).
FUNCTIONS OF THE PARABOLOID OF REVOLUTION
The results of the two following sections comprise only a small part of
the formulas which arise from boundary value problems of Aa-f k2& = 0 for
the paraboloid of revolution. The whole subject has been thoroughly
investigated by Buchholz; the formulas in sections 8.7 and 8.8 indicate which
type of results can be found in the papers to which a reference is made.
8.7. The solutions of a particular confluent hypergeometric equation
If in 8.1 (8) k9 pt, A are arbitrary complex constants, we have a
differential equation which is equivalent to the confluent hypergeometric
equation. However, if k and A are real and 2[i is an integer, 8.1 (8)maybe
reduced to
(1) 4^"+ rl^z+ <4fZ -P*C - 4r) a = 0

8.7
PARABOLIC CYLINDER FUNCTIONS
127
where
(2) p = 0, 1, 2, ... , and r, £, real.
Equation (1) has the solutions
(for the notations see sec. 6,9). They are connected by the relations
(4) e-X^r+V MiTiip{i{)~ e^1"*" M_iTi,Ap{-iO
p! exp[7rr- Va i 7r(p+l)]
(5) r« + xp-«ir) "'^ ^
p! exp[7rr+ %i7r(p + l)]
+ r(y2+1/2P + t» ^.^Gf)
where £ denotes a real, positive variable, and arg ±i£= ±%tt. For £-» <»
and fixed values of r, p we have:
(6) ViT,xPW -€ire-Xie-X«r [1 + Otf"')]
(7) r_iT(Kp(-»f)-e'iTe}"f"!iwTtl + 0<r1)].
The corresponding expression of theJf-functions in (3) can be derived
from (4), (5), (6), and (7).
The function
(8) e-^^M.^tiO-e^^ni^yJ-it)
is real for real, positive values of £ (if r, p are real).
If p and £ are fixed and r is large, 6.13(8) gives the following
asymptotic representations:
(9) IT±.Ti!<p(+^)=2^e^i'Te+iTr±<T(^)-!ie-^T
x cosh[(r- 2{tO% ±V*in\ [1 + 0(r^)],
(10) IT (±»f)-2*e±iTr+iT(£r)-Ke~*WT
+ ir,HP
x cosfTif—.2(r£)X -*ir] [1 + OCr"*)],
where r, £, are real and positive.
Erdelyi (1937) investigated the case where \r\ and |£| are both large
but where r/£ is a fixed negative number. This result is
(11) il/_iT#M[jV(2sinh^)2]=r(2M+l)e^w(^
x sin [r(sinh2 /B + 2 j8) -(/£- X)ir] [1 + 0 (r-*)]
l/)r. AtanhffV

128
SPECIAL FUNCTIONS
8.7
where r, /S, ft + Vi, are real and positive.
For the solution of certain boundary-value problems, the following
functions are needed. Let £Q be a fixed real positive number. Then there
exists a sequence of real numbers rn, n = 1, 2, 3, ... , such that
t,<t2< t3 .- and Mir y ii£0) = 0.
n r
The functions
(12) (— J M. u HO
are orthogonal in (0, £Q). In order to compute the rn for a given £0 and
in order to find the normalizing factors for the functions (12), Buchholz
(1943) gave the formula
(13) HO'*'*'*lTtvpiiQ
r(Ji+xP) ,£0 r(z+i+y2P)
11 L1 + (t + « + «p)1.J
z!
x
r =0
^i-xKI^ + f^W*!^]
where r, £, are real, r > 0, £ ^ 0, and also similar formulas for the partial
derivatives
<9r' <9£' <9r<9£
of the function (13).
8.8. Integrals and series involving functions of the paraboloid of
revolution
As a consequence of 6.15(15) we have
ei{x+y) -i rioo ds
x + y 2Uy)* -£oo ' ' cos ns

8.8
PARABOLIC CYLINDER FUNCTIONS
129
This is the representation of a spherical wave with the center in the
focus of the paraboloid in terms of the functions of the paraboloid in
revolution. Formula (1) was first proved by Meixner (1933) who also
derived the formula
2ffp!p! Gbv)x^+i)
(2) (^^r(P + 1 + 2l'a)r(P + 1-2ia)(^j^^^^+y)
-/~rftp + K + »(a+r)]r[Kp+K+»(a-r)]
— oo
x rKp + lA -i(a + r)] r[V2p + K - *(a - r)]
X Mia+ir,%p & Mia+ Kir, XpW d*
Re%>0, Rey>0, p = 0, 1, 2, ... .
The integral representations for more complicated types of waves
with a singularity at the focus of the paraboloid of revolution were given
by Buchholz (1947). One of his results is
(-l)p~' (n + p)\ ro- + »°°
=■— rr:—T7 -T r(s+y2P+y2)r(-s+y2P+y2)
2ff p\pl(n-p)\ <r-i<*>
where
x>y>0, oKK+Vip,
and where 3F2 stands for
3F2= 3F2(-ri + P>n + p + l,-s+y2+y2p;p + l,p + l; 1),
[see 4.1(1) for the definition of the generalized hypergeometric series].
If n ~ p, (3) becomes equivalent to 6.15(15). It should be noted, that
H( * is an elementary function, see 7.2 (6)«
The expression of a spherical wave, the center of which is at an
arbitrary point, in terms of the functions of the paraboloid of revolution
was also given by Buchholz (1947). If
# = {[(%1-y1)-U0-y0)]2 + 4%0y0 + 4%T yT
-^oyox^y^)lA cos(0t-02)}*

130 SPECIAL FUNCTIONS 8.8
and if xQ, y0, xy , yt, are real and positive and xQ > x} , y0 > yy , then,
for a real o < %,
W <W,r,>*^ = -2 S (2-»t,).-'<*'-'*')
# p=o °'P p\p\
x(2ir d_I Ja+i~r(s + y2 + y2P)r(-s +■« + JiP)
a-too
where S0 0 = 1 and S0> = 0 if p > 0.
For tine plane wave Buchholz (1947) gives a mixed series and integral
representation:
(5) exp[i(x —y) cos 6 + 2(xy)y> sin 0 cos <£]
-, x* ■ ^ X -SiH (P cos (p<f>)
(xy)% sind Li pi pi
p= o l l
x (2IT*)-1 Ja + £~ T(s + H + Kp) T(-a + % + KpHtan y20)2s
There correspond certain series expansions to the integral
representations in this section. In the simplest case, the formula corresponding to
(l)is
x + y UyK2 n=o n ^'° -ir%o
For a large number of other series and integrals see Buchholz (1943,
1947, 1948, 1949).

PARABOLIC FUNCTIONS 131
REFERENCES
Appell, Paul, and M. J. Kampe de Feriet, 1926: Fonctions hyper geometriques
et hyperspheriques. Polynomes d'Hermite. Gauthier-Villars.
Auluck, F. C, 1941: Proc. Nau InsU Sci., India 7, 133- 140.
Bailey, W. N., 1937: Quart. J. Math. Oxford Ser. 8, 51-53.
Buchholz, Herbert, 1943:~Z. Angew. Math. Mech. 23, 47-58, 101-118.
Buchholz, Herbert, 1947: Z. Physik, 124, 196-218.
Buchholz, Herbert, 1948: Ann. Physik (6) 2, 185-210.
Buchholz, Herbert, 1949: Math. Z., 52, 355-383.
Cherry, T. M., 1949: Proc. Edinburgh Math. Soc. (2), 8, 50-65.
Darwin, C. G., 1949: Quart. J. Mech. Appl. Math. 2, 311-320.
Dhar, S. C, 1935: /. Indian Math. Soc. (N.S.) 1, 105-108.
Erdelyi, Arthur, 1936: Math. Ann. 113, 347-356.
Erdelyi, Arthur, 1937: Akad. Wiss. Wien. S.-B. Ha 146, 589-604.
Erdelyi, Arthur, 1938: /. Indian Math. Soc. (N.S.) 3, 169-181.
Erdelyi,'Arthur, 1941: Proc. Royal Soc. Edinburgh, 61, 61-70.
Humbert, Pierre, 1920a: C. R. Acad. Sci. Paris, 170, 564.
Humbert, Pierre, 1920b: C. R. Acad. Sci. Paris, 170, 832.
Humbert, Pierre, 1920c: C. R. Acad. Sci. Paris, 170, 1482.
Humbert, Pierre, 1920<1: C. R. Acad. Sci. Paris, 171, 428.
Ince, E. L., 1944: Ordinary differential equations, Longmans.
Langer, R. E., 1932: Trans. Amer. Math. Soc. 34, 447-480.
Magnus, Wilhelm, 1940: Jber. Deutsch. Math. Verein, 50, 140-161.
Magnus, Wilhelm, 1941: Z. Physik, 118, 343-356.
Meijer, C. S., 1934: N. Archiv. V. Wiskunde (2), 18, 35-57.
Meijer, C. S., 1935 a: Proc. Kon. Akad. Wetensch. Amsterdam 38, 528-535.
Meijer, C. S., 1935b: Quart. J. Math. Oxford Ser. 6, 241-248.
Meijer, C. S., 1937a: Kon. Akad. Wetensch. Amsterdam, 40, 259-262.
Meijer, C. S., 1937b: Kon. Akad. Wetensch. Amsterdam 40, 871-879.
Meijer, C. S., 1938: Kon. Akad. Nederland. Wetensch. 41, 744-755.
Meijer, C. S., 1938a": Proc. Kon. Nederland Akad. Wetensch. Amsterdam 41, 42-44.
Meijer, C. S., 1941: Proc. Kon. Akad. Wetensch. Amsterdam 44, 590-598.
Meixner, Joseph, 1933: Math. Z. 36, 677-707.
Mitra, S. C, 1927: Proc. Benares Math. Soc. 9, 21-23.

132
SPECIAL FUNCTIONS
REFERENCES
Mitra, S. C, 1946: Proc. Edinburgh Math. Soc. 7, (2), 171-173.
Schwid, N., 1935: Trans. Amer. Math. Soc. 37, 339-362.
Shanker, Hari, 1939: /. Indian Math. Soc. (JV.S.) 3, 226-230.
Shanker, Hari, 1939: /. Indian Math. Soc. (N.S.) 3, 228-230.
Taylor, W. C, 1939: /• Math. Physics, 18, 34-49.
Tricomi, Francesco, 1947: Ann. MaU Pura Appl. (4), 26, 283-300.
Varma, R. S., 1927: Proc. Benares Math. Soc. 9, 31-42.
Varma, R. S., 1936: Proc. London Math. Soc. (2), 42, 9-17.
Varma, R. S., 1937: /• Indian Math. Soc. (N.S.) 2, 269-275.
Watson, G. N., 1910: Proc. London Math. Soc. 8, 393-421.
Wells, C. P., and R. D. Spence, 1945: /. Math. Phys. Mass. Inst. Tech. 24,
51-64.
Whittaker, E. T., and G. N. Watson, 1927: A course of modern analysis,
Cambridge.

CHAPTER IX
THE INCOMPLETE GAMMA FUNCTIONS AND
RELATED FUNCTIONS
9.1. Introduction
A considerable number of functions occurring in applied mathematical
work can be expressed in terms of the incomplete gamma functions,
(1) y(a,x)=f*e-tta-*dt Rea>>6,
o
(2) rU*) = re-U*'* dt~T{a)-y{a,x),
X
which in their turn are closely connected with the particular case a - 1
of the confluent hyp ergeome trie functions $(a, c; x) and ^(a, c; x\ By
6.5(1), 6.5(2), and 6.5(6) we have
(3) ■yU*)«a-|*ae~**(l, 1 + a; x) = a"1 *a$ (a, 1 + a;-*),
(4) rUx)e*V'?(l, l + as^-c^Td-a, 1-a;*).
When a = 1, the confluent hypergeametric equation 6.1(2) has the
elementary solution
ex xy"c
so that the special confluent hyp ergeome trie functions to be discussed
in this chapter satisfy simple differential equations of the first order.
In many ways it is advantageous to adopt the slightly modified function
—a
(5) y*U*) = ~ j% e"Ha" dt
i{a) °
e'x 1
= — 4(1, l + a;x)= — -4 (a, 1 + a;-*)
T(l + a) T(l + a)
as the basic function because this is a single-valued entire function of
both a and # and is real for real values of a and x.
The following functions are expressible in terms of the incomplete
gamma functions1: the exponential and the logarithmic integral, sine and
cosine integrals, error functions and Fresnel integrals and their
generalizations. Definitions and notations of these functions vary considerably.
133

134
SPECIAL FUNCTIONS
9.1
The notations to be used here will be explained in the sections dealing
with these functions*
THE INCOMPLETE GAMMA FUNCTIONS
9.2. Definitions and elementary properties
The incomplete gamma functions were first investigated for real x by
Legendre (1811, Vol. 1, pp. 339-343 and later works). The significance
of the decomposition
(1) r(a) = y(a,x) + r(a,x)
was recognized by Prym (1877) who seems to have been the first to
investigate the functional behavior of these functions (which he denotes
by P and Q).
There are several notations for these functions. At present the most
frequent notation besides the one adopted here is the notation used in
astrophysics and nuclear physics,
En(x) - J™ e"xu u'n du = x»-} HI - n, x).
The alternative notation K (x) is sometimes used. For the formulas in
ft
this notation see Placzek (1946), Le Caine (1948), and Busbridge (1950).
The older theory of the incomplete gamma functions is presented, and
references to the literature are given in Nielsen (1906a, especially in
Chap, XV, and 1906b). A more recent account is found in Bohmer (1939).
It is customary to define the incomplete gamma functions by the
incomplete Eulerian integrals of the second kind 9.1(1) and 9.1(2).
However, in order to avoid convergence difficulties in 9.1 (1) when Re a<0
we shall adopt 9.1(3) and 9.1(4) as the definitions of the incomplete
gamma functions with the remark that %aand *P are defined uniquely by
the conventions of Chap. VI. Apart from the notation, 9.1(2) was known
to Legendre. While y*(a> x) is an entire function of both a and x, the
function y(a> x) itself fails to be defined for a = 0, — 1, — 2, ... . The
function Via, x) is an entire function of a, but in general, except when
a is an integer, it is a many-valued function oi x with a branch-point
at x = 0.
The recurrence relations
(2) y(a + ly x) = a yia, x) -xae~x,
(3) T(a + 1, x) = aHa, x)+xae-ii,
are simple consequences of the definitions and can be derived from the
incomplete Eulerian integrals of the second kind by integration by parts.

9.2 INCOMPLETE GAMMA FUNCTIONS 135
They can be used as an alternative definition of the functions under
consideration.
We have the convergent expansions in ascending powers of x9
a + n oo / \n. a + n
(4) y(a.x)-e-'l — J LLf_-,
w (a)„+1 ^ ra ! a + n
V (-)"
(5) ru«)-rw- 2 —
n=0 nT| n=0
7i ! a + n
(\n ^a + n
n= 0
valid for all x, and a £ 0, - 1,-2, ... , with
r (a + n)
(a)0 = 1, (a)n = ■—= a(a+ 1) • • • (a + rc - 1)
Al = 1, 2, «« . ,
and the asymptotic expansions in descending powers of x,
I m +oci*rjr)
|*'| ^ oo, - 3 tt/2 .< arg x< 3 tt/2, « = 1, 2, ,.. ,
(7) 7(a,x)=r(a)-x«-' e-*f ^ lll^L + 0(|^| ^)1 .
Either from the power series expansion or from the definitions one
obtains the differentiation formulas
dy(a, x) dTia, x)
= x~ ' e ~,
(8) -/"»-^_" "»~'_„fl-i„-*
(9) —— &Ta.yU *)] = (-)nx"a^ y(a + n, x\
dxn
(10)—- [e'yUx)]B(-)B(l-(i e*.y(a-n,*),
(11)—— [*~a TU *)] - Wn^~a"n T(a+ n, x),
dxn
j n
(12) —[eTU *)] = (-)n(l-a) exF(a-n,x),
dxn n
the last four for n = 0, 1, 2, •.. .

136
SPECIAL FUNCTIONS
9.2
The continued fraction expansion
(i3) ru*)« —
e'xxa
1-
x +
1 +
2-a
x + -
1+ ...
is due to>Legendre and can be derived from (3). Other continued fractions
have been obtained by Schlomich (1871), and Tannery (1882).
Whenever a is a positive integer, the confluent hypergeometric
functions $(a, c; x) and *P(a, c; x) may be expressed in terms of incomplete
gamma functions by means of the formulas
a dn
(14) *(n + 1, a+l;x) = —■ [exxn'ay(afx)] n = 0, 1, 2
n ! dxn
(15) ¥Gi + l,a+l;*)
1 d*
n-I(l-a) <9xn
b4Bxn"ar:(of *)] n = 0,l,2,
The first formula is meaningless for negative integers a, but it retains a
meaning if it is divided by T(a + 1) before a approaches a negative
integer. The second formula looses its meaning when a is a positive
integer.
9.2.1. The case of integer a
In this section
n
(16) e(x) = I ?—- n= 0,1,2
is the truncated exponential series, and E (x) is the integral defined in
sec. 9.2. We have
(17) y(l + n, x) = n ! [1 - e~z e B(*)],
(18) r(l + 7!, x)=n le~xe (x\
n
(19) r(l-n, x)=x,_B£ (*).
/I
By repeated integrations by parts we also have

9.3
INCOMPLETE GAMMA FUNCTIONS
137
(20) r(-»,*)=L^£/*)-e-*n2 (-)"^rr
^ s= JL, £) o, «• • •
The function y(a, *) does not exist when a = — n9 but we have from 9.1 (5)
(21) y*(-n, *)«*».
It may be pointed out that for positive integer a and integer c, the
confluent hypergeometric functions $(a, c; #) and W(a, c; x) may be
expressed in terms of the functions discussed here. For $, with a = 1 +rc
and c a 2, 3, ••• , this follows from (14). For other integers c, we have
to divide (14) by T(c + 1), and write (14) in terms of y* before letting c
be an integer. For V with a ** \ + n and c = 1, 0, — 1, — 2, ... , we have
(15) and (19). The case c = 2, 3, ... , can be reduced to the former one
by applying 6.5 (6).
When a is close to an integer, we may obtain useful approximations to
incomplete gamma functions by evaluating their derivatives with respect
to a for an integer a. By manipulating the integral representation 9gl(5)
one can prove
(22) Y * «-log*-£f(*)f
da |a-o
and other results follow by application of the recurrence relations.
9*3. Integral representations and integral formulas
The basic integral representations are the incomplete Eulerian integrals
of the second kind, 9.1 (1) and 9.1 (2). The first of these fails to converge
when Re a < 0. It may be replaced by a loop integral
(1) yUjc) = - (2» sin Tra)"1 x a J*" e"xu (-u)a~} du
where — n < arg(— u) < n on the loop of integration, x is arbitrary, ^ 0, and
a is not an integer. With the unit circle — u = cos 0 + i sin 6, — n < 6 < tt,
as the path of integration one obtains
(2) yia, x)=xa cosec na j7* excc*e cos (a6 + x sin 6) d$.
A real integral for Re a <0, x <0 may be derived from 6.11 (13).
For Via, x) the basic integral representations are 9.1 (2) and
- a) J x + t
(3) r^*)=*(i~ x / -Z—l—dt.

138 SPECIAL FUNCTIONS 9.3
The latter integral is obtained when 6.5 (2) is applied to the last ¥
function in 9.1(4). 'Legendre's continued fraction 9.2(13) is a consequence
of (3).
Other integral representations are
(4) y(nx)-x*lS~e"Uis*~'-J[2(xt)ii]dt Rea>0,
0 Ct
(5) r (a, x) = ~* 6 /" e-H-^KMxt)*] dt Re a < 1,
1 U - a)
(6) r(2-2«)r(a,-Jx)rUa)
= 2re-xtrJ_i_ F(lt i.i_a._!_\
J° L* + 2i 2 ' \ 2'2 t+2i/
+ ihi ^(l'b32 -«»7T2r)]* Rea<1' Re*>0-
The last of these is due to Tricomi (1950 a).
Some of the more important integral formulas are
/(«f) *= aa(r iA «* (iy a+* °+1; tti)
(7) J" e-***^'.y(
0 avi + s) -->- \
Re (a + 0) > 0, Re 5 > 0,
,8) /r«-^-.r(,,w,.^llgF/,(1,0 + ft^1,Ii-)
Re£>0, Re(a+/S)>0, Res>-^,
(9) jx e "*.y(* tz) dt = 2' ~ar (2a)r'e5 */8 n (2-X s)
ReO-% a/4 0, Re s > 0,
(10) T(a) *a^ J' e-***"^-' y(ft * - *t) A
0
-T (0) T(a - j8) yU *) Re a > Re j8 > - 1, afi £ 0.
The hypergeometric function reduces to an elementary function if )3 = 1
in (7) or a = 1 in (8); in (9), D is the parabolic cylinder function. It may
be noted that (3) to (9) are Laplace integrals. For other integrals see
Nielsen (1906 b, c), Le Caine (1948), and Busbridge (1950).
9,4. Series
The power series and continued fraction expansions were mentioned in

9.4
INCOMPLETE GAMMA FUNCTIONS
139
sec. 9.2. Using the expansion
* > 0, Re x > 0
1 v (->.
in 9.3 (3), we obtain the expansion in inverse factorials
ob
(1) r(n*)-e-'iB y ^ Re*>0,
vhere
" r(i-a) Jo » r(i-a) Jo U/
From 9,1 (IX ^e have
y(a,* + y)-y(a,a:) = e~x*a"1 Jy e"u (1+-1 ) du.
-nr
If |y| < |#|, we may expand (1 + u/x)a~l in the binomial series, integrate
term by term, and use 9.1 (17). Thus we obtain Nielsen's expansion
(2) r (a, x) - T (a, x + y ) = y (a, * + y) - y (a, a;)
= <T**a-' f (1 - a)n (-*)"" [l-c^C||(y)] M<M,
n = 0
which is useful for numerical computation.
Incomplete gamma functions occur in a large number of series
expansions, many of which may be obtained by specializing parameters in the
expansions of Chap. VI and will not be given in full. It is noteworthy
that with h = 0, a = — 1, the coefficients in 6.12 (7) can be expressed in
terms of the truncated exponential series; 6.12(6) becomes
(3) y(M)=rWe-"^ 2 eB(-l)**"7 (2**),
n= 0
and is rapidly convergent for all x A 0 provided that a is not a negative
integer. In the expansion 6.12(11) the coefficients may be expressed in
terms of Laguerre polynomials.
If x and y are positive and x > y9 we have
(4) Ha, x) yU y) = a-*"' (*y)a 2, , '!' ^a>(*) ^ ffr)-
n^0 U + 1) (
The limiting case as y -> 0 of this expansion is

140
SPECIAL FUNCTIONS
9.4
(5) rUx)-e'*xal _2-ll X>Q
„=o w + 1
and it coincides with the particular case a = 1 of the expansion 6.12(3)
of the ¥-function in a series of Laguerre polynomials.
For other expansions see Nielsen(1906a, sections 82, and 83).
9.5. Asymptotic representations
For a-> oo, x = o(|a|), the first series 9.2(4) is an asymptotic
expansion; for x -+ oo and a = o(|*|), we have 9.2(6). If x and a are of the same
order of magnitude, an expansion may be obtained from 6.13(17), but it
is not at all easy to find the general form of that expansion or to discuss
conditions under which it represents yia, x) asymptotically as both a and
x increase. Considerable complications arise when x and a + 1 are nearly
equal, more precisely if a-» .<» and x = a + 1 + o (\d\).
Tricomi (1950b) has made a through investigation of the problem. He
introduces the parameter
a*
(1) zwm
x - a
and distinguishes two cases according as z is small or large.
If z->0 and |argz| < 377/4, he proves that r(l + a, x) is asymptotically
represented by
(2) e-'x*** 2 Z (a)7i!(*-arn-1
n =0
where the coefficients
(3)
n ! Z>> -i-TT- ie-at(l+t)a] \ = L la~n)(a)
are certain polynomials of degree [ti/2] in cu These polynomials have
been studied extensively CTricomi 1951). In particular, we have
(4) r(H-»s)= e '**" [l-.-?-^ 2aW0(l\a\2\x-dr)\
x-a L I* - ft) \x - a) J
If z -+ 00 (when x and a are nearly equal), one has to distinguish two
cases according as Re a is positive or negative. In the latter case
Tricomi uses the function
(5) ytU*) = rU)*ay*U-*) *>0.
He then finds when £-* + «> and y is bounded,

9.6
INCOMPLETE GAMMA FUNCTIONS
141
(6) yfl + o,a + (2 id* y] = T(l + a) [5i+ ir"x Erf (y) + 0(<T*)]9
(7) r(a)yt[l-a, a+2(2o)*y]
= - 77 ctn (air) + 2ir* Erfi (y) + 0(a"*).
For a = 7i we have in particular
(8) en fe + (2ii)K y]
= exp [n + (2»)* y]. [y2 - it"* Erf (y) + 0(»"x)].
See also Furch(1939) and a contribution by Blanch in Placzek (1946)t
9.6. Zeros and descriptive properties
Information about zeros for real a and x may be derived from the results
of sec. 6,16t It turns out that y(a9 x) has
(i) no real zeros (apart from x = 0) if a > 0,
(ii) one negative zero x and no positive zero if
1-2ti < a < 2 - 2ti, where (n = 1, 2, 3, • • • , )
(iii) one negative zero xf and one positive zero x" if
- 2n < a < 1 - 2n, n = 1, 2, • • • •
The general behavior of these zeros as functions of a can be seen from
the altitude chart (p. 142) of y*.
Approximations to the zeros for large a have been obtained by Tricomi
(1950b); he proves that
(1) x ' =- (1 - a) [1 + 2* (1 - a)"* y*(a) +0(|ar,)l,
* l + r(-a7r/2)X
(2) x" --ra-- log : h 0 [\a\'' (log |G|)2].
1 + t sin a 77
Here y*(a) is the unique positive root of the equation
(3) Erf (y)=U/2)* ctn (ay),
and t- 0.278463 • • • is the unique positive root of the equation
(4) 1 + x + log x = 0.
If a > 0 is fixed, clearly y(a9 x) is a monotonic increasing function
of x for x > 0, and increases from zero to T(a) as x increases from zero
to oo. It can be shown that for a fixed x > 0, the function T (a, x)/T (a) is
a monotonic decreasing function of a for a > 0, In the other quadrants
of the real a, x, plane the incomplete gamma functions were investigated
by Tricomi (1951), who puts
(5) rU *) = - a"1 e'xxaG(a9 x\ a < 0, x > 0,
(6) y^ia, x) = a~* ex xag^(a,x) a>0, * < 0,

142
SPECIAL FUNCTIONS
96
Altitude chart of y* (a, x)

9.7
INCOMPLETE GAMMA FUNCTION
143
(7) y*(-a, -*) = T(a+ 1) e*&U *) a>Q, * > 0,
and proves
dG dG dg, dg.
<0, <0, -Sl<0, -^>0, \k\<l
dx da dx da
throughout their domains of definition, \k\ < % for a > 1, and
furthermore that ^ as a function of x has only one maximum or minimum if
0 < a < 1, while it has two maxima or minima if a > 1.
The altitude chart (p. 142) is taken from Tricomi's paper. It shows
the curves y*(a, x) = constant.
SPECIAL INCOMPLETE GAMMA FUNCTIONS
9.7. The exponential and logarithmic integral
The principal functions to be considered are
(i) et(x) = -Ei(-%) = £°c"*r,it = no,x) = e"«?a i,*).
(2) £*fe) = -£%"**"'rft *>0,
(3) \i(x)=[ —^- = Ei (log %) = -£_(-log*).
/ log*
In (2), the integral is a Cauchy principal value, i.e.,
lim (J'~€ + I°°) as^O, £>0;
this function is denoted by Ei (x) in Jahnke-Emde (p. 2). We have the
following relations between the functions defined in (1) and (2)
(4) -£T(%e±i77) =£*(%) ±in *>0.
The following formulas, and some others, can be obtained by making
a -> 0 in the results of the first part of this Chapter:
(5) Ei(-*)=y+log%+ 2, —;
n = t n ! n
oo X
= y + log x - e ~x 2 (1 + Vi + • • • + Vn) ,
n = i n !
oo
(6) £*(*) = y + log x + 2,
%
n=t * !*

constant of sec. 1,7.2,
'~x\ I ^F + 0(W",')J
L « = o
144 SPECIAL FUNCTIONS 9.7
where y is Euler's constant of sec, 1,7.2,
(7) £,(x)=%-' e"
H-»~, -3;7/2<arg*<3n/2, M = 1, 2
x -» oo, x > 0, W = 1,.2, ... ,
(9) d 21' ^ = (->""' <» ~ 1} ' *"" e"* en-. <*> »-l,2
(10) £= =e*Ei(-x)+ ^ ^TTT" »-l,2f....
m= 0
Re ]8 > 0, Re 5 > - &
To these we add Raabe 's integrals
/sin (pet) 1 „ *,
Y <^= — [e«gf(o«)+e-"g (ax)]
a2+*2 2a T
(13)
0
j £ cos (xt) r *• xn
—5 prf^^e^.laxj-e^S (ax)]
Jo a +t
a > 0, x > 0,
a > 0, x > 0,
both of which may be deduced from (l) and (2), and
(14) J°° (6 +*)"1 e"ct dt = e6c £, [(a + i) c] fiec> 0,
a
(15) f~e~xtlog tdt = %~T £,(*) Re*>0,
(16) J00 ta'1 E^ (t) dt = a"T[r (a, *) - xaE f (*)] Re x > 0, a ^ 0.
For other integrals see Nielsen(1906, especially Chapters II and IV),
Le Caine (1948), Busbridge (1950),
From 9.4(5) we have
V I to
(17) E,{x)^e'x > —2—- *>0,
^* /i + 1

9.8
INCOMPLETE GAMMA FUNCTIONS
145
and from 9.4 (2)
(18) £,(* + ?)«£,(*) +e""* f n!(-x)"n"![l-c"yc (y)]
n= 0
\y\ < 1*1-
The formulas for li (x) may be derived from those for E l (x).
Certain generalizations of the exponential integral function occur
in the investigation of wave propagation in a dissipative medium. A
typical example is
f e~uu~*dt where u = (a2 + t2YA.
For this and related functions see Harvard University (1949b).
9.8. Sine and cosine integrals
The definitions used in modern tables are
f sin t 1
(1) si*=/ dt =—[Ei (ix) -Ei(- ix)],
] t 2i
j Sin t 7T
2) Si x = / dt = — + si x,
J « * 2
/ COS t 1
(3) Ci x = / cb =— [Ei (&) + Ei (- ix)\
Joo t 2
(4) Ei (± ix) = Ci x ± i si x.
Here ±i = exp (± %J7r). Nielsen (1906) uses the same definition of si,
and writes ci instead of Ci. Some authors define the symbols Ci, Si,
somewhat differently.
Si x and also si x are entire functions of x9
(5) Si (-%) = - Si (x\ si (- x) - - 7T - si x.
Cixisa many-valued function, with a logarithmic branch-point at x = 0.
However,
/* 1 - cos £
(6) Ci % = y + log x - I dt,
Jo l
so that Ci x — log x is an even entire function of x. Li particular, we
have
(7) Ci(%e±£7T) = Ci% ±in, » > 0.
(2)

146
SPECIAL FUNCTIONS
9.8
The following formulas, and many others, are obtained by
straightforward manipulation of the definitions or of results in the earlier parts
of this chapter:
v Mn*2n+I
(8) Si% = %7r + six= > — r,
i^ (2/i+ l)!(2n + l)'
(9) Ci x = y + log x + 2,
n= 0
n _2n
(-)" x2
£, (2n)!(2i»)'
(10) si x = — cos x
[*?,' (_)« (2m)! 1
2 -^J-+0(W'2Af_,)J
In*[ | —^r + <>(\*\ W)J
- n- < arg * < jt,
f fff (-)'(2m-l)! I
Li,—^— J
[if -1 (_)«(2 ro) i n
m = 0 -J
8 = 0
r ^
+ sin
- n < arg * < 77, M, N =1, 2, ... ,
T "^ W- (2m -
(11) Ci x = cos x
+ sin x
I £ x^"1
L- m = 0
- 77 < arg x < n, M, N= 1,2,...,
(12) /o°°c-**Ci(t)(/t--— log(l + s2) Res>0,
(13) Jo°° e^'si fc) dt tan"1 s, Re s > 0,
(14) Jo°° e^"1 log(l + t2) dt = [Ci {s)]2 + [si ts)]2 Re s > 0,
sin x si x dx = I cos x Ci x dx = - — ,
0 Jo 4 '
(16) / si x Ci x dx = - log 2, / (si x)2 dx = f (Ci x)2 dx = Krr.
0 00
For other integrals see Nielsen (1906b, especially Chap. IV)t
The notations
rx dt
(17) Shix=J sinlu —= - i Si (ix),
0 t
(18) Chi x = y + log % + 1 ^Zll 1 dt = Ci fcr) - Yiin
C cosh £ - 1
log*+ —

9.9
INCOMPLETE GAMMA FUNCTIONS
147
are also used. The generalizations
(19) f'smu—, u = (az + t2)y>
0 u
and other similar generalizations have been discussed (Harvard University
1949 a).
9.9. The error functions
The principal functions in this group are
(1) Erf x = j* e'*2 dt = Ky(%, x2) = x 0(1/2, 3/2; -*2)
= *e-*20(l, 3/2; x2\
(2) Erfc x = f~e-*2 dt = %n* -Erf x = V2r{%, x2-)^%e^2 <P(K2> Vv,x\
(3) Erfi % = -i Erf (w) =f*e*2dt=-x 0(1/2, 3/2; %2),
(4) H(x)=2n-lA f* e-*2 dt = 2n-% Erf * = 1 - 2t7~* Erf*,
(5) a(*)=-(2/ir)*/'e^ cfc =-2ir"* Erf(2"* *>.
o
The first three are the most convenient for mathematical work, and (2) is
the function, although not the notation, originally introduced by Kramp
(1799% The function (4) is more convenient for numerical work, and (5)
arises in statistics where it is frequently used. There is a great variety
of notations.
All the error functions are entire functions; Erf x and Erfi x are odd
functions of x* Most of the following formulas are either straighforward
deductions from the definitions or else specializations of earlier results
of this section:
~ ( \n „2n + 1 „ °° ^2n + 1
v*\ \—) X 2 \~\ X
n= 0
* 2
2n + I , J^> /_\i»x2n + 1
(7) Erfi* = y -!L = e* Y
£* nl(2n + l) L,
..--. (3/2)„
(8) Erfc * = Vie'
L" = ° Rex>0, *•>«, M-l, 2
(9) Erfi*=-W^ + Xe*2 J ^L + 0(|«r«*-V)
%>0, x -* oo, M=l, 2, ••• ,

148 SPECIAL FUNCTIONS 9.9
(10) J"0"° e "a h 2~bt dt - a -' exp f —5- J Erfc ( — J Re a > 0,
(11) Je°EtHflt)e-*tdt = s"1 exp(-i_ ] Erfc(_]
0 \4a / \2a/
|arga| < & 77, Re s > 0,
(12) J"°°Erf (at)* e~stdt = K(air)* s"'(o + s)_*
0 Res>0, Re(a + s)>0,
(13) J00OErfc(af-,^)e-st^=^ff^s-, e-2as^ |arga|<^^, Re s > 0,
(14) f°°Erii(at)e-*ttZ-stdt =
u 0
iiexpfe) Ei\-h)
Res>0, |arga|</£n.
(15) /'e^2'2 -^- = ea2 Kir-Grfa)1] Re a > 0,
0 l + tz
(16) /'Erf trf« = xErfx-y2(l-e'a:\
rfn+1Erfx 2
(17) = (-)ne-* tf (*) n - 0, 1, 2, ... ,
where Hn is the Hermite polynomial of Chap. X.
A series of Nielsen's type is given below:
(18) Erf[(*+y)x]
2xHn^Q 2.4-~(2n) *"
Sre en«W\ |y| .< |*|.
Expansions in series of Bessel functions (Tricomi, 1951) follow:
(19) Erf(*) = KWe-*2 2 e (-1) x" I ^(2x),
(20) Erf(**)-(X>r)* I (-I)Mt „(*),
(31) &«(**)-■&*)* I {-D^^ In+ii(x)

9.10
INCOMPLETE GAMMA FUNCTIONS
149
The first of these expansions is a particular case of 9*4(3), the other
two can be verified by means of the Laplace transformation.
The most re cent monograph on error functions is that by Rosser (1948)
who discusses the double integral
(22) J* e"P2y2 dy J* c"*2 dx » = 1, 2, .•. .
as afunctionof the complex variablesp, zandalso other related integrals.
Repeated integrals of the error function have been investigated byHartree
(1936) who puts
(23) i° erfc x = 2n~% Erfc x, inevicx = f°° in~* erfc t dt,
9.10. Fresnel integrals and generalizations
Fresnel's integrals are
C(*)-(2ir)-x Jox T* cos tdt,
S(x) = (2ir)~x f,VX sin tdt.
Instead of these, we shall consider the more general integrals introduced
by BShmer (1939)
(1) C(x, a)^00^1 cos tdt
= ^^i7jaru ix) + y2e+*i7rar(a,-ix),
(2) Six, a)= J0°° ta'} sin tdt
2i 2i
The same functions, with a different notation, have been discussed by
Bateman (1946). Clearly we have
(3) T (a, ix) = e*i7Ta[C(x, a) - i S(*, a)].
Fresnel's integrals are:
(4) C (x) = (2/fr)'A f* ^cos (t2) efc = K - (2 *)"* C (x, V2)
= (2n)-K [e-^£7rErf(e+K£7rx^) + e^i7rErf(e-^£7rxy!)],
(5) 5 (x) = (2/ir)* J* sin (t2) <fc = V2 - (2 ff) "X 5 (*, X)
0
= f(2ff)-x [e-*i7rErf(e*i7rx*)-e*£7rErf(e-*i7rx*)].

150 SPECIAL FUNCTIONS 9.10
There follows a brief collection of formulas:
(6) C(x, a)=r(a) cosMan) - > /n xt/ ,
w^o (2/7i)! (2/7i+ a)
(7) S{x,a) = r{a)SinMan)~l - —- ^,
/*.<, (2m + 1) ! (2/71 + 1 + a)
(8) C(x, a) = -xa[P(x)sinx+Q(x)cosx],
(9) S(*, a) = *a[P(%)cos%-()(*) sin*],
where
ao)PW= J (")[11+Ta)2° +o(|«|-^-1)
m = 0
and
e«. y 2~ te~' +o(Nr2tf~2)
x -» oo, - 7T < arg x < n, M = 1, 2, . • • ,
(11) /" e""C(j, a) rfl = s"' T(a) [cos (Hair) - K(* + i)~a- Vi(* - i)~*I
0
Re 5 > 0, - 1 < Re a,
(12) i~ e-°*S(t, a)dt=s~i r(a) [sin Wajr) _ iAi(s + i)-*+y2i(s _ ;)-«j
0
Res>0, -l<Rea,
(13) /0°° ^-' C (t, a) dt = /3-' r (a + 0) cos[H(a + |8) vl
Re£>0, 0<Re(a + £)<l,
(14) J"o"«^"'S(tf tt)«fc-/3~,r(a + j8)sin[X(a + /8) 77]
Re/S>0, 0<Re(a + /S)<l,
(15) C (x) = J^ (x) + <75/2 (x) + c/^ (x) + • • • ,
(16) S (*) = J3/i (x) + J7/2 (x) + Ju/2 (x) + • • • .
An integral representation of
[C(x, a)]z + [S(x, a)]2
follows from 9.3 (6).

9.10
INCOMPLETE GAMMA FUNCTIONS
151
The curve represented parametrically by
(17) £ = C(t, a), rf = S{t9 a) t>0
for a fixed a, 0 < a < 1, is a spiral and has been investigated by Bohmer
(1939). It reduces to Cornu's spiral when a = %. It may be of interest
to note that this spiral has a simple "intrinsic equation"
(18) p = (aS),-,/a
where p is the radius of curvature and s is the arc length.

152
SPECIAL FUNCTIONS
REFERENCES
Bateman, Harry, 1946: Proc. Nat. Acad* Sci. 32, 70-72.
BShmer, Eugen, 1939: Differenzengleichungen und bestimmte Integrale, Leipzig,
Busbridge, I. W„ 1950: Quart. J. Math. Oxford Ser. (2) 1, 176-184.
Fureh, R., 1939: Z* Physik 112, 92-95.
Hartree, D. R., 1936: Manchester Memoirs 80, 85- 102.
Harvard University, 1949a: Annals of the Computation Laboratory, Vols. XVIII
and XIX. Generalized sine - and cosine - integral functions. Parts I and II,
Harvard University Press, Cambridge, Mass.
Harvard University, 1949b: Annals of the Computation Laboratory, Vol. XXI,
Tables of the generalized exponential - integral functions, Harvard University
Press, Cambridge, Mass.
Jahnke, Eugen and Fritz Emde, 1945: Tables of functions with formulas and
curves, Dover Publications, New York.
Kramp, Christian, 1799: Analyse des Refractions, Strasbourg and Leipzig,
Le Caine, J., 1948: National Research Council of Canada, Division of Atomic
Energy, Document No. MT- 131(NRC 1553), 45 pp.
Legendre, A. M.. 1811: Exercises de calcul integral, Paris.
Nielsen, Niels, 1906 a: Handbuch der Theorie der Gammafunktion, Leipzig, 326 pp.
Nielsen, Niels, 1906b: Theorie des Integrallogarithmus und verwandter Trans-
cendenten, 106 pp., B. G. Teubner, Leipzig.
Nielsen, Niels, 1906c: Monatsch. Math. Phys. 17, 47-58.
Placzek, George, 1946: National Research Council of Canada, Division of Atomic
Energy, Document No. MT- 1, 39 PP»
Prym, F. E„ 1877: /. Math. 82, 165- 172.
Rosser, J. B., 1948: Theory and application of
J"2e^*and j* e^ dy jy e^ dx
Jo Jo J Jo
Mapleton House, Brooklyn, New York.
SchlSmilch, Oskar, 1871: Z. Math, Phys. 16, 261-262.
Tannery, Jules, 1882: Comptes Rendus 94, 1698- 1701, 95, 75.
Tricomi, F.G., 1950a: Boll. Un. Mat. Ital. (3) 4, 341-344.
Tricomi, F. G„ 1950b: Z. Math. 53, 136-148.
Tricomi, F. G., 1951: Ann. Mat. Pura Appl. (4) 28, 263- 289.
Tricomi, F. G., 1951: /. D'Analyse Math. 1, 209-231.

CHAPTER X
ORTHOGONAL POLYNOMIALS
The standard textbook on this subject is the book by Szego (1939) to
which we shall refer frequently. There is a systematic bibliography up to
1938, by Shohat, Hille, and Walsh (1940). Although the present chapter is
concerned witn orthogonal polynomials only, in the introductory sections
we consider more generally systems of orthogonal functions. For further
information on this latter topic the reader may be referred to books by
Kaczmarz and Steinhaus (1935) and by Tricomi (1948), and by Vitali and
Sansone (1946).
10.1. Systems of orthogonal functions
With an interval (a, b) and a weight function w (x) which is non-
negative there, we may associate the scalar product
(1) (<Pii<P2)sJ' WW?,W%W^
which is defined for all functions cpfor which w* (pis quadratically inte-
grable in (a, 6). More generally, a scalar product may be defined by a
Stieltjes integral
(2) (cp,, q^) = fab q>i(x) y2(x)da(x)
where a(x) is a non-decreasing function. If a(x) is absolutely continuous,
(2) reduces to (1) with w (x) = a\x). On the other hand, if a(x) is a jump
function, that is constant except for jumps of the magnitude w . at x = x ■,
then (2) reduces to a sum
(3) (<p1f cp2) = 2wicp1Ui)cfcUi)
i
which is the appropriate definition for functions of a discrete variable.
The above definitions refer to real functions of a real variable, and
to this case we shall restrict ourselves throughout this chapter. If the
functions in question are complex-valued, or else if the domain of inte-
153

154
SPECIAL FUNCTIONS
10.1
gration is an arc in the complex plane rather than a segment of the real
axis, then Cp2 (x) in all these definitions must be replaced by the
conjugate complex quantity.
Except in the last few sections (where we use definition (3)), we shall
use definition (1) mostly, and shall assume moreover thatwix) is positive
almost everywhere and integrable* It should be mentioned, however, that
many of the results of the introductory sections hold for the definition
(2), and therefore also for the definition (3), of a scalar product.
Two functions are said to be orthogonal if their scalar product
vanishes. A family of functions is an orthogonal system, on the interval
(a, b) and with the weight function w (x) (or distribution a(x)), if for any
two distinct members of the family, (cpt , cp2) = 0, Since the space of
quadratically integrable functions is separable, it follows that an
orthogonal system consists either of a finite number or at most of adenumerable
infinity of elements. Thus an orthogonal system can always be written as
a (finite or infinite) sequence, <po(x), cpl (x), ••• or shortly I q>n(x)\, and the
orthogonal property is then expressed as
(4) (cpA,cp^) = 0 h^k.
We shall assume that i(pn(x)l does not contain any null function, i.e.,
that (qpA, yh) is positive for all A. It is then easy to see that the functions
of any finite subset of an orthogonal system are linearly independent,
that is that a relation of the form
(5) c0 % (x) + ct cpt (x) + .... + ck yk {x) = 0
cannot be valid almost everywhere in (a, b), except when cQ = ct = ••• =
c h = 0. (Form the scalar product with (p^O*) for h = 0, 1, ••• , &•)
The functions lep (x)\ form an orthonormal system if
(0 if h + k,
(6) «*.%>- {, .f A = i,
Every orthogonal system can be normalized by replacing <ph(x) by
A (finite or infinite) sequence }0n(%)} of linearly independent
functions can be orthogonalized with respect to the scalar product (2) by
the formation of suitable linear combinations. For instance we may put
recurrently
(7) <po(*)=0o(*)
<p, (*)=/* 10<p0(x)+ x/>x(x)
»•• ••• ••• •••.

10.1
ORTHOGONAL POLYNOMIALS
155
and see that lcpn(x)i is an orthogonal system if we take
(8> ^rw "^n' (P.)/((?., CpJ /7l= 0, 1 II -1.
Alternatively, we may put
(9) cpBW-ABO0oU) + ABl0IU) + ;.. + A„,0Il(x) Am^O
and determine the A's so that Icp (x)| is an orthogonal system. One
possible determination leads to
(10) <i (*) =
(0O,0O) (0O,0,)
(0,,0O) (0,,0,)
0O(«) 0!^)
0„(x)
It is clear that \<f>n(x)\ is an orthogonal system, for (10) is orthogonal to
0O(%), \Jj y (x), ... , 0n-i (%) an(l hence to <£R (x) for all m < n. Moreover,
any orthogonal system of the form (9) is a constant multiple of \<p (%)i.
In order to normalize the system (9), we introduce Gram's determinant
G n which is the cofactor of *An + 1 (*) in the express ion (10) for <f>n + ] (%)• Gn
is also the discriminant of the positive definite quadratic form
/' [f0 tf0(*)+ - + £n4,n(x)Yw(x)dx
in f , ... , f , and hence positive. We also put G = 1. The orthonormal
system of the form (9) with A > 0 is then uniquely determined as
(in cPnu) = (Gn_1cnr%n(x).
Furthermore the following integral representation can be established
(12) ^>)=[U-1)!]-' J-°V,^o £,.-,>V£o.
x w{£0)... ">(£„_,)<*£„ ... if B_,
where the integral is an n-tuple integral over (a, b) and
0o(xQ) 0, (xQ) .» 0„(*o)
(13) V>0, .»,*„)= '
ra = 1, 2, •••
■W *,<*„>•••*„<*„>
(See Szego, 1939, sec. 2.1.)
In this chapter we shall be concerned with the orthogonalization, in
the form (9), of the functions ip ix) =xn. Thus we obtain a sequence of

156
SPECIAL FUNCTIONS
10.1
orthogonal polynomials ipn(x)\,n = 0, 1, 2, ••• where pk (x) is a polynomial
in x of exact degree k9 and (ph, pk) = 0 for A, k = 0, 1, 2, ... and A ^ &.
The interval and the weight function (or distribution) determine the
system of orthogonal polynomials up to an arbitrary constant factor in
each pn(x)% The polynomials may be standardized by the adoption of
additional requirements.The three most frequently used additional
requirements are: (i) \pn(x)\ shall be an orthonornial system and the coefficient
of xn in p {x) shall be positive; (ii) the coefficient of xn in p (x) shall
have a prescribed value (usually unity); (iii) for a given x (for instance
xQ = a), pn(xQ) shall have a prescribed value.
10*2. The approximation problem
2
Let L be the class of all functions f(x) for which the (Lebesgue)
integral
fbw(x)\f(x)]*dx
a
exists and is finite, and let Iqp (x)\ be an orthonornial system in L2 • In
approximating any function f(x)oiL2 bya linear combination
coCpo (*)+»• +c„<pB(*),
we regard
d> In(ch)=f'w(x)\f(x)-c0cp0(x) cn%(x)YdX
as the measure of accuracy of this approximation. It is easy to see that
the best possible choice for c , is that of the Fourier coefficients
(2) <*h=(f9 <Ph).
In fact, expanding [••• ]2 in (1) we find
In(ch) = Jabw(x)[f(x)]*dx+ t c\-2 i ahch
h—o n= o
.= Jhw(x)[f(x)]2dx- i a\+ 1 (ch~a)\
a h= 0 h— 0
that is the best approximation is the (n + l)st partial sum of the
(generalized) Fourier series
(3) a0<Pok) + ai<P|k)+ "•
of f (x\ and the measure of the accuracy of this approximation is
(4) In{a A) - J* w (x) [fix)]2 dx- f a\.

10.3
ORTHOGONAL POLYNOMIALS
157
Since In(aJ> 0, it follows that 2 a* is convergent and we have Bessel's
inequality
(5) 1 a\<jh w(x) [f(x)]2 dx.
h=o n a
It may happen that Parseval*s formula
(6) 2 a\= fbw(x)[f(x)]2dx
/»= o a
holds for every function f(x) of L2. Then the orthonormal system {(p (*)}
is said to be closed in L . In this case clearly
(7) fbwb)[f(x)- 1 a.cp.(x)]2dx^0 as n -* «,
a /» = i a a
and we say that the partial sums of the Fourier series (3) converge in
the mean to /(%)• In L*, every closed orthogonal system is also complete,
i.e., if (/, <pfc) = 0 for all h9 then f(x) vanishes almost everywhere. This
is a consequence of the Riesz-Fischer theorem (cf. for instance Kaczmarz
and Steinhaus, 1935, or Tricomi, 1948, sec. 3.3).
For a finite interval (a, b) every function of L* can be approximated
arbitrarily closely, in the mean, by a continuous function, and by the
theorem of Weierstrass the continuous function can be approximated by a
polynomial. Thus for a finite interval and 0 (x) = t", or cp (x) = p (x\
we may make In(ah) arbitrarily small by making n sufficiently large. In
other words, any system of orthogonal polynomials for a finite interval is
closed. This need no longer be true if the interval (a, b) is of infinite
length (Szego 1939, sec. 3.1).
10.3. General properties of orthogonal polynomials
A weight function w (x) on an interval (a, b) determines a system of
orthogonal polynomials \p (x)\ uniquely apart from a constant factor in
each polynomial. The numbers
(1) c = J w(x)xndx
are the moments of the weight function, and with ifj (x) = *n we have
(2) WB,<A„) = c.+B.
In the notation of sec. 10.1 we then have

158
SPECIAL FUNCTIONS
10.3
(3) Gm
c c ,. ••• c„
. *
1 *n "■• *n
0 0
1 *, ... *?
r> *
1 x -. *n
n n
If the (undetermined) coefficient of xn in p (x) is denoted by h , we have
C1 C2 "" °n+1
n—1 n 2n —1
* ... *
(5) p.W--^=- /" n(fr-f.)« ft [<*-f>tfvWfvl
n —1 */
Since 1, x, «•• , xn~* are orthogonal to p (x), we have
(6) ^=(Pn.P„) = ^
C-,
For the normalized polynomials k - (G _f/G ) , but we shall not
standardize our polynomials at this stage«
Any polynomial of degree m < n is a linear combination of P 0(x\
pt (*), ••• , p n (x) and hence orthogonal to p (*)• This leads to a simple
proof of the following theorem on the zeros of orthogonal polynomials.
All zeros of p (x) are simple, and located in the interior of the interval
(a, b). For if pn(x) changed its sign in (a, b) only at m <n points, we
could construct a polynomial n (x) of degree m so that p (x) n (x) > 0
in (a, 6), and this contradicts (p , n ) - 0« It can also be shown that
between two consecutive zeros of p (x) there is exactly one zero of
P +f (x\ and at least one zero of p {x) for each m> n (Szego, 1939, sec.
3?3).
Any three consecutive polynomials are connected by a linear relation*
We use the following notations: k is the coefficient of xn, and k^ the
coefficient of xn~\ in p (x); r = A'/A , and h - (p , p )• We shall then
prove the recurrence formula
(7) Pn+1U)=Un.+Bn)Pn(x)-Cnpn_|W
/i — 1, 2, 3, •••

10.3
ORTHOGONAL POLYNOMIALS
159
in which
(8) A =k .Jk , B =A (r . -r),
n n + r n n n n + 1 n
C =A h /(A h ,) = ft ., ft , A /(ft2 A ,).
n n ff n-1 n-J n + 1 n-J n n n-?'
To prove (7), we remark that with the value (8) of A , the expression
P +i(*)~~ A xp (x) is a polynomial of degree n or less, and consequently
of the form
From the orthogonal property of the p (x)9 we find that y2 = y = ••• =
= yn = 0, and
Now, xp (x) - (k A/k ) p (x) is a polynomial of degree n - 1 or less,
and hence
- A h k Jk = yt h ,
n n n —1 n ' 1 n~1
or y. = C • Lastly, the value of B follows on. comparing coefficients of
xn on both sides of (7). The recurrence formula (7) remains valid for
7i = 0 if we put
(9) P_,(*)=0.
This convention will be retained throughout this chapter.
It may be noted that conversely, a system of polynomials satisfying a
recurrence relation (7) with positive A and C , is an orthogonal system.
From (7) we easily obtain the Chris to ffel-D orb oux formula
do) 1 A--P|/U)P|/(r)-I^rP-^)p-(y)-p-C')p-(y)
v= o v " k ,.h x -y
n+1 n J
and for y -* x,
(ii) i A-'[P>)]2 = i^[PnwPn'+1w-Pn'U)pn+1(x)].
n + 1 n
Let {p (#)1 be the system of orthogonal polynomials for the weight
function w (x)9 and let p(x) be a polynomial of degree I which is non-
negative in (a, b) and has simple zeros at x ]9 xz, .♦• , x ^ The orthogonal
polynomials q n(*), belonging to the weight function p(#) ">(*) are then
given by Christoffel's formula

160 SPECIAL FUNCTIONS 10.3
P» Pn+i*** - Pn + l^ I
P.**!* Pn + 1(*i> - P. + W
I 9
in which c is an arbitrary constant factor (Szegd*, 1939, sec. 2«5). If
some of the zeros of p(x) are multiple zeros, (12) must be replaced by a
confluent form.
Orthogonal polynomials have some important extremum properties.
The first of these can be derived from the result at the beginning of sec.
10.2 and reads: The integral
(13) Jb\n (x)\2w(x)dx
a n
in which n (x) denotes any polynomial of degree n with the leading term
xn becomes a minimum if and only if n (x) = ck~y p (x) where e is a
n n l n
constant and |e| = 1. The second property involves the polynomials
(14) Kn(x,y)= t h~m'p 6)p fy)
which are defined for complex x, y (x"is the conjugate complex of x). We
may remark here that for finite xQ, a and for xQ < a, the polynomials
Kn(xQ, x) are orthogonal with respect to the weight function (x-x Q)w(x)
(cf« (10) and (11)). The extremum property in question may be formulated
as follows (Szego, 1939, theorem 3.1.3). Let n (x) be an arbitrary
polynomial of degree n with complex coefficients such that the integral (13)
is equal to unity. For any fixed (possibly complex) xQ the maximum of
\n GO|2 is reached if and only if
*nU) = e[Kn(*0,*0)r*Kn(*0,*)
where |e| = 1. The maximum itself is K (x , x 0).
10.4. Mechanical quadrature
Many interesting properties of orthogonal polynomials depend on their
connection with problems of interpolation and mechanical quadrature. In
this section we can give no more than a brief description of some of the
basic results, and refer to Szego's book (1939, sec. 3.4* chapters XIV,
XV) for further information.
Let x}, x2, ... , xn be n distinct points of the interval (a, b) and let
(12) cnp(x)qn(x)-

10.4
ORTHOGONAL POLYNOMIALS
161
(1) tfnW = (* ~ xy)(x -x2) ••• (x -*n\
lv(x)= (x - xv)~* TTnM/nfaj) v= 1, ... , n.
The lv(x) are the fundamental polynomials associated with the abscissae
x , ... , x in the Lagrangean interpolation
(2) L(x)= I f(xv)lvM
v— 1
of the function /(%)•
If the integral
(3) I=Jbw(x)f(x)dx
a
is to be computed for a function whose values at the x are given, it
seems natural to use (2) and compute
(4) J=fbw(x)L(x)dx= £ fix )Jbw(x)l (x)dx
in the expectation that J will be an approximation to L Actually, for any
x\ 5 •• 9 x 9 we have I~ J for all polynomials /(re) of degree < n — L
However, if we choose the xy to be the n zeros of p nipc\ the orthogonal
polynomial of degree n associated with the weight function w{x), then
I- J for all polynomials f(x) of degree <, 2n — 1* For in this case f(x)
— L (*r) is a polynomial of degree < 2n — 1 vanishing at all the zeros of
p (x) and hence of the form Pn(*) ^-t (*) where tt . (#) is a polynomial
of degree < n — 1. Then
I-J= J bw{x) [fix) -L (*)]<& = (p , it t) = 0.
It is customary to write
(5) J=Jbw(x)L(x)dx= £ A, /(* )
where the A are called the Christoffel numbers. They are connected
Vn J J J
with the moments of w (x) by the relations
(6) lxh\ =ch h = 0,1 ii-1
obtained by choosing /(re) = x \ The Christoffel numbers are positive,
and the following formulas hold:
rb wMpU) J rb , r P>) 1
(7) W.F^^H."wbt*^>J
2

162 SPECIAL FUNCTIONS 10.4
k ., h /k i
If we denote by *tn, *2n, ... , *nn the rc zeros of pn(*) and by y t n, ... ,
ynn the * numbers in (a, i) defined by
(9) fv"w(x)dx=k,n+... + \vn = Avn
a
then we have a number of separation theorems
<10> *v-I.n <*v.n + 1<*x,,n
(id yv.t(n<yv.n+t <rv.n
(12) »: <y <* .f
(13) A f < A Al < A
10.5. Continued fractions
The recurrence formula 10.3(7) suggests the continued fraction
(i)
1| C,| CJ
\AQx + B0 \A,x + B, \A2x + B2
where An, Bn, Cn are given by 10.3 (8)« The rctn convergent R n/S n is
defined as the finite fraction obtained by stopping at the term A _t x + Bn_^
in (1) so that
(2) R0 = 0, S0=l; K, = l, St=40x + B0 = p,U)/p0(x).
Both /? and S satisfy the recurrence relation
(3) *.=G4 x + B)X -C X ..
s ' n + t n n n n /j-1
The initial conditions are
(4) for R n : XQ = 0, *f = 1; for Sn : ,¥Q = 1, X% = p, (*)/p0 (*).
Referring to 10.3(7) it is seen that
(5) Sn = Pn{x)/p0{x) = k-0'pn(x).
In order to express also R , we introduce the associated polynomial
(6) g («)-/" ^-^-a)*

10.6
ORTHOGONAL POLYNOMIALS
163
which is a polynomial of degree n - 1. From 10.3(7)
xnti n n * n n * n I
= -Anfa pn(t)w(t)dt = 0 11=1,2,....
Moreover, qQ (x) = 0, q f (%) = J"a 4^(0^ = 4,0 , and hence
(7) *.-(*,c0 )-,>).
We thus see that /? /S is a rational function of x with simple poles
at x = x • The residues at these poles can be computed from
lim (x-xJ-JL—-—- J^_ w(t)dt =\vn,
see 10,4(7), and we have the decomposition in partial fractions
(8) A=3l_ J Jbi. .
o k . c „ , x — x
On expansion of the sum in descending powers of x the relation 10.4 (6)
shows th
formally
shows that*the first 2n coefficients are the moments c^. Hence we obtain
(9) lim-i=7^- Z TFT-
n ! 0 n= 0
For a finite interval (a, 6), and for any x in the complex plane cut
along the segment (a, b) of the real axis, Markoff proved that lim R /S
exists and (9) is valid. Moreover,
(10) lim —i =-^L_ / «fe
in this case (Szeg<V, 1939> sec. 3.5). Intervals of infinite length present
formidable difficulties which are discussed in the theory of (Stieltjes
and Hamburger) moment problems. For these see Shohat and Tamarkin
(1943).
10.6. The classical polynomials
The orthogonal polynomials belonging to the intervals and weight
functions listed in the following table arise very frequently and have

164
SPECIAL FUNCTIONS
10.6
been studied in great detail. They are known as the classical orthogonal
polynomials.
CLASSICAL ORTHOGONAL POLYNOMIALS
a b w(x) NAME
-1
-1
-1
— oo
0
1
1
1
oo
oo
1
a-x*)k~*
(l-x)a(l+z)£
exp(-x2)
xa e'x
Legendre or spherical
Gegenbauer or ultra spherical
Jacobi or hyper geometric
Her mite
(generalized) Laguerre.
Ail these polynomials have a number of properties in common of which
the three most important ones are:
(i) \p^(x)l is a system of orthogonal polynomials;
(ii) p (x) satisfies a differential equation of the form
A{x)y" + B(x)y' + kny = 0
where A (x) and B (x) are independent of n, and A is independent
of x;
(iii) there is a generalized Rodrigues' formula
(i) pJx>>=t-^ -^rM*Un]
K w(x) dx
where K is a constant and X is a polynomial in x whose
coefficients are independent of n.
Conversely, any of these three properties characterizes the classical
orthogonal polynomials in the sense that any system of orthogonal
polynomials which has one of these properties can be reduced to a classical
system. For (i), this has been proved by Hahn (1935) and Krall (1936);
for (ii) by Bochner (1939)((in this case there are some trivial exceptions);
and for (iii) by Tricomi (1948a). We shall briefly indicate the argument
in this last case.
Let \p {x)\ be a sequence of polynomials, p (x) of exact degree n, for
which (1) holds for every n = 0, 1, 2, ... , the polynomial X being of
degree k% Note that it is not necessary to assume that the p (x) are
orthogonal polynomials or that w (x) is a weight function. From (1), with
n = 1, we have
(2) KlPi{x) = X '+Xw*(x)/w(x).

10.6
ORTHOGONAL POLYNOMIALS
165
'(*) * -r
First let k =0. Then X is a constant and w '/w is a linear function of
x% By a linear change of the independent variable we may make w */w
- —2x, hence w = exp (— x2), and the polynomials are the Hermite poly-
nomials, see 10.13 (7)« Next let k = 1. Then a linear change of x brings
(3) -TT=-L^
W\X) X
into the form w'/w = -1 + a/x, so that X - x9 w = xae~x9 and we have
the Laguerre polynomials, see 10.12 (5)«
We now discuss k > 2. In this case we may take
(4) X= n (x-a)
r= \ r
and at first we assume all the a different from each other. From (3)
h
n 2 —
w(x) r==1 x-ar
so that (1) becomes
k -a dn k n+a
P,W = X n (x-a) r-—-[ n (x-a ) '],
r = 1 a# r = 1
and for n = 2 this fails to be a quadratic polynomial except when k = 2.
The case of repeated factors in (4) can be excluded by a similar
consideration, so that in (4) we must have k~%a^a% By a linear change
of x we may make a = — 1, a ~ 1, and write
x = a-x)\ w(x) = a-x)aa + x)P
so that this case leads to Jacobi polynomials, see 10„8(10)«
It may be mentioned that Hahn (1949) has extended these results
considerably. He replaced the differential operator df(x)/dx by the more
general linear operator
rrf \ f^X + <*)-f{x)
LfM=- -7
\q - l)x + co
and showed that in this more general case each of conditions (i), (ii),
(iii), and of two further conditions, characterizes the same family of
orthogonal polynomials. The classical polynomials are limiting cases
of Hahn's polynomials, and so are the polynomials of sections 22-25 •

166 SPECIAL FUNCTIONS 10.7
10.7. General properties of the classical orthogonal polynomials
Many important properties of the classical orthogonal polynomials
follow easily from the generalized Rodrigues' formula 10.6(1). We assume
a > — 1 in the Laguerre case and a > — 1, jS > — 1 in the Jacobi case.
In each case we have in 10.6(1) a w(x) which is non-negative and
integrable in (a, b)% Moreover, since all derivatives up to and including
the (n - l)st of w (x) Xn vanish at a and b, we may integrate by parts n
times in
(/, Pn) = K-' Jb fM^r bv(x)xn] dx,
a dx
obtain
(f,Pn) = (-\YK-< jhfV{x)w(x)Xndx
n a
and hence (f9 p ) = 0 if f is a polynomial of degree < nm In other words,
the polynomials 10.6(1) form an orthogonal system in the interval (at b)
with the weight function w{x\ and all the results of the previous sections
are valid for these functions. In particular, we have the recurrence formula
10.3(7) with the notation 10.3(8) which we shall use again in the present
section.
In deriving the differential equation from 10.6(1) we shall write D
instead of d/dx9 From 10.6(1) and from Leibniz' formula of the
differentiation of a product we have
Dn+'[XD(wXn)] = K [XDHwp )+(n+l)X'D(wp)
-I- l/2 n{n + 1) X " wpn]m
On the other hand, using 10.6(3),
Dn+t [XD(wXn)] = Dn+1 \[K} p , + (n - 1) X '] wXn \
= Kn\\Kx pt + (n - 1) X '] D (ivpn)+(n + l)[K} p, '+(n-l) X ] wpj
since K p + (n — l)X'is at most a linear function of x% Comparison of
the two results yields the differential equation
for y = p (x), where
(2) An = -ra[A, A', + y2(n-l)X"].

10.7
ORTHOGONAL POLYNOMIALS
167
The self-adjoint form of the differential equation is
dyl
Xw (x) — + X w (x) y = 0.
dx]
d
dx
For the details of the proof see Tricomi (1948 a, p. 2,10-212). Since X is
at most a quadratic polynomial, and pt(*) is a linear polynomial, the
differential equation (1) can be reduced to the hypergeometric equation
or to one of its special or limiting cases.
For the classical polynomials we also have the differentiation formula
(4) X^^- = («„ + l^X" x)Pn(x) + /3n Pn_, (*)
where
(5) an = nX'(0)-KX" ra, A n £„ = - C nUt, K, + (n - %) X" ],
and An, Cn$ k 9 r have the same meaning as in sec. 10.3. By means of
10.3(7), the right-hand side of (4) can be expressed in terms of p and
The proof of (4) in Tricomi (1948a, p. 212-215) is based on the fact
that
Xp\x)- VinX" xpn(x)
is a polynomial of degree < n and hence of the form
flnPnW + Pn Pn-t {X) + ?Z Pn-Z{x) + - + ^n P0 <*>•
The coefficients a , ... , y are then determined by the orthogonal
property. In the determination of )S the differential equation (3) is also used.
Finally we note that by n successive integrations by parts as at the
beginning of this section,
(6) h =(p ,p )=(-l)nk n\K-x JbXnw(x)dx,
n l n ' n n n a
from 10.4(8), 10.3(7), and (4)
(7) Ai/<B-^._1A..IWUViliy/8.][p._1(*Vi.)]-«
and from (6)
(8) (-1)"* K >0.

168
SPECIAL FUNCTIONS
10.7
Each of the following six sections is devoted to one of the principal
families of classical orthogonal polynomials. Each of these six sections
is organized on the following plan:
(i) Standardization of the polynomials,
(ii) Computation of the ten constants
(9) hn,kn,rn,An,Bn,Cn,Kn,\n,an,Pn
given by 10,7(6), 10,3(8), 10.7(2), 10.7(5).
(iii) Statement of the recurrence relation, differential equation, and
other relations, except that whenever these relations are
cumbersome, it will be left to the reader to substitute the
values of the ten constants (9) into the general formulas of
this and the previous sections,
(iv) Connection with functions of the hypergeometrie type and
complete integration of the differential equation,
(v) Generating function or functions,
(vi) Integral representations.
(vii) Addition theorems, series expansions, and miscellaneous results.
Asymptotic properties, zeros, expansion problems will be discussed in
later sections.
We shall use the notation
^ d
(10) D = —
ax
and shall put
r(a+7i) .
(11) (a)0 = 1, (a)n= = a(a+l) ... (a+n - 1).
L\Ct)
Accounts of the classical orthogonal polynomials are given in the
works referred to in the introduction, and also in the book by Magnus and
Oberhettinger (1948, Chap. V).
10.8. Jacobi polynomials
We shall use Szego's notation P ta'^ (x) for the suitably standardized
orthogonal polynomials associated with
(1) a=-l, 6 = 1, w(x)=(l-x)a(l+x)P, X=l-xz.
In order to make the weight function non-negative and integrable, we
assume
(2) a>-l, j8>-l.
Many of the formal relations are valid without this restriction.

10.8 ORTHOGONAL POLYNOMIALS 169
(i) Standardization.
i~ a\, s (n + a\ (a+1)
(3) Pia'PKl)= ( ) = 2-.
\ n J n\
(ii) Constants,
(4) (2rc+a + £ + l)rc!r(rc + a+0 + lHa=2a+£+1 r(n + a+l) r(n + j8+l)
(5) k =2~n ( J , r = —
a \ n J n 2n + a+0
(6) 2U + DU + a + 0 + l)4a=(27i + a+0 + l)(27i + a+0 + 2)
(7) 2(b + DOi + a + 0 + D(2ji + a + j8) 5a = (a2 - 02)(2n + a + j8 +1)
(8) (n + l)(n + a + 0+l)(2rc + a + 0)Cfi = (7i + a)(n + j8)(2n + a + j8 + 2)
(9) K =(-2)nn!, A=/i(7i + a+0 + l), a =r ,
(2n + a + )8) j8n = 2(n + a)(n + )8)
(iii) Rodrigues9 formula.
(10) 2n n! pb'P\x) = (-1)" (1 -*)-« (1 + x)~0Dn[a - x)a+n (l + x)P+n].
Recurrence formula:
(11) 2(ji + l)0i + a + 0 +1)(2ji + a + j8) P^}U)
= (2ji + a + j8 +l)[(2ji+a+j8)(2n + a+j8+2)« + a2 - 02] P^Hx)
— 2(n + a)(n + /3)(2rc + a + ^3 +
2)p£'/3)(*).
From (10) we obtain the explicit expression
(i2)p<^>(*)=2-» f (n+a) (n+fi) u-i)—(*+«■
„f0 \ « / \n-mj
which shows that
(13) P<a'£>(-*) = (-l)nPf 'a)(*).
Differential equation:
(14) Q-x2)y" + [|8-a-(a + |8 + 2)x]y'+n(iH-a+|B + l)y=0.

170
SPECIAL FUNCTIONS
10.8
Differentiation formula:
j
(15) (2n + a + j8)(l-*2)— P(a-P\x)
dx n
= n[(a-p)-(2n + a+ jS)x] P <a^'(x) + 2(n + <z)U + /3) P„(f ^'(x).
(iv) Hypergeometric functions. Equation (14) can be reduced to the
hypergeometric differential equation 2.1(1), and the Jacobi polynomial
is that solution of (14) which is regular and has the value (3) at x = 1.
From the formulas of sec. 2.9,
(16) P{*'0\x)= r + C) F(-n,n + a + jB + l;a+ 1; J* - Hx)
= (-l)n f J F(-n, * + a+j8 + l; 0 + 1; % + %x)
fn + a\ / x - 1 \
From this we find the further differentiation formula
(17) 2" D* P(a-PKx) = {n + a+B + 1) /><«+«.£+« >(x)
/ft = 1, 2, ... , n
which confirms statement (i) of sec. 10.6.
It follows from 2.9(14) that the function <?*a,y5)(%) defined by
(18) r(2n + a+j8+2)^J1^)(x)= —-— jr—t
x F[rc + 1, n + a+1; 2rc + a+0 + 2; 2(1-a;)"1]
is a second solution of (14)« It is known as the Jacobi function of the
second kind. This function is not a polynomial, but it satisfies the same
recurrence formula (11), and the differentiation formula (15), as the
Jacobi polynomial (except that n = 0 is not admissible with the Q);
it vanishes at infinity when Re (a + /3) > — n — 1. For the various
transformations of the hypergeometric series in (18)> an(l for its analytic
continuation, see sec. 2.1.4*
Jacobi polynomials and Jacobi functions of the second kind are
connected by several relations. From the connection between various
solutions of the hypergeometric equation, see sec- 2.9, we have

10.8
ORTHOGONAL POLYNOMIALS
171
(19) (?<a^)(x) = -y2ffcosec(a»7)P);a^)(x)
+ ^,rwruVi>
r(n + a+ j8+ 1)
x F (n + 1, -7i - a - j8; 1 - a; Vi - lAx\
There is also the integral relation
(20) ()^^,(x)=K(x-ira(x+ir^J_\(x-0",(l-Oa(l + ^Pja^\Oe/*
valid for all points x in the complex plane cut along the segment (-1, 1).
This segment is a branch cut, and Q^a^ 'assumes different values
according as x approaches a point £ on the branchcut from the upper half-plane
(£+i0) or from the lower half-plane (£-i0). The values of Q^M (<f ±i"0)
may be computed from (19)> taking arg (x - 1) = n for x = £+ iO, and
arg (x - 1) = -77 for x = £- i'O. In particular,
(21) Qb.0\£+iQ)-Ql*'PK£-iQ)
'-t^.,.(J)^>rh;<,;»(i-fr-a.fr.
TU + a+ £ + 1)
xF0i + l,-n-o-j8; 1-a; >4-Hf) -Kf<l.
On the cut itself, one may use the function
(22) Q<a^)(^) = ^[<?<a'/3)(^+J0)+<?<a^'(^-t0)] -1<£<1
which is real when a and fi are real. From (19)
(23) Qta^\{) = - Hi? cosectair) P<a^>(<f)
l2^-wa,)r,(;)ru+> + " (i-^a + ^
ru + a + zs + i)
xF(n + l,-re-a-/3; l-a;1/*-^) -1<£<1.
Jacobi functions of the second kind are also connected with the
polynomials
(24) <7<a^>(x)=j"_'((«-%)"'a-tra+tfiP^^M-p^^Xx^dt
associated with Jacobi polynomials according to 10,5(6), for clearly
(20) may be rewritten as
(25) Q{°-P\x) = -VAx-l)-a{x+irPq(a-PKx) + Q^'^MP^-PKx).

172
SPECIAL FUNCTIONS
10.8
Other relations connecting the P and Q are
(26) P ia-P\x) Q {af\x) - P {afKx) Q toPKx)
- 2^- (2» + a + /g)1^^"^^'^ fe - D-a (x + IT?
nirdi+o + jS+l)
(27) P^x) — P^x)-!?^*) — Pk'^GO
= _2a+P r(n + g+l)r(n + /9 + l) (x _ i)-<*-i (x + i)-/3-i
n!r(n + a + /3 + l)
and from these it follows that Q (a"^' satisfies the same differentiation
formula (15) as P(a^> .
n
From the theory of hypergeometric functions one obtains integral
representations {ov.Q^a*^\ The simplest of these is
(28) Q^'^Kx) = 2-n~* (x - l)'a(x + l)-£
x /Jf (* - ^)-^! (1 - *)n+a (1 + /)n+^*,
valid when x is in the complex plane cut along the segment (—1, 1)«
(v) Generating function.
(29) f P{a'P\x)zn = 2a+PR-x(l-z+Rra(l + z+RrP
n=0 n |*| <1
where
(30) R = (l-2xz + z2yA
and R = 1 when z = 0, For several ways of proving (29) see Szego (1939,
sec. 4»4)« For particular values of a, )8 there are other generating func-
tionst
(vi) Integral representations. From Rodrigues' formula (10), we have
where x ^ ± 1, the contour of integration is a simple closed contour, in
the positive sense, around t = x% The points t — i 1 are outside the
contour, and [(1 - t)/(l - %)]aand [(1 + t)/(l + x)]P are to be taken as
unity when t = x%
Further integral representations may be obtained, from integrals
representing hypergeometric functions, by means of (16)«

10.8
ORTHOGONAL POLYNOMIALS
173
(vii) Miscellaneous results. We may apply Christoffel's formula
10.3(12) to w{x) = (1 - x)a(l + x)0, p(x) = (1 - x). In virtue of (3) we
obtain
(32) (n + Jio+XiS+Dd-*)?^1^^)
= Oi + a + 1) P^^k) - (n + 1) P £f }(*)
and similarly
(33) 0i + Ka+fcj8 + 1)(1 + x) P <a^+1 >(*)
= Oi + 0 + 1) P {a^Kx) + (n+ 1) P (a.f >(*).
These- are examples of relations between contiguous hypergeometric
functions (see 2.8(31) to 2.8(45)): other relations of this nature are
(34) (X-x)P {a+u/3)(x) + (1 + x) P (a»*+,) (x) = 2P <*•*>(*)
(35) (2b + a + /3) P <a-,l/3)(*) - (n + a+ /3) /> <a^)(*)-(re+/3)P^a_f'(x)
(36) (2b + a+ /3)/><a^-,)U) = (b + a+ /3)/><a^)U)+U + a)/><a_f >(*)
(37) P <a^"' >(*) - P <a~''*Xx) = P <a_f>(*).
Repeated application of these formulas results in the expression of
p (a+h.fi+k ){x) for any integers h, k in terms of P ^^Hx).
From Rodrigues' formula (10) we have
(38) 2b f* (1 - y)a(l + y^P^^Ky) dy
= /»(«+«./s+»(o)-a-*)*+' (i+ *)£+' /><tv,/s+I)w.
Toscano (1949) found a counterpart of Rodrigues' formula in terms of
finite differences. We define the difference operator by
(39) AaP(a) = F(a + 1) - F(a), AnaF = Aa(A"a-' F)
and write Toscano's result in the form
(40) b ! T (a + £ + b + 1) P <a>/3)(*)
(-l>T(q + » + l) rr(a + £ + B + l) 1
-(K-K*^« 4 r(a+i) (/2"/2^ J*
Lastly we quote the important limit
,41) Hn,_ [.-|.W.(.-I)] - 1» [ .-I'M' (l~)]

174
SPECIAL. FUNCTIONS
10.8
where c/ais the Bessel function of the first kind. This formula holds for
arbitrary a and $ uniformly in any bounded region of the complex z-plane.
10.9. Gegenbauer polynomials
We use Gegenbauer's notation C (x) for the suitably standardized
polynomials associated with
(1) a = -l, 6-1, ii;(*) = (l-*2)X"X, X = l-x2.
These polynomials are also known as ultraspherical polynomials and are
often denoted by Pn (*). Clearly, Gegenbauer polynomials are constant
multiples of Jacobi polynomials with a = /3 = A — %. In order to have a
real and integrable weight function we assume
(2) A > - K,
although many of the formal relations are valid without this restriction.
For these polynomials see also sec. 3«15«
(i) Standardization.
(3)
x v / * + 2A-l\ (2A)
\ n / n\
By comparison with 10«8(3)
(4) (A + y2)nC^(x)=(2\)nP{«'aKx) a = A-M.
The standardization (3) fails when 2 A is zero or a negative integer. The
only exception in the range (2) is A = 0 and for this we standardize
according to
(5) C°(l)=l, C°(l) = - .. = 1,2
and have
A, - - <* ~ Wl
(6) C°(*) = lim A"' C\x)=2-—— ?<-*•-*>(*).
n
In many formulas of this section A = 0 must be excluded. This case will
be considered in sec* 10.10.
(ii) Constants.
(7) (n + A) n! T (A) h = nA (2A) T (A + K)
(8) n!An=2"(A)n, rn = 0, (2A)BXI1 = (-2)"(A+K)„
(9) (n + lM = 2(n + A), 6 = 0, (n + 1) C = re + 2A - 1

10.9
ORTHOGONAL POLYNOMIALS
175
(10) An = n (n + 2 A), an « 0, j8n = n + 2A-l.
(iii) Rodrigues' formula.
(11) 2n/i!(A+y2) Q-*2)^cH*)-(-l)n(2A) Dn[(l-*2)n+^]
n n n
(12) C^(*)=l, C*<*)«2A*.
Recurrence formula
(13) (n + 1) C\, (*) = 2(n + A) *C*(*) - (n + 2A - 1) Ck , (x).
Differential equation
(14) (l-*2)yw-(2A+l)*y'+ii(n + 2A)y = (>.
Differentiation formula
(15) (l-*2) Ck(x) = -nxC\x)+(n + 2X-l)Ck Ax)
dx n n n_l
= (n + 2 A) *C^(*) - (n + 1) C^+, (*).
Parity
(16) Ck(-x) = (-l)n C^x).
Explicit representations
(17) C\cOS0) = J —^5 2f2- COS U- 2 77l)0
n ^ m\(n-m)\
m = 0
L^J (-i)m (x)
/« = 0
f0 if /i is odd,
(19) <^(0) = <j
^(-l)m (A)m//w! if n = 2/ra is even.
(iv) Hyper geometric functions. The differential equation (14) can be
reduced to the hypergeometric equation, and C (x) is that solution which
is regular at x = 1 and has the value (3) there. Moreover, in the case of
Gegenbauer polynomials the hypergeometric series in question admit of
quadratic transformation, see sec. 2.1.5, and we obtain the following
representations:

176 SPECIAL FUNCTIONS 10.9
(20) n\Ck(x)=(2\)nF (-b,b + 2A; A + %; xA-%x)
= (-1)" (2A)n F(-n, n + 2\;K+1A;1A+ %x)
= 2" (A) (x - 1)" F (-n, -n - A +—; -2b-2A+1; )
" \ 2 1-x/
■(2A).G+r)v(""--x+fx+v^r)
(21) C^(x)=(-l)"-^-= F(-m, m+\;V2;x2)
(2 A)
'2m
(2m)!
(A)
F (-m, m + A; A + %; 1 - % 2)
<K)
i p ^-X.-X)(2x2_l)
(22) C* .,(*)= (-1)" -%tl 2xf( -m, m + A+1;-;*2)
2B+' m! \ 2 /
x F(-m, m + A + 1; A + %; 1-x2)
(2X)
(2ib + 1)!"
J}Ll±LxP^-X'X)(2x>-1).
From these representations in conjunction with (13) and (19) one
obtains
(23) Dn c\x) = 2* (A) CX_+mU) m = l,2, ...,n
(24) DCX Ax) = xDCX(x)-nCX(x)
n*~ i n n
(25) »C\, (x) = xDCX(x) + (n + 2\)C X(x)
(26) 20i + A) fCX(x) dx = C^+, (x) - CX_} (x)
. „ if b is even,
(27) DC^(0)= _
,/m! if b = 2m + 1 is odd.
U-l)"(A)n+l/

10.9
ORTHOGONAL POLYNOMIALS
177
A second solution of the differential equation (14) can be obtained
from the work of 10.8(iv)by means of the connection, (4), (6), (21), or
(22), between Gegenbauer and Jacobi polynomials. No generally accepted
notation or standardization seems to exist in this case,
(v) Generating functions. From 10.8(29),
(28) f (A + %)" cHx) z" = 2^~* JT' (1 - xz + R)X~k
L>n (2 A)
\z\<l, R = (l-2xz + z2)'A, R = 1 when z = 0;
but in this case there is a simpler generating function, viz.
(29) I C\x) z " = (1 - 2xz + z z)~k \z\<l
n= 0
which can be verified by putting x = cos 0, factorizing the right-hand
side as (1 - e z) (1 - e~ z) , expanding in the binomial series, and
using (17). A third generating function
(30) 1 c)(x)-— = r(\+%)ez':°s0(1AzsmeYA-XJk_y(zsmd)
n=0 (2A)
n
is connected with (29) by means of the Laplace transformation.
(vi) Integral representations. Each of the generating functions leads
to a contour integral representation of Gegenbauer polynomials. In
addition, we have the real integrals
(31) C^(x)=2 ,J^ + ra) fn [x + (xz-lYA cosq^Uintp)2^-'^
n I [IvA)J /
Jo
(32) C^(cos0 = 5 —-S (sintf 1_2M ^r dy.
it* nITU) JQ (cosy-cosOY-^ f'
both for A > 0. For (31) see 3.15(22) and Seidel and Sza'sz (1950).
Equation (32) is Mehler's integral 3.15(23); there is a second integral obtained
by replacing <f> and 6 by n — <f> and n — 6 respectively. Mehler's integral
suggests a functional transformation which will carry ultraspherical
polynomials into powers.
(vii) Miscellaneous results. From the connection with Legendre
functions,
(33) n! Ck(x) = T(A + H)(2A)n [%(x2 - 1)]*~'AkP*£ (x),

178 SPECIAL FUNCTIONS 10.9
we have the addition theorem
iini/r cos qp)
[(A).]2
(34) C^(cos0 cos«/r + sin 6 sim/r cosqp)
= 2 2"(2A + 2m-l)0i-m)! , _, .
(2A-l)n+.+)
x (sin 0)" C^_+; (cos d) (sin 0)" C £.+■ (cos 0) C^"H (cos <p).
Relations between contiguous hypergeometric functions are
(35) 2A(1 -*2) Ck" {x) = (2A+ n - 1) Ck Ax)-nxCk{x)
= (n + 2A)* C^ (*) - (n + 1) C^+t (*)
(36) (n + x.) c^;; (*) - a -1) [c*+l (*) - c^_, (*)i
The differentiation formula
(37) (*2-l)X+^Dnt(*2-irX] = (-l)n/..'^[*U2-l)-^]
n
follows from (11) and a linear transformation of the hypergeometric series
in (21) and (22)t It is due to Tricomi (1949)« We note also Gegenbauer's
integral
(38) nlfVeUamaCk (cos0)(sin0)2\*0
o n
= 2M ru + y2)(2k)n r z-kJk+n(z)
and the expansion in a trigonometric series
(A). r(n+m + 2K)
• o
(39) rU)CX(cos0)«2 I AiL "+,yi + 2A Cos[U+2m + 2A)fl-Afr]
« = o ml rOi+in + A+1)
0 < A < 1, 0<0<rr
(§zego, 1939, p. 95).
10,10. Legendre polynomials
Legendre polynomials P (x) are the suitably standardized polynomials
associated with
(1) a--lf 6 = 1, u>(%)« 1, X=l-*2.
These polynomials are also known as spherical polynomials • Clearly
they are Jacobi polynomials with a = )S = 0, and also Gegenbauer
polynomials with A = %• Legendre polynomials, and more generally Legendre
functions have been studied extensively (eft chapter !!!)•

10.10 ORTHOGONAL POLYNOMIALS 179
(i) Standardization.
(2) J\,U) = 1.
Hence
(3) Pn(x)-C*(x)-P<;0'0)(?c).
(ii) Constants.
(K)
(4) A.-U + Ji)"1, An = 2"gn = 2"_-iL, rn = 0
n I
(5) if = (-2)"n!, (n + l-M =2B + 1, B = 0, (n + l)C --»
(6) \n = n(n + 1\ an = 0, j8n = n.
(iii) Rodrigues9 formula.
(7) 2nrc!P (x) = Dn[(*2-l)n]
n
(8) P0(*)=l, P, (*) = *, P2U) = i*2--.
Recurrence formula
(9) (n + 1) Pn+1 U) = (2» + l)xPn(x) -»Pn-i (*).
Christoffel-Darboux formula
(10) 1 (2m+l)Pii(x)Pii(y)=-^[Pn+,(:c)Pn(y)-Pn(*)Pn+|(y)].
Differential equation
(11) (l-x2)yu -2xy'+n(n + l)y = 0.
Differentiation and integration formulas
(12) (1 - x2) Pn'U) = n [?„., (*) -xPn{x)] = {n + 1) [*PnGc) - Pn+I (*)]
(13) xP'n(x)-P'n_^x) = nPn(x)
(14) Pn'+1(*)-xPn'W=U+l)PnW
(15) (2n + 1) fPn(x) dx = Pn+1 (*) - Pn_, (*).
In these formulas P '(x) = dP (x)/dx.
n n

180
SPECIAL FUNCTIONS
10.10
Explicit representations, parity, special values
«*'.«-? «-»■(:) cry-
* = o N '
(17) P (cos<9) = £ g g cos (rc-2^)0
«= 0
(is) Pn(-x) = (-iypn(x), pn(± i) = (±i)n
(19) ^(ONt-lVg., P2.+1(0) = 0
(20) /,2'n(0) = 0, /,2'B+I(0) = (-D- (2m+l)gm.
Here
m! \ m /
(iv) Hypergeomettic functions. See also 10.9 (iv).
(22) ?„(*)- F(-n, n + 1; 1; V2 - K2*)
= 2"gn*nF(-^re, J£-J£n; &*~2)
(23) Pn(cose») = F(-ra, ra + 1; 1; sin2 V2 6) = (-1)" F(-ra, n + 1; 1; cos2^)
(24) P2. (*) = (-1)' ga F(-m, m + V2; %• x2)
(25) ^2.+,(*)=(-!)" (2m + \)gmxF (~m,m+—,—,x2J
(26) P(x) = 2»m\g C"+,/i{x) n > m.
dxm n 6- »-«
Information about a second solution of Legendre's differential equation
(11) may be obtained from 10.8(iv). Such a second solution is the Legendre
function of the second kind
(27) <?>) = <?<°'o)0O.
In the complex rc-plane cut along the segment (— 1, 1)
(28) 2_n(2n + l)!(n!r2(?n(a;)
= (*-l)~"_l F[n + l, n + 1; 2n +2; 2(1-*)"']
= (*+ l)-n-' F[n + l, n + 1; 2n + 2; 2(1 + *)"']
= *-"-1 F(l/2+n/2, 1 + n/2; 3/2 + n; a;-2).

10.10
ORTHOGONAL POLYNOMIALS
181
The Legendre function of the second kind is not a polynomial: it satisfies
the same recurrence relation (9) and the same differentiation formulas
(12) - (15) as the Legendre polynomials, except that n = 0 is inadmissible
in these formulas when written for Q*
(29) <?n(-x)=(-l)n + '<?„(x)
1 x + 1 1 x+1
(30) <?0(x)=_log -, £,(*)=--* log --1
2 x-1 2 x-1
(31) Qn(x) = 2""-' J^ (1 - 12)n (x - t)-»"' dt
(32) <?„(x) = J"0°°[x + (x2 - I)" cosht]-"-' rft
(33) <?n(cosh£) = C [2(coshz -co8h^)]'x e~{n+Wl dz
Re 2 > Re {, Im z = Im £
(34) <?„(x) = H /_', (x - 0"' Pn(t) dt.
<35) <?.(.)-<?.«.) ^.W-l'"'^^li-^ P„.ut,(«).
fc = 1
The last formula is equivalent to the special case a = )S = 0 of 10.8.25;
for the proof in the form (35) see Hobson (1931, pp. 53-54)« The point at
infinity is a zero of multiplicity n + 1 of Q (x); and this function has no
other zero in the cutx-plane.
The segment of the real axis from — 1 to 1 is a branchcut of Q (x), and
(36) Qn({ + i0)-Qn({-i0) = -7TiPn(O -1<£<1.
On the branchcut we may define a second solution of Legendre's equation
by
(37) Qn(<f)=^<?n(<f+i'0) + ^<?n((f-i0) -1<£<1.
We then have
(38) Q„(.f) = K2£',(£-;)-' Pn(t)dt -1<£<1
where the integral is a Cauchy principal value, that is
-i-e . ri
lim(J?_e + T ) as e>0, f-»0.

182
SPECIAL FUNCTIONS
10.10
(v) Generating functions.
(39) 1 P (x)zn=a-2xz + z2)-* -!<*<!, |z| < 1
n= 0
oo
(40) T -V Pn(coSd)zn = e*coa6J0(
^ n\
z sin
9)
n= 0
(41) V P (cos 60 a;2"*1 ^FtsinKfl, cp)
n~° x = tan%(£ 0<cp<%77, 0<0<n.
The first two formulas are particular cases of 10.9(29) and 10.9(30).
The last formula may be derived from (39), and F (A, Cp) denotes Legendre's
incomplete elliptic integral of the first kind with modulus k
(vi) Integral representations.
(42) Pn(cos0)=^' J7(c os 0 + £ sin 6 cos cp)n c^cp
= 7t"} J (cos 0+ *sin0 cosCp)"""1 dcp
(43) Pn(cos 6) = 2^ tT1 Jq (cos Cp- cos 0)~* cos (n + JO cpdcp 0 < 6 < n
(44) Pn(x) = (2™)"' J(° + ) (1 - 2xz + z2)-* z-»-} dz
(45) pnu)=(-2^n(27^£^, /^^l^^^-u-^r^1 d«.
Equation (44) follows from (39), and (45) from Rodrigues' formula. The
integral in (45) is known as Schlafli's integral. Laplace's first and
second integral, (42), may be deduced from (45) when the contour of
integration is taken to be the circle
and Mehler's integral (43) may be deduced from Laplace's integral
(Whittaker and Watson 1940, sections 15.23 and 15.231).
(vii) Miscellaneous results* With the notation
(46) P" (cos 60= (-2)* m\g (sin0)* Cm+*(cos60
for the associated Legendre function of the first hind [see 3.4(1) and
3.15(4)] we have from 10.9(34) the addition theorem of Legendre
polynomials

10.11 ORTHOGONAL POLYNOMIALS 183
(47) Pn (cos 6 cos 0 + sin 6 sin 0 cos cp) = Pn (cos 6) Pn (cos if/)
n
V (n - /ti)!
+ 2 > •: - P"(cos<9)P"(cos0)cos7icp.
,r|U + ™)! n n
We note the expansion in a trigonometric series
(48) P . (cos 0) = T " ' f' sin[(n+2m)fl] n = 2, 3, ...
and the integral formulas
23/2
(49) r(i-xr^BW(b-—-
r« A / , (-U'OS-JSA)
(52) J, .*!■. „<.»*- 1(w)tMjM BeA>-2
and the bilinear expansion
(53) Y "* lX P>)PJy)=2 1og2-l-log[(l-*)(l + y)]
_^. n(n + 1) n n
-1< * .< y < 1.
n= 1
10.11. Tchebichef polynomials
Sometimes (especially in the French literature) orthogonal polynomials
in general are called Tchebichef polynomials. There are also several
special systems of orthogonal polynomials called Tchebichef polynomials.
In this chapter we shall reserve the name Tchebichef polynomials of the
first and second kind for the suitably standardized orthogonal polynomials
associated with
(1) a = -l, 6 = 1, w(x) = (l-x2)TlA, X = l-x2.
Clearly these polynomials are multiples of Jacobi polynomials with
a = /3 = -lA for the polynomials of the first kind T n (x) and with a = /3 = lA
for the polynomials of the second kind Un(x)* Also, the Jacobi
polynomials in question are ultraspherical polynomials with k = 0 for the

184
SPECIAL FUNCTIONS
10.11
polynomials of the first kind and with X = 1 for the polynomials of the
second kind*
The orthogonal relationship for Tchebichef polynomials of the first
kind reads
J* T (x) T (x)(l-xz)~lA dx = 0 m£n.
If we substitute x = cos 6 and note that cosn 6 is a polynomial of exact
degree n in cos 6, we see that T (x) must be a constant multiple of
cosn 6; and we show in a similar manner that Un(x) is a constant multiple
of csc# sin (n + 1)6, We standardize our polynomials by putting
™ x v sin(rc + 1)0
(2) T (cos 6)= cos n$, U(cos6)= .
sin 6
Many identities involving Tchebichef polynomials are paraphrases of
well-known trigonometric identities. As an example, we mention the
connection between the two kinds of Tchebichef polynomials,
(3) Tnix)=Vj#)-xVn_^x)
(4) (1 -x2) Un_, (x)=x TnW-Tn„ {x).
Tchebichef polynomials are ultraspherical polynomials with X = 0, 1.
From 10.9(23) it is seen that C„(*) can be expressed as a derivative of
a Tchebichef polynomial whenever X is a positive integer.
(i) Standardization, This is given by (2). It follows that
(5) Tn{x)=Y2nCHx) = (gnr,P^'~)i)(x) n=l,2, ...
(6)' {/„(*)= CjtO^g^,)-' *>«•*>(*) n-0, 1, ...
where C° is defined by 10.9(6) and gn by 10.10(21).
(ii) Constants. For T {x)
(7) hQ = „, AB-Hir i»-l, 2, ...
(8) *„=2»-', rn = 0, Kn=(-l)»2%!gn
(9) An = 2, Bn = 0, C„=l
(10) X =n\ a =0, fl-i».

10.11 ORTHOGONAL POLYNOMIALS 185
Fori/ (x)
n
(U) hn = %n, A.-2", rn = 0, Xn = (-l)"2"+'n!gn+I
(12) 4=2, 6=0, C = 1
(13) kn = n(n + 2), an=0, j8n = « + l.
(iii) Rodrigues9 formulas.
(14) 2n(«) r (*) = (-!)" (l-*2)*Dn[(I-*2)""*]
n n
(15) 2n+,(,/2)n+I £/„(*) = (-l)nU+l)(l-*T*Dn[(l-*2)n+*].
Recurrence formula [2 (%) is either T (x) or U (x)]
n n n
(16) zn+l{x)=2xzn{x)-zn_t(x).
Christoffel-Darboux formula.
(17) 2' z U) * (y)-U-y)-Hz.+,(*)* (y)-*(*)* +I(y)]
• = 0
where 2 is either T or £/ , but in the case of 71 , the first term (m = 0)
of the sum must be halved•
Differential equations.
(18) (l-*)y" -xy'+n2y = 0 for y=Tn(*)
(19) (l-*)y" -3*y'+rc(rc + 2)y = 0 for y = tfn(*)
Differentiation formulas (primes denote differentiation with respect to x)
(20) (1 - x2) rU) = n [7"B_tU) - * Tn(x)]
(21) u-*2) £/'(*) = (n + i)£/ ,W-mj;w.
Explicit representations
/ , ,. , x n [f] (-D" (»-»-Dl , x .
(22) T (*)=— A TT JT^l (2*)n_2n n = 1, 2, ...
2 „t?0 m\{n-2m)\
fV] (-l)"(n-m)!
(23) ">>- Z ,, 9 „ (2*)n"2"-
/=0 ml (n -2m) !

186
SPECIAL FUNCTIONS
10.11
(iv) Hyper geometric functions*
(24) rW-F(-B,»;X;»-^)
(25)
( 3 1 x \
U (x) = (n + 1) F I -a, n + 1; - ; - - - J .
\ 2 2 2/
From these relations and from 10.9(iv)
(26) Dm T (x) = 2*~l (m-1)1 nC\ (x) n>m
n n n —
(27) D" 1/ (*)=2" m!C" + ,(:r) n>m
(28) ?"(*) =/ȣ/.(*).
(v) Generating functions.
(29) 1 + 2 J r.fe),"-^
(30) 1 + 2 2 n_l 7" (*) z" =-log(l - 2xz + z2)
n= 1
(31)* 2 0 (*)zn=(l-2*z+z2)-'
(32) I g 7" (*)*"-2'* Je_I(l-**+*)*
n = 0
(33) 1 g +l 0 U)*"^-**-1 (1-«+*)"*.
In all five formulas
-1 <* < 1, \z\ < 1.
In the last two formulas
R = (l-2« + z2)*.
Equation (31) is a special case of 10.9(29), and (30) is a limiting case
of the same relation: (29) can be derived from (30)t R = 1 and log R =0
when z - 0. Formulas (32) and (33) are special cases of 10,9(28)«
(vi) Integral representations* Contour integrals which represent
Tchebichef polynomials follow from any of the generating functions.

10.11 ORTHOGONAL POLYNOMIALS 187
(vii) Miscellaneous results.
(34) 2Tm(x)Tnb)-Tn+m(x)-Tn_a(x) n>m
(35) 2(*2 - 1) U _. (x) U_. (x) = T. (x) - T (x) n>m
(36) 2 7^ (*) £/n_, (*) = f/n+._, (*) + Vn_m _, U) n > m
(37) 27;(*)£/._,(*)=» £/„+._,(*)-£/„_,,_,(*) n>m
(38) 2[Tn(x)]2 =1+2 T^U), 2fnU) Un_t (x) = £/2n_, (*)
(39) 2(1-*2) [£/„_,(*)]* = 1-27^00
(40) £ 7'2ii(x)=1/2 + 1/2[/2n(x), "£' T2 .,(*)= lAU.(x)
m. = 0 « = 0
(4i) 2d-*2) £ phw-i-rh+lw
n = 0
(42) 2(l-a;2) "S' ^+IW=*-7,fB+I(x).
m = 0
All these formulas are paraphrases of trigonometric identities.
Mehler's integral 10.10(43) may be interpreted as a relation between
Legendre andTchebichef polynomials. Inverting this relationship, Tricomi
(1935) found
(43) in + K)(i + x)* £* (* - *r* Pn(t)dt = rn(*) + rn+t w
(44) U + JOd -*)* /x! (*-*)-* PBWi = rn(*)- rn+t (*).
From 10.9(21) and 10.9(22) we obtain
(45) P^\2x2-l) = gnU2n(x)
(46) xP^\2x*-l) = gnT2n„(x).
Finally we note the principal value integrals
(47) /_' (y - *)"' (1 - y2T% Tniy) dy = n Un_, (x)
(48) ^ (y - *)"' (1 - y2)* £/„., (y) <*y = -ir f, (*) n = 1, 2, ...
which are paraphrases of trigonometric integrals and are of importance
in the theory of the integral equation sometimes called the airfoil equation.

188
SPECIAL FUNCTIONS
10.12
10.12. Laguerre polynomials
The polynomials La (x) are the suitably standardized orthogonal
polynomials associated with
(1) a=0, 6 = oo9 w(x) = e-*xa, X = x a>-l.
Instead of L° (x) it is usual to write L (x)% This is the polynomial
introduced by Laguerre. The La(x) are often called generalized Laguerre
polynomials, but we shall call them Laguerre polynomials simply.
Equivalent polynomials have also been discussed by Sonine (1880, p. 41).
(i) Standardization, We shall adopt the standardization hn = {-Vp/nU
The standardizations k = (— l)n and, less frequently, k = 1 are
sometimes used*
(ii) Constants,
(2) n ! hn = r(a + n + 1), nlkn= (-l)n,
nr = - (ti + a), # = n !
n n
(3) (n + DA =-1, (ii + 1)B = 2n +a + lf (n + l)C =n + a
n n «
(4) An=n, an=n, /3B = —(n+a).
(iii) Relationships.
(5) n!L*(»)-e**"aDB(e "**"+*)
(6) L*Ge)-l, L*(*)=a+1-*
(?) l*u)» y ( )——
(8) 0» + 1) La+1 (x) - (2/i + a + 1 - *) La U) + (» + o) L* . (*) = 0
(10) *y" +(a+l-*)y'+»y-0, y~L*{x)
(11) (*z')'+ ( n+^1--—-— Jz=0, z = e-** *** £,*(*)

10.12
ORTHOGONAL POLYNOMIALS
189
d
(12) x— Lan{x) = nLan{x)-in + a)Lan_y{x)
= (n + 1) L* + 1 (x) - (n + a + 1 - *) L* (x)
„, x /ii + a\ (a + 1)
(13) L*(0)= f ^ U__i.
(iv) Hyper geometric functions. Laguerre polynomials are connected
with the confluent hypergeometric functions of Chapter VI. From the
explicit representation (7)
(14) Lan(x)= r1*" J <D(-rc,a+l;*)
(-1)*
= ——¥(-11-0, l-a;%).
From this we have
confirming statement (i) of sec. 10.6,
(16) 4- &*(*)-£»+,<*)]=£*<*).
ax
and many other formulas which are instances of relations between
contiguous confluent hypergeometric functions.
The general solution of Laguerre's differential equation (10) may be
obtained from the theory of confluent hypergeometric functions.
(v) Generating functions.
oo XZ
(17) 2 La(x) zn = (1 - z)-a-y exp |2| <1
n=0 z - 1
(18) f [T(ii + a+l)rILa(x)2" = U2rXae*«7[2(xz)X]
n= 0 a
(19) f La~nU)zn = e~X2(l + z)a |z|<l
n=0 n
oo ^
-<i-,>--„,(-..iii) <*,,-»•<, [2igL] „,<i.

190
SPECIAL FUNCTIONS
10.12
The function on the right-hand side of (17) is the most common
generating function and may be established by means of (7)* Equation (18)
is due to Doetsch and follows from (17) by means of the Laplace
transformation. Equation (19) follows from (7) and is due to Erdelyi. Equation
(20) is a bilinear generating function and is known as the Hille-Hardy
formula (see also Myller-Lebedeff, 1907)«
(vi) Integral representations. Contour integrals representing Laguerre
polynomials may be obtained from the Rodrigues formula (5) and from any
of the generating functions in an obvious manner* In addition, the
connection (14) with confluent hypergeometric functions may be exploited (cf.
sec. 6.11)% We mention only the following integrals
(21) nlL«(x)=e*x-*a f°°e'*tn+*aJa[2(tx)*] dt
*dz
(22) 277* 2* Lan (x) = (-1)" e** J° +)<T*" (—) U ~ **)**'
k = n + %a+ %.
The first of these is a consequence of 6.11(5); and the second is due to
Tricomi.
(vii) Miscellaneous results. The number of results under this heading
is tremendous, and many of them were discovered several times. We give
a small selection only and do not attempt to credit the formulas to their
original discoverers.
Contiguous polynomials. In addition to the recurrence formula (8) we
have
(23) x La+1 (*) - (n + a + 1) La(x) - (n + 1) La+1 (x)
(24) L^'(*)=!*(*)-I *_,(*)
(25) (b + a) La~' (*) « Gi + 1) La., (x) - Gi + 1 -*) La(x).
n n t 1 n
Differentiation formulas and indefinite integrals. In addition to (5)
and (12) we have
(26) Dn[x~a-y exp(-*-,)]=(-l)nn!*-a-n-1 L*(x-1) exp(-x"')
(27) D"[xaLa(x)] = (n-m + a+l)xa-' L*~* (*)
(28) n!D" [e~* xaLan(x)]= U + ») I e~z xa~* !"+"(*)

10.12 ORTHOGONAL POLYNOMIALS 191
(29) /"e ~y L«(y) dy = e~* [L«(*) - L«_, (*)]
(30) r (a + j8 + n + 1) f* (x - y)M yaLan(y) dy
= r(a + n + l)r{p)xa+PL«+P(x) Rea>-1, Re £ >0
(3D J0XLm(y)Ln(x-y)dy = JZoLm+n(y)dy=Lm+n{x)-Lm+n+t(x).
Further indefinite integrals follow from the product theorem of Laplace
trinsforms.
Laplace integrals. With the notation
Z[F(t)l = f°°e-stF(t)dt
0
we have
T(a + n + l)(s-l)n
(32) Z[taLa(t)]= -r-Xf Rea>-1, Res>0
(33) n\r(a+l)£[tPLan(t)]
= T (j8 + 1) T (a + n + 1) 5 -P-' F (-n, j8 + 1; a + 1; *"f)
Re£>-1, Res>0
(34) C[^a+B Ja(2(At)K)] = Ji!A*as~a~,,~f e-V* L*(k/s).
Limit formulas.
(35) La(x)= lim P(a*P\l-2x/p)
(36) lim [n-aLa(x/n)] = x-1AaJ (2x*A).
n->oo n a
Finite difference formula. With the notation
^af(a) = f(a + 1) - f(a), A£+' f(a) = Aa(A£ f(a)) n = 1, 2, ...
we have
A^(a)= 2 (-1)""" (n)f(a + m)
m = 0 \/7l /
/i = 1, 2, •••
id h
and hence
, a , x , v T (a + n + 1)
(37) £"(*)-(-!)" — A"
n\xa a r(a+l)

192
SPECIAL FUNCTIONS
10.12
Finite sums. In addition to those already recorded we have
(38) S I"U)-I"+I(*)=*-I[(*-i»H"(*) + (a + i»Ha .<*)]
(39) £*(*) = 2 (/nir1 (a-jS) I*. (%)
n «=o ■ n "
n / _i_ V
(4o)L^u*)=y r a) \n-«a-\)'La (X)
«=o \ m / " *
(41) £ L?(x)LP (y) = La+0"(x+y)
■ = o n
(42) /i!L*(*Ua(y)
= r(a + n + i) i [m!r(a + /7l + i)r1 M" £a+2" <* + y).
Infinite series: generating functions have already been given [(17) to
(20)], Bessel function expansions are in sec* 10.15, and other examples
of infinite series involving Laguerre polynomials are in sec* 10.20*
10.13 * Her mite polynomials
Hermite polynomials are orthogonal polynomials associated with the
interval (—oo, oo) and an exponential weight function. Unfortunately, the
notations adopted by different authors vary a great deal. The simplest
form of the exponential weight function seems to be exp(-#2), but for
applications in mathematical statistics exp (— %x2) is preferable* Ofthe
most important books on the subject, Courant-Hilbert, Doetsch, Sansone,
Szego use exp(-#2), and Appell and Kampe' de Feriet, Jahnke-Emde,
Magnus-Oberhettinger, Polya-Szego, and Tricomi use the weight function
exp(-%#2)* In this chapter we shall adopt Szego's (1939) notation and
regard Hermite polynomials, H (x)9 as the suitably standardized
orthogonal polynomials associated with
(1) a = -oo, b = oo, w(x) = exp(-#2), X = 1.
The orthogonal polynomials associated with the weight function
expi—YiX2) may be denoted by He (x). These polynomials may also be
expressed in terms of parabolic cylinder functions [see 8*2(9)]*
(i) Standardization. We shall adopt the standardization K n = (~l)n*
This agrees with the standardization adopted, among other authors, by
Courant-Hilbert, Feldheim, Hille, and Szego* The standardization Kn = 1
is used by Doetsch, Erdelyi, Sansone, and others*

10.13
ORTHOGONAL POLYNOMIALS
193
Hermite polynomials, so standardized, have been expressed by Szego
and Koschmieder in terms of Laguerre polynomials.
(2) ffgll (*) = (-D" 22" m!ZT*(x2)
(3) ff2ji+, (*) = (-D" 22"+' m!*L*(*2).
These expressions show that // (x) is an even or an odd function of x
according as n is even or odd. These formulas are analogous to (and are
in fact limiting cases of) 10.9(21) and 10.9(22).
(ii) Constants,
(4) h =nA 2nn\, k = 2n, r =0
(5) £ =(-lV, A =2, B = 0, C =2n
(6) A = 2w, a = 0, j8 = 2w.
(iii) Relationships.
(7) tf Or)=(-lVe*2Z>ne-*2
(8) ff0U)=l, H,(x)=2x
V (-D" (2x)"-2»
(9) ff"W = n! ^ .(n 2m), •
Here [n/2] is n/2 or (rc - l)/2 according as n is even or odd.
(10) H .,(x)-2xH (x) + 2nH , (x) = 0
nn Y HM».W H^(x)Hn(y)-Hn(X)Hn+,(y)
Jio 2" m\ 2n+1 n\(x-y)
(12) y" -2xy'+2ny = 0, y=ffa(x)
(13) z" + (2n + l-x*)z=0, z = exp(-y2x2)Hn(x)
(14) ff „(-*)=(-IV fl„(*X ff;U)-2iiff„_,(*)
(15) #2ji(0)=(-lV (2ot)!/ot!, ff2K+I(0) = 0.

194
SPECIAL FUNCTIONS
10.13
(iv) Hyper geometric functions. Hermite polynomials are connected
with parabolic cylinder functions which are special confluent hypergeo-
metric functions
(16) tfn(*)=2*nexp(/2*2)Dn(2**)=2n>H-/2n, lA; x2)
(17) ml H2m (*)= (-1)" (2m)! <D(-m, V2; x2)
(18) mlH2m+l(x)=(-l)m (2m + 1)! 2x 4>(-m, 3/2; x2)..
The general solution of Hermite's differential equation (12), or of the
self-adjoint form (13) (which is virtually Weber's equation), may be
obtained from the theory of parabolic cylinder functions.
(v) Generating functions.
(19) I H (x)zn/n!=exp(2xz-z2)
n= 0 "
(20) 2 (-1)* tf2ji(x)22"7(2/n)!=exp(22)cos(2**2)
R= 0
(21) 2 (-l)"^2ji+1U)22"+1/(2m+l)!=exp(22)sin(2^x2)
■ = 0
V $*)" , u (2xyz-(x2 + y2)z2>|
n^O "i [^ L - Z J
Equation (19) is the well-known generating function, (20) and (21) may be
derived from (19), and (22) is Mehler's formula*
(vi) Integral representations. Contour integrals follow in the usual
manner from (7) or from any of the generating functions. In addition, the
connection with parabolic cylinder functions may be exploited (see sec.
8.3)« We note explicitly
(23) e'%1 Hn{x)=2n+} iT* Jfe'^r cos(2x* - V2n n) dt.
(vii) Miscellaneous results. See remarks under 10.12(vii)*
Limits.
cos x
sinx.
Integrals.
(26) JZ e~yZ Hn(y) dy = //„_, (0) - e-** //n_, (*)
(27) J"*//n(y)<fy=[2(n+ !)]-' [#„+,(*)-//„+,«»]

10.13 ORTHOGONAL POLYNOMIALS 195
,~oo _ 2 ,, (2m)\
(28) / e~y H. (xy) dy = ^ —- (x2 - 1)«
-oo m!
rxe^\,I2a+,(xy)dy = „*{^^-x(x*-l)-
(29) jrooe~y2y"Hn(xy)dy=n/' n\ P Jx).
Here P (x) is the J.egendre polynomial.
Gauss transforms.
Qux[F(y)]=(2nu)-* J~x F(y) exv[-(x - y)2/(2u)]dy
is the Gauss transform (with parameter u). We have
(30) G^[//B(y)] = (l-2B)5J"//B[(l-2B)"5J*] 0<u<>/2
(3D Q*[//„(y)] = (2*)", Q|||y"]-(2*r"«„(&).
Connection with Laguerre polynomials. In addition to (2) and (3) we
have
!= 0 ^
(33) S°°e-y2[H iy)]1 cos (2'A xy)dy = nlA 2n~* n\ L (x2)
(32)
k
(34) r(n + a+l)Jf (l-^)a-M//2aU^)^ = (-l)a77^(2«)!r(a+^)L^(x)
Re a>-%.
The first two of these formulas are due to Feldheim, the last one to
Uspensky (1927).
Finite sums. In addition to those already given we have, among
others, the following relations.
n
(35) 2 (2- m!)-,[ffBU)]2 = (2n+,n!)-,tW„+1U)]2-ffnU)ffn+2U)j
(36) £ (-2)'*! (J (J *._*<«>*.-*«-*.+»
k= o
min(a,n) / ™\ / n\
(37) z 2 ^J^J*.*.-.*^-*.^".^
J (")ff.(2X«)ff...(2*y)-2»-fli|U+y)
* = 0
(38)

196
SPECIAL FUNCTIONS
10.13
(39) J0 (22mk)H»{2'Ax),/>->k(2'/y)
= 2"-,[//2B(* + y) + //u(«-y)]
m m
1 r
(a? + ... +a2)^n ra.xf+...+ax "1
= —! : // ! ! r r
n\ "L («? + -+«*)* J '
Equation (35) is due to Demir and Hsu. The last three formulas are
addition theorems and can easily be proved by means of the generating
function (19). In (40) the sum is extended over all non-negative integers
mx , ... , m r whose sum is n%
Infinite series. Generating functions have already been given [(19)
to (22)]. For expansions in spherical Bessel functions see sec. 10.15,
and for other infinite series involving Hermite polynomials see sec. 10.20.
10.14. Asymptotic behavior of Jacobi, Gegenbauer and Legendre
polynomials
The behavior of Jacobi polynomials as n -+ oo and at the same time
x -> 1 in a suitable manner, is given by 10.8(41). The corresponding
behavior as x-*-l follows from 10.8(13), and the behavior of Gegenbauer
and Legendre polynomials may be obtained by means of 10.9(4) and
10*10(3). The behavior of Jacobi polynomials as /3 -> oo and x -> 1 in a
suitable manner is given by 10.12(35).
For die investigation of the convergence of infinite series of Jacobi
polynomials, and for many other purposes, it is desirable to determine
the behavior of Jacobi polynomials for fixed a, jS, x> and for n -> oo. The
example of Tchebichef polynomials
T (cos 6) = cosrc #, x = cos 6
suggests that the asymptotic behavior will be different according as x
is on the interval (—1, 1) (0 real) or outside it (0 complex). Also the end-
points of the interval have to be considered separately. In the present
section we shall entirely omit the case in which x is outside (—1, 1) and
refer the reader in this respect to Szegff (1939, Chapter VIII). We shall
give certain results for — 1 <x< 1; and all estimates given in this section
hold uniformly in any interval -1 + € < x < 1 - e (e > 0). We shall also
give important results for the case that x is in the neighborhood of ± 1.

10.14
OKTHOCONAL POLYNOMIALS
197
Proofs of the results to be given are based either on explicit series
or integral representations (with integral representations the method of
steepest descent is frequently used), on generating functions (Darboux's
method), or else on the differential equations (Liouville's method and its
later developements).
Darboux has proved (from the generating function)
V £ O/2) co8[{n-m + }i)0-(l/2m + l4)rr]
'» >.<«■«-*«. Ioort^r» —5*5^
a — o n
+ 0{n~M~v') 0<d<n
where g is defined by 10,10 (21).
A similar formula was obtained by Stieltjes in a manner permitting to
estimate the remainder.
2*v "!« cos[(n+in + J4)0-O4i» + JOw] „//n
(2) P (cos* — > ^- —rr. + RA6)
0 <d<n
where
2 re!
«J
<3) |R«<9,|-<7i.77xu-^i„«'^
and
(4) 4 = 2sin# if sm2d>lA
4= Icosel"1 if sin2d<%
so that in any event 1 < A < 2.
If 2 sin # > 1, i.e., for 77/6 < 6 < 5n/6, we may make M -* 00 in both (1)
and (2) and obtain convergent trigonometric expansions of Legendre
polynomials.
For the neighborhood of x = 1 we have Ililb's formula
(5) P„(cos#) = (tfcscfl)* JQ[(n + lA)d] + 0(rT3/2)
valid uniformly forO <6<n- e (e > 0). For more precise bounds for the
error term see Szego (1939, p» 189). For an expansion of Legendre
polynomials in series of Bessel functions see Szego (1933). If 10.14(11) is
specialized for a = /3 = 0 and the confluent hypergeometric function there
is expanded in a series of Bessel functions by means of 6.12(6), the
result is

198 SPECIAL FUNCTIONS 10.14
(6) Pn(*) = [4/(* + 3)]n+' e-£/i2n+])[J0(2Z1A)+JL J2{2{*) + 0(n-3)]
where
2(* + 3)£=(l-*)(2rc + l)2.
Some of these results can be extended to Cegenbauer polynomials and
to a lesser extent also to Jacobi polynomials.
A) ml
n n\ n = o \n - m + ,
cos[{n-m + \)d- % (/ft + A) ;r]
(2sin0)^
+ «
+ 0{n-»'-X)
A^O, -1,-2, ... , O<0<tt
x x r(2A + n) v d-A)
(8) Ck{cob8 =2 -—— > =
cos[(n + m + X)6-1A(m + X.)n]
x -t-t +RA6) 0<A<1, Q<e<n
(2sin0r+» *
(9) \R»m± 2lfW (M + A)n+%! (2sin^*
and 4 is given by (4).
, ., n cos\[n+V2(a+^ + l)]d-(1Aa+1A)7T\
<10) F^(™n -^MH-VIMW *°b '
a, 0 real, 0 < 0 < tt.
A formula of Hilb's type for Jacobi polynomials has been given by
Szego and by Ran: see Szego (1939, p. 191). Tricomi (1950a) obtained the
expansion
nfafli, x -, A + *Y*Y T(/V +m)r(a + n+l)
" \ 2 / « = o n!r(/V)r(a + m + l)
x 4 _ (A, Ho + ^2)I>/(2A)]B <M-n - |8, a + m + 1; z)
where
(12) k = n + %a+l/2, N-n + a+p + 1, * = l-4z/(2£ + z) |z|<2|A|,

10.15
ORTHOGONAL POLYNOMIALS
199
0 is the confluent hypergeometric series and the A are the coefficients
defined in sec. 6.12. Using the expansion 6,12(6)> a Bessel function
expansion may be obtained for Jacobi polynomials. In the particular case
a = /3 = 0 it leads to' (6).
10.15. Asymptotic behavior of Laguerre and Hermite polynomials
The general remarks of the preceding section apply here too, but the
situation is more involved on account of the infinite interval. The
polynomials are oscillatory in part of the interval, and monotonic outside this
part.
The asymptotic behavior of Laguerre and Hermite polynomials as
n -> oo and at the same time x -> 0 in a suitable manner is given by 10.12
(36), 10.13(24), and 10.13(25).
Fur real a and fixed x > 0, or uniformly inO < t <x < co <°°, we have
FejeVs formula
(1) L*(x) = ,Txex* «-**-* ii*1-* cos[2(n*)* -W -%„] + 0{nAa~%)
which has been generalized by Perron (see Szego, 1939> P» 192). Sansone
(1950) gave a two-term approximation with an estimate of the error. This
formula fails for small x, but there is a formula of Hilb's type
(2) e-*' «** L\ fa) - ^(" tT+t1} JJlvx)*] + 0(n*a-*)
valid for a > -1 uniformly in 0 < x < co <<*>• The notation
(3) v=4n + 2a+2
is used in (2)> and will be regained throughout this section.
The behavior of Laguerre polynomials when rc-> oo and x is unrestricted
has been investigated by several authors (see sec. 6.13). We shall
confine ourselves to a brief survey here, based on a memoir by Tricomi
(1949)» Tricomi distinguishes four cases according as x is near 0, in
the oscillatory region, near vf or in the monotonic region.
The expansion
(4) n\e'** La(x) = r{a+n + l)(vx/4y1Aa f A*(x/v)*m J . ([i/*]*)
m = 0
which is a particular case of 6.12(11) and in which
(5) 4*=1, 4* = 0, A* = V2a+y2
(m + 2)A* . =U + a+ l)A* -VivA* , m = 1, 2, ... ,
is tz ra oi~i * '

200 SPECIAL FUNCTIONS 10.15
converges uniformly in any bounded region of the complex variable x.
By considering the order of magnitude of successive terms, one sees
that (4) has the character of an asymptotic expansion as/i->oo provided
that x = O(n^) with A < 1/3. This establishes the behavior of Lan(x)
"near" the origin.
A similar expansion,
(6) n\{ux)*ae-hxLa{x) = T{a+n+\) I A (h)(x/uYAmJ . (2[ii*]*)
Tl _ 01 CLTBI
at = 0
with appropriate coefficients is due to Toscano (1949), in case u = n to
Tricomi (1941)*
In the oscillatory region, 0 < x < i/, Tricomi puts
(7) x=vcosz0, 0<0<1/27T, 40 = v(2<9-sin2<9) + 77
and proves that for a fixed 6
(8) e^LaU)=2(-l)n(2cos^)-a(7n/sin2^r^
x [ *£' A (a)(0)(!^sin20r* sin(0 + 3imr/2) + OU~")]
. in
m =0
whe
(9) i*>(0) = l, ,4<a>(0)=-L _JL^-(l-3a2)sin20-l .
0 ' 12 L4sin20 J
For the general expression for the A (a'see Tricomi (1949).
Near the transition point i/,
(10) e-*' L"^)- rAji')*^-)'" \j A-M
^s^/ _row,\ "I |
10 \ 2r(2/3) /J J
where
(11) f = (4v/3)-'/3(v-x),
(12) ^,=(-l)"2-a [^v-"^:^ £l^v-'+0(n-^) ,
1 |_ 10 T(2/3) J
(13) ^ (t) = U/3)(t/3)* l^ [2(t/3)3/2] + J,/3[2(t/3)3/2]}
is the Airy function, and A'(t) is the derivative of A (t).

10.15
ORTHOGONAL POLYNOMIALS
201
Finally, in the monotonic region
(14) x = vcoshz6, 0>O, 4© = i/(sinh2<9~ 2 6)
(15) e'1Ax La(x)= (-l)n e-&(2cosh6)-a(7Tvsinh2dr*
x[M2 (-1)* A^ (<9)0^sinh2<9r* +00*-*)]
«=o
vh
wnere
(16) 4^(0)=*!, A[a]{6)=— [ 5 -(l-3a2)sinh2fl+ll
0 ! 12 L4sinh2(9 J
In the following summary of the corresponding results for Hermite
polynomials we use the abbreviations
(17) /V=2n + 1,
J Vm if n is even,
\V2r1 - Vi if rc is odd.
For a fixed real * (or uniformly in any bounded interval)
(18) r(y2n + l)exV{-V2X2)Hn{x)
= r(n + l)[cos(N1Ax-y2n7T) + 0(n-1A)].
Szego (1939, P» 194) gives a second term explicitly, and also the general
form of the asymptotic expansion*
For the behavior of Hermite polynomials for n -> <*> and unrestricted x
we have the Plancherel-Rotach formulas (Szego 1939, p* 195)* Tricomi's
work covers this case too if use is made of 10*13(2) and 10.13(3)* The
Bessel functions involved in (4) when a = ± Vi are so-called spherical
Bessel functions and can be expressed in closed form* They serve as
asymptotic expansions provided that for some A < 1/3 the quantityn~\
is bounded as n -> <*>•
The oscillatory region is 0 < \x\ < 2mr9 and here the expansion (8),
with a ~ ± /2> niay be used* In the neighborhood of the transition points
x = ± 2m" we have (10), and in the monotonic region \x\ > 2m/1 we have
(15)*
The basic expansions in series of spherical Bessel functions are
particular cases of the more general expansions given by Tricomi (1941):
(19) e~hx #2 (*)=(-1)" 22"+,(K) x* 2 (2m)'-rC' Q .(2m**)
r— 0
(20) e^ H^^ix) = (-!)« 22"+1 (3/2) m x S (2m)"r C." Qr(2m**)

202 •
SPECIAL FUNCTIONS
10.15
where
(21) Q0(*) = z~T sinz, G_f(z)=z"2cosz
(22) (]r + 1U) = (2r+l)(Jr(z)-z2Qr.1(z)
7* — Uj ly 2l) ••• y
and the coefficients C^ also satisfy certain recurrence relations. The
expansions (19) and (20) are convergent. They can also be used as
asymptotic representations asm-> oo, and for this purpose it is convenient
to take h = Vi%
10.16. Zeros of Jacobi and related polynomials
Let us define Jacobi polynomials for all values of a, j8, x by 10.8(12),
and let us denote by N}(a9fi)the number of zeros of P{a*P\x) in the
interval (-1, 1). If a > -1 and jS > -1, Jacobi polynomials are orthogonal
polynomials associated with the weight function 10.8 (1)» and by sec.
10,3 all their zeros are simple and located in (—1, 1). For other real
values of a and j8 the number of zeros in (-1, l)is indicated in the figure
p
a = -n a = -n + l a = -n + 2 a=-2 a=-l
71-1
n - 2
n — 3
i-
71-1
71-2
Ny (a, j8) for real a and jS .
■0—1
0=-2
/B--3
0 = -n+ 1
0=-n

10.16
ORTHOGONAL POLYNOMIALS
203
We see from 10.8 (12) that for negative integer a, P (a'^'(x) has a zero
of order \a\ at x = 1, and for negative integer 13 it has a zero of order
\P\ at x = -1. In the interval (-«>, -1) there are ^(1-0-/8-2/1, 0)
zeros, in the interval (1, oo) there are N (l-a-$-2rc, a) zeros. All
zeros not accounted for in this enumeration occur in conjugate complex
pairs •
Gegenbauer polynomials are defined by 10.9(18) f°r all values of
A, x% They are orthogonal polynomials, and all their zeros, are simple
and in the interval (-1, l), if A > —Vi* For other real values of A, the
number of their zeros can be deduced from the result on Jacobi
polynomials by means of 10.9(4).
The location of zeros of orthogonal Jacobi polynomials, and of their
particular cases, in (—1, 1) has been investigated by many authors. We
refer the reader to Szego (1939, Chapter VI), and for more recent work in
particular to papers by Gatteschi; Geronimus; Lowan, Davids, and
Levenson; and Tricomi listed at the end of this chapter.
We assume
(1) a>-l, 0>-l, A>-%, x = cos# 0<d<n,
and arrange the zeros in a monotonic sequence,
(2) p^Hcosem) = o, o < 0f .< e2 <... < en < 77.
(3) Pf^)UB) = 0, -Kxn<xn_i <-. <x, <l, x^costf^.
For ultraspherical polynomials
(4) x +x =0
and hence it is sufficient to investigate the positive zeros (1 <m <}Ati).
For Jacobi polynomials
xm =xni(a, ft n)
and for Gegenbauer polynmials
x* = '*.(A, n^- x (A - %, A- Vi, n\
If m and n (and in the case of Jacobi polynomials also one of the
parameters a, /S) are fixed, we have the following monotonic properties:
(5) x (a, jS, n) I - 1 as a -> <», t 1 as j3 -* <»
m = 1, ... , n
(6) *R(A, n) ± 0 as A -> <» m = 15 ... 5 [%rc].

204
SPECIAL FUNCTIONS
10.16
The last of these relations, for instance, means that the m-th (positive)
zero of a Gegenbauer polynomial is a strictly decreasing function of X
(for X > -%) and tends to 0 as X -* «, From (5) and (6) there follow
corresponding statements for 6m . Since we know 6m (± %, ± %, n) from 10,11 (5)
and 10.11(6), we have the following inequalities
(7) (2m - 1) 77 < (2n + 1) 0m (a, jS, n) < 2m;7
-1/: < a, £<%, l<m<n
(8) (m-V2)7T/n<dm(\9 n)<mn/(n + 1) 0 < X < 1, 1 < m < %n9
For further results see Szego (1939, Chapter VI). Tricomi (1947)
pointed out that the asymptotic behavior of the zeros of any function can
be deduced from the asymptotic behavior of the function itself and has
applied this principle to many functions, among them orthogonal
polynomials (see Tricomi 1950, Gatteschi 1949, 1949a). It transpires that
the asymptotic distribution of the zeros towards the middle of the interval
depends on the zeros of trigonometric functions [see 1.14(5)] and the
zeros near the end-points depend on the zeros of Bessel functions [see
the remark following 10.14(12)].
Asymptotic formulas for the Christoffel numbers may be derived from
asymptotic formulas for the zeros by means of 10.7(7).
For numerical values of the zeros and Christoffel numbers of Legendre
polynomials see Lowan, Davids, and Levenson (1942, 1943).
10.17. Zeros of Laguerre and Hennite polynomials
The polynomial defined by 10.12(7) for all values of a and x9 has n
positive zeros if a > -1, [n + a] positive zeros if —n < a < -1, no positive
zeros if a < —n; it has a zero of order k at x = 0 when a =■ -A, k = 1,2,..., n;
and it has one negative zero if (a + l)n < 0* All the zeros not accounted
for in this enumeration occur in conjugate complex pairs. The Hermite
polynomial of degree n has n real zeros which are situated symmetrically
around the origin.
For detailed information on the location of the zeros of orthogonal
Laguerre polynomials (i.e. for a > -1) and of Hennite polynomials we
refer to Szego (1939, Chapter VI), and to papers by Greenwood and Miller
(1948), W. Hahn (1934), Salzer and Zucker (1949), Spencer (1937), and
Tricomi.
We assume
(1) a>-l, x>0

10.18
ORTHOGONAL POLYNOMIALS
205
and arrange the zeros of La(x) in increasing order so that
(2) LHx ) = 0, 0 <*. <*<... <xn, x = x (a,n).
For fixed m, n we find again that x^ is an increasing function of a.
For bounds for the zeros see Szego (1939, Chapter VI), and W. Hahn
(1934)% The asymptotic representations of Laguerre and Hermite
polynomials can be used to find approximations for the zeros (Tricomi 1949)%
It is clear from sec* 10% 15 that we have to distinguish three cases. The
"first" zeros are those for which m remains bounded while n -* «>: these
are investigated by means of 10.15 (2). The "middle" zeros are those
for which \m — Vm\ remains bounded while n -* «>: these are deduced from
10.15(8). The "last" zeros, for which n — m remains bounded as n -> °°f
are deduced from 10.15(10)* The resulting approximations give
satisfactory numerical results even for moderate values of rc,for instance n= 10«
Asymptotic formulas for Christoffel numbers may be derived from
10.7(7).
For numerical values of the zeros and Christoffel numbers of Laguerre
polynomials, L (x)9 see Salzer and Zucker (1949).
10*18. Inequalities for the classical polynomials
For inequalities for general orthogonal polynomials and for their
application to the classical polynomials, see Szego (1939, Chapter VII).
In the notation of sec. 10.3, there is the following result for monotonic
weight functions (Szego 1939, Theorem 7.2). If w (x) is non-de creasing
[non-increasing] and b [a] is finite, then [m>U)]^ |pn(*)| attains its
maximum in (a, b) at b [a].
Application of this to those of the classical orthogonal polynomials
whose weight function is monotonic, at once leads to the inequalities
(1) |P U)| < 1 -1<*<1
(2) [(l-*)/2]^+* \Pf-°\x)\<l -1<*<1, a>-V2
(3)" e-*x\L (*)| <1 *>0.
Another fruitful source of inequalities is the Sonine-Polya theorem
(Szego 1939, Theorem 7.31.1 and footnote). If in the differential equation
(4) I*(*)r']'+£(*)y-o
k(x) and </>(x)are positive and continuously differentiable, and if k (x)<f>(x)
is monotonic, then the successive (relative) maxima of |y| form an
increasing or decreasing sequence according as k(x) <f>(x) is decreasing or
increasing.

206
SPECIAL FUNCTIONS
10.18
The following results can be proved by constructing the differential
equation satisfied by the functions involved, and applying the Sonine-
Polya theorem.
The successive maxima of |P (x)\, n > 2, as x increases from 0 to 1
form an increasing sequence. [This confirms (1) J The successive maxima
of (sin 0) \P (cos 6)|, n > 2, as 6 increases from 0 to % n form an
increasing sequence. As an application, it can be proved that
(5) (sin 6)% \P n (cos 6)\ < (Y2 7m)_t 0 <6<tt.
Furthermore,
(6) \P'n(x)\< lAn(n+l) -1<*<1.
For Gegenbauer polynomials
(2A)
(7) max |C^(*)| = C^0) = 2- A >0
(A).
(8) -,m<a,x<, lc^)|-|c^o)|-
- \ mi \
— m < A < 0, A not integer
(9) max |C^+1U)|<2[(2ni+l)(2X + 2ni + l)rx \(k)m^\/ml
-7n-%<A<0, A not integer.
(10) (sin0)^|C^(cos0)| <(l/zn)k-1 [r(\)T} 0 < A < 1, 0< 6 < n.
For Jacobi polynomials we put
(li) <7 = max(a, 0)
and obtain
(12) m<ax< \P^>$\x)\=maxP^K±l)= r + q~J
a>-l, j8>-l, q>-l/2.
If — 1 < a, jS < — %, the largest maximum of \P ^a,^Kx)\ is one of the two
nearest to xQ = (/S - a)/(a + /3 + 1), and this maximum is of the order of
n" as n -> °°. Among the numerous estimates for large n we mention only
j m
(13) P{a^Kx)=0(nq\ q = max(2/n + a, 2m + j8, m - l/2)
dxm n
n -> oo.

10.18
ORTHOGONAL POLYNOMIALS
207
For the special Laguerre polynomial L° we already have (3). Bounds
for Lan may be obtained from this by using 10.12(39) with jS = 0. The
result is
(14) |L*(*)|<(a+l)> !)-'<> a>0
(15) \Lan(x)\ <[2-{a+l)n(niy* ]e** -Ka<0.
The following results can be proved by applying the Sonine-Polya
theorem to the differential equation satisfied by the functions involved.
For any real a, the successive maxima of
e-'Axx'Aa+'A |La(a;)|
form an increasing sequence provided that 2n + a+l>l and
x > max{0, (a2 - l)/(2n + a + 1)}.
The successive maxima of
1 n ■
form an increasing sequence provided that x > 0 and
*2>max(0, a2 -%).
The successive maxima of
e'1Ax \La{x)\
form a decreasing sequence when a > — 1,
0 < x < (2 a + 1)(2n + a + l)/(a + 1)
and an increasing sequence when a> — 1,
x>{2a + l){2n + a+ l)/(a+ 1).
The successive maxima of
e~*x xAaLa(x)
form a decreasing sequence if
0<%<2rc + a+l,
and an increasing sequence when
x>2n + a+l>0.
AH these statements are contained in the following more general result.
For real a and jS, the successive maxima for x > 0 of
e-*xx0\LaM\
1 n '

208 SPECIAL FUNCTIONS 10.18
form an increasing or decreasing sequence according as
4/SC/3 - a)(a - 2)3) + (2n + a + 1)(2 a - 4)8 + 1)% - (a-2/8 + 1) x2
is negative or positive*
For an asymptotic estimate see Szego (1939, Theorem 7.6.4);
improvements of this estimate maybe derived from Tricomi's expansion 10.15(4)-
Bounds for Hermite polynomials may be derived from (14) and (15) by
means of 10.13(2) and 10.13(4)* See also Sansone (1950a).
(16) exp(-y2x2)\H2Bi(x)\< 2*« m\(2-gm)
(17) %-f exp(-%x2) \H2^(x)\ < 22*+2 («+ D!gj|+|
where
(18) gn=(K)I/n! = (lrn)-5< +0(n"3/2).
H. Cramer has proved
(19) exp(-y2X2)\Hn(x)\<k2*n(nl)X
where A; is a constant for which Charlier (1931) gave the approximation
1.086435. Sansone (1950) gave bounds valid for complex values of the
variable.
From the Sonine-Polya theorem it may be proved that the successive
maxima of \Hn(x)\, and likewise those of exp (— %x2) \H (x)\, for x > 0
form an increasing sequence.
Let /zr n be the r-th (relative) maximum of fix) \pn(x)\9 where f(x) is
a fixed non-negative function and }p (x)\ is a sequence of orthogonal *
polynomials. The results derived from the Sonine-Polya theorem state
monotonic properties of /zr as r increases while n is fixed. The study
of numerical tables led John Todd to some conjectures about monotonic
properties of \i for fixed r and increasing n.The following results were
subsequently proved. For
/(*)= i, Pn(*) = />>),
and counting maxima from x = 1 (to the left), Cooper (1950) proved that
\l is a decreasing function of n for sufficiently large n, and Szego
(1950) proved that this true for all n > r + 1. For
/U)=l, pn(x)=C^(x),
Sza*sz (1950) proved that n ! \l /T (n + 2 A) is a decreasing function of n.
For
/(*) = e-**, pn(x)=Ln(x).

10.19
ORTHOGONAL POLYNOMIALS
209
J, Todd (1950) proved that fi is an increasing or decreasing function
of n as t is odd or even.
P. Turan observed that
u =P (x) -1<*<1
n n — —
satisfies the inequality
(20) ^-un_,un+!>0.
Szego (1948) gave several proofs of this inequality and showed that it is
also satisfied by
u = cHx)/cHl) = n! Ck(x)/(2X) -1< x < 1
n n n n n —• —
«B=L;W/L:(0) = B!rU)/(a+l)n x>_0
These results have been reproved, refined, and generalized; determinants
whose elements are orthogonal polynomials have been considered, and
other related investigations have been carried out by Madhava Rao and
Thiruvenkatachar (1949), Sansone (1949), Szasz (1950a, 1951), Becken-
bach, Seidel, and Szasz (1951), Forsythe (1951). See also J. L. Burchnall
(1951, 1952).
10.19. Expansion problems
The expansion of a given, "arbitrary" or analytic function in a series
or orthogonal polynomials has been discussed often and in great detail.
The subject is somewhat outside the' scope of the present survey, and a
brief indication of some of the more important results must suffice. For
further information see SzegS (1939, especially Chapter IX), Kaczmarz
and Steinhaus (1935)«
Let \pn(x)\ be a system of orthogonal polynomials belonging to the
weight function w (x) on the interval (a, i). We assume that the
assumptions of sections 10.1 and 10.2 are satisfied, and denote by L? , p > 1
the class of functions f(x) for which the (Lebesgue) integral
J*\f(x)\Pw(x)dx
exists and is finite. We put
(1) hn = j\Pn(x)Yw{x)dx

210
SPECIAL FUNCTIONS
10.19
and call
the Fourier coefficients,
(3) 2onPn(*)
the (generalized) Fourier series of f(x) with respect to the system \p (x)\
of orthogonal polynomials. We shall say that the series (3) converges in
LI to fM it
(4) jb\f(x)-s(x)\Pw(x)dx + 0 as n -> oo,
a
where s (#) is the n-th partial sum of (3)»
Approximation in L* has already been discussed' in sec. 10.2, and
from the results described there it follows that in case of a finite interval
(a, b) for any function f(x) of Lf9 (3) converges in L* to f(x).
Convergence in Lp has been investigated by Pollard (1946, 1947, 1948, 1949)
and Wing (1950). For Jacobi polynomials, given by 10.8 (1), Pollard
proved convergence in Lp when
(5) a>-%, &>-%
and
(6)
/ o+l j3 + l\ / o+l /S + l \
4 max I , I < p < 4 min I , J .
\2a + 3 2/3+3/ ^ \2a + l 2/8+1/
For Gegenbauer polynomials we have 10*9(1)and convergence in Lp when
2A+1 2A + 1
(7) A>0, --—— <P< —.
A+ 1 X
Lastly, for Legendre polynomials w (x) = 1 and we have convergence in
L p when
4
(8) - < p < 4.
3
It has been pointed out in sec. 10,2 that infinite intervals present
additional difficulties. Nevertheless, convergence in L^ for Laguerre
polynomials with a > — 1, and for Hermite polynomials, has been proved
when p = 2.
The series (3) is said to converge to f(x) for a fixed x, or in an
interval, if for that x, or for all x in that interval

10.19
ORTHOGONAL POLYNOMIALS
211
s (x)-> f(x) as n -> <»,
where s n(x) is again the rc-th partial sum of (3). This type of convergence
(sometimes called "point-wise convergence") requires much more
restrictive assumptions on f(x) than convergence in Lp •
Rau (1950) has investigated the convergence of the expansion of a
function f(x) in a series of Jacobi polynomials with a > -1, /S > -1.
Assuming that f(x) is continuous and has a piece-wise continuous
derivative, he proved that the expansion converges to f(x) uniformly in every
interval — 1 + e < x < 1 — e, e > 0.
The Abel summability of series of Laguerre polynomials was
investigated by Caton and Hille (1945) by means of Laplace integrals.
Asymptotic formulas such as 10.14(1), (7), (10), and 10.15(1), (18)
suggest a connection between the convergence of orthogonal expansions
and that of certain related Fourier series. This is the source of the so-
called equiconvergence theorems. As a sample, we shall give an equi-
convergence theorem for Legendre polynomials (Haar, 1918)»
Let |/U)|2 be integrable in (-1, 1); let s (x) be the rc-th partial sum
of the expansion of fix) in Legendre polynomials, and let o (6) be the
the rc-th partial sum of the Fourier cosine expansion of /(cos 6). Then
s (cos0)-er (6)-+0 as n -> <», 0 < 6 < n.
Such equi converge nee theorems, in combination with conditions for
the convergence of Fourier series,enable one to discuss the convergence
of orthogonal expansions. Equiconvergence theorems for Jacobi, Laguerre,
and Hermite polynomials were given by Szego (1939, Chapter IX). Szego
also gives some results regarding the behavior of such series at the end-
points of the basic interval.
The expansion of analytic functions presents rather different problems^
A series of Jacobi polynomials converges in an ellipse whose foci are
at ± 1, and every function which is analytic in such an ellipse may be
expanded in a series of Jacobi polynomials (a, f3 > -1) there. A function
which is analytic outside such an ellipse, and vanishes at infinity, may
be expanded in a series of Jacobi functions of the second kind, QvL*P\
there (a, /S > -1, see Szego sec. 9*2).
In the case of Laguerre polynomials the region of convergence is a
parabola around the positive real axis, with its focus at the origin: in
the case of Hermite polynomials the region of convergence is a strip
whose central line is the real axis. In both cases the region of
convergence is unbounded and an analytic function which is to be expanded in
a series of Laguerre or Hermite polynomials must satisfy certain growth
conditions in addition to being analytic in an appropriate region.
Expansions in series of Laguerre polynomials were investigated by Pollard

212
SPECIAL FUNCTIONS
10.19
(1947a), series of Hemiite polynomials by Giuliotto (1939), and Hifle
(1939, 1939 a, 1940).
10.20. Examples of expansions
In this section we list some series of orthogonal polynomials whose
sum canbe given in closed form. Except in the case of Legendre,Hermite,
and Laguerre polynomials, not many such series are known, and some of
the following examples have been developed by Tricomi to fill this gap.
The computation of the coefficients of such an expansion is based on
10.19(2), where one may often take advantage of Rodrigues' formula
(or its generalizations) to simplify the integral by integration by parts
in the manner explained in the second paragraph of sec. 10.7»
In the following formulas we shall freely use the notations for
confluent hypergeometric and related functions which have been introduced
in Chapters VI, VIII, IX.
SERIES OF JACOBI POLYNOMIALS
Notations as in sec. 10.8. We always assume a, /8 > -1, and use hn
as defined in 10.8(4)«
(1) sgn*=c0+ 2 pW*p+i\0)PW\x) -1<*<1.
Here
(2) sgnx =
and
•■{:.
when x > 0,
when x < 0,
c0° ri(a+f+i2) ■ 2-«-*-/'[(i-*)«(i+*y>-(i+*)au-*)?i«**.
0 r(o+i)r(/8+i) °
Note that only the terms corresponding to odd values of n actually occur
in the summation in (1).
(3) (l-*y-2T(o+p+D
x f r(2n + a + ff + i)r(n + a + ff + i)(-P)n p(a
n4> r(B + a+l)rG» + a + /3+p + 2)
-p < min(a+ 1, V2a + %), -1<*<1

10.20
ORTHOGONAL POLYNOMIALS
213
-1<X<1
where
A = Vi (a - 0), m = n + % (a + 0 + 1).
For a generating function see 10.8(29); for a bilinear generating function
see Watson (1934), Erdelyi (1937 a) and Bailey; for another expansion in
products of Jacobi polynomials see Bateman (1904, 1905)»
SERIES OF GEGENBAUER POLYNOMIALS
Notation as in sec, 10,9, The constant hn is defined by 10,9(7)«
oo
A>-%, -Kx<l.
(6) (i-x)^ = 22^ff-^r(A)r(A + P + ^)
(n + A) (-p)„ ^
n^0 r(n + 2A + p + D n
-1<*<1, -p.<Ji(A+D if A>0, -p<y2 + \ if -^2<A<0
(7) e"> = r(A)(1/2yr^ 2 in(n + A)Jn+A(y)C^U)
-!.<*<!, A>0
(8) (y sin 0 sin 605* ""^^(y sin0 sin 0) e * c-* COB0
L, (2A) r(2n+A)
n— o n
x Jn+^(y) C^-(cos 0) C^(cos 0) 0 < 0, 0 < ,7, A > 0.
(9) ft)"A-CA(ft)) = 2A-ra) 2 (n+A)2"^Z-^
n= 0
where
\ze±i<^\<\Z\, co2 = z2 + Z2-2zZ cos 0,

214 SPECIAL FUNCTIONS 10.20
and
C^(o>) = c t J^ico) + c 2 J^co)
is any cylinder function in the sense of Sonine and Watson (Watson, 1922,
sec. 3.9). In the case c 2 - 0 the restrictions on z, Z may be omitted.
Some expansions in series of Gegenbauer polynomials have been
noted in sec. 10.9: for a bilinear generating function see Watson (1933 b).
SERIES OF LEGENDRE POLYNOMIALS
Notations as in sec. 10.10. The constant g is defined in 10.10(4).
All expansions valid for -1 < x < 1, or 0 < 0 < n9 respectively, unless
stated differently.
>-l
(11) |*|'sgn*= J (-l)-(2m + %)i^_Ma_ P (*) p>-l
■^ (1+/2p)-+' crce^-U!
V 2n+l (~p)
a«(i-.')--^[»t-||B.4.-);1.^),:p,M]
(14) ^e^^'[cos2(^<?i)-cos2(1/2^rK= X e^PJcosd)
n = 0
0 < <f) < 0 < 7T
(15) log[l + csc(1/2l9)]= S (/i + l)"1 P (cosfl).
n= 0 n
Series involving Bessel functions may be derived from (7), (8), and (9)
by putting A = %. Generating functions are listed in 10.10(v) and (viii).
SERIES OF LAGUERRE POLYNOMIALS
Notations as in sec. 10.12. We always assume a> -1, x > 0.
°° (- )
(16) x^=r(a + p+D J , P n -LHx) -p<l + min(a, Via-1/*)
£n r(a+n+l) "

10.20 ORTHOGONAL POLYNOMIALS 215
oo
(17) 0(o+l)-log*-r(a+l) 2 ""* ' La(x)
n= , r(a + n + 1) "
(18) -eI+*Ei[-max(*, y)] = 2 (n + l)-1 L (x) L (y) *, y >0
n= 0 " "
(19) ex*"ar(o,*)= 2 (n + \)~x L«{x)
n— 0
(20) ez+>Urrar[a, max (%, y)] y[a, min(%, y)]
= 2 (n "A (g) *:wjw
(21) (*y)"ae*+,ir[o,max(*,y)]-r(o,x)r(o,y)/r(o)l
00 !
= I ;—rrr, -La{x)Laiy) *,y>o
^ Oi + l)r(n + a+l) " J
n= 0
(22) «■»»<*•»>-1+ 2 tLnW-Ln_,U)][Ln(y)-Ln_I(y)]
71= 1
(23) {pey)~,s*e-*<>e+y)e-*triy[at e ^min(x, y)]
OO
= 1 7 T7T, ^L>>L>) Rea>0
„~n (n + a)r(n + a+l) "
n= 0
(24)
r(a+l, x) ., V 0»-D!
r(a+l) ,ti r(o+B + l) n~'
0< %, y.
In (24), H(z) = 0, H, 1 according as z < 0, = 0, > 0.
(25) x^^'y-k(a+^eVa^t2Uy)«]= f ^ "' L>)Lf "(y)
(26) T(a)V(a, o + l;%)= f U + a)"1 La(%) a .< X
n=0 n
(27) (l-y)-*(.,C;^)-jo^-.y-L:-U)
x > 0, |y| < 1, c > 0.

216
SPECIAL FUNCTIONS
10.20
Other series of Laguerre polynomials in 10,12 (v) an^ (vii). The
expansion 10,14(11) is an expansion in Laguerre polynomials when /S
is an integer > —n.
SERIES OF HERMITE POLYNOMIALS
Notations as in sec, 10,13.
(WW-—S-2 «-W-S^»..M '>"'
«= 0
(2m)!
mw^JJ^Ll^.%z^H_M p>.x
7T* «=o (2/n+l)!
*, y>0
r(-v) ffl^0 (m-../)22"+1 ml
(33) (l + y)-« (o.fc^LLj- Y Jj-S- y- # (») |y| < 1
\ 1 + y / ^o (2m)!
/ 3 x2y\ v (a)
Other series of Hermite polynomials are in 10,13 (v).
The following key indicates the derivation of these examples; it also
gives references to further material on infinite series of classical
polynomials.
Series of Jacobi polynomials. The coefficients were computed by
integrations by parts. For other examples see Brafman (1951).
(5) From (1) by 10,9(4).
(6) From (3) by 10,9(4).
(7) From (4) by 10,9(4); Watson (1922, p. 368)«
(8) Watson (1922, p. 370).
(9) Watson (1922, p. 365).

10.21
ORTHOGONAL POLYNOMIALS
217
Series of Legendre polynomials. Many examples may be obtained from
series of Jacobi polynomials or series of Gegenbauer polynomials by
means of 10„10(3). Numerous other examples are found in books on
Legendre functions. For some examples, see Tricomi (1936, 1939-40).
(16)
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
(25)
(26)
(27)
Tricomi (1948, p. 332).
Toscano (1949).
Neumann (1912).
9.4(5).
9.4(4).
Watson, (1938).
Tricomi (1935), Doetsch (1935).
Erde'lyi (1936).
Tricomi (1948).
Toscano (1949).
6.12(3).
6.12(5).
For some examples of series of Laguerre polynomials
1938).
(28),
(30)
(31),
(33),
(29)
(32)
(34)
From (16) by 10.13 (2) and (3).
From (3) by 10.13 (2) and (3).
Tricomi (1950a).
From (27) by 10.13(2) and (3).
see Erde'lyi (1937,
10J21. Some classes of orthogonal polynomials
Beside the classical orthogonal polynomials there are other classes
of special orthogonal polynomials which have been investigated in detail.
In this section we shall describe some of these, mentioning very briefly
those discussed in Szego's book, and giving fuller details about those
not otherwise conveniently accessible.
POLYNOMIALS OF S. BERNSTEIN AND G. SZEGO
These polynomials belong to the interval (-1, 1) and their weight
function w (x) is of one of the forms
a - *2r* [P(*)]-\ a - *2)* [p(*)r\
[(l-zVd + ^tpU)]-1
where p(x) is a polynomial of exact degree Z, and positive for — 1 < x < 1.
ChristoffePs formula 10,3(12) suggests a connection between these
polynomials on the one side, and certain Jacobi polynomials on the other
side.
The polynomials were encountered by Szego (1921) and investigated
by Bernstein (1930, 1932). See Szego (1939) sec. 2,6,

218
SPECIAL FUNCTIONS
10.21
(2) 2 0(*)(x - y)-j-i + [(* - y) 0 *(*)- 20(*)]-^-
POLYNOMIALS OF E. HEINE AND N. ACHYESER
Heine's polynomials belong to the interval (0, a) and to the weight
function
(1) w(x)=[x(a-x)(b-x)T* 0<a<b.
They are related to Jacobian elliptic functions.
Heine (1878-1881, vol. 1, p. 294-296) showed that the polynomial of
degree n satisfies a differential equation of the form
^+[U-y)0'U)-20(*)]^L
ax' dx
+ [a+/8*-n(2n- l)x2]y = 0
where
ijj (x) - x (a -x)(b - x)
and a, )8, y are certain constants. This differential equation has four
singularities of the regular type and hence is an instance of Heun's
equation.
Achyeser (1934) investigated the orthogonal polynomials associated
with the interval (— 1, 1) and the weight function
f\c-x\[(l-x2)(a-x)(b-x)r* -\<x<a or b<x<\
w(x) = 4
I 0 a <x < t.
Here — 1 < o < 6 < 1, and c depends on a and 6. These polynomials are
also related to elliptic functions.
POLYNOMIALS OF F. POLLACZEK
Recently, F. Pollaczek defined certain families of orthogonal
polynomials which are generalizations of classical orthogonal polynomials.
The weight functions associated with Pollaczek's polynomials fail to
satisfy certain conditions which it is customary to impose in the general
theory (roughly speaking, they vanish too strongly at the end-points of
the interval), and thus these polynomials are important, and readily
accessible, examples of certain "irregular" phenomena in the general
theory of orthogonal polynomials.
Finite interval* Let a, b9 A be real parameters,a > |6|, A > -1. We put
(3) -l<* = COS(9<l O<0<77,
and use the abbreviation
(4) t = (a cos 6 + b)/sin6=(ax + b)ti-x2VlA.

10.21 ORTHOGONAL POLYNOMIALS 219
The polynomials P (x; a, b) are defined recurrently.
(5) /^,=0, Pk=l
(6) nPk-2[(n-l + \+a) x + b] Pk_, + (n+2\-2)Pk = 0
Jl = 1, ^, ... .
These polynomials were defined by Pollaczek (1949a, for A = %,
1949 c, for Re A > 0) and studied by Szego (1950a). Some'related
polynomials were also studied by Pollaczek (1949b, 1950a).
Multiplying (6) by zn and adding, one obtains a simple differential
equation of the first order for the generating function, and hence
(7) 1 P^(x;a,b)zn^(l-zeie)-K+it(l-ze-ierK-it \z\<l.
n= 0
Comparison with 10*9(29)and 10* 10(39)shows the relation to Gegenbauer
and Legendre polynomials
(8) PK(x; 0, 0) = cHx), />*(*; 0, 0) = Pn(*).
The polynomials are orthogonal on the interval (4), the weight function
being
(9) w^\x;a, i) = ^"' 22^' e {20~7r)t (sin 0)2^' |r (A + it)\ 2 .
The asymptotic behavior of P (x; a, b) when x is fixed, between -1
and 1, and n -> <*> was investiaged by Szego.
Either from the generating function (7), or from the recurrence relation
(6) it may be proved that
(10) n\ P^(x; a, b) = (2A)r e in6 2F} (-n, \+it; 2 A; 1 - e'1 ie).
This expression in terms of hypergeometric polynomials is the source
of many further formulas for Pollaczek's polynomials. It should be noted
that t depends on x so that P does not satisfy any differential equation.
Formulas connecting P for different values of A follow from (10) as
instances of relations between contiguous hypergeometric series.
In a later paper (1950 c), Pollaczek introduced a more general system
of polynomials which depends on the real parameters a, i, c, A where
(11). either a>\b\9 2A + c>0, c>0
or a> \b\9 2A + c > 1. c >-l.
With the notations (3) and (4), Pn(x; a9 i, c) satisfies
(12) ^=0, P^=l

220
SPECIAL FUNCTIONS
10.21
(13) (n + c)P^-2[(n-l + \ + a+c)x + b]P^+(n + 2\ + c-2)P^2 = 0
71 = 1 , Z, ••• •
The generating function of these polynomials has been obtained by
Pollaczek, who also proved that these polynomials are orthogonal on the
interval (4), the weight function being
,x, x (2sin^)2A"1e(2^)t
(14) w <M(*; a, b, c) =
277 i(2A+c) r(c + 1)
x |r(A + c+i*)|2 l/Jd-A + fc, c;c + A + **; e2i&)\'2 .
The recurrence relation (13) is a difference equation for P as a
function of n. This equation serves to express P^ix; ay b> c) in terms of
hypergeometric functions. The expression is fairly complicated and the
hypergeometric series appearing in it are no longer polynomials. Putting
_ r(2A + c+n) » _ntX+it.2k;1_e-^X
" r(c+n+l)r(2A) 2I
r(2-2A)
x tFI(l-2A-c-n, 1-A+if; 2-2A; l-e~zie)
the resulting expression is
X , A.. B-AB.
In this form, it is valid when 2A is not an integer. An alternative form,
valid for integer values of 2A is available. A_y = 0 when c = 0, and in
this case (15) reduces to (10).
Infinite interval. For the infinite interval -«» < x < «>, Pollaczek
(1950b) has the system of polynomials P„(x; cp) where
(16) A>0, 0 < cp< 77
are parameters,
(17) P^, = 0, ?£-=l,
(18) nP^-2[(n-l+X) cos cp+ x sin q>] P^_, + (n-2 + 2A) P^_t = 0
n — 1, z, ••• .

10.22
ORTHOGONAL POLYNOMIALS
221
Clearly, these polynomials may be obtained from those defined by (6)
by replacing 6 by (p and t by x. The generating function is
(19) S Pk(x; ^z^d-zeWr^^a-ze-W)-^1* Izl < 1,
and the weight function is
(20) w^(x; q>)=n-' (2 sin cp)2^' e"^-2r-P)x |r(A+,-*)|*.
These polynomials may also be expressed in terms of hypergeometric
series in the form
(21) n\P^(x; cp)=(2A)neinCP/l(^n, k+ix; 2\; 1 - e~2iV ).
These polynomials were mentioned by Meixner (1934) and by W. Hahn
(1949)* They have a representation in terms of finite differences, an
analogue of Rodrigues' formula (Toscano, 1949). Setting
8F(x) = F(x + %i)-F(x-%i)
SkF{x)=S[Sk-1 F(x)] k = 2, 3
we have
/99\ p^(r* rri) —
n l
where
ra-A+w)
U V.A + y2 n, x)
GU,x)
■e2?1.
10.22. Orthogonal polynomials of a discrete variable
In the remaining sections of this chapter we shall briefly list a few
systems of orthogonal polynomials for which the distribution function a(x) of
sec. 10*1 is a jump function, and the appropriate definition of the scalar
product is 10„1(3)« The points at which the jumps of a(x) occur are
x i9 and we shall use the jump function j(x\ the jump of a(x) at x = x .
being /(*.)• Thus, the appropriate definition of the scalar product is
(1) (cpff <P2) = S /(*;) (?,(*.) cp2{xt\
and the jump function corresponds in some measure to the weight function
of the earlier sections. We always assume that the jump function is
positive and that 2 j(x .) is finite. Many results of the introductory
sections of this chapter hold for scalar products of the form 10.1(2) and
hence remain valid for the definition (1) of a scalar product.

222
SPECIAL FUNCTIONS
10.22
The x^will be taken as integers, a <x . <_6. The intervals and jump
functions listed in the table below are those of most frequent occurrence.
The orthogonal polynomials associated with them correspond to the
classical orthogonal polynomials of a discrete variable, and most of them
have been studied in some detail.
POLYNOMIALS OF A DISCRETE VARIABLE
" b f(x) NAME
0 N - 1 1 Tchebichef
-a
0 A' PX ? ( ) Krawtchouk
ru + i)
(/3)x (y),
x\ (S)
Charlier
Meixner
W.Hahn
All these polynomials have a number of properties in common, among
which we mention only the finite difference analogue of Rodrigues' formula
(2) pn(x)=[Knj{x)r' &n[j'(x-n)X(x)X(x-l)>» *(*-* +D]
where K is a constant^X {x) is a polynomial in x whose coefficients are
independent of n9 and A is the operator of forward differences,
(3) Af(*) = f(* + l)-f(x)9 An+1 fix) = A [A" fix)] n= 1,2,
Conversely, this property characterizes the above orthogonal polynomials
in the sense that any system of orthogonal polynomials possessing a
Rodrigues' formula can be reduced to one of the systems listed above
(Hahn 1949, Weber and ErdeTyi 1952)» The proof is analogous to that
given in sec. 10.6 and will be omitted.
The proof of the orthogonal property of these polynomials may be based
on (2) and "summation by parts". Alternatively, the method of generating
functions may be used.

10.23
ORTHOGONAL POLYNOMIALS
223
10.23. Tchebichef's polynomials of a discrete variable and their
generalizations
Tchebichef's polynomials t (x) arise in the graduation (fitting) of
data by least squares. For an account of their properties see Szego(1939,
sec. 2*8), Jordan (1921 and 1947, Chapter VIII), and the references given
in these places.
Definition and orthogonal property.
«".-'-a-[C) (*:*)]
n = 0, 1, .... N - 1
(2) *S' t (x)t Gc)=(2n + 1)-'/V(/V2-l2)(/V2-22)... (/V2-n2)S
. n n mn
%— 0
m, n = o, i,..., yv - i.
Symmetry and "central values".
(3) tn(N-\-x) = {-\)ntn{x)
(4) «ta^-x)-(-i)-te»)i(2;) (*N~**m)
Difference equation
(5) (* + 2)(*-/V + 2)A2t (x)+[2x-/V + 3-n(n + l)]At (*)
n
Recurrence formula
(6) U + 1U +1U)-(2n+l)(2x-/V+l)* (x) + rc(/V2-n2n , U) = 0
ft = I, Z, ••• •
Connection with Legendre polynomials.
(7) lim /V~n* GV*) = P (2*~1).
A generalization of Tchebichef's polynomials may be obtained by the
definition
, 1 x\(S) f (j8) (y) "1

224
SPECIAL FUNCTIONS
10.23
In particular,
(9) P.hU-l-.-U-^^OCr)]'
and in this form it is immediately seen that
(10) pn(x;l,l-N,l-N)=tn(x).
Certain polynomials introduced by Bateman (1933) are also particular
cases of (8). The polynomials (8) were introduced by Hahn (1949)» They
belong to the jump function
(11) ibiP'Y'S)-^]'^' .
x\ (8)x
The explicit formula
OS) (y)
(12) pjx; 0, y, 8) = H " / " Fz(-n, -x,p + y-8 + n; j8, y; 1)
n I
and a recurrence relation were given by Weber and Erdelyi (1952).
There is a connection with Jacobi polynomials,
(13) lim y-»p (y*;a+l,y, y -/S) = (" + "\ p k.0>(2* + 1).
y->oc \ & /
10.24. Krawtchouk's and related polynomials
The orthogonal polynomials associated with the binomial distribution
in probability theory were introduced by Krawtchouk (1929). They were
studied by Aitken and Gonin (1935), and an account of their properties
is found in Szego's book (1939, sec. 2«8„2).
We assume
(1) p>0, q > 0, p + q - 1, /V a positive integer.
Definition, jump function, orthogonal property.
n = 0, 1, ... , N.
f , x (-1 )nxl(N-x)l Tp ? 1
(2) k U) = r AnU-L
nlp*qN-* [_(*-*)!(#-*)! J
(3) /(*)«(*)?*?""*
(4) 1 j(x)hn(x)hn(x)=( )pnqnSmi m, n = 0, 1, ...,N.

10.24
ORTHOGONAL POLYNOMIALS
225
Explicit representation, generating function*
(5) kn(x) = qn(X) F{-n,x-N;x-n;-p/q)
(6) 1 kn{x)zn = a+qz)x(l-pz)N-x.
n= 0
The explicit representation shows the connection with the Jacobi
polynomials; the (limiting) relations with Hermite polynomials and with
the Charlier polynomials are given in S2ego(1939, p. 35, 36).
The special case p = q = % has been studied by Gram (1882) and
Greenleaf (1932).
The polynomials
(7) mn(x;P,c) = ^-c-*-'l&n
4
*-n)lJ
were investigated by Meixner (1934), Gottlieb (1938, /3 = 1), and other
authors. (See references in Hahn 1949, p. 32), They are generalizations
of Krawtchouk's polynomials.
(8) pnmn(x;~N, -p/q) = n\kn(x).
Explicit representation, jump function, orthogonal property.
(9) iBn(*;/3,c)=(/S+*)nF(-ra,-*; 1 - /S - n - x; c"')
= (/3)nF(-n,-*;/3;l-c"')
(10) Hx)=c*(P)i/x}
(11) S /(*)mn(*;/3f c) »!,(*; j8,c) = n!(/3) c"n(l-c)"^S2
j8 > 0, 0 < c < 1.
Symmetry, generating function
(12) (p)xmH<?iip,c)-(p)nmtb;P,c)
(13)
X in (*;/3,c)—= 1 I (\-z)-*-P \z\ <min(l, \c\
n= 0 nl V Cl

226
SPECIAL FUNCTIONS
10.24
The explicit representation (9) leads to the following connections with
Jacobi, Laguerre and Charlier polynomials.
(14) mn(x;/3, c) = n!Pf-'-^-"-V-i-l)
(15) limmn(-^—;(3,c)=nlLP-'(x)
c-* 1 \ C — i /
(16) Hrn^ (- —) rnn(x; P> ^jr) \ = n\ Lx^n{a) = {-a)n c n{x; a).
A recurrence relation and a difference equation have been given by
Meixner (1934).
10.25 • Charlier's polynomials
The polynomials introduced by Charlier are the orthogonal polynomials
associated with Poisson's distribution of rare events in probability theory.
They have been investigated by several authors among whom we mention
Meixner (1934, 1938) and Doetsch (1933). For an account of their
properties see Szego (1939> sec. 2.8.1) and Jordan (1947, sec. 148)»
Jump function, definition, orthogonal property.
(1) j(x) = e-aax/xl a>0, x = 0, 1, 2, ...
x\ f ax'n 1
(2) cnfelo)=_A.|_—j
(3) 2 j(x)cm{x;a)cn{x;a) = a-nn\8m.
x- 0
Explicit representations, .generating function.
,4) «.«->-i#«-»r(;)C)^
x [(-a)-"
(5) c (x;a) = - — $(-n, x-n+ 1; a)
(x -n)l
(6) T cAx;a)^- = e-t (l-—) fre e**HcU! \z\<a.
„=o «! \ aJ
A bilinear generating function was given by Meixner (1938)«

10.25 ORTHOGONAL POLYNOMIALS 227
Symmetry, recurrence relation,difference equation.
(7) cn{x; a) = cx(n; a)
(8) <*c n + } (x; a) + (x-n-a) c n(x;a) + nc n_} (x; a) = 0
(9) acn(x + l; a)+{n-x-a) c n(x; a) + xc n(x-l; a) = 0.
From the explicit representation (5) follows the connection with
Laguerre polynomials
(10) cn{x;a) = (-a)-nn\L*n-n(a).
The connection with Meixner's polynomials has already been given in
10,24(16).

228
SPECIAL FUNCTIONS
REFERENCES
Achyeser, N., 1934: Comm. Inst. Sci. Math. Mec. Univ. Kharkoff (Zapiski Inst.
Mat.Mech.) (4)9, 3-8.
Aitken, A.C. and H. T. Gonin, 1935: Proc. Roy. Soc. Edinburgh 55, 114- 125.
Bateman, Harry, 1904: Messenger of Math. 33, 182-188.
Bateman, Harry, 1905: Proc. London Math. Soc. (2) 3, 111-123.
Bateman, Harry, 1933: Tohoku Math. J. 37, 24-38.
Beckenbach, E.F., Wladimir Seidel and Otto Szasz, 1951: Duke J. 18, 1-10.
Bernstein, S., 1930: Comm. Soc. Math. Kharkoff (4) 4, 79-93.
Bernstein, S., 1932: Comm. Soc. Math. Kharkoff (4) 5, 59-60.
Bochner, Salomon, 1929: Math. Z. 29, 730-736.
Brafman, Fred, 1951: Proc. Amer. Math. Soc. 2, 942-949.
Burchnall, J.L., 1951: Proc. London Math. Soc. (3) 1, 232-240.
Burchnall, J.L., 1952: Quart. J. Math. Oxford (2) 3, 151- 157.
Caton, W.B. and Einar Hille, 1945: Duke Math. J. 12, 217-242.
Charlier, C.V.L., 1931: Application de la theorie des probabilites aVastronomie,
Gauthier - Villars •
Cooper, R., 1950: Proc. Cambridge Philos. Soc. 46, 549-554.
Doetsch, Gustav, 1933: Math. Ann. 109, 257-266.
Doetsch, Gustav, 1935: Atti Accad. Naz. Lincei Rend. CI. Sci. Fis. Mat. Nat. (6)
22, 300-324.
Erdelyi, Arthur, 1936: Atti Accad. Naz. Lincei Rend. CI. Sci. Fis. Mat. Nat. (6)
24, 347-350.
Erdelyi, Arthur, 1937: Math. Z. 42, 641-670.
Erdelyi, Arthur, 1937a: /. London Math. Soc. 12, 56-57.
Erdelyi, Arthur, 1938: Akad. Wiss. Wien. S.-B. Ha 147, 513- 520.
Forsythe, G. E., 1951: Duke J. 18, 361-371.
Gatteschi, Luigi, 1949: Boll. Un. Mat. Ital. (3) 4, 240-250.
Gatteschi, Luigi, 1949a: Rend. Mat. e applicazioni Roma (5) 8, 399-411.
Geronimus, J., 1944: Doklady Akad. Nauk. S.S.S.R. (N.S.) 44, 355-359.
Giuliotto, Virgilio, 1939: 1st. Lombardo, Rend. 72, 37-57.
Gottlieb, M.J., 1938: Amer. J. Math. 60, 453-458.
Gram, J .P., 1882: /• Math. 114, 41-73
Greenleaf, H.E.H., 1932: Ann. Math. Statistics 3, 204-255.
Greenwood, R.E. and J.J. Miller, 1948: Bull. Amer. Math. Soc. 54, 765-769.

ORTHOGONAL POLYNOMIALS
229
REFERENCES
Haar, Alfred, 1918: Math. Ann. 78, 121-136.
Hahn, Wolfgang, 1934: Jber. Deutsch. Math. Verein 44, 215 -236.
Hahn, Wolfgang, 1935: Math. Z. 39, 634-638.
Hahn, Wolfgang, 1949: Math. Nachr. 2, 4-34.
Heine, Emil, 1878- 1881: Handbuch der KugeIfunktionen, second edition, Riemer,
Berlin.
Hille, Einar, 1939: C. R. Acad. Sci. Paris 209, 714-716.
Hille, Einar, 1939 a: Duke Math. J. 5, 875-936.
Hille, Einar, 1940: Trans. Amer. Math. Soc. 47, 80-94.
Hobson, E.W., 1931: The theory of spherical and ellipsoidal harmonics, Cambridge.
Jordan, Charles, 1921: Proc. London Math. Soc. 20, 297-325.
Jordan Charles, 1947: Calculus of finite differences, Chelsea Publishing Co.
Kaczmarz, Stefan and Hugo Steinhaus, 1935: Theorie der Orthogonalreihen, Warsaw.
Krawtchouk, M., 1929: C. R. Acad. Sci. Paris 189, 620-622.
Krall, H.L., 1936: Bull. Amer. Math. Soc. 42, 423-428.
Lowan, A.N., Norman Davids and Arthur Levenson, 1942: Bull. Amer. Math. Soc. 48,
739-743.
Lowan, A.N.. Norman Davids and Arthur Levenson, 1943: Bull. Amer. Math. Soc. 49»
939.
Madhava Rao, B.S. and V.R. Thiruvenkatachar, 1949: Proc. Indian Acad. Sci. Sect.
A 29, 391-393.
Magnus, Wilhelm and Fritz Oberhettinger,l948: Formeln und Satze fur die speziellen
Funktionen der Mathematischen Physik, Springer, Berlin.
Meixner, Joseph, 1934: /. London Math. Soc. 9, 6-13*
Meixner, Joseph, 1938: Math. Z. 44, 531-535.
Myller-Lebedeff, Wera, 1907: Math. Ann. 64, 388-416.
Neumann, Richard, 1912: Die Entwicklung willkilrlicher Funktionen noch den Hermite
schen und Laguerreschen Orthogonal funktionen auf Grund der Theorie der Integral^
gleichungen, Breslau.
Pollaczek, Felix, 1949a: C. R. Acad. Sci. Paris 228, 1363-1365.
Pollaczek, Felix, 1949b: C. R. Acad. Sci. Paris 228, 1553-1556.
Pollaczek, Felix, 1949c: C. R. Acad. Sci. Paris, 228, 1998-2000.
Pollaczek, Felix, 1950a: C. R. Acad. Sci. Paris 230, 36-37.
Pollaczek, Felix, 1950b: C. R. Acad. Sci. Paris 230, 1563-1565.

230
SPECIAL FUNCTIONS
REFERENCES
Pollaczek, Felix, 1950c: C. R. Acad. Sci. Paris 230, 2254-2256.
Pollard, Harry, 1946: Proc. Nat. Acad. Sci. U.S.A. 32, 8-10.
Pollard, Harry, 1947: Trans. Amer. Math. Soc. 62, 387-403.
Pollard, Harry, 1947 a: Ann. of Math. (2) 48, 358-365.
Pollard, Harry, 1948: Trans. Amer. Math. Soc. 63, 355-367.
Pollard, Harry, 1949: Duke Math. J. 16, 189-191.
Rau, Heinz, 1950: Arch. Math. 2, 251-257.
Salzer, H.E. and Ruth Zucker, 1949: Bull. Amer. Math. Soc. 55, 1004-1012.
Sansone, Giovanni, 1949: Boll. Un. Mat. Ital. (3) 4, 221-223 and 339-341.
Sansone, Giovanni, 1950: Math. Z* 53, 97-105.
Sansone, Giovanni, 1950a: Math. Z. 52, 593-598.
Seidel, Wladimir and Otto Szasz, 1951: /. London Math. Soc. 26, 36-41.
Shohat, J .A., Einar Hille and J.L. Walsh, 1940: A bibliography on orthogonal
polynomials, Washington.
Shohat, J.A. and J.D. Tamarkin, 1943: The problem of moments, Mathematical
Surveys 1 , New York.
Sonine, N., 1880: Math. Ann. 16, 1-80.
Spencer, V.E., 1937: Duke Math. J. 3, 667-675.
Szasz, Otto, 1950: Boll. Un. Mat. Ital. (3) 5, 125-127.
Szasz, Otto, 1950a: Proc. Amer. Math. 5oc, 1, 256-267.
Szafez, Otto, 1951: /. d'Analyse Math. 1, 116-134.
Szego, Gabor, 1921: Math. Z. 12, 61-94.
Szego*, Gabor 1933: Proc. London Math. Soc. (2) 36, 427-450.
Szego, Gabor, 1939: Orthogonal polynomials, New York.
Szego, Gal)or, 1948: Bull. Amer. Math. Soc. 54, 401-405.
Szego, Gal)or, 1950: Boll. Un. Mat. Ital. (3) 5, 120-121.
Szego, Gabor, 1950a: Proc. Amer. Math. Soc. 1, 731-737.
Todd, John, 1950: Boll. Un. Mat. Ital. (3) 5, 122- 125.
Toscano, Letterio, 1949: Boll. Un. Mat. Ital. (3) 4, 398-409.
Tricomi, Francesco, 1935: Boll. Un. Mat. Ital. 14, 213-218; 277,-282.
Tricomi, Francesco, 1935a: Atti Accad. Naz. Lincei Rend. CI. Sci. Fis. Mat. Nat.
(6) 21, 332-335.

ORTHOGONAL POLYNOMIALS
231
REFERENCES
Tricomi, Francesco, 1936: BolL Un. Mat. ItaL 15, 102-105.
THcomi, Francesco, 1939-40: Atti Accad. Sci. Torino CL Sci. Fis. Mat. Nat. 75,
369-390.
Tricomi, Francesco, 1941: Giorn. 1st. ItaL Attuari 12, 14-33.
Tricomi, Francesco, 1947: Ann. Mat. Pura Appl. (4) 26, 283-300.
Tricomi, Francesco, 1948: Serie Ortogonali di Funzioni, Torino.
Tricomi, Francesco, 1948a: Equazioni differenziali, Torino.
Tricomi, Francesco, 1949: Ann. Mat. Pura Appl. (4) 28, 263-289.
Tricomi, Francesco, 1950: Ann. Mat. Pura Appl. (4) 31, 93-97.
Uspensky, J.V., 1927: Ann. of Math. (2) 28, 593-619.
Vitali, Giuseppe and Giovanni Sansone, 1946: Moderna TeoriadelleFunzioni di
Variabile Reale, parte seconda, Bologna.
Watson, G.N., 1922: A treatise on the theory of Bessel functions, Cambridge.
Watson, G.N., 1933: /. London Math. Soc. 8, 189-192.
Watson, G. N., 1933a: /♦ London Math. Soc. 8, 194-199.
Watson, G. N., 1933b: /• London Math. Soc. 8, 289-292.
Watson, G. N., 1934: /• London Math. Soc. 9, 22-28.
Watson, G.N., 1938: Akad. Wiss. Wien. S.-B. Ha 147, 151- 159.
Weber, Maria and Arthur Erdelyi, 1952: Amer. Math. Monthly, 59, 163-168.
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Wing, G. Milton, 1950: Amer. I. Math. 72, 792-808.

CHAPTER XI
SPHERICAL AND HYPERSPHERICAL HARMONIC POLYNOMIALS(1)
11.1. Preliminaries
11.1.1. Vectors
We shall define a point in (p + 2)-dimensional Euclidean space
(p = 1, 2, 3, • • • ) by a vector
(L) £ = (*,, xz, ... , xp+2),
and shall write u (j) for a function u of xy, *2, ... , * . The length
of £ will be denoted by ||y| | or r. Explicitly, we have
||?:||=r-(*J+*|+...+**„)*.
In sec. 11.7 we have both vectors with three and vectors with four
components. We then shall write ||£||3, ||^||4 to indicate the number of
components of £, fc), respectively.
A point on the unit-hyp ensphere 0, i.e., on the hypersurface r = 1, in
(p + 2)-dimensional space can be defined by a unit vector
(2) f-i-' ?-(£,, {2 fp+2),
we shall reserve the letters f, 77, £ for unit vectors of p + 2 components.
•H § - (>V 3^2' * •' > y p+2^s a secon(l vector, the inner product of £ and
fc) is denoted by
(3) (& $ = *f yt + *2 y2 + • • • + *p+2 yp+2.
For unit-vectors £ 77 making an angle #we have (£, 77) = cos #.
We shall encounter matrices (i.e., linear operators which are applied
to vectors). For full definitions and an outline of the theory, see Birkhoff
and MacLane (1947). Only square matrices will occur. If M is a matrix
with the general element \l jk (/, k = 1, 2, ... , p + 2) the determinant of M
will be denoted by
det M = det \l >k.
(1) In preparing this chapter, the unpublished notes of a course given by
G. Herglotz have been used. The idea and the arrangement of many of the
proofs are due to him.
232

11.1.1
SPHERICAL HARMONICS
233
The identity or unit-matrix will be denoted by 7; a matrix 0 will be called
orthogonal, if
(4) 0' 0 = 1,
where 0' denotes the transposed matrix of 0. From this it follows that
0 0' is also the identity. The vector resulting from the application of a
matrix 0 or M to a vector £ will be denoted by 0 £ M £. A matrix 0 is
orthogonal if and only if for all £
(5) (Of, 0?) = (y, 0.
Jean be defined by the property that Jy = £ for all £.
A function of x^, #2, . .. , x +2 will be called a function of £ and will
be denoted by /"(£). (A function of two or more vectors is defined in an
analogous manner.)
A function f ($) will be called an orthogonal invariant if for all £ and
for all orthogonal matrices 0
(6) f(0$ = f($.
Similarly, a function of two variable vectors is an orthogonal invariant if
f(0$, Ofc)) = /"(£, t)) for all y, t) and for all orthogonal matrices 0.
Sometimes we shall use hyperspherical polar coordinates
r, 0t , ... , 0p, 0,
defined by
^ = r cos 0t,
*2 = r sin 0t cos 02,
#3 = r sin 0t sin 02 cos 0 ,
(7) ••• ••• ...
x - r sin 0t sin 02, ... , sin 0 _t cos 0 ,
x +| = r sin 0t sin 02, ... , sin 0 cos 0,
x +2 = r sin 0t sin 02, ... , sin 0 sin0,
where r > 0
(8) 0^0^ 77(7 = 1,2, ... ,p), 0^0^ 2t7.
In these coordinates, the (p + 2)-dimensional volume element is given by
(9) dV = r'+1 (sin 0,)' (sin 02)p_1- .. (sin 6p)drd 0f... c/0p^0,
and the surface element dQ, becomes
(10) rfO = (sin fl^Uin 02K~1 . •• (sin0 ) d6y • • • c/0p d<£.

234. SPECIAL FUNCTIONS 11.1.1
The total area co of Q can be computed either from this or from the remark
that
/".••• r.«p<-«?—*p2+2w*, •••^p+2=(r00e-2^)"+2
= / f/exp (- r 2) dV = co J™ r*+' e ~r* dr
which gives
(11) co= .
ru + Xp)
Here and in the whole of this chapter we shall use three, two or one
integral signs to denote integrals taken over a (p + 2), (p + 1) or p-dimen-
sional manifold respectively.
A function which is defined on Q can be considered as a function
F(£) of the components of the unit-vector £. The expression
C12) J jF(OdQ(£)
denotes the (p + 1)-tuple integral which will be obtained if we substitute
for the components of £ the expressions in terms of 6.,. .. , 6 , 0,
from (2), (7) and for dQ,(£) the corresponding expression from (10).
If F (£), Fz (£) are two functions which are defined on G, and if
n
exists and is zero, we shall say that F (£), FA£) are orthogonal on
Q(£). We shall write fl instead of Q(£) if the context indicates which is
the variable vector.
If not stated otherwise, Laplace's operator A will refer to the
components of <c, i.e.,
dxx dx£ dxp+2
We have
. , rm(m-l) (fa, fa) /(Z+p + 2m)l ,
(14) A [r1^, *))»] = ——±JL+ L rl(?f t>f.
L (?. &) r J
The operator A is invariant under orthogonal transformations, i.e.,
where 0 denotes an orthogonal matrix.

11.1.2
SPHERICAL HARMONICS
235
In polar coordinates (3) we have
(15) A„ = r-P-'±(rP*«jL») +r-W.a,)-^ [(sin*t)^»]
+ r-2(sin0,r2(sina2),-P-^ [(sin^)""' -^ ul
dd2 I dd2 J
+r~2(sin0, sin 02)"2(sin 03)2-p ^- | (sind,)1'"2-^- J + •«
<?03 L a#3 J
+f-2(sin01 ^:sindp^)'2(sindpyl— [(sintf/ — J
Pl p J
<92
+ r"2(sin^1 -M.sin# )~2 r a.
1 p d<f>2
11.1.2. Gegenbauer polynomials
The polynomial Cv(x) of degree n which is defined by the generating
function
(i6) a-2xt + tzrv= f cv(x)tn v^o
is called the Gegenbauer polynomial or the ultras pherical polynomial of
degree n and order v. Szego (1939) denotes it by P^\x)* Gegenbauer
(1877, 1884, 1890, 1891, 1893) has investigated these polynomials for
arbitrary values of v. An account of his theory is given in sec. 3,15.
We shall need here only the case where 2v is an integer, 2v = p =1,
2, 3, ••• • In this case we have
2lH dl 2~nll dn+21
where 1=0, 1, 2, ... ,
(19) Pn(x)=^- -^ (*2-l)n = 2F,(-n, » + l; 1; K-Kx)
n I dxn
is the Legendre polynomial of degree n9 and
(20) Tn(x) = y2\[x + ia-xzYA]n+lx-i(l-x2)X]n\
(21) = ^(-n, *;>£,«-#*)
(22) = cos (/i cos *)

236 SPECIAL FUNCTIONS 11.1.2
is the Tchebichef polynomial of degree n. Tchebichef polynomials take
the place of the ultraspherical polynomials for v = 0; their generating
function is
(23) -1/2log(l-2^ + i2)^ I (* +I)"1 T +f (x)tnU.
n = 0 nT
From (20) we have for n - 0, 1, 2, • • • ,
(24) [* + »(i-*2)*]"+l = r w + ioi + D"1 (i-**)*-£ r+f(*).
n ' dx n+1
We also have for an arbitrary v £ 0:
(25) c*« = (-a-» u -*vv+*, (^}»—£L (i -**)»+*-x.
(v+M)nn! d*n
Here
(a)0 = 1, (a) ^ = a (a + 1) .. ♦ (a + n - 1) n = 1, 2, ... .
Equation (25) is a consequence of 3.15(3) and 2.8 (23).
Between the numbers a> in {6\ h{n, p) in 11.2(2), the square of the
normalization factor for the Gegenbauer polynomial
(26) N = J" [C*'(*)]* (1 -^y-'"8 & = 22'Pgr(B + P) ,,
-' n n\{p + 2n)\r(Y2p)Y
the total area of the surface of the unit-sphere in (p + l)-dimensional
space
(27) <u = ,
and the value
there exist the relations
, % co'N o>C*p(l) 4w,+to
C^Q) A0i,p) (2n+p)r(1/2p)#
Proofs for the formulas in this section are given in Appell-Kampe
de Feriet (1926).
For an investigation of the C^p which starts from particular solutions
of Au + k2u =0 [waves in (p + 2)-dimensional space] see A. Sommerfeld,
(I943)and also W. Magnus (1949).

11.2 SPHERICAL HARMONICS 237
11.2. Harmonic polynomials
A polynomial Hn($) of degree n in x%, x,f .. , , x +2 which is
homogeneous of degree n, so that H n(X$) = An #n(?)> and satisfies Laplace's
equation A #n(?) = 0, is known as a harmonic polynomial of degree n.
Clearly, r~n Hn($ «= Hn(£) is a one-valued continuous function on the
hypersphere fl, or r = 1, and can also be expressed as a trigonometric
polynomial in 0|f ... , 6 , <£. For the notations see sec. 11,1,
A partial differential equation of the form Au + f(r) u = 0, where
f(f) is a given analytic function of r only, and u = u (£), has solutions of
the form u = #n(r) Hn(£)9 where H (£) is an arbitrary harmonic polynomial
of degree n9 and/? (r) is a solution of the ordinary differential equation
d2R p + 1 dR
rtr* r ar
We shall now show that there are
(n + p - 1) !
(2) AOi, p) = (2n + p) £-
pin!
linearly independent harmonic polynomials of degree n of the p + 2
variables x}9 x2, .. . , *p+2.
To prove this, we compute first the number g (n, p) of linearly
independent homogeneous polynomials of degree n of p + 2 variables. Clearly,
(3) g (n, p) = g (n, p - 1) + g (n - 1, p - 1) + • • • + g (0, p - 1),
(4) g(», 0)=n + l,
and g(n, p) is uniquely determined by (3) and (4).
/«r\ / \ ^+P + 1>! /p+"+l\
(5) g(/l, p)«— 7x^ = 1 J..
n! (p + 1) ! \ n J
Now, Laplace's equation imposes conditions upon the coefficients in Hn,
since A H is a homogeneous polynomial of degree n — 2, there are at
most g(n — 2, p) independent conditions and
(6) h (n, p) £ g fo, p) - g (n - 2, p).
On the other hand, the g(n — 2, p) linearly independent polynomials
where P denotes any homogeneous polynomial of degree n - 2, do not
satisfy Laplace's equation, so that
(7) h (n, p) < g(n, p) - g(n - 2, p),
and this proves (2).

238
SPECIAL FUNCTIONS
11.2
Except for n = 0, there is no harmonic polynomial which is invariant
under all orthogonal transformations/1' But there exists an H (£) which
is invariant under all those orthogonal transformations which leave one
point of the unit-sphere fixed. Since (0$ 77) = (£, 77) for all orthogonal
transformations which leave 77 invariant, it is sufficient to prove
LEMMA 1. For each unit vector, rjf there exists one and only one
harmonic polynomial H (f) such that
(i) Hn(^) depends only on r and (£, 77);
(ii) Hn(rt)-l.
This polynomial is given by
(8) H (x) r- C"P[(^)]
Fl
where £« jp/r and where C p is given by 11.1 (16).
Since C p (x) can be expressed in terms of even or odd powers of x
according as n is even or odd, the right-land side of (8) is a polynomial
of x , ... , x + , although rn is not necessarily one. Since C"p(l) ^ 0,
(8) satisfies (ii). Therefore we have to show now that (i) determines
H (£) apart from a constant factor. Since Hn($) is homogeneous and of
degree n it must be of the form
where C „, ... , c are constants.
Since AH = 0, we find from 11.1 (14) the relations
n
(9) (n - m) (n - m - 1) c + (m + 2) (2n - m - 2 + p) c +0 = 0
for m = 0, 1, 2, ... , and
(10) cf = 0.
Therefore H is uniquely determined by cQ and c 1 = c3 = • • • = 0. To
construct H , we observe that we have from 11.1 (14) Ar p = 0 and
therefore
(11) A \\TV-z\\-r^A[P£Z (TVk-xk)z]-*r = 0
k = 0
for all values of r. With r = t~}, we find that the coefficient of tn in the
expansion of
(12) [l-2(Z,r,)rt+r2t2]-Xi>
satisfies Laplace's equation. This completes the proof of Lemma 1 for
(1) G» Polya and B. Meyer (1950) have investigated the harmonic polynomials
of three variables which are invariant under any given finite subgroup of the
orthogonal group.

11.2
SPHERICAL HARMONICS
239
p = 1, 2, #tt # In the case p -0 we can start with 11.1(23) instead of
11.1 (16) and we find instead of (8) that for p =0
(13) rnTn[(^rj)]
is the polynomial whose existence is stated in Lemma 1.
We can now construct a complete set of linearly independent harmonic
polynomials of degree n. Let
denote any homogeneous harmonic polynomial of degree m which is
independent of x1, ... , x fe_j . It can be verified that
(15) A[a-2x,t+r2tT"'~iipH ,] = 0
for all values of the parameter t9 and this enables us to find all
homogeneous hannonic polynomials of p + 2 variables if those of p + 1
variables are known already. From the h (m, p + 1) linearly independent
polynomials ff 2 we obtain h {m, p - 1) linearly independent polynomials
H (.#) which are of degree n - m with respect to x 1 , namely,
(16) r'-mCm*^"{xt/r)Hm,,
where m = 0, 1, • • • , n. Since it follows from (3) and
A (*» P) = g fo p) - g U - 2, p)
that
(17) h(n, p)=h{n, p - 1) + h{n - 1, p- 1) + ..? + h (0, p -1),
we obtain all the Hn($) from (16).
Since
as) (xp+i±i*p+2r
form a complete set of linearly independent H , we obtain by induction
THEOREM 1. Let mQ, ... , m be integers such that
(19) Tl = mQ > 772, > ••• > 772 >0,
and let rkbe defined by
(20)rt=(*2t+1+^+2+...+^+2)*
where k = 0, 1, ,.. , p and r = r. Then
(21) H (mk, ±; $)~ H{n, m} , .. . , m p^ , ±mp; x}, ... , *p + 2)
±m"rmP "n r7*^+'C4+'+*">*<*4+l/r.)
fr+i^)

240
SPECIAL FUNCTIONS
11.2
form a complete set of h (n, p) linearly independent harmonic polynomials
of degree n. Of course, H (mk, + ; $) = H (mk, - ; $) it m = 0.
Corollary. In hyperspherical polar coordinates 11,1 (7) we have
(22) H(mk, ±; $)=rnY(mk; 6k, ± <f>)
where
(23)Y(mk;ek,±<?>) = e ' U (sin *4+I) *+I Cm *« P (cos^+t).
fe= 0 fe fe + 1
11.3. Surface harmonics
If // (y) is a homogeneous harmonic polynomial of degree n, we call
(1) r-nHn{?) = Hn(£) = Yn{d,<f>)
a surface harmonic of degree n. Here 6 stands for 0 ,... , 0 and £
denotes again f/r. The surface harmonics are one-valued continuous
functions on fi (the unit-hypersphere r = 1). In particular, we have from
11.2(22) and 11.2 (23) the surface harmonics of degree n = mQ
(2) r~n H(n, mx , ... ,±mp;xt , ... , %p+2) =r~n H(mk, ±; £)
= //(», w,, ... , ±/np; £f> ... . €p+z) = H(mk> ±; £)
(3) = y(n, to,, ... , /wp; (9,, ... , 0p, ±0)= YUfe; ft ±0).
We shall now state the orthogonal property (compare sec. 11.1 for the
definition) of the functions (2), (3). With the notations
(4) Ek(l,m)
7r2k-2m~pr{i + m+P + i-k)
= (/ + y2p + i/2 _ i/2&) (/ - m)\\T(m + Mp + % - K^)]2
for any integers /, 771 where / > m > 0, and
P
(5) N(mQ, 772, , . . . , 772^) = 277 II £ fe fa fe_ f , 772 fe )
where 7720, 772,, ... , m satisfy 11.2(19), we have:
THEOREM 2. Any two distinct functions in (2) or (3) are orthogonal on
0 unless they are conjugate complex. In the case of conjugate complex
functions [or in the case of the square of a real function (2) or (3)] we
have:
(6) JnS \H{mk, ±; 0\2 dQ^ J^f\Y{mk', 6, ±<f>)\2 dQ

11.3
SPHERICAL HARMONICS
241
= N{mQ, 772,, ... , mp) = N{mk).
In particular, any two surface harmonics of different degrees are
orthogonal on the unit-hypersphere.
The functions in (2) or (3) form a complete set of orthogonal
functions on Q. We shall prove:
THEOREM 3. A function f(£) which is continuous everywhere on
Q and is orthogonal, on fi, to all the functions H(mk, ±; £) vanishes
identically on O.
To prove this we assume that f(rj) = 2a > 0, where 77 is a fixed
unit-vector (i.e., a point on 0). Since f{£) is continuous, we may
assume that f{£) > a for all £ satisfying ||£ - rj\\ < 8 where 8 is a
sufficiently small positive number, or f(£) > a if 1 - (£, jj) <. Y282.
According to Weierstrass* theorem on polynomial approximation (cf.
Widder, 1947, p. 355) applied to the function
'fix) = 1 - (1 - xWAS2) I-x<y28z,
= 0 i-x>y2s2,
we have that given any e > 0, there exists a positive integer n and a
polynomial F {x) of degree n such that
\Fnix)-<f>(x)\<c -!<*<!•
Then
n
where a* is a positive number depending on a and 8 but not on n and
6, and hence
(7) lim JJ/(£)FJ(6ij)]rfn = a*
e^ ° n
Since f(£) is orthogonal to all functions in (2) or (3) and, according
to theorem 1, C%p[(£9 77)] is a linear combination of these functions,
f(£) must be orthogonal to C^p[(£, 77)] for each k. Moreover, since
C±p{z) is precisely of degree k in z, Fn(z) is a linear combination of
the C\p{z), k - 0, 1, ... ,71. Hence f(£) is orthogonal to Fj(<f, 77)]
and this contradicts (7) and proves theorem 3.
From the proof of theorem 3, we can obtain a statement about the
approximation of a special class of continuous functions by surface
harmonics. We have:
LEMMA 2. Let F (x) be a function of the real variable x which
is continuous for - 1 < x < 1. We define for n = 0, 1, 2, ... ,

242
SPECIAL FUNCTIONS
11.3
(8) 0n[(6i,)]- 2 aaCXP[({,v)]
m = 0
where
(9) C*> (1) 4 U, p)an-J JC*> [(£ V)] F [($ ,)] rffi (£),
n
and
(10) i4(n, p) Cjj'(l) = / / IC*'[(£ r?)]}2 rfG(£).
n
7%e7i .F[(£, 77)], which is a continuous function of £ on Q will be
approximated by the <f> in such a way that
(11) lim J J\F[(^rj)] - 0B[(£ij)]|2rffi = O.'
Incidentally, the A (n, p) do not depend on the fixed unit-vector 77;
their values are given in 11,4(13).
To prove this lemma we choose in (10) the coefficients a so as to
minimize the integral in (11). Since C^[(<f, 77)] and C*p[(£ 77)] are
orthogonal on fiwhen k ^ m (cf. the remark after theorem 2), we find
precisely the values (9) for the a^. On the other hand we know from
Weierstrass* theorem on polynomial approximation that for a suitable
choice of the a n and a sufficiently large n the integrand in (11) can be
made arbitrarily small* Therefore the minimum of the integral in (11)
must tend to zero as n -* 00.
The problem of the expansion of a function which is given on 0 in
a series of surface harmonics has been investigated by several authors.
For p = 1 see Hobson (1931), where many references are given. The case
p = 2 has been investigated by Kogbetliantz (1924), Koschmieder (1929),
and the case of an arbitrary p has been treated by Koschmieder(1931).
The expansion of a function in a series of surface harmonics is
sometimes called its Laplace-series. In general, one does not know much
about the convergence of the Laplace series of a continuous function
but its Cesaro-summability (of a sufficiently high order) can be proved.
11.4. The addition theorem
For a fixed 77, the surface harmonic C^p[(£, 77)] can be expressed in
terms of the S (mk , ±; £) where mQ=n. More generally we have:
THEOREM 4. Let S*(£), I = 1, 2, . •. , h be h = h (n9 p) linearly
independent real surface harmonics of degree n, and let the S nbe ortho-
normal on £1 so that, for I, m = 1, 2, • • • , h,

11.4
SPHERICAL HARMONICS
243
10 if n 41 m
(i) JJsln(€)s*ni€)da-l
^ (l if n = m
then for any fixed unit-vector rj
For notations compare 11,1(11), 11.1(12), 11.2(2), 11.1(16).
As a special case of (2) we have from theorem 2:
(3) C*P[tf,r,)]
1 p co ^\ dm )
2 (2rc + p) ^ /VU^) * *
+ ff(roJk,-;£)ffUJk,+;??)]
where the sum is to be taken over all integer values of mk such that
n = mn > m< > • • • > m > 0 and where
(4) e(0)=l, £(i»)=2 m>0.
From 11.2 (21) we find that S (zn fe, +; £) vanishes if the last p + 2 - /
components of £ vanish, i.e., if
except when
771 »= 771 j , . = • • • = 771 =0.
Therefore, if we put
£ = (cos p, sin p , 0, ... , 0)
rj = (cos a, sin a, 0, • • • , 0)
(3) becomes for p > 1
(5) C *p (cos p cos o + sin p sin <r) = C ^ [ cos (p - a)]
r(p-i)c*'-*(i) v
« _JL—L_el—Li ) 5 (sin p) - c ■ +KP (cos p)
r(XP) r(XP) ffito n' n"°
x (sina)" C*+^(cosa)
where
(6) Bn>R = 2** Oi - m)! (p + 2/7i - 1) [r(in + ^p)]2 [T(p + * + w)]"1.

244
SPECIAL FUNCTIONS
11.4
If we put in (3)
£= (cosa, sin a cosp, sin a sinp, 0, %%% , 0),
77 = (cos^S, sinjS cos a, sinjS sin a, 0, ... , 0),
we obtain from (5) with p — a = <j> for p > 1
(7) C^(cosa cos/3 + sin a sin/3 cos0)
r(P-n
X *».>*«>■
[r^p)]2
n= 0
x C*+*P (cos a) (sinjS)"1 C^'tsina) C **"* (cos<£)
where Z^ ^ is given by (6)% For p = 1 we find
(8) P„ (cos a cos 6 + sin a sin jS cos <£) = P (cos a) P (cos 8)
V (n - m) ! _
+ 2 > — P"(cosa) P■ (cosj8.) cosing
^ (n + m) I n n
m = 0
where
(9) P>)=C*(*)
is Legendre's polynomial and
(10) P"(«)-(-l)" »r~X r(m + K)2" (l-*2)*1" C+*(*)
n n A
is an associated Legendre function*
Usually, (7) or, in the case p = 1, (8) are called the addition theorem
of ultraspherical polynomials• We can obtain (3) (but not the whole
theorem 4) by a repeated application of (7) and (8)% In a modified form,
(7) and (8) are also valid for a general Cv where 2v is not necessarily
an integer; for this see 3.15(19) and 3.11(2).
The proof of theorem 4 will be based upon the fact that C%p[(£, 77)]
is an orthogonal invariant of £, rj (see sec. 11.1,1 for the definition).
We shall show first that apart from a constant factor CM(£f rj)] is the
only orthogonal invariant which is a surface harmonic of degree n. To do
this we need
LEMMA 3. Let F (£, ^) be a polynomial in the components of £ and ^
and let
(11) F(0?, Ot)) = F(?, t»
for all orthogonal transformations 0 (compare sec. 11,1,1). Then there
exists a polynomial $ (u9 v, w) in three variables u, v9 w9 such that

11.4
SPHERICAL HARMONICS
245
(12) F(£, $=*[(?,?), (£,*)), (*),*))]
identically in the components of £ and fc).
Proof: If £, fc) are fixed, we can find an orthogonal coordinate
system such that
£=(a, 0, 0, ... , 0), ^ = (/8, y, 0, ... ,0),
(?, ?) = «2, (?, ^) = aft (*), t))=/S2 + y2,
and therefore
a=uM, p = v/u*, y = (uw -v2)K/u.
Since F is an orthogonal invariant this shows that it can be written as a
polynomial
F=F*(a, j8, y)=F*[uy\v/u%9(uw-v2)*/u]
in a, /S, y. Since there exist orthogonal transformations which have the
effect that
a-*- a, /3->-/S, y ~* y
or
a* a, /S->/S, y->-y»
we find that F*is a polynomial in y2, a2, /82, ajS and that we can write
F* in the form
(13) F*= u~* $*(u, v, w),
where /n is an integer and $ * is a polynomial of u> v, w.
Interchanging the role of £ and fc)
(14) w-ky(u, v, w)=u'* Q*(u, v, w),
where k is an integer and ^ is a polynomial. Since u9 v9 w are
algebraically independent, we can conclude from (14) that u~* $*isa
polynomial and this completes the proof of lemma 3.
LEMMA 4. Letg,rj,£ be arbitrary unit-vectors in the (p + 2)-c?£-
mensional space* Then
(15) J J C*'[(&*)] CJJ'Ki/, &] rfO(,) -A(n, p) C*>[(€, Ol
n (r,)
where
(16) A (b, p) = C*p(l) .
v n h(n,P) (n + y2p)r{y2p)
Lemma 4 is of the nature of a convolution theorem for the basic surface
harmonic C^p[(£, 77)].

246 SPECIAL FUNCTIONS 11.4
To prove this lemma, let £ and % be any two vectors, £ = ?/||y|| ,
£=3/||a||. Since
are harmonic polynomials in the components of £ and % respectively, we
see that ||^||n ||^||n times the left-hand side of (15) is a harmonic
polynomial both in £ and %, of degree n in each set of variables.
Moreover, this harmonic polynomial is an orthogonal invariant in £ and £, for
it remains unchanged if any orthogonal transformation is applied
simultaneously to £, % and 77 (and therefore to £, £ and 77) and the integral
remains unchanged if any orthogonal transformation is applied to 7/ •
Thus by lemma 3, our harmonic polynomial is a polynomial in ||£||2,
H2H2, and (£, %) = \\$\\ ||#|| (£, 0 • Therefore we find from lemma 1
that it is a multiple of
\\*\\* \\i\\n c*\kt, o\
and this proves lemma 4. We can determine the factor A (n, p) by putting
f=4>U*o °)
which gives
(17) A (n, p) CJf'U) = ©' J"+' [C*"(*)]2 (1 -*2)*"-* «fa,
where gj ' denotes the area of the hypersphere in the (p + l)-dimensional
space. From 3,15(17), 11.1(26), 11,1(29) and 11.2(2) we obtain (16).
Now we can describe the effect on the surface harmonics of an
orthogonal transformation of £.
LEMMA 5. Let S^(^), I = 1, 2, ... , h be a complete set of ortho-
normal surface harmonics of degree n9 so that (1) holds, and let 0 be an
orthogonal transformation of the (p + 2)-dimensional space,, Then
as) s^oo- i slkskn(0,
k = 1
where the matrix G of the h2 elements g lk is an orthogonal matrix of
h = h(n9 p) rows and columns, i.e.,
(19) G'G = GG'=L
Here G 'is the transposed matrix of G, and Jis the unit-matrix of h (n, p)
rows and columns.
Proof: Since Laplace's operator is invariant under orthogonal
transformations (compare sec. 11.1), Sln(0£>) is a surface harmonic of
degree n, and so can be expressed, in the form (15), in terms of the
complete system Sn(£\

11.4
SPHERICAL HARMONICS
247
Since the integrals in (1) remain unchanged if £is replaced by 0 £, it
follows that also the S {0 £) form an orthonormal system, and this gives
GG' = J. But it is well-known that from this we also have G G = I
(see, e.g., Birkhoff and MacLane 1947, Chapter IX).
Now we can prove theorem 4 by showing that
(20)
i,5^)5>= J, sln(oesln(On)
is an orthogonal invariant of £ and 77 . This follows from Lemma 5 and in
particular GG = J. From the proof of Lemma 4 we see that (20) must be
a multiple of CM(<f, 77)]. The constant factor can be determined by
integrating the square of (20) with respect to 77 over the whole area C On
account of (1) this gives
(21) i [sln(0]z.
1= 1 n
On the other hand we can see that it must be a certain multiple of C^p(l)
by making £ = 77 in (12). By integrating (21) over fl(£) we obtain h
because of (1), and this leads to (2) in theorem 4.
From theorem 4 we have that for every surface harmonic 5 (£) of
degree m
I 0 n £ mf
(22) JJclp[({,r,nsm(Odn(€)=l
n(^' (Sco/h)Cfa)Sp(r,) n=m.
From Lemma % in particular from 11.3 (8), 11*3 (11), we find by an
application of Schwarz's inequality:
lim J J\F[^9rj)]^cf>n[(^rj)]\S (£)<ffi(£)=0,
where F9 <f> are defined in 11„3, 11*3 (8). H we combine this with (22) we
obtain (cf. Funk, Hecke, 1916, 1918):
FUNK-HECKE THEOREM: Let F (x) be a function of the real variable
x which is continuous for — 1 < x < 1 and let S (£) be any surface
harmonic of degree n. Then for any unit-vector 77
(23) J J F[(^r,)] Sn({)dn({) = An 5n(r,),
where the integral in (23) is taken over the whole area of the unit-
hypersphere Q, and where

248
SPECIAL FUNCTIONS
11.4
n
Here Co denotes the total area of the unit-hypersphere in the (p + IV
dimensional spacef
, 27T1Ap+1A to' (in)*'nirOLp)
~r^P + I/2)' c^d)" (b + p-di '
Erdelyi (1938) has shown that it is sufficient to assume that |FU)| and
|F(#)|2 are Lebesgue-integrable for —1 < x < 1, and he also has shown
that
X^i^nV^Pf^ r*> Jn^p(t)f(tUt,
where
/W-(2ir)"f Jjt e-k'FMdx.
Here J denotes a Bessel function. Note that
(-*74)«
'~*p<w<>='n2~n~*'2
m! r Gi + 77i + 1 + ]i p)
n = 0
is a one-valued function of t*
11.5. The case p - l9 h(n, p) = 2n + 1
11.5.1. A generating function for surface harmonics in the
three-dimensional case
(1) 5F-(*|f *2>*3)
denotes a vector with three components. We define the polynomials
ff«(?)by
(2) [*2 + w,-2*, t-=(x2-w,) **]"=■«" £ #;(?)*'.
If we substitute -r"1 for t we find
(3) Hmn = {-D* #~\
where a bar denotes the conjugate complex polynomial. The left-hand
side of (2) can be written in the form (u, $)n where
(4) u« (~2t, l-tz, i+.it2).
From (u, u) = 0 and from 11.1(14) we find that both sides in (2) satisfy

11.5.1
SPHERICAL HARMONICS
249
0
2rr
r(%)r(n +
T(n +3/2)
T)_
(2°)
\ m + n/
m 7
m =
Lm'
t
= m
Laplace's equation for all ty i.e., Hm($) is a homogeneous harmonic
polynomial of degree n. The linear independence of Hn follows from the
algebraic independence of
x2 + ix3, -2x^, ^{x2-ix3).
With r = ||jp||, £= £/r, the functions
(5) r-»ff»(?) = S»(£) m«0, ±1, .... ±n
form a complete set of .linearly independent surface harmonics of degree
n • From (3) we have
(6) S;'(£)-M)"5B "•(£).
The orthogonality relations
(7) SSS\{OS*(&dQ-
n
m, m = 0, + 1, • • • , ± n,
in which the integral is to be taken over the whole area of the unit-sphere
C, can be proved by introducing
(8) b = (-2s, 1-s2, i + is2)
and considering
(9) {fin,on(*>ondmo
n
which is an orthogonal invariant of U and b (cf. the proof of Lemma 4 in
sec. 11. 4) • According to Lemma 2 it must be a polynomial in (u, u),
(b~, b), (u, "b), and since (u, u) = (b, b) ^ 0, (9) must be a multiple of
(u, b)n. If we introduce (2) [and the corresponding expansion of (b, £)n
into (9)] we find
(10) (ts)n £ tlsm J JSln(£)Smn(£)dQ
= ^(u> ^)n= ^2"(l + s«)2n
and here we can compute p by putting s = t = 0 and
(11) £ = (cos 6, sin 0 cos <£, sin 0 sin <£), dQ = sind d 6 d <f>,
which gives
(12) 2V = J*nd<f> J"d0(sin0)2n" -2irr(K)ii!/r(ii + 3/2).

250 SPECIAL FUNCTIONS 11.5.1
By comparing the coefficients of t ls* on both sides of (10) we obtain (7).
To obtain an explicit expression for 5* (£) we apply Cauchy's formula
to (2) and obtain
(13) #"(?)=— /<0+>(u, $yrn~*-'dt
n 2ni
^{27rirl{-l)n{x2-ix3)nJi0+)i[t+xi/(x2-ix3)]z
-rz(x2-ix3r2\nrn-Bi-it dt.
If we put
t +x}/(xz-ix2) = r, *,/(*2 ~ix^ = a>
this gives
(14) H'n(z) =(2ni)-< (ix3-xz)n Jia + hr2 -r2(x2- ix3rz]n(T-o)-n-'-} dr
(-1)- , dn+" T , /ra\2 1"
(15) —{x 2-ix3V—— \rz-(—)
{n+m)l 2 3 drn+a I \*,/ J
rn /x-ix, \" dn+"
(16) -u7riT(-*r-A)«*--(1-«>" ^"'/r-
If we define the associated Legendre's functions Pa(#) by
n
(17) Pa (*) = (-l)n+a 2'n(n\r' (l-*2)*" — (l-*2)n
n dxn+m
m « 0, ± 1, ... , ± nf
we find that
(18) S'n{0=r-nH~n{$
= (-1)n+"£^7T tft-i£>"(i-*?>-*■ p'^.).
\n+ m) I
and for the corresponding functions in spherical polar coordinates (see
sec. 11,3),
(19) Yffl(fl, 0)=5ffl (£) = (-!)"** , *' e^P'-tcosfl).
U + ro)!
According to (3) and (18) we have
&i-m)!
P"'W*(-1)'7 ^P">V
Oz + to)! n
The addition-theorem has been stated as equation 11.4(8).

11.5.2 SPHERICAL HARMONICS 251
The orthogonality relations (7) give
(20) CV^jl^.
1 2n + 1 (n - m) I
From (2) we obtain the generating function
(21) [I^s*cos0-%(1-*2) sinflF1
= f 2 (nI/fe!)P*"n(cos^sn**.
n=0 fe= 0
For other properties of the P" see sec. 3.6.1.
11.5.2. Maxwell's theory of poles
Let x t, x , a: 3, be independent variables, let r = (x 2 + % * + x 2) ^ and
define the differential operator D k by
(22) Z)fe = k = 1, 2, 3,
Since
(23) Ar"1 -(Df + D'+Dpr"1 =0,
clearly D" D% D* r"1 satisfies Laplace's equation. Moreover this is
clearly of the form of a homogeneous polynomial of degree n = a + b + c
multiplied by r"2n-t. Lastly, it can be verified that for every
homogeneous polynomial H of degree n, the statements
A# =0 and A# r"2n^ = 0
n. n
are equivalent. Thus we find
(24) D°Db2Dl r-1 = ffn(*,,*2,*3)r-2n-1 „ = a + b + c.
It is a consequence of this observation that to every homogeneous
polynomial of degree n of three quantities D}, D2, D3 for which
(25) D* + D22+D23 = 0
there corresponds a harmonic polynomial of x^, x2, x3 of degree rc.
Comparing this with the remarks after 11.7(12), it seems plausible that
we can obtain all harmonic polynomials from (24). Actually, it can be
shown that (see Hobson, 1931, Chap. 4, Nos. 85-92)
1 (-l)"-»(rc-m)! .. ,
(26) D?-B(Z)2±*D3r-= —^ e±tm*P*(cosQ
m = 0, 1, • • • , n9

252
SPECIAL FUNCTIONS
11.5.2
and
(27) x t = r cos 0, x 2 = r sin d cos 0, x 3 « r sin 6 sin <£ •
According to (19) this shows that all spherical harmonics can be obtained
from (24).
For geometrical reasons, the surface harmonics in (26) are called
zonal if m = 0, sectorial if m = n and tesseral if 1 _< m ^ ti — 1. For this
and for the following remarks on Maxwell's results see Hobson (1931)
and Maxwell (1873, 1892).
Let
(28) yk = {ak, 0k9 yk) A = 1, 2, ... , n
be unit-vectors which therefore define points on the unit-sphere. These
points will be called poles. Then the surface harmonic of degree n with
the poles rj k is defined by
(29)S>t) = (-l)"r»+'[tni(atD,+/3tD2 + yiD3)]r-'.
Introducing n parameters, 11, ... , tn, we find that this is the coefficient
of ty t2 »m £ in the expansion of
e
(3D r2= i «4«,(?», ?,), £=(^,^,^)
*. 2= i " * " \r r r /
(30)
where
and where the sum in (30) is to be taken over lc = 1, 2, ... , n. This is a
function of the cosines of the angles between the vectors £ r/t ,.. * , r/fc.
The standard surface harmonics (26) are obtained when the vectors r\k
coincide with some of the axes of the coordinate system.
Van der Pol (1936) and Erdelyi (1937) have extended (26) to solutions
of the wave equation Aa + &2a = 0by showing that
(32) *n""r^) Jn+%{kr)P*n(cos6)eim<*>
= k K^'^rj P»" \k dxj kr >
where PU) denotes the m-th derivative of the Legendre polynomial Pn,
where P* is defined by (17), Jn+% denotes the Bessel function of the
first kind and of order n + lA and r, d, <f>> *,, x2, xi are connected by
(27).

11.6 SPHERICAL HARMONICS 253
11.6. The case p —2, h(n, p) - (n + D2
From now on let ^ be a vector with four components
and let
(2) 77=VP. P-Ufcll.
We introduce the vectors
(3) u= (i - its, - it -is, -t + s9 1 + ts)
(4) b = (i - i ra9 -if- ia, —-r + .a* 1 + tit)
for which we have
(5) (u, u) = (b, b) = 0, (u, b) = 2(l+ir)(l + sa).
From (5) we find again as in sec. 11.5.1 that the (n + l)2 polynomials
Hk'l($) defined by
sZ
(6) (u,$)»« J f * ) Hkn'K$tk
are harmonic polynomials of degree n.
By the same argument as in sec. 11t5.2 we find that
— 2,~n772 —
(7) f f (u, ^(M)B«W= — (u, *>)n,
fid,) n + 1
and therefore the surface harmonics
(8) S*'K7/) = ?-"//*„•'03)
form an orthogonal set of h(n, 2) = (n + l)2 linearly independent surface
harmonics where
k£k' or Zj^Z,'
(9) J JS*. %)SkJl'(V)rfO-,'
2»r2
C)/C)
71+1
From (6) we also have
do) skn- %)-i-i)k*lsnn-k>n-l(v).
In order to find explicit expressions for the S ' we introduce
(11) o=-y4 + £y|f i=y3"iy2, c = -y3-*>2, rf=-y4-iyf.

254 SPECIAL FUNCTIONS 11.6
Then
(12) p= H^ll = (arf^ic)^, (u,J}) = a + 6s+(c+<Zs)*,
and we obtain from (6) that
(13) 2 Hk'l(fysl=(a + bs)n-k(c+ds)k,
l = o n
(14) H k-lity=— fi0 + \a+bs)n-k(c+ds)ks-l-*ds.
n 2ni
Putting
(15) a~.-s(bc-ad)/bd9
(16) a0= ad/'(ad - be) = (y* + yzJ/{y\ + y\ + y* + y*),
and expressing a, i, c, d in terms of the y^,
L 7 ("I)4 LJ.7 7 L ^°"o +> L , da
(17) tf *- zty) —- 4>nWP)k+l-» (b/P) l'k J ° a"-*(W rrrr
(18) .fcj^p^-^.
where a0 is given by (16). Here the Z-th derivative can be expressed by a
hypergeometric function (which is a Jacobi polynomial) and our final
result is [cf. 2.8(27), 2.1(2), (1) and (2)] as follows.
If n>k + l
(19) S*'Z(7/)-p-n#*'Z0j)
M-1)*Cz^)(r74 + '??l)n^Z(r73 + l,r72)*"Z
x2F,(-/, n - I + 1; n - k - I + 1; i\\ + r)*)
(20) =(-l)*0?4 + /v?,)n"*"Z(v?3+*v?2)*-Z
xpC--*-U-l)(j|. + ,*.,*_,*).
If n <k + 1
(21) S*'l(77) = p-nff*'i(^)
a(-1>"",(Ji*z)<l4-'l.)'+l""<1a-»1«)1"'
x F (Z — n, /+l;Z + A-n-Kl;?if+?if)
2 1 4 ' 1

11.6 SPHERICAL HARMONICS 255
(22) S*'J(77) = p_Bff *•'($)
= (-l)n-l(Vi-iVl)k + l-n(V3-ir,2)l'k
where P^a'^'denotes a Jacobi-polynomial (see Chap. 10)«
If we introduce polar coordinates, the expressions (20), (22) for the
Sk'1 became rather complicated and it is better to use the functions
11.2(23) (in the special case p = 2) for this purpose. But for the
transformation of spherical harmonics the S ' (with an even value of n) are
very useful; they also satisfy some relations which do not have an
analogue in the cases where p f 2. These relations (which will be proved
in sec. 11.7) are the following ones (written in terms of the H^1
instead of the S^ l).
Let t), % be two vectors with four components each, and let to be a
vector with the components
(23) wt-y%*4 + y4Zi~-yz*s + ys*z
w2=y2zA + yAz2~y*z<i + r, s
M73 = r3^4 + y4^3-r1 *2 + y2*,
w***y**A-r\ z^-y2z2-y3z3^
If we introduce quaternions (see Birkhoff and MacLane, 1947, Chap. VIII,
5) this can be written in the form
(24) w A + iwz + jw3 + kw T
= (*4 + *'z2 + /z3 + *2f) (y4 + iyz + ;y3 + Ay,),
where 1, i9 /, h are the fundamental units. Then we have the addition
theorem:
(25)#£,z(to)= \ Hk±mtyHB-l{$).
ft = 0
The matrix
(26) [ff*;l(t>)] K 1 = 0, 1, ...,2n
where A; denotes the rows and I denotes the columns has the determinant
(27) (yf + y^+y|+ypn(2n+,),
the characteristic roots
(28) A; A2.,""" m-0, 1, ...,2n

256 SPECIAL FUNCTIONS 11.6
where A1 , A2 are the roots of the equation [cf* (4)]
I a -A, b I
(29) I =0,
I c, a - A |
and the trace
(30) lo^'^^l7^^
where ^'+1 denotes the derivative of the Tchebichef-polynomial 11J(20)%
11.7. The transformation formula for spherical harmonics
Let £ be a vector with three components and ^ be a vector with four
components. We use the notations
(1) ll?ll,-r, ||$||4-p, f-Jp/r, 1=VP-
We shall now show that every orthogonal transformation 0 of £ with the
determinant+1 can be uniquely described by a unit-vector r/.If detO=+l,
there exists a vector £Q ^ 0 (the axis of rotation) such that
(2) ?0 = 0?0.
The transformation Ois completely defined if £Qand the angle of rotation
ift are given. Since — £Q is also an axis of rotation we can choose £ in
such a way that 0 < i/j < 77. If x// is zero, every vector £Q is an axis of
rotation, and in this case we put £Q = 0. We may assume therefore that
(3) ||£0||3 = sin%<A 0<i£<77
which means that the components xQ f , xQ 2 , xQ 3 of £0 are given by
xQ ^cosajsin1^ I = 1, 2, 3,
where a^ is the angle between the axis of rotation and the xl axis.
Now we define the four-dimensional unit-vector
(4) 17 = (cos a t sin % t/f, cos a2 sin lA t/s cos a 3 sin Vi t/s cos % t/0
and put ^ = pr/. Then the orthogonal matrix 0 can be written in the form
(5) 0 = (yAI-A)(yAI+A)-* = (l/p2)(p2I-2yAA + 2A2)
x /y^f-y^y** 2r,r2-2r3y4' 2yiy3+2y2y4
(6) =7^ ( 2riy2+2y3y4' y!+y*-y?-y3» ^3-^4
\.2r,r3-2r2r4, 2y2y3+2y,y4> yj+ys-yf-yl,

11.7
SPHERICAL HARMONICS
257
where
/l 0 o\ A y3 -^
(7) 2-0 10 J, A =( -y, 0 y,
V01/ v. -y- °.
This is Cayley's representation of the orthogonal group (see H. Weyl
(1939), p. 169ff). In the form (6) itis valid without exceptions, i.e., even
if the determinant of yA J+ A vanishes.
With the notations (1), (2), (4), (5), 11,5(18), 11,5(19), 11,6(8) we
have the
TRANSFORMATION FORMULA OF SPHERICAL HARMONICS
(8)
s><0(). j_n «.„.♦. (;;,)/(;;,) sr'-w »;«>.
This formula shows the effect of an orthogonal transformation 0 of the
three-dimensional space upon the surface harmonics on the sphere, and
it gives the coefficients of the linear transformation of S in terms of
surface harmonics in four-dimensional space with Cayley's parametersof
0 as variables.
A formula equivalent to (8) has been proved by Adam Schmidt (1899)
(see also Hoenl, 1934). In an unpublished note left by Bateman, it is
shown that the coefficients of the S in (8) can be expressed by a hyper-
geometric series. In its present form, (8) is due to Herglotz, whose proof
will be given here.
In order to prove (8), we show:
(i) We can map the harmonic polynomials H* ($) upon the product of
the powers of two variables w ^, w 2 by putting
(9) xl=w}w2, x2+ixz=w*, -xz+ixz=wzz
because then 11,5 (2) becomes
(10) (u;f-2ii;| wzt + w\t2)n= ± H\{& *n+*
and therefore
(11) Hn (£) = (-l)*( U ) wn-"wn2+«.
Although this implies a relation between x ^, x 2, x , namely,
(12) x\+x\+x\=0,

258
SPECIAL FUNCTIONS
11.7
[cf. 11.5(25)], we see from (11) that the complete set H "(£) of linearly
independent harmonic polynomials is mapped upon the set of linearly
independent products of powers of w 1 and w 1%
(ii) If we define a, b9 c, d by 11.6(11), then the linear substitution
(13) w'} = aw y + bw2, w'z = cw]+dwz
leads to the substitution for w f w2 , w2, w2 given by
(14) uj'h w'2 = (ad + be) w 1 w2 + ac w2 + bd w2
w'y2 = lab wy w2 + azwz + b2w\
w22 = 2cd w 1 wz + c2 w2 + d2 w2
and if we put w[ w'z ='X\*W\Z = x '2 + lx 3» w 'z = ~~X2 + *x 3 anc^ assume
that
arf - 6c = y* + yI + y\ + y\ =1,
(14) is precisely the linear substitution
(15) ?' -0$, $' ~{x\tx[,x'z)
where 0 is given by (6). This is the representation of the ternary
orthogonal group by unitary binary substitutions (cf. Van der Waerden, 1932,
Chap. Ill, 16).
(iii) With the expressions in 11 j6(ll) for a9 b, c, d and with s = wjw ^
we obtain from 11.6(13)
(16) 2 Hk^%)wflwlz ^(aw^bw^-Hcw^+dwjK
If ||i} |] = 1 we obtain the transformation formula (8) from (11), (13), (14),
(15), (16) and (6).
The formulas 11,6(25) to 11.6(30) are consequences of the fact that
(8) can be considered as a representation of the orthogonal group (cf.
H. Weyl, 1939, for the concepts used here). In particular, 11.6(30)
follows from the fact that the characteristic roots of an orthogonal matrix
0 for which det 0=1 are completely determined by the angle of rotation,
i.e., by J Jp% Since the characteristic roots of a matrix JJ corresponding
to 0 in a representation of the orthogonal group depend only on the
characteristic roots of 0, the trace of JJ (which is the negative sum of
the characteristic roots of JJ) must depend on J Jp only. According to
lemma 1, the expression on the right-hand side of 11.6(30) and its
multiples are the only surface harmonics satisfying this condition.
Y. Sato (1950) expressed the transformation 0 as a product of three

11.8
SPHERICAL HARMONICS
259
simple transformations, proved equation (8) for these transformations and
gave a table of the coefficients in (8) for n < 7.
11.8. The polynomials of Hermite - Kampe* de Feriet
A different approach to the investigation of surface harmonics has been
made by Hermite, Didon and Kampe de Feriet. The far reaching and
important theory as developed by these authors has been fully presented
in the second part of the book by Appell and Kampe de Feriet (1926).
Rather than giving all the results obtained there in detail we shall
confine ourselves to a short indication of what can be found there and refer
the reader to the book itself for a full account of the theory.
Generalizing Maxwell's construction of surface harmonics in the
three-dimensional space we define the following functions of the p + 2
components of a vector £,
0) w". -, w--,i^T. t tttStt ™
r I p OX • • • OX F
where r = ||3p|| and where the non-negative integers m%, ... , m satisfy
(2) m + mz + • ♦ • •+ m = rc.
The function on the left-hand side of (1) satisfies Laplace's equation;
it is the coefficient of
(3) a^a^ ... a/
in the expansion of
(4) [(xt-at)2 + ...fUp-ap)2+-^+I+^+2r^
into a series of products of powers of a x, ... , a .
Then
is a surface harmonic of degree n which depends on the first p
components of $/iv As a generating function, we have
(6) (l-2o, £f 2ap^ + at2 + -.. + ap-^
where the sum is to be taken over all non-negative integers m %, . .. ■, m •
Explicit expressions and expressions in terms of hypergeometric
functions of p variables for the functions V have been given by Appell-
Kampe de Feriet (1926). The connection with the ultraspherica 1 poly-

260
SPECIAL FUNCTIONS
11.8
nomials is given by
771 , 771
(7) So' '».*>Vm m (£,,...,£ )
(a t + ••*
p' » [ (of+....+op* J
where the sum is taken over all non-negative integers m 1 , ... , m
satisfying (2). From this, recurrence formulas can be obtained.
With the definition
, ,...,771 , 0, ...,0^1' •'•• > £<?' ••"• ' ^7+*-1^
9
m * .....77i
where s, q - 1, 2, 3, ••• , it is found that the functions
U-tf f,|)X'e1"*FW+" (ft.....*,)
*1 ' —»'p
(9)
~F / /
p
form a complete set of linearly independent surface harmonics of degree
n, if the non-negative integers Z, ZJf ... , I satisfy
(10) 1 + l} + ••• + lp^n
and
(id ««*-(^+I+i^w)d-^-..,-#-*
-tW'W^ + W-
The functions in (11) do not form an orthogonal set on the unit-sphere;
the integral
J qJ ! ^P 'tf-^'p "»!>•••> Wlp
vanishes only if either
Z1 + • •• *f Z f 77i t + »» + m
or all the differences Z - m. , ... , I — m are odd numbers. For this
i i y p p
reason, a second set of functions U is introduced by means of the
generating function
771 , 771
(12) la ' ...a p UM (P ( )
1 » '"'"'p
= [(o,^+... + an4-l)2 + (af+». + ap2)(l-^ <f*)]*1.

11.8
SPHERICAL HARMONICS
261
These functions are surface harmonics in p + / + 1 dimensional space,
and the U and V together form a biortkogonal system bo that
(13) / / (1 - {* tyi~* V\l) . U<» m dQ = 0
unless m! =/,, m2 = /2, ... , m = 7 . Thus the functions U can be used
to determine the coefficients in the expansion of a function on the
hypersphere, and in particular of a hypersurface-harmonic of given
degree, in terms of the functions (11).
For many other results about the functions U and V9 in particular
for partial differential equations, expressions in terms of Lauricella's
generalized hypergeometric series and expansion of arbitrary functions
in terms of the U and V compare Appell-Kampe de Feriet (1926). A
generalization of the V„ m for values of / which are not a positive
flij, ««« , mp r
integer, see A. Angelescu.
Generalizations of surface harmonics connected with operators other
than Laplace's operator have been investigated by M. H. Protter.

262
SPECIAL FUNCTIONS
REFERENCES
Angelescu, Aurel, 1916: Sur les polynomes generalisant les polynomes de
Legendre et d'Hermite et sur le calcul approche des integrates multiples.
These no. 1579, Paris.
Appell, Paul and J. Kampe de Feriet, 1926: Fonctions hypergeometriques et
hyperspheriques, Polynomes dJHermite, Gauthier-Villars.
Birkhoff, Garrett and Saunders MacLane, 1947: A survey of modern algebra, New
York.
ErdeTyi, Arthur, 1937: Physic a 4, 107-120.
Erdelyi, Arthur, 1938: Math. Ann. 115, 456-465.
Funk, Paul, 1916: Math. Ann. 77, 136-152.
Gegenhauer, Leopold, 1877: Akad. Wiss. Wien., S.-B. lla, 75, 891-905.
Gegenbauer, Leopold, 1884: Denkschriften Akad. Wiss. Wien. Math. Naturw. Kl.
48, 293-316.
Gegenbauer, Leopold, 1888: Akad. Wiss. Wien., SrB. lla, 97, 259-270.
Gegenbauer, Leopold, 1890: Denkschriften Akad. Wiss. Wien. Math. Naturw. Kl.
57, 425-480.
Gegenbauer, Leopold, 1891: Akad. Wiss. Wien., S.-B. lla, 100, 225-244.
Gegenbauer, Leopold, 1893: Akad. Wiss. Wien., S.-B. lla, 102, 942-950.
Hecke, Erich, 1918: Math. Ann. 78, 398-404.
Hobson, E. W., 1931: The theory of spherical and ellipsoidal harmonics,
Cambridge.
Hoenl, H., 1934: Z. Physik 89, 244-253.
Kogbetliantz, Ervand, 1924: /. Math. Pures Appli., IX Ser., 3, 107-187.
Koschmieder, Lothar, 1929: Math. Ann. 101, 120-125.
Koschmieder, Lothar, 1931: Math. Ann. 104, 387-402.
Magnus, Wilhelm, 1949: Abh. Math. Sem. Univ. Hamburg 16, 77-94.
Maxwell, J. C, 1873, 1892: A treatise on electricity and magnetism, Vol. 1,
Chapter 9, Oxford, Third edition 1892.
Nielsen, Niels, 1911: Theorie des Fonctions Metaspheriques, Gauthier-Villars.
Polya, George and Burnett Meyer, 1950: C. R. Acad. Sci. Paris, 228, 28-30,
1083-1084.
Protter, M. H., 1949: Trans. Amer. Math. Soc. 63, 314-341.
Sato, Yasuo, 1950: Bull. Earthquake Res. Inst. Tokyo 28, 1-22, 175-217.
Schmidt, Adam, 1899: Z. Math. Phys* 44, 327-338.
SommerfeId, Arnold, 1943: Math. Ann. 119, 1-20.
Van der Pol, Balthasar, 1936: Physica 3, 385-392.

SPHERICAL HARMONICS
263
REFERENCES
Van der Waerden, B. L., 1932: Die gruppentheoretische Methode in der Quanten-
mechanik, Berlin,
Weyl, Hermann, 1939: The classical groups, Princeton University Press, Princeton,
New Jersey.
Widder, D. V., 1947: Advanced calculus, New York.

CHAPTER XH
ORTHOGONAL POLYNOMIALS IN SEVERAL VARIABLES
12.1. Introduction
Let R be a region inn-dimensional Euclidean space in which x ,,„,*
are Cartesian coordinates, and let w>U) - w(x{ , ... , x ) tea non-negative
weight function defined in R* For any two functions /(*,, ... , x ) and
g(*,, ... , xn) we put
(1) (/", g) = /••• //Uff ... » *B)g(*,t — » *B)w(*1,...,*||)rf*1»-rf*|i
if
and call this the scalar product of / and g: it is defined whenever / and g
are defined in R and the integral exists* Two functions are called
orthogonal (with respect to the weight function w) if their scalar product
vanishes.
Given a weight function and any sequence of linearly independent
functions 0|f 02, ... for which all scalar products (i/r., \jj .) are defined,
the process of orthogonalization described in sec. 10.1 may be carried
out with respect to the scalar product (1), and leads to an orthogonal
system which is determined uniquely up to a constant factor in each
function. This is no longer true of a multiple sequence of functions.
Before proceeding to orthogonalize a multiple sequence, it is necessary
to rearrange it as a simple sequence. To every possible rearrangement
there corresponds an orthogonal system, and in general different
rearrangements will lead to different orthogonal systems. Thus, a multiple
sequence does not, in general, determine an orthogonal system
(essentially) uniquely; moreover, in most cases, the rearrangement destroys
the symmetry of the multiple sequence. For these reasons it is often
preferable, in the case of a given multiple sequence
tV'jTj 9 ••• 9 m ^X 1 9 •'•• 9 X n'>
of linearly independent functions, to construct two multiple sequences
264

12.2 ORTHOGONAL POLYNOMIALS IN SEVERAL VARIABLES 265
which form a biorthogonal system, i.e., for which the integral
(ch , v * * )
^m^ ... , m^' ^^p ... , m n
vanishes except in case m} = m^9 m2 = m', ... , /7i - m' % Biorthogonal
systems give a greater freedom of choice which may be utilized to
preserve symmetry.
These remarks are pertinent when dealing with orthogonal polynomials.
In order to orthogonalize the multiple sequence of monomials
m. m m
(2) x% x2 — xnn my,m2, ... , ma= 0, 1, ... f
it is necessary to order monomials in a simple sequence* Except in the
case of very special regions and weight functions, there is no
(essentially) unique system of orthogonal polynomials, and any system of
orthogonal polynomials obtained by an ordering of the monomials (2) is
necessarily unsymmetric in the x t, ••• , x • The equal standing of the
variables may be preserved by adopting a biorthogonal system of
polynomials.
There does not seem to be an extensive general theory of orthogonal
polynomials in several variables. Special biorthogonal systems,
corresponding to the classical orthogonal polynomials in one variable, are
known, and have been investigated in some detail. The book by Appell
and Kampe de Feriet gives a comprehensive account, and an extensive
bibliography, of these investigations up to about 1925*
In the present chapter we shall give a brief account of the general
properties of orthogonal polynomials in two variables, and then discuss
in somewhat greater detail those systems of biorthogonal polynomials
in two and more variables which correspond to, and are generalizations
of, the classical systems of orthogonal polynomials in one variable.
There are many points of contact with Chapters 10 and 11.
12.2 General properties of orthogonal polynomials in two variables
The general properties of orthogonal polynomials in two variables
have been investigated by Jackson (1937) who also considered orthogonal
polynomials in three, and in two complex, variables (Jackson, 1938,
1938a). In this section, and in sec. 12.3, we restrict ourselves to the
case of two (real) variables. The corresponding properties for orthogonal
polynomials of n variables will suggest themselves to the reader.

266
SPECIAL FUNCTIONS
12.2
Given a region R in the x9 y-plane and a non-negative weight function
w(x9 y), both fixed, we shall assume in the case of a bounded region
that w is integrable over R9 and in the case of an unbounded region R
that all integrals
(1) J~ J~ w(x, y) x* yn dx dy m, n = 0, 1, ..«
R
converge. Orthogonal property, normalization, etc. will be understood to
refer to the scalar product
(2) (f, g) - J J fix, y) gU, y) w{x, y) dx dy.
R
Since / and g will be polynomials, the integral in (2) certainly exists.
The monomials xa yn will be ordered as follows:
(3) xn yn is higher than xk yl if
either m + n > k + I
or m + n = h + I and m > k%
The ordered sequence of monomials is
(4) 1, x, y, xz, xy, y2, x3, x2 y, ....
The ordering (3) induces a partial ordering of the polynomials in x, y.
A polynomial q {x9 y) will be said to be higher than p (x, y) if the highest
monomial (with non-zero coefficient) in q is higher than any monomial
(with non-zero coefficient) in p.
It is to be noted that the ordering (3) is arbitrary, and is not symmetric
in x and y% The orthogonal polynomials to be described below will be
based on (3): in general, a different ordering will result in a different
system of orthogonal polynomials.
Applying the process of orthogonalization described in sec* 10*1 to
the sequence (4), the scalar product being determined by (2), we obtain
a sequence of orthonormal polynomials which will be written as
(5) ?00> 9'°> q"> q™> ?21. ?22> ?30' ?31i —
so that q (x, y) is of degree n in x and y, and of degree m in y alone,
n = 0, 1, 2, **„ , m = 0, 1, *„„ , n. The orthonormal property is
<6> ten.,9iki)=84ll8fc
where 8 ^ 0 if r ^ s, and = 1 if r = s; and ^ is higher than qH if
either n > h or n ~ h and m > I.

12.2 ORTHOGONAL POLYNOMIALS IN SEVERAL VARIABLES 267
There are n + 1 polynomials of degree n in x and y, viz*,
Any polynomial of degree rc which is orthogonal to all polynomials of
lower degree is a linear combination of <7 „,..., <7 • Note that such a
polynomial is not necessarily orthogonal to all lower polynomials [lower,
that is to say, in the sense defined in (3)]«
With any real orthogonal constant matrix [c ..], where
n
(7) ^cucki=Sik »»*s0,l n,
j= o J J
the polynomials
n
(8) pni(x, y)= 2 cuq .U, y) j = 0, 1, ... , rc
; = 0 J J
are orthogonal to each other, normalized, and orthogonal to all
polynomials of lower degree (but not to all lower polynomials). Conversely,
any n + 1 mutually orthogonal, normalized polynomials which are
orthogonal to all polynomials of lower degree, may be represented in the form
(8) where the c.. satisfy (7)% Note that in Pni(x9 y)9 the subscript n
indicates the degree in x and y, but the subscript i does not indicate the
degree in y.
Suppose there is an affine tranformation
(9) x'=ax + /8y, y'= yx + 8y, a£-/3y=l
which maps R onto itself, and leaves the weight function invariant. For
each n9
form a system of n + 1 mutually orthogonal and normalized polynomials
which are orthogonal to all polynomials of lower degree. Thus, the
p {ax + /3y, yx + 8y) may be obtained by a real orthogonal
transformation of the q ni{x, y) and hence of the p ni(x, y). An affine
transformation (9) under which R and w are invariant induces, for each n,
an orthogonal transformation of p , ... , p n. Different systems of pni
(for the same R, w, n and a, j8, y, 8) undergo similar transformations; to
a group of affine transformations (9) which leave R and w invariant there
corresponds, for each n9 a group of orthogonal transformations. For further
details and for a reference to work by A. Sobczyk, see Jackson (1937)%

268
SPECIAL FUNCTIONS
12.2
If R is a rectangle,
(10) a<x <b, c <y < d,
and w (x9 y) = u (x) v (y), then we may take
(11) Pni(x,y) = pn-ik)q.(y) i = 0, 1, ... , n; n = 0, 1, ...
where \pn\ is the system of orthogonal polynomials associated with the
weight function u on the interval (a9 b) and \q \ the system of orthogonal
polynomials associated with the weight function v on the interval (c, d)9
12.3. Further properties of orthogonal polynomials in two variables
Let lpniU, y)\ be a system, of the form 12*2(8), of orthonormal
polynomials for the weight function w on the region /?. For each i9 p .(x9 y)
is a polynomial of degree n in x and y, and any polynomial of degree n
may be expressed as a linear combination of the p . (x, y), 0 < i <m9
0 < m < n. Several of the general properties of orthogonal polynomials
in one variable (see sec. 10.3) have their analogues in two variables,
although the corresponding formulas are less simple.
First, we shall prove the existence of a recurrence relation, expressing
(ax + by) pni(x, y) as a linear combination of polynomials of degree
n + 1, n, and n - 1. The proof is analogous to the proof of 10,3 (7)» For
fixed 7i, i9 the product
(ax + by) p ni(x9 y)
is a polynomial of degree n + 1, and hence of the form
n + 1 m
(1) (ax + by) p . (x9 y) = 2 2 . ym - pmj (x9 y\
m= 0 /= 0 J
(2) ymj. = / / (ax + by) p ni (x9 y) pnj (x9 y) w (x9 y) dx dy.
R
Since (ax + by) p m . (x9 y) is a polynomial of degree m + 1, and pniis
orthogonal to all polynomials of degree less than n, we see that
(3) yB/ - 0 m = 0, 1, ... , n - 2.
Thus, in (1), only terms corresponding to m = n — 1, n, n + 1 actually
occur.
It does not seem to be known whether the p ., that is to say the c ..
in 12,2(8), may be chosen so as to result in simple recurrence relations;
nor does it seem to be known under what conditions a system of
polynomials satisfying a recurrence relation of the kind described here, is a
system of orthogonal polynomials corresponding to a non-negative weight
function [compare the remark following 10,3(9)].

12.4 ORTHOGONAL POLYNOMIALS IN SEVERAL VARIABLES 269
As m the case of one variable, the recurrence relation may be used to
derive a relation which corresponds to the Christoffel-Darboux formula,
With the pnias in 12,2(8), we form
(4) K (*, y, u9 v)= £ 2 PkA*' y)PkM> v)
k=o i=0 l
(5) Ln(x9 y, u, v) = Kn(x, y, u, v)-Kn_y U, y, u, v)
(6) Mn(x9 y, u, v, r, s) = Ln+1 (u, i;, r, 5) £nU, y, r, s)
- Ln(u, v, r, 5) Ln+,U, y, r, 5).
Note that although the p ni are arbitrary to the extent of an orthogonal
transformation for each 1, the polynomials defined by (4) to (6) are
uniquely determined by the weight function w {x, y) and the region /?.
The "Christoffel-Darboux formula" is
(7) l(au + bv) -(ax + by)] Kn(x, y, u, v)
- I I ^ar + bs) M (x, y, u, v9 r, s) w(r9 s) dr ds •
For the proof see Jackson (1937)«
For the minimum properties of orthogonal polynomials in two variables
see Grobner (1948).
ORTHOGONAL POLYNOMIALS IN THE TRIANGLE
12.4. Appell's polynomials
Let T be the triangle
(1) x>0, y>0, * + y<l,
and
(2) t(x) = xy-*yy'-ia-*-y)a-w'
the corresponding weight function. The weight function is integrable if
(3) Rey>0, Rey'>0, Re a > Re(y + y') - 1,
but many of the formal results are valid without this restriction*

270 SPECIAL FUNCTIONS 12.4
Appell (1881) introduced the polynomials
(4) 3Ja,y,y;x,y) = (l-i-y)^
*-y vl ~y
* X ' V
—n J
-a
X •
dx;"1 dyn
[%y+--tyy+»-i (i_*_y)<*+«+*-y-y ]
which are analogous to Jacobi polynomials [cf. 10«8(10)]» Here, and
throughout this chapter,
(5) (o)0 = 1, (a)n = aU + 1) ... (a + n - 1) n = 1, 2, ...
(a)v-r(a+i/)/r(a).
For a detailed study of these polynomials, and for references to the
literature, see Appell and Kampe de Feriet (1926, Chapter VI and the
bibliography).
From equation (4) it is seen that 3 is a polynomial of degree m + n
in x and y. The expression of 3 in terms of Appell's hypergeometric
series Fz is given in 5.13(1).
Adopting the region (1) and the weight function (2) in the definition
of the scalar product 12.1(1), we see that
//
Pk§ y) d " [xy+«-' yr'+*-i(i_x_y)a+*+*-y-y']^y
dxn dyn
T
and repeated integration by parts shows that 3 is orthogonal to all
polynomials of degree < m + n. In particular,
(6) (3^,3^=0 m + ntk + l.
On the other hand, by repeated integrations by parts
(7) {-'dkl)-(y).(y\ &"*"
X
J J xy+*-x y?'+n-' a-x-y)a+"+n-'y ~y' dx dy
T
r(y)r(yor(a+m+n+i-y-y0( d'+n 3H
r(a + 2m + 2n + l) dx* dyn
m + n =k + I,

12.4 ORTHOGONAL POLYNOMIALS IN SEVERAL VARIABLES 271
and since this does not vanish, the polynomials <5 m do not form an
orthogonal system* No orthogonal or biorthogonal system of polynomials
seems to be known for the weight function (2).
The system of partial differential equations satisfied by
(1 - x - y)a~y "y om(a, y, y\ x, y)
maybe derived by means of 5-13(1), 5.11(8), 5,9(10). With the notations
_ dz _dz _ dzz dzz _ dzz
dx' dy dx2' dx dy dy2
it reads
(9) x(l -x)r-xys + [y- (2y + y'- a~n + l)x~} p
- (y + ™)yq - (y + tn)(y + y'~ a-m -n) z = 0
y(l - y) t - xys + \.y'-\y + 2y'~ a- m + 1) y] q
— (y'+ n\xp — (y'+ n)(y + y'— a — m — n) z = 0.
When a= y+ y', the weight function (2) simplifies to
(10) tQ(x) = xy-] yy'~] Rey, Rey'>0.
For this weight function Appell (1882) considers two systems of
polynomials
(ID Fm(y,y',x,y)=3Jy+y'>Y>y'>*>y>>
= - (**+■-* yy+»-»(i-.*^ y)»+«]
= Fz(-m -n, y+ro, y'+n, y, y'; x, y)
(12) EmW> y'> xty)= F2(y+y'+m + n,-m, -n, y, y'; x, y)
where F2 is the series defined in 5.7(7). The partial differential
equations satisfied by F m and E m may be derived by means of 5.9(10). They
are
(13) x(l - x)r -xys + [y - (y - n + l)x] p - (y + m) yq
+ (m + n)(y + m) z = 0
F
mn
y(l - y) t -xys + [y'~ (y'— m + Dy] q - (y '+ n)xp
+ (m + n)(y'+ n)z = 0

272
SPECIAL FUNCTIONS
12.4
(14) x(l -x)r -xys + [y-{y+y'+n + l)x]p
+ myq + m{y + y '+ m + n)z = 0
Ml
y(l - y) t - xys +[y'-{y+ y'+ m + Dy]q
+ nxp + n(y+y'+m + n)z = 0
Adding each of these two pairs, it is seen that both F and E satisfy
the partial differential equation
(15) x{l-x)r-2xys + y(l-y)j+ [y-(y + y'+ l)x]p
+ [y'~ (y + y'+ Dy]<7 + (m + n)(y + y'+ m + n)z = 0,
and this partial differential equation may be used to prove that
(16) J f xy< yr-t FJy, y\ x, y)Ekl{y, y', x, y) dx dy
T
vanishes except when m = k and n - Z. This shows that the two systems
of polynomials (11) and (12) form a biorthogonal system for the region
(1) and the weight function (10).
The formula
(17) J J %y~' yy'-' FJy, y', *, y) EH(y, y', *, y) dx dy
T
8^ 8nl mini (m + a)! T(y) r(yQ
y + y '+ 2m + 2n (y)R (y *)nr(y+y'+m + n)
is proved in Appell and Kampe de Feriet (1926, p. 110, 111)% It may be
used to compute coefficients in the expansion of an arbitrary function in
a series of the F , or in a series of the E • Two examples of such
win nn
expansions are
V (k + l)\ (y+m), {y'+n)j
(18) FJy,y',*,y)= I ',\ '* ' g„W,«,r)
(19) (l-*-y)X-1= 1 1 (-l)"+n(y+y'+2m + 2/i)
»= 0 n- 0
77i!n!(m + ti)! T(y + y'+ A+ m + n) m

12.5 ORTHOGONAL POLYNOMIALS IN SEVERAL VARIABLES 273
(Appell and Kampe' de Feriet, 1926, p. 112, ll3)« In (18) summation is
extended over all non-negative integers k and I for which k + I = m + n.
For the case y = y'= 1, a - 2, when the weight function is constant,
see Grobner (1948» sec. 5)*
ORTHOGONAL POLYNOMIALS IN CIRCLE AND SPHERE
12.5. The polynomials V
In this section and in the following section we shall use notations
similar to those of Chapter XI.
(1) ?-(*,, ... ,xn)
will be a vector, with (real) components x 9 ... , x in n-dimensional
(real) Euclidean space, and
(2) ||?||=r = (** + ...+**)*
will be the length of this vector. With two vectors
(3) a « (a j , ... , an), y = (x ], ... , x n)
we associate the scalar product
(4) (a, £) = a x. + ... + a x
and the angle 0, where
cos 0 =
IKII ll?ll
[The scalar product (4) of two vectors is to be distinguished from the
scalar product of two functions occurring in (17), 12,6(4), and similar
relations.] The unit sphere, ||y|| < 1, in our space will be denoted by S,
the element of volume by dx, so that
Jftydx
s
will be written for
/••• J f(x\* ••• > xn) dx\ •" dxn*
x\ + ••• + x1 < 1

274
SPECIAL FUNCTIONS
12.5
We shall consider orthogonal polynomials in the region S with the
weight function
(5) (1 -r*)**-K = (1 - x\ x*)**-*.
1 n
For n = 2, the region is a circle in the plane, for n = 3, a sphere in three-
dimensional space, and for n > 3, a hypersphere.
Polynomials
«> VJ*)-Vmi,mt „_(*,.*,.....*„>
will be defined by the generating function
(7) [l-2(a, 5)+-||a||*r*"'**+*
= 2 a !.«a n F5 (* a; ).
1 71 K77Jf 771 V*l » "■ 9 *n>*
l n
In this sum, and in all similar sums, summation will be understood to
take place over all non-negative integers m|f ... , mn. Clearly, Vs (£) is
a polynomial of degree mk in xk, being an even or odd polynomial in x,
according as m, is even or odd; and
(8) m = my + ••• + mn
is the degree of this polynomial.
For n = 1, a comparison of (7) and 10«9(29) shows that
(9) F*(*)=C**(*) n = l.
For n = 2 and s = 0, 2, the polynomials (6) were introduced by Hermite
(1865,1865 a), for any n by Didon (1868)% There is a detailed presentation
of these polynomials and of related matters in Part Two of the book by
Appell and Kampe* de Fe*riet (1926) where there is also an extensive
bibliography. Additional references are listed at the end of this chapter
under Angelescu, Appell, Brinkman and Zernike, Caccioppoli, Chen,
Dinghas, Erdelyi, Koschmieder, Orloff, and Schmeidler.
The expansion in powers of a , ... , a^ of the generating function
(7), by the multinomial theorem, leads at once to the explicit representation
(10)
/n + S-n ft/'-x/
V™1 mn(*.'-'*n> = ^ —J m ,...„ ,
N ^77i I n
( m\ mn 1-ffti l~nn
\~T"" T'~Y~'"" 2 '
n + s — 3
- /n —

12.5 ORTHOGONAL POLYNOMIALS IN SEVERAL VARIABLES 275
(11) FB(at, ... , an, /3,, ... , fin, y; z |f ... , z n)
-2
<«,>«.-<«»>» (0i>».-OA
77l1 ! ••• 772
! (y)
71 V/7J
... Z
771 j + ••• +771
is one of Lauricella's hypergeometric series of n variables (Appell and
Kampe de Feriet, 1926, Chapter VII). There are also representations of
Vs in hypergeometric series of ascending (rather than descending) powers
of the xk, these representations being different according to the parities
of the 772 k [see also 10,9(21) and 10*9(22)].
If one puts ak = tb k in (7), and compares coefficients of t* on both
sides, the relation
(12) [|6||»C*"+**-* [i^JLl
1,1 ■ L -ii'ii J
771 + ••• + 771 =771
1 n
771 , 771
!>, ' ... 6 " Vi
i n 77i.
\xJf ... , %n)
is obtained.
It may be verified from the explicit formula that the polynomial
defined by (10) satisfies the following (hypergeometric) system of partial
differential equations
(13)
d ( dV I" n dV 1 ")
!7~ ]1T."Xi I (m + n + s-l)V+ I xk — \f
+ (m.+ 1)
BV
= 0
(m + n + s-l)V+ > x k
k=y dxk
j = 1» .•• » n,
where m is the degree given by (8). Adding these n equations, we see
that all polynomials of degree m satisfy the partial differential equation
(14) (m + n)(m + s~ l)V

276
SPECIAL FUNCTIONS
12.5
There is a remarkable symbolic representation of our polynomials,
(15) F'(5p)-.
»,!
x 0F, (-B/2 - s/2 + 3/2 + m; A 74) (* , ' ••• x„ ")
where F, is a generalized hypergeometric series [see 4« 1 (1)] and
(16) A=—— + -..+■
dx* dx2
I n.
is Laplace's operator. This representation is derived by means of the
connection between the polynomials V 5 and hyperspherical harmonics
(see sec. 11.8). The same connection may be used to show that the
integral
(17) |(Nr2)MF:Wr^)&
s
vanishes if m ^ m ', and also if m - m ' and some of the differences m .— m^
are odd numbers. Since the integral does not vanish when m = m 'and all
differences m. — m' are even numbers, the Vs do not form an orthogonal
system of polynomials.
The formula corresponding to Rodrigues' formula [equation 10,9(11)]
is
(18) »,!...»|il(l-r«)*-+-+-'>F; m fe, «„)
1 n
■3771
_ ( i)m q _ r2\^(n+s-I)
771 f 771
where, on the right-hand side,
(19) y. = *.(l-r2)-* i = l, ... n
are the independent variables, and
(20) l-r2 = (l+ ll^ll2)-1.
The formula may be derived from the generating function (7) by the
substitution (19) and upon replacing a ^by a f(l - r2r.

12.6 ORTHOGONAL POLYNOMIALS IN SEVERAL VARIABLES 277
The generating function is also the source of the integral representation
(21) irx" m ,! - mn\ r(H«) V\(i) - i" Oi + s - 1). rOin + K2s)
* J ^'-^"^-^'"Ulall+ *'(?. a)]-""-**1^
For other integrals see Dinghas (1950).
Recurrence relations, differentiation formulas, and similar relations
also follow from the generating function and are recorded in Appell and
Kampe* de Feriet (1926, sec, LXXVI).
12.6. Tne polynomials U
A second system of polynomials,
a) W = <£ »(«,.....*.)
l n
will be defined by the generating function
(2) l[(a, 0-l]2+||a||2(l-|:^||2)r^
mt . ™n rrs
= 2 a/ -ann Usm^ m (x%9 ... ,xn).
1 n m, , ...,m_ 1 ' ' n'
For n = 1, we have
(3) U9mix) = C*'(x) n=l.
For n = 2, s = 1, 2, these polynomials were introduced by Hennite; for
any n see the literature quoted in sec. 12,5•
The most important property of these polynomials is the biorthogonal
property which connects them with the Vs. The integral
(4) /(l-r2)**-* V($)U\{$)dx
s
vanishes, except when m } = ly 9 ... 9 m = I ; and
(5) /(l-r2)**-* V>{z)U*(z)dx = h*
2n%n r«s+l) (s)m
2m + n + s-\ r(y2n + Y2s -V2) 77i f!..;../7in!
This biorthogonal property may be proved from the generating functions
(see the corresponding proof for Hennite polynomials in sec. 12,9)»
Conversely, Kampe de Feriet (1915) postulated the biorthogonal property
and deduced the generating function from it.

278
SPECIAL FUNCTIONS
12.6
The theory of the polynomials U resembles that of the polynomials V
and we shall simply list some of the relevant formulas*
Explicit representation
(6)
f/i m (x., ... , x) =
m ., ... , 771 1 ' ' 71
771 , 771
771 . J ••• 771 !
I 71
( m , m 771, 772 S + 1
v J? I I n 1 I I n ____ .
*\" T' T 2"* ~T~'
l-r2 1 - r2 ^
2 ' •" ' 2 J
with corresponding series in ascending powers of
(7) t(4, ?)t+||6tr(l-r^
" \[(*>, ?)2 +||i||2 (l-r2)] V
771 . 771
2 6, ' -bnnU° m («, * )•
m,+... + m =m ' " mi mn ' "
i n
The polynomial £/ satisfies the system of partial differential
equations
(8) (l-r2) -_ _+«( m£/- 2 *4 - )
+ m.(l-r2) (,ȣ/- ^ *kjf)
_(s- 1) x. +xz ( mU- Y *,, ) -m.£/ =0
] — -Lj ••• 9 71 •
All polynomials of degree m satisfy the partial differential equation
(9)
V d f dU
(m + n){m + s -l)U + / —' <
/-> dx. dx.
-x. {s-l)U+ T x. > =0
3 L i, *** J J

12.6 ORTHOGONAL POLYNOMIALS IN SEVERAL VARIABLES 279
which is obtained by adding the n ecpiations (8), and is identical with
the corresponding equation 12,5(14) for the Vsn*
The symbolic representation may be written in the form
(10) U'a (?) =—^ r 0F, ft* + X; -XU - r2) A2](*7 ' •-> **"),
i n
where the 4-th power of (1 -r2) A2 is to be taken as (1 - r2)k A2*, There
is also a relation corresponding to 12,5(17) but it is of little importance.
The analogue of Rodrigues* formula is simpler in this case than in
the case of Vs •
(11) 2™ (-ill) »,!..., „Kl-r«>K-X0: U, «,)
= (-ir (s) ————— a-=r*r **•-*.
771 771 . m
dx\* "' dxn"
Koschmieder (1925) obtained expressions for the Us in terms of partial
derivatives with respect to x . ,
The integral representation corresponding to 12,5(21) is
(12) 7T1Anm^\-mn\r(y2s -y2n + y2) usm(%)
= (s)mr(y2s + y2)fs u-r*)**-**-*
x[*,+ 13,(1-Hall2)*ff - [»„ + &„(!-Hall2^fBifa.
The two systems of polynomials, i/* and Fs are connected: the
connection may be expressed in two equivalent forms,
(13) (2-2»-»-«).(r'-l)^y:^(r^1)XJ
/ra +s- l\
= 2"f 1 f/2_ "'S(?)
(M)2-(-»-^B(r»-l)*-l/:[TFi^]
-(«). p*-2*1-"-*(?).

280
SPECIAL FUNCTIONS
12.6
The biorthogonal property has already been stated in (4) and (5). Another
connection, closely related to the biorthogonal property, is given by the
circumstance that the system of partial differential equations satisfied by
which can be derived from (8) and is
(15)
(m + s - l)R - ) x- ——
dx. \dx. >l
r, x v m i
+ m.\ (m + s-1)R- > x. =0
'L i, '**.J
}
} — 1, z, »tf», n,
is easily seen to be adjoint to the system 12*5(13) of partial differential
equation satisfied by Vs (£),
12.7. Expansion problems and further investigations
The biorthogonal property of the U and V suggests the expansion of
an "arbitrary" function f($) in either of the two series
a) s «:#;(?)
(2) SiJF'te).
From 12,6(4) and (5) one obtains the expressions
(3) *;«;-/ a-r*)*'-*f(?)vB(?)dx
(4) hsB j;=J(l-r2)Mf(y)[/;(y)&.
A general discussion of such expansions is contained in the book by
Appell and Kampe' de Feriet (1926, Part II, Chapter V). More precise
results were obtained by later writers.
In studying the expansion problem it is usually assumed that 5 is a
positive integer in (1) and (2). Koschmieder calls (1) and (2) Appell
series if s > 2, Didon series if s - 1, and shows that an Appell series
in n variables may be reduced to a Didon series in n + 5 — 1 variables.
Moreover, it is usual to rearrange the multiple series (1) and (2) as a
simple series, by grouping together all the terms of equal degree. Thus,
(1) is interpreted as

12.7 ORTHOGONAL POLYNOMIALS IN SEVERAL VARIABLES 281
(5) f [ 2 a* m V* m (*f, * )],
i rt
and there is a similar interpretation of (2)* The rearranged series may
then be related to the Laplace expansion of a function on the surface
of the unit hypersphere in n + 5 + 1 dimensions, and this connection has
often been used.
Convergence of the series (1) and (2), rearranged as described above,
has been investigated for n = 2, s = 1 by Caccioppoli (1932), and by
Koschmieder (1933)« Caccioppoli summed the series and discussed its
convergence by means of a singular integral, proving convergence for
continuously differentiable functions. Koschmieder used the theory of
integral equations and proved absolute convergence for twice
continuously differentiable functions.
The case of general n and (positive integer) 5 was investigated by
Koschmieder (1934). Adopting the interpretation (5) for (1), and a
corresponding interpretation of (2), Koschmieder showed that these series
are equiconvergent with certain expansions in Gegenbauer polynomials.
Koschmieder (1934 a) also obtained an equiconvergence theorem for
Laplace's expansion, with a Fourier series as a comparison series.
The Cesaro summability of Laplace' series has been discussed by
Chen (1928) and Koschmieder (1929). The results have been applied to
Appell's series by Koschmieder (1931).
The Appell series of a function f($) which is integrable in 5, is
(C, 8) summable to /(£) almost everyhwere in 5, and certainly on the
Lebesgue set of / in 5, when
(6) 5 > n + « - 1.
Moreover, the Appell series is (C, 8) summable also for
(7) y2(n + s - 1)< 8<n + s - 1
at all those points ^ for which
iif-$irK<"+<-«> irt^i
is an integrable function of £ in S.
The following examples of expansions are taken from Appell and
Kampe de Feriet (1926, sections LXXXVIII and XCI).

282
SPECIAL FUNCTIONS
12.7
(a e,,^. v'-"" <-*>-'* w
xa7'...^"||a||*-m^(?),
where Ms a positive integer, and summation is over all those values of
m , ••• , m for which i - m is a positive even integer;
(9) exp [i (a, 0] = 2*rt+^-l/* r(Y2n + Ks -H)
x V ,* (,+ " + *"!l) a7'...«"-||a|r-*->«*+M
xJm+y2n+Hs^(||a||)^(?)
(10) r(s + ^)exp(a, $)Jy2S_H[||a||(l-r2)]
= D<||a||(l-r«)]»-« ^ ^- «7« ... ,£» f/* (?).
In the last two expansions, summation is over all non-negative m , ...,m ♦
The case n — 2 has been investigated in greater detail [see Appell
and Kampe'de Feriet (1926, Part II, Chapter VII), and the papers quoted
in sections 12„5- 12,7 of the present chapter]. An alternative approach
to orthogonal polynomials in spherical regions was suggested by Brink-
man and Zernike (I935)and by Grobner (1948)« Polynomials connected with
the partial differential equation A9F = 0 in spherical regions were
studied by Giulotto (1939) who obtained a biorthogonal system for this
case. Devisme (1932) introduced polynomials defined by the generating
functions
(H) (l-3a*+ 3a2y-a3rv, [1 - Sax + 3(a2 - b) y - a3]"v
which arise in the study of the partial differential equation
dzu d3u d3u d3u
(12) A *= - + -+ --3 = 0.
3 dx3 dy3 dzz dxdydz
A generalization of the polynomials Us, Vs may be defined by means
of a fixed quadratic form, <£($), its reciprocal form iJ/($, and the bilinear
form <£(£, fc)) [see 12.8(6) to (8)1 • The generating functions are

12.8 ORTHOGONAL POLYNOMIALS IN SEVERAL VARIABLES 283
(13) \[</>(a, y)- l]2 + 0(a) [1 -^(?)]}-*s= £ a™' — o™" U* (?)
(14) [1 - 2(a, 5) + ^(a)]-1^71-1/^ +* = £ a™ ' ... a* » l)« (?).
These polynomials have been introduced by Hermite and were studied
by Angelescu (1916). If <£($) = (& y) = ip($), the polynomials defined
by (13) and (14) are [/* and F* respectively.
HERMITE POLYNOMIALS OF SEVERAL VARIABLES
12.8. Definition of the Hermite polynomials
As in the preceding sections,
(1) £ =(*,, ... , xn)
will be a (real) vector,
(2) ||?|| = (*? + ••• + **)*
the length of £, and
(3) (a, j) = of %f + ... + aA^A
the scalar product of two such vectors. C will be a fixed positive definite
symmetric square matrix of real elements, i.e.,
(4) C = [c£.] i, /= 1, ... , n
c..= c. real, X c..#.x.>0
£^0.
The reciprocal matrix will be denoted by C"1: its elements are y ./A,
where
(5) A = detc i,/ = 1, ... ,rc
is the determinant of C, and y^. is the cofactor of c^ in A. With C we
associate the positive definite quadratic form
(6) ^(e) = (ce, ?) = (?, Cf)= i *..*,*,
i, / = I J J
and the symmetric bilinear form

284
SPECIAL FUNCTIONS
12.8
(7) <£(&*})= (C& *)) = ($ Ci))= S c..*.v..
i,; = i lJ J
We also have the reciprocal form
(8) 0(f) -^(C-1 ?) = (C_I ft f) -(ft C-' ?),
which is also a positive definite quadratic form, and the reciprocal
symmetric bilinear form
(9) <£(¥, $) =(£-'& $)-(jp, C"1^),
These forms are connected by a number of relations.
(10) #(?+$)« #(0 + 2#(&$)+#($)
^(?+ $)->(0 + 2 «/,(& $) + t/j(i,)
<£(0=^(C0, 0(0 = #(£-'0
(11) (?S(?:+C-,^) = ^(?;) + 2(ft $) + #($)
(12) #(?+C$)« 0(0+ 2($ $) + #($.
Lastly we mention the integral formula
(13) J" exp[-J/2(£(?)+(a, f)J(fa-=(2lr)}{"A-x exp[K2«£(a)]
where integration is over the whole space, dx stands for dx • •• eke ,
and Ct is a constant vector. The formula may be proved by using (11)
and then transforming the quadratic form <£(y + C*1 a) into a sum of
squares*
The notations introduced above will be used throughout this section
and the following sections*
Hennite polynomials of several variables are a biorthogonal system
of polynomials associated with the weight function
(14) n>(jf)= A* (2ir)-Xn expl-H<£(£)],
the region being the entire rc-dimensional space* The relation
(15) Ju»(f)i = l
is a consequence of (13)* These polynomials are clearly rc-dimensional
generalizations of the orthogonal polynomials defined by 10« 13 (1)« They
were introduced by Hennite (1864) and since then studied by many authors.
Appell and Kampe de Feriet (1926, Part III) give a detailed presentation
of the theory, as of 1926, and a bibliography* Additional references are

12.8 ORTHOGONAL POLYNOMIALS IN SEVERAL VARIABLES 285
listed at the end of this chapter under Caccioppoli, Erdelyi, Feldheim,
Grad, Koschmieder, Mazza, Picone, Thijssen, and Tortrat. Extensions
to infinite-dimensional spaces are due to Cameron and Martin (1947),
and Friedrichs (1951, see in particular p. 212ff)«
Two systems of polynomials,
(16) CB(y)=Gm(_>mn(*,,... ,*„)
HmW~Hmt,...,m C* ,,...,*„)
\7 * n
will be defined by means of the generating functions
(17) cxp[(C a, f) - y2<f>(a)] = exp\%<f>($) - %<£(?- a)]
m
i-l in.!
m, m
m,! mn\
(18) exp[(a, ?)-X0(a)]-wpDi^yj-K^p-C"1 a)]
77i, m
ir7-trc«(f)'
which are extensions to several dimensions of the generating function
10.13 (19). In all sums m) , ••• , m n run through all non-negative integers,
unless other regions of summation are explicitly stated. The polynomials
defined by (17) and (18) are of degree m { in x jf and their (total) degree
is
(19) ro = to, + ••• + to •
In these definitions we followed Appell and Kampe' de Feriet (1926,
sec. CXVIII)« For n - 1 we have the Hermite polynomials defined in
sec. 10„ 13 ii we take cu = 2.
771 ! 771
If the coefficients of a ] ••• a^n in the generating functions (17) and
(18) are computed by means of Taylor's theorem, one obtains
(20) #m(?)= (-ir expK^(y)] — — exp[-J^(?)]
<9*, ' •••<9*n"
(21) Gm.(C-' ?) = (-ir expfttf (?)] —^-2 — exp[-X0(?)]
l n

286
SPECIAL FUNCTIONS
12.8
corresponding to 10.13(7). Koschmieder (1925) has given an alternative
expression in terms of partial derivatives for certain Hermite polynomials
of two variables. Either (17) and (18) or (20) and (21) may be regarded as
definitions of Hermite polynomials in several variables.
An • alternative notation, in case of a special quadratic form 0(y),
has been proposed by H. Grad (1949)*
12.9. Basic properties of Hermite polynomials
The most important feature of Hermite polynomials is the biorthogonal
property
(1) /•»(?) Gl (?) HJ$) dx=St m -. St m «, ! - mn \,
11 n n
where w($) is the weight function defined by 12.8(14), S is defined
in sec. 12*2, and
I = I + ...;+ I ,
1 n
To prove the biorthogonal property, we remark that by 12*8(14), (17)
(18), the integral on the left-hand side of (1) is the coefficient of
11 I _ m f m
(2)
1
1 ~n v 1 ~ n
a.' a." b ' ft"
/. ! L\ m,! m„!
1 n 1 n
in
(3) (2 n)-*n A"* / expt-1/^^) + (a, f) -■}£ 0(a) + (C 6, f)- Ji0(6)]cfc.
By 12.8(13), the expression (3) is equal to
(4) exp[H0(a+Cfc)--H0(a)-X0C6)l
and by 12.8(12) this is
r, *n X Ml^ K^)^
(5) ~r/- ^- x 1 ! n n
X(a,b J
—H-
772, I
The coefficient of (2) in the series (5) gives the right-hand side of (1).
A bilinear generating function, corresponding to Mehler's formula
10.13(22), may be obtained in a similar manner, computing the integral

12.9 ORTHOGONAL POLYNOMIALS IN SEVERAL VARIABLES 287
(6) (nnt ... * r' f f exph 2 U^ + v!)A.+ J4^(y)
- %<£>(£- u- ib) + %0(ij)- %0(ij--u+ ib)] du dv
for sufficiently small positive £ , ... , £n in two different ways, once
by using 12,8(13), and another time by first using 12„8(17) and (18),
and then integrating directly. Putting
(7) 0.(0= £ %V*, + 0(5)
02(0= 2 X*/t -<!>($,
noting that for sufficiently small positive t , ... , t the quadratic
forms 0^(j), k = 1, 2, are positive definite, denoting the determinant
of 4>k ky &k9 and the reciprocal quadratic form by iffk($)9 the result is
m m n
(8) Z ITT^TT^-^^-^
= u,.- ^r1 (a, A2r* expcx^/cy+c^-x^^cf-c^].
In this form the result was obtained by Erdelyi (1938 a) together with
the corresponding result for the generating function of H (y) G (^),
thus extending results by Koschmieder (1938, 1938 a) who gave explicit
formulas for n - 2. The bilinear generating function was also discussed
by Tortrat (1948, 1948 a).
The system of partial differential equations satisfied by H (£)
may also be derived from the generating function. The function on the
left-hand side of 12.8(17) satisfies the system of partial differential
equations
A dzF A y (9F dF
l = J., Z>) ••• , 71

288
SPECIAL FUNCTIONS
12.9
where A is the determinant of the c .. and y . is the cofactor of c .in A.
\) % T ij t j i
Expanding in powers of a., we obtain the following system of partial
differential equations for H (p)
n
(9) I
J- 1
dzH
dx .dx
n
~ 2 cikxk
dH
i t= i
j -i
m.Aff-0
£ — Lj auw j " •
The partial differential equation
(10) 1 1 yij^—z—A I xk-j-
£ £ * dx.dx kf} dxk
- mMi = 0
is obtained by adding the n equations (9) and is common to all
polynomials of the same degree /n.
The proof of the system of partial differential equations
(id I
d2G dG
v.. Ax + ro.AG=0
/ j • - -
13 dx .dx .
1 dx.
i — ij tfMI) ^ n
ac
i = t
J=1
> xA + mAG = 0
*=1
satisfied by G^ (£) is similar.
Recurrence and differentiation formulas may also be obtained from
the generating functions. For n = 2 they are recorded in Appell and
Kampe de Fe'riet (1926, sec. CXXIl).
There are many connections between Hermite polynomials in one and
those in several variables. Replacing a by £ain 12*8(17) and (18), and
expanding in powers of t by 10*13 (19) we obtain
(13)
i
1 n
i,\ m ! mt mn '
K^a)]14"
H.
ml

12.10 ORTHOGONAL POLYNOMIALS IN SEVERAL VARIABLES 289
(14)
I
f
7 n
— Cm
.,m <*,.■■•.*»)
[y2^(a)]1/2m
#,
/(a, y) ^
\[2^(a)]V '
For other connections between Hermite polynomials of one and those of
several variables see the book by Appell and Kampe' de Feriet, and the
papers by Feldheim, listed at the end of this chapter. Note that Feldheim's
notation differs from our notation.
An addition theorem for Hermite polynomials in two variables was
obtained by Koschmieder (1930 a).
12.10. Further investigations
By a comparison of the generating functions it is easy to see that
Hermite polynomials of several variables are limiting cases of the
polynomials defined by 12*7(13) and (14).
(1) lim s^aU*(_L)= 1 H (?)
x ' 1 n
(2) lim s~*a U*(-4- ) = i G (y).
For the further investigation of Hermite polynomials one may use the
multi-dimensional Gauss transform
(3) G£ [F{b,)\=Ay>(2 7Tw)-'Anf F(b>)exp \-~ ^(?-
^ L 2u
[see equations 10,13(30), (31)]. The first of the formulas
A;
*3)
dy
(4) (J"[ff (A &)]=(!-A*B)*"ff,
(1
\± 1
-AM* J
n n
(5) (Jlw (^)]= n ( s c..x.)
*<C m
i=l j=l lJ J
n , n
(6) Q^iM.^.y,)]^-"-//^)

290
SPECIAL FUNCTIONS
12.10
may be proved from the generating function 12«8(17), and is an integral
equation satisfied by Hermite polynomials; the second is a limiting
case of the first; and the third, which is also a limiting case (A■-> <*>) of
the first, is an integral representation of Hermite polynomials. The
corresponding formulas for G are
(7) Q^[G.(^)]Ml-X2u)^G^(i^2^j
(8) Q^[Gm(i3)]= fl x.
(9) Gjjfi^.l-r-c.(*?).
Feldheim (1942) used a more general definition
(10) G"[F(lj)]- -5-
^ (a j ••• anr2
and investigated the behavior of Hermite polynomials under the functional
transformation defined by (10).
The biorthogonal property 12 & (1) shows that an "arbitrary" function
/"(£) may be expanded in series of Hermite polynomials in either of the
two forms
(11) ZanGJ?)
(12) Si.fl.f?)
where
(13) m I • •• m
(14) in,!-mjfc. =Jw($)f($)GJ$)dx.
The convergence of such expansions was discussed by Thijssen (1926,
1927) for the case n = 2 and functions /(£) which vanish identically
outside a bounded region, and satisfy certain continuity requirements in
that region. The problem of approximations in mean square (see sec.
1092) was discussed by Caccioppoli (1932a) for functions of the class
L 2 , that is to say for functions for which the integral

12.10 ORTHOGONAL POLYNOMIALS IN SEVERAL VARIABLES 291
J 1/(01* erp[-X*(?)]<**
is convergent. The approximation to arbitrary functions in unbounded
regions has also been discussed by Picone (1935)*
Mazza (1940) has also discussed Hermite polynomials and constructed
an orthogonal system* Devisme (1932) defined systems of polynomials
which are, in some measure, analogous to Hermite polynomials• The
generating functions are
(15) exp(ax - azy + a3/3), exp [ax -(a2 - b) y + a3/3].
The polynomials generated by (15) are related to certain partial differential
equations involving the differential operator 12,7 (12).

292
SPECIAL FUNCTIONS
REFERENCES
Angelescu, Aurel, 1915: C.R. Acad, Sci. Paris 161, 490-492.
Angelescu, Aurel, 1915- 16: Bull. Math. Soc. Roumaine Sci. 4, 30-35.
Angelescu, Aurel, 1916: Sut des polynomes generalisant les polynomes de
Legendre et d'Hermite et sut le calcul approche des integrales multiples,
Thesis, Paris, 140 pp.
Appell, Paul, 1881: Arch. Math. Physik (1) 66, 238-245.
Appell, Paul, 1882: /. Math. Pures Appl. (3) 8, 173-216.
Appell, Paul, 1901: Arch, Math. Physik (3) 1, 69-71.
Appell, Paul, 1903: Arch. Math. Physik (3) 4, 20-21.
Appell, Paul and Joseph Kampe' de Feriet, 1926: Fonctions hypergeometriques et
hyperspheriques, Polynomes d' Hermite, Gauthier-Viliars, Paris.
Brinkman, H.C. and Frits Zernike, 1935: NederL Akad, Wetensch, Proc. 38, 161-
170.
Caccioppoli, Renato, 1932: Rend. Sem, Mat. Univ, Padova 3, 163-182.
Caccioppoli, Renato, 1932a: Giorn, 1st. Ital. Attuari 3, 364-375.
Cameron, R.H. and W.T. Martin, 1947: Ann. of Math. 48, 385-389.
Chen, K.K., 1928: Sci, Rep, Tohoku Imp.Univ. Ser. I, 17, 1073-1086.
Devisme, Jacques, 1932: C.R. Acad. Sci. Paris 195, 437-439, 936-938.
Didon, Francois, 1868: Ann, Sci. Ecole Norm, Sup, (1) 5, 229-310.
Dinghas, Alexander, 1950: Math, Z, 53, 76-83.
Erdelyi, Arthur, 1938: Math. Ann, 11, 456-465.
Erdelyi, Arthur, 1938a: Math, Z, 44, 301-311.
Feldheim, Ervin, 1940: Ann. Scuola Norm. Super. Pisa (2) 9, 225-252.
Feldheim, Ervin, 1941: C.R. (Doklady) Acad. Sci. URSS CV.S.) 31, 534-537.
Feldheim, Ervin, 1942: Pont, Acad, Sci. Comment. 6, 1-25.
Friedrichs, K.O., 1951: Comm. Pure Appl. Math, 4, 161-224.
Giulotto, Virgilio, 1939: 1st, Lombardo Sci. Lett, Rend, Cl, Sci, Mat. Nat. 72,37-57.
Grad, Harold, 1949: Comm. Pure Ai-pl. Math. 2, 325-330.
Grobner, Wolfgang, 1948: Monatsh, Math. 52, 38-54.
Hermite, Charles, 1864: C.R. Acad. Sci. Paris, 58, 93-100, 266-273.
Hermite, Charles, 1865: /. Reine Angew, Math. 64, 294-296.
Hermite, Charles, 1865a: C.R, Acad. Sci. Paris 60, 370-377, 432, 440, 461-466,
512-518.

ORTHOGONAL POLYNOMIALS IN SEVERAL VARIABLES 293
REFERENCES
Jackson, Dunham, 1937: Duke Math. J. 2, 423-434.
Jackson, Dunham, 1938: Duke Math. J. 4, 441-454.
Jackson, Dunham, 1938 a: Ann. of Math. (2) 39, 262-268.
Kampe de Feriet, Joseph, 1915: Sur les fonctions hyperspheriques, Thesis»Paris,
111pp.
Koschmieder, Lothar, 1924: Math. Ann. 91, 62-81.
Koschmieder, Lothar, 1925: Jber. Deutsche Math. Verein 34, 57-64.
Koschmieder, Lothar, 1926: Revista Mat. Hisp.-Amer. (2) 1, 97-107.
Koschmieder, Lothar, 1929: Math. Ann. 101, 120-125.
Koschmieder, Lothar, 1930: Revista Soc. Mat. Espanola (2) 5, 1-14.
Koschmieder, Lothar, 1930a: Revista Soc, Mat. Espanola (2) 5, 274*280.
Koschmieder, Lothar, 1931: Math. Ann. 104, 387-402.
Koschmieder, Lothar, 1933: Monatsh. Math. Phys. 40, 223-232.
Koschmieder, Lothar, 1934: Math. Ann. 110, 734-738.
Koschmieder, Lothar, 1934 a: Monatsh. Math. Phys. 41, 58-63.
Koschmieder, Lothar, 1938: Math. Z. 43, 248-254.
Koschmieder, Lothar, 1839a: Math. Z. 43, 783-792.
Koschmieder, Lothar, 1940: Anz. Akad. Wiss. Wien. Math.-Nat. Kl. 41-43.
Mazza, S.C., 1940: An. Soc. Ci. Argentina 130, 137-148.
Orloff, G.A., 1881: On some polynomials in one or several variables, Thesis, St.
Petersburg, 124 pp.
Orlow, G.A., 1881a: Nouv. Ann. (2) 40, 481-489.
Picone, Mauro, 1935: Giorn. 1st. Ital. Attuari 5, 155-195.
Schmeidler, Werner, 1941: /. Reine Angew. Math. 183, 175-182.
Thijssen, W.P., 1926: Verslagen Amsterdam 35 (2) 1100-1111.
Thijssen, W.P., 1927: Nederl. Akad. Wetensch, Proc. 30 (1), 69-80.
Tortrat, Albert, 1948: C.R. Acad. Sci. Paris 226, 298-300.
Tortxat, Albert, 1948a: C.R. Acad. Sci. Paris 226, 543-545, errata 758-759.

CHAPTER XIII
ELLIPTIC FUNCTIONS AND INTEGRALS
13.1. Introduction
Elliptic integrals were encountered by John Wallis in 1655-59. They
were known to Euler who, in 1753, obtained their addition theorem.
Legendre* whose work on elliptic integrals stretches over several
decades, introduced the normal forms which are still in use. Jacobi, in 1828,
introduced elliptic functions obtained as inversions of (indefinite) elliptic
integrals; he also studied systematically theta functions* Abel obtained
some of Jacobi's results independently, and he also studied what is now
called hyperelliptic and abelian integrals* Weierstrass showed how the
theory of elliptic functions fits in with the theory of functions of a
complex variable, and developed the general theory of doubly periodic
functions*
The history of elliptic functions is given in the article by R* Fricke in
the Encyklopadie (1913). This article also contains a list of references
up to 1913« The more important books on elliptic functions which appeared
since 1913 are listed at the end of this chapter; for the older literature
the reader may be referred to Fricke's article*
The present chapter consists of two parts, one on elliptic integrals,
and the other on elliptic functions* In the second part, both Jacobian and
Weierstrassian functions are treated, the former on account of their
usefulness in connection with numerical work, the latter on account of
the symmetry and simplicity of the basic relations* It may be mentioned
here that Neville (1944) developed a systematic notation for Jacobian
elliptic functions which simplifies the formulas to a considerable extent:
in the present chapter we shall adhere to the traditional notation for the
sole reason that it is still generally used* Theta functions are also
included in the second part, and there is also a brief section on elliptic
modular functions* For further information on modular functions see
Chapter XIV,.
294

13.2
ELLIPTIC FUNCTIONS AND INTEGRALS
295
PART ONE: ELLIPTIC INTEGRALS
13.2. Elliptic integrals
The simplest (indefinite) integrals are integrals of a rational function.
The next simplest type consists of integrals of the form
(1) I=jR(x,y)dx,
in which R is a rational function of its two variables, and y is an
algebraic function of x, that is to say, is given by an equation of the form
(2) P(x,y) = 0
where P is a polynomial of degree n, say, in its two variables. Such
integrals are called abelian integrals.
One of the striking features of the theory of abelian integrals is the
fact that the behavior of the integral (1) depends not so much on the
nature olR as on the nature of P9 or rather on the nature of the algebraic
curve C in the #,y-plane represented by equation (2)« For the theory
of abelian integrals, algebraic curves of degree n are classified according
to their genus (or deficiency\
(3) P = (n~21)-*,
that is the difference between the largest possible number I J of
double points of a non-degenerate curve of degree n, and the actual
number of double points, d, of the curve in question. The genus is a
birational invariant, that is, it remains unchanged if the curve is
subjected to a birational transformation
(4) * = /?,(£,,,), y = R2{{,v),
where the rational functions R ^ and R are such that two further rational
functions R 3 , R 4 exist so that
(5) f=R3(*f y\ 77 = fi4(*,y).
Curves of genus zero are unicursal (or rational) curves. It is known
that for such curves x and y can be expressed as rational functions of a
parameter. Since rational functions are single-valued, this parameter is
a uniformizing variable for the curve. On introduction of this parameter as
a new variable of integration in (1), the integrand is a rational function
of the parameter, and the integral may be evaluated in terms of elementary
functions (of the parameter). The parameter itself is an algebraic function

296
SPECIAL FUNCTIONS
13.2
of x, and hence abelian integrals of genus zero may be expressed in terms
of elementary and algebraic functions.
For algebraic curves of genus unity, Clebsch (1865) proved that x and
y can be expressed as rational functions of two parameters £and 77 where
77 is a polynomial in £ of degree three or four. Introducing £ as a new
variable of integration in (1), it is seen that every integral of genus unity
can be reduced to a form in which the equation (2) becomes
(6) y2 = a0x4+4a1x3 + 6a2x2 + 4a3* + a4
where either aQ ^ 0 or aQ = 0 and at ^ 0. Integrals defined by (1), (6) are
called elliptic integrals, and we have proved that abelian integrals of
genus unity may be reduced to elliptic integrals by a rational change of
the variable of integration. We shall see later, in sec. 13„14, that in
equation (6), and hence for any algebraic curve of genus unity, x and y
may be expressed rationally in terms of single-valued elliptic functions
of a variable z which is a uniformizing variable for the curve in question.
For p > 2 the situation is much more involved. We have here hyper-
elliptic integrals for which equation (2) takes the form
(7) y2 = a0xn + na]Xn-] + .-.+an,
but it is no longer true that every curve can be transformed, by a birational
transformation, to the form (7). Accordingly, hyperelliptic functions do
not suffice for the uniformization of algebraic curves of genus p > 2, and
automorphic functions must be used: see also sec. 14.9.
In this chapter we restrict ourselves to elliptic integrals defined by
(1), (6), and to the elliptic functions associated with such integrals. The
polynomial on the right-hand side of (6) will be denoted by GA(x) when
a 4 0, and by G z (x) when a = 0, a £ 0. If the polynomial on the right-
hand side of (6) has a double zero, the integral J may be evaluated in
terms of elementary functions. Thus we may assume that GA (or Gz, as
the case may be) has no double zero.
13.3. Reduction of elliptic integrals
It has been stated in the preceding section that for the behavior of the
elliptic integral
(1) J= fR(x,y)dx, y2 = aQx4+4a1x3+6a2x2+4a3x + fl4
the polynomial aQxA + "* + aA'is more important than the rational function
R. This statement is justified, and is given a precise meaning, by

13.3
ELLIPTIC FUNCTIONS AND INTEGRALS
297
the following theorem due to Legendre. The elliptic integral (1) may be
expressed as a linear combination (with constant coefficients) of an
integral of a rational function of x and of integrals of the following types:
/v T C dx f lAa x2 + a x C dx
(2) I,= / — , -£«/ °- '—dx, I3=/- —
J y J r J (x-c)y
where c is a constant parameter, and
r x dx
(3) ^* =
■/
is interpreted as the integral I% corresponding to the case c = «>. The
reduction will be effected in several steps.
Since even powers of y may be expressed as polynomials in x, we may
write R in the form
(4) fl(*,y)«
Mf (x) + M2(x) y \JII% (x)+M2(x)y][N} (x)-N2(x)y] y
N}(x) + N2(x)y " \[Nt{x)]2-[Nz(x)y]z\y
where M , M2 , Nt , ^V are polynomials in x, and this may be written as
(5) R{x,y) = R(x)+*^-^
y
with two rational functions, R and /?2, of *, thus completing the first
step in the reduction process.
As a second step we remark that R 2(x)9 being a rational function of
x, may be decomposed into a polynomial in x and a sum of partial
fractions. Thus,
(6) 1= J R U, y) dx = JR (x)dx + 2< a / —dx
1 n V y
C dx
and it is sufficient to consider further the integrals
r xn
(7) J = / dx n = 0, 1, 2, ...

298
SPECIAL FUNCTIONS
13.3
The third step is based on certain recurrence relations for Jn and H •
Let us define b Q, ♦♦♦ , b 4 by means of the identity
(8) ofl«4 + 4a «3+6a *2 + 4o k + o^
0 1 Z 3 4
= bAx-c)* + 4b(x-c)3+6bAx-cV+4b(x-c)+bi
0 1 Z 3 4
in x% We then have the following identities
(9)
d 1 T 1 d(y2) 1
(x* y) = mxn -' y + xn y '^—Lx* ~l y2 + — xn —
dx y [ 2 dx J
77i% Dl~1 (a0#4+4a1#3 + 6a2#2 + 4a3# + a4)
+ — *"(4a0%3 + 12a1%2 + 12a2* + 4a3)
1
y
= (m+2)aQ + 2(2/n + 3)a
y y
L +1
+ 6(/ra+l)a,
. x' x"
+ 2(27n + l)a,— + ma a —
3 4
d (*-c)"+3 (*-c)*+2
(10)-— [U-c)"y] = (/7i + 2U +2(2/n + 3) 6 .
a% y
U-c)»+1 / (*-c)B
+ 6(w + l)6. + 2(2/n +1) b, + /7i6,
y
U-c)"+l
y y y
Putting /7i = 0, 1, 2, ••• in (9\ m = -1, -2, -3, ••• in (10) and integrating,
we obtain successively
(11) 2a0c73 + 2- 30,^2 + 6 • la2J, + 2- la3c/0=y
3o0J4 + 2 • 50^3 +6 • 2o2J2 + 2 • 3a3J1 +o4c70 = %y
4o0J5 + 2 •7o1J4 + 6 •3a2J3 + 2-5a3J2 + 2aAJ} = x2y

13.3 ELLIPTIC FUNCTIONS AND INTEGRALS 299
rix-c)2 r x-c
(12) bQ I dx +2-1. 6f A & -2 . 1.63i?f-1.64ff2 =
-2.1.6fJ-6.1.^#-2.3.6#-2.6tf 7
y
x — c
10 3 1 3 2 4 3 k_c)2
y
-Vo-2'3'6!^-6'2'^!-2'5'^-3'^-
(*-c)2
Now,
(13)
/ dx = <7t-c<70, / d* =<72- 2c<7t + <70
and hence equations (11) and (12) serve to express all J and H in terms
of J , Jj , c^, #t , and certain rational functions of x and y% Moreover, a
comparison of (7) with (2) and (3) shows that
(U)J0 = I}, c7,=I*, a0J2 = 2J2-2a,V. H} = I3,
and thus proves Legendre's theorem*
If aQ = 0, and hence also bQ = 0, there is a slight simplification* In
this case
(15) I, = a, I* % = 0
and hence all integrals reduce to a linear combination of Iy , J3, £*♦ Also
from (11) and (12) it is seen that in this case all J and H may be
expressed in terms of only JQ , Jy , Hy , and rational functions of x and y*
The integrals Iy , i"2, I may be called elliptic integrals of the first,
second, and third kinds, respectively*
A linear fractional transformation of the variable of integration in (1)
changes the polynomial y2, and an appropriate transformation of this
kind may be used to reduce the polynomial to a standard form (see sec*
13J5)« There are two such standard forms in use, and we shall give the
more important results of the present section for each of these two
standard forms, adding a brief note on a third form*
Weierstrass9 form* Here
(16) y2 = 4*3 -g2x-g3.
The integrals of the first, second and third kinds are, respectively,

300 SPECIAL FUNCTIONS 13,3
C dx
(i7>i'--,°-y (4.--,..-,.)«
dx
s2*-g3)%
dx
I.
3 ' / ^_c)(4*3_g2*_g3)*
The first few recurrence relations are
/xz dx l/o x^-^ t
= (4*3_g *-g )*+ —g, J,
(4*J-g i-j/ 6 12
/* x3dx
J^J (ix3-g2x-
1
^3^
)X =^"*(4a;3-«2a:-^3^
*2 = j u^>
2(4*8-g2*-g,)*
2(J)-cJ0)-(6C2-1/2g2)ff|-(4*3-g2*-g3)*(*-cr'
4c3-g2c -g3
Legendre's form. Here
(19) y2 = (l-*2)Q-£2*2).
It is customary to define the corresponding elliptic integrals of the first,
second and third kinds respectively, as
r dx _ r d<j>
(20) F=J [(1-*2)<1-A2*1)]* ~J (l-kzsinz<f>)'A
E= f (l~k_ZX2)' dx = J(l-ki sin2 </>)* d<j>
x = sin<^>

13.3 ELLIPTIC FUNCTIONS AND INTEGRALS 301
dx
(20)
'I
2/cz)[(l-x2)(l-k2x2)]X
d<f>
{\-c-2sm2cf>){\-k2 sm2<f>)1A
x = sin <j> .
The basic integrals of the general theory are
(21) IX-J0 = F
(22) I2 = lA(F-E)
(23)
"■■J
f
,.f.
U8-c*)t(l-*
-c2)[(l-*2)(l
x dx
2)u-
-h*x
■kzx
:2)]*
2)]*
— 11
c
(24) 1
The first integral on the second line of (23), and the integral in (24), may
be evaluated in terms of elementary functions so that everything may be
expressed in terms of E9 F9 II. The recurrence relation for the J are
(25) 2k2Jz-{\ + kz)Jy = [(\-x2){\-kzx2)-\*
Sk2 JA - 2(1 + kz) J2 + JQ = x[(1 - *2)(1 - k2xz)¥
4^2J5-3(l + ^2)J3 + 2Jt=x2[(l-^2)(l-^2x2)]^
ajid the recurrence relations for the H maybe obtained from equation (12)»
A third canonical form,
(26) y2 = *U-/rc)U- 1)
has been suggested by A.R. Low (1950)» In a sense it is between Weier-
strass' and Legendre's form and has some of the advantages of both.
It may be obtained from Weierstrass' form by a translation and
normalization, or from Legendre's form by the substitution
*2=l/,f, y2 = r?2/r.
The latter derivation shows that the parameter m corresponds to kz in
Legendre's form.

31)2
SPECIAL FUNCTIONS
13.4
13.4. Periods and singularities of elliptic integrals
We shall now consider
(1) Hx)-SxRit,v)d€,
a
where
(2) r?2 = G(£)=a0£4 + 4aI£3+6a2£2 + 4a3£+a4,
and. regard I(x) as a function of x, the lower limit, a, being fixed (and
the integrand regular at £= a).
The integrand is a two-valued function of £ whose branch-points
coincide with those of 17; and we shall study the behavior of I(x) on the
Riemann surface of [GOO] rather than in the a>plane. If a £ 0, let a ,
a2, a3> aA be the four (distinct) zeros of G A(x); if aQ = 0 (and ay ^ 0\
let ax , a2, a3 be the three (distinct) zeros of G3(x), and let a4 = <» . In
either of these cases, join a, end a2 by an arc c, and a and aA by an
arc c 'which has no point in common with c. Cut two copies of the
complex x-plane along the arcs c and c ', and join them crosswise along the
cuts, thus obtaining a model of the Riemann surface, R, of [G(x)] . The
integrand, R (x, y), is a meromqrphic function on R, that is to say R (xf y)
is a single-valued function of x on R, and is analytic, except
possibly at a finite number of points where it has poles• On the other hand,
I(x) is a many-valued function on R,
since there are closed curves, T, on R
which cannot be deformed into a point,
and for which fp R d££ 0. The closed
curves y and y3 of the figure are such
curves. (The curve y crosses the branch-
cuts and its dotted portion lies in the
"second sheet" of the Riemann surface*)
In addition there is a closed curve
encircling each pole at which the residue
is 4^ 0. Let b. be one of the poles of /?,
and let ri be the residue at b i of /?.
Given any closed curve, C, on R, it
follows by deformation of contours that
there are integers m9 n9 p{ (positive,
negative, or zero) such that
J «(f,^f=«J Rd£+nf Rd£+ 2p.2«r*r...
c yx y3 i
Im x

13.4
ELLIPTIC FUNCTIONS AND INTEGRALS
303
This means that IQ (x) being one of the possible values of I(x), any
other of the possible values of this function is of the form
(3) I(x) = I Ax) + m, fl, + /n Q„ + ••• + m, fl,
0 II Z Z R R
where mx , ... , mk are arbitrary integers and Ot, ... , flfe are certain
complex numbers independent of x% They are known as the periods or
moduli of periodicity of I(x)%
Every elliptic integral has at least two periods (for instance, the
periods corresponding to y} and y3). The integrands of I} and I in
13«3(2) have no non-zero residues in the cut plane, and hence elliptic
integrals of the first and second kinds have exactly two (independent)
periods. On the other hand, x = c is a simple pole with residue [G(c)]
of the integrand of J3, and accordingly elliptic integrals of the third kind
have three (independent) periods.
We are now in a position to describe the singularities of elliptic
integrals of the first, second, and third kinds. They all have
branchpoints at x = a, a, a, a, and their values at these branch-points are
finite, with the single exception of the point a = °° for J£ in the case
aQ = 0. In addition we have the following behavior of these integrals.
Elliptic integrals of the first kind are analytic on R, except at x = a ,
a , a 3, a • They are finite at every point of R. This is clear from the
behavior of their integrand.
Elliptic integrals of the second hind are analytic on R except at
x - a ,a2,a3, aA,and oo. At °° they have poles if aQ^ 0. (If aQ = 0, then
aA ~ °S an(l I* ^as a hranch-point, and becomes infinite there.) There
are two poles at infinity if a ^ 0, one in each of the sheets of R, and
the residues at these poles are zero*
Elliptic integrals of the third kind are analytic on R except at x = a ,
a2, a3, a.9 and c. TTtey have logarithmic singularities at x = c. There
are two points % = c, one in each sheet of R, and the behavior of I in
the neighborhood of these points is like that of
±[G(c)T% log(*-c).
The different behavior of these elliptic integrals shows clearly that
in general (i.e., apart from special values of c otx), an elliptic integral
of the third kind cannot be reduced to integrals of the first and second
kinds.
Another interesting feature of elliptic integrals of the third kind is
expressed by the interchange theorem* Let

304
SPECIAL FUNCTIONS
13.4
Then
I3 (*, c) - I3 (c, x) = Iy (c) I2 (*) - I, U) J2 (c) + (2m + 1) *.
For the proofs of the statements presented in this section and for
further details, see Tricomi (1937).
13.5. Reduction of G(x) to normal form
In considering elliptic integrals it is convenient to reduce the
polynomial
(1) G(*) = a0*4+4al*3 + 6a2*2 + 4a3* + a4 = y2
to one of the two standard forms given in sec. 13«3» The reduction is
achieved by means of a linear fractional transformation of x% For
Weierstrass' form, one of the zeros of G (x) is mapped on 00, and then the
centroid of the remaining three zeros is taken as the origin. For the
Legendre form, a pair of points is chosen which is apolar with respect to
(forms cross-ratio -1 with) each of two pairs of roots of G (x), and
these points are mapped on 0 and 00. The four roots of G (x) can be
grouped in two pairs in three distinct ways, and accordingly there are
three distinct ways of the reduction to Legendre's form of any given G (x).
Weierstrass' form is more symmetric, and hence more suitable for
theoretical investigations; Legendre's form is more highly standardized and
hence more suitable for numerical computations. Most of the existing
numerical tables have been computed for Legendre's form. We shall
describe briefly the reduction to each of the two standard forms.
Reduction to Weierstrass9 normal form* If a Q£ 0, we reduce G (x) to a
cubic by the transformation
1 Y
(2) *-4-p y=JT
where a4 is one of the zeros of G(x). This transformation changes (1) into
(3) 4AxX3 + 6A2Xz + 4A3X + At=Yz
where
(4) A , = %G'(aA) = aQ a\ + 3a, a\ + 3<*2a4 + a,,
A*=l2 G"(a4) = O0a4 + 2oia4 + (V
^3=^G"'(a4) = a0a4+at, A 4 ~ G" («4) = «0 .

. 13.5
ELLIPTIC FUNCTIONS AND INTEGRALS
305
If aQ = 0, then (1) is already of the form (3) and no preliminary
transformation is needed.
Next, we eliminate the quadratic term by the transformation
(5) X.±^L, y-__l_
which changes (3) into Weierstrass' form
(6) 4{3-gJ-g3 = VZ
where
(7) Sz = ZA\-*AxA3, g3 = 2A,A2A3-Al-A*AA.
From (4) and (7) it is seen that
(8) g2 = a0a4 + 3a2-4ata3
are invariants of the quartic G (x); see, for instance, Burnside and
Panton (1892, sec. 160) where the expression of these invariants as
symmetric functions of the roots is given. It should be noted that the
final form (6) is independent of the zero, aA, of G (x) which was selected
for the transformation and that the coefficients in (6) are rational
functions (actually polynomials) of the coefficients of (1). In particular, if
°o» ••• » °4 are reaU then also g2 andj^ are real.
Reduction to Legendre's normal form. We first show that G (x) may be
factorized in the form
(9) GU)= [B,(«- j6)2•+..C,(* - f)*][B^x - /S)2 + C2U - y)2L
In fact, G (x) may certainly be factprized as
(10) G(x)=Q^x)Q2(x)
<?1(x) = p1x2 + 2g1x + r1> Q2(x) = pzx2 + 2qzx + rz.
With a constant multiplier A, Q — A@ will be a perfect square if
(11) (P, - Ap2) Ir, - Ar2)- (g, - A?2)2 = 0.

306
SPECIAL FUNCTIONS
13.5
Let Aj, A2 be the two roots of this equation. Then
(12) e,-A,e2-(p,-AlP2)(*-/B)2
and hence
(13) C^fljU-jS^ + CjU-y)2
Q2 = B2(x-p)2 + C2(x-y)2
with certain constants By, ... , y; and this proves (9). Moreover, if
oQ, ... , a4 are real and G (x) has at least one pair of complex roots,
let Q y (x) have complex roots. Then the left-hand side of (11) is > 0 when
A = 0, and < 0 when A = p ,/p2, so that A, and A2 are real, /8 and y in
(12) are real and so are 5, , ... , C2 in (13), If aQ, ••. , a4 are real and all
zeros of G(x) are real, the factorization (10) may be arranged so that
the zeros of Q y (x) do not interlace those of Q 2 0*0, and in this case it
is easy to see that By , ... , y are real. Thus for a real G (x) there is
always a real factorization of the form (9). Furthermore, this factorization
is valid both for G . and G • in the latter case either B. + C. = 0 or
4 3 7 11
In (9) we put
and obtain Legendre's normal form
(15) (W2)(l-*2£2)=772
where
(M) *..££.
The quantity k is the modulus. Clearly we may take \k2\ < 1, and if
|&2| = 1, a different grouping of the zeros may be used to make \k2\ < 1,
except in the so-called equianharmonic case when —k is a complex
cube root of unity. This exceptional case arises when the zeros of
(1 - £2)(1 - k2 g2) lie at the end-points* of two diameters of the unit
circle and the angle between the two diameters is 77/6.
We shall now give more specific reduction formulas for the case that
the coefficients in (1) are real and that G(x) > 0 in the interval of
integration. It will be seen that in this case the reduction may be effected

13.5
ELLIPTIC FUNCTIONS AND INTEGRALS
307
by a Teal transformation in such a manner that 0 < k2 < 1% We shall give
the reduction to trigonometric form [the variable (f> in equations 13,3(20)],
(17) y2=cos2(f>(l-k2 sin2(f>).
By division by a positive number we may make the leading coefficient
(a in G4, or a in G3) ± 1, and we shall assume that this has been done
so that
(18) G(*)=±nU-a.)
i
where i = 1, 2, 3, 4 or i = 1, 2, 3, according as G is GA or G • We shall
use the abbreviations
(19) a = a - a
v ' rs s r
a-y /3-S
(20) (°, /S, y, S) =
(21) fi =
\ G(x) )
a-S P-y
* dx
d<f>
where fz is a constant and
dx dd>
(22) r^, v.* -E-
[G(*)]* ^(1-A2sin2<£)
so that /z occurs in the conversion of the elliptic integrals of the first
kind.
Table 1 gives the transformation formulas for the case that all roots
of G (x) are real, it being assumed that
(23) at > a2>a3>aA
(in the case of G3, omit a4)* For each of the two possible leading
coefficients, 1 and — 1, of G (x), the table lists the intervals in which
G (x) >_ 0, the transformation formulas, some corresponding values of x
and <£, the values of A;2 and fi.
Table 2gives the corresponding transformations in the case that there
are complex roots• In the case of G , the real root is a , and the
complex roots are
(24) b ±ic c >0.

308
SPECIAL FUNCTIONS
13.5
TABLE 1. TRANSFORMATION TO
All zeros
G(x)
zeros
kc*)
four
real
zeros
\Gz(x)
three
real
zeros
Leading
coefficient
+ 1
-1
+ 1
-1
Interval
a\ — x
or
— 4
"Z<X<*Z
aA<x<a3
az<x <«t
a3<x<a2
ay<x
x<a3
a2<x<a]
Transformation
x =
ata42~a2a41 ^^
a42_a4I Sin2<^
Q3Q42^a4a32Sin2<i6
a42"a32Sin2^
a4a31+aiQ43SiD2<ft
Si +a43Sln2<56
g2a31-a3g21 ****</>
a31-a21Sin2<£
a3 + a32 sin2<ji
a1-a2sin20
1 — sin 0
^ °3'
' sin2<j<,
g2g31-g3g21 8inV
a31-a2, sin2 «^

13.5
ELLIPTIC FUNCTIONS AND INTEGRALS
309
LEGENDRE'S NORMAL FORM
of G (x) real
sin 0 =
Corresponding
values
4>
Yin
Vm
0
VlTT
0
0
VlTT
0
VlTT
(a,a2aAaJ
(G3G2G4ai)
"32
(a3l CLAL)A
«-*>"

310
SPECIAL FUNCTIONS 13.5
TABLE 2. TRANSFORMATION TO
G (x) has
CW
zeros
c4«
two real
and two
complex
zeros
two
complex zeros
\GA(x)
four
complex zeros
k>*2
G4W
four
complex zeros
C1>C2
Leading
coefficient
1
-1
1
-1
1
Interval
a 1 - x
or
x <a2
az<x<ax
a <x
* <<*i
— oo < x < oo
Transformation
a, + a2 ay-a2 v-cos<fi
2 2 1 — V cos0
„ cos# a-x
(tun K rh)2 - 1
cos 0 x - a„
2 2
c _1 — cos 0
.£ — CL - ■ "' . . - —
cos# 1 + coscfc
(tany20)2= !•(<!,-*)
c
x^fcj + c, tan(0+}40g + }404)
tan(0 +X0s+ 1/2(94)= (*-6f)/cf
* = 6 — c. ctn 0
tancS= 1—
6,-c

13.5 ELLIPTIC FUNCTIONS AND INTEGRALS 311
LEGENDRE'S NORMAL FORM
complex zeros
Auxiliary
Quantities
Q acute
# obtuse
\d}902 acute
6 obtuse
0. acute
acute
^3=^4=^77
Corresponding
| values
X
°2
oo
—oo
oo
*
0
77
0
77
-yin-yie3
-HO,
-1Ad3-1AdA
1An-yId3
-HO,
k2
[•inXi0,-e2)]*
[sinUid t + y<n)]2
sin20s
-w
M
(-cos^ cos02)^
c
(COB^COS^I
c
(-COS 6^
V c /
-CT
fcosd5\A
w
1
c1

312
SPECIAL FUNCTIONS
135
In the case of G A with two real and a pair of complex roots, a. > a ate
the real roots and equation (24) represents the complex roots. In the case
of G A with two pairs of complex roots, the roots are
(25) by±icy, b2±ic2 Pi^b2> c,>0' c2>°>
In this table, the transformation formulas, k \ and \i are expressed in
terms of certain auxiliary quantities defined as follows
at — i a^ — b
(26) tan 0, =—! , tan $2 = —^
i/=tan(^02-^0t)tan(%02+ J£0f)
(27) tan 0, =—! i, tan 0, =--! S_
i-*2
4
(tan %05)2 = cos <93 /cos 04 .
The transformation formulas given in these tables remain valid when
the zeros of G (x) do not satisfy the conditions given in the first column
of the tables and equations (23) to (25); however, in this case the
transformations, and k 2, will in general be complex*
There are several integral tables, textbooks, and works of reference
which give tables of reduction formulas for elliptic integrals to normal
form. We mention Grobner and Hofreiter (1949 sections 241 to 246, 1950
sections 221 to 223); Jahnke-Emde (1938, p. 58, 59); Magnus and Ober-
hettinger (1949, Chapter VII); Meyer zur Capellen (1950, sec. 2«3);
Oberhettinger and Magnus (1949, sec. 2), and Tricomi (1937, p. 76, 77)•
The tables given here are adapted from Tricomi's book. See also a
forthcoming book by Bfrd and Friedman. $f£ CtttcV*;
For the evaluation of elliptic integrals by means of elliptic functions
see sec. 13.14; for the evaluation in terms of theta functions see sec.
13.20.
13.6. Evaluation of Legendre's elliptic integrals
In sections 13 «3 and 13,5 the reduction of any elliptic integral to
elliptic integrals of the first, second, and third kinds in normal form has
been described. The evaluation of integrals in Weierstrass' normal form
by means of Weierstrassian elliptic functions will be given in sec.
13«14; in the present section we discuss the evaluation of Legendre's
elliptic integrals.

13.6
ELLIPTIC FUNCTIONS AND INTEGRALS
313
First we make the definitions 13.3(20)more specific by setting
(1) FU9 k) = f*(l - k2 sin2*)"* dt
o
(2) Efa k) = f4>(l-k2sin2t)lAdt
o
(3) Il(<56, v, *)= jAl + i/sin2*)"1 (l-k2 sin21)'^ dt.
We also recall that, with the exception of the equianharmonic case, the
reduction may be performed in such a manner that
(4) \k\ < 1.
The integrals of the first and second kinds may be evaluated by
binomial expansion of the integrand*
(5) F(0,A)= J —r^SU) |A|<1, |sin<£|.<l
n- 0
/ l/\
(6) E(<f>,k) = Y —JLi A^S-U) \k\<l, |sin0| < 1
n = 0
where
(7) (a)0 = 1, (o)n = a(a + 1) ... (a + n - 1) -
r (a + n)
ru)
(8) S2n(</,)= J^ (sin*)2" dt = 2"211 [" QnV + | (-1)" ^J\
sin
(2m<f>T\
m - I
Thus in the real case there is always a convenient convergent series
for computing F andE* When the modulus A; is near unity, the convergence
of the series is slow, and alternative, less simple, expansions must be
used* Some such expansions were given by Radon (1950), who also gave
the expansions of F and E as trigonometric series. There are extensive
numerical tables of elliptic integrals of the first and second kinds; see
Jahnke-Emde (1938> p. 52-89); Fletcher, Miller, and Rosenhead (1946,
sec* 21)«
Elliptic integrals of the third kind present a far more formidable
computational problem on account of their dependence on three
parameters • The analogue of equations (5) and (6) is
(9) 11(0,,/,*)= 2 (-i/)"fiJ-*)(A7i>)S8ii(0)
n= 0
|A|<1, M<1, |sin^|£l

314 SPECIAL FUNCTIONS 13.6
where
n
(10) B<a)(*)= J
a = 0
is the truncated binomial expansion. The condition \v\ < 1 in (9) limits
the usefulness of this expansion* For alternative expansions see Radon
(1950).
For the computation of 11(0, i/, k) by means of theta functions and
Jacobian elliptic functions see sec. 13.20.
We note here that
(id n(<£, o, k) = F(<t>,k)
(12) (l-k2)U(cf>9 -k2, k)=E(<?>, k)-(l-k2sin2cf>)-^k2smcf>coscf>
(13) (1 - k2) 11(0, -1, h) = (1 -k2)F(ct>,k)-E (0, h)
+ tan<£(l- k2 sin2<f>YA
13.7. Some further properties of Legendre's elliptic normal integrals
The integrals
(1) K = K{k) = F(y27T,k), E = E(k)=E(l/27T,k)
are the complete elliptic integrals of the first and the second kinds,
respectively. With the complementary modulus
(2) k'={\-k*)*
we also have
(3) K'=K'(k) = F(V2JT, k'\ E'=E'(£)=E(K27r, k'\
The incomplete elliptic integrals F(<f>9 k) and E {<f>, k) are many-
valued functions on the Riemann surface R of the function y defined by
equation 13.3(19). The branch-points are x = sin<f> = ± 1, f&^.The
periods may be evaluated in terms of complete elliptic integrals.
Integrals Periods
F(<p, h) 4K, 2iK'
E{<f>,k) 4E, 2;(K'-E').
In each case the first of these periods is called the real, the second the
imaginary, period (because they are respectively real and imaginary when
0 < h < 1).
Although F(0, k) and E{<f>, k) are many-valued functions of x = sin0
on K, E considered as a function of F is single-valued on K provided
:)■■

13.7
ELLIPTIC FUNCTIONS AND INTEGRALS
315
that corresponding values of E and F are obtained by integration over
the same path. This gives rise to Jacobi's function E (u), see sec. 13.16.
Elliptic integrals, like elliptic functions, possess addition theorems.
Given <f> and 0, determine y from the equations
(4) (1 - k2 sin2 0 sin2 0) sin y = sin0 cos 0(1 - k2 sin2 0 )**
+ sin0 cos0(1 - k2 sin20)^
(1 - k2 sin2 0 sin20) cosy = cos 0 cos 0
-sin0sin#(l-*2sin20)K (1 - &2 sin2 0)*
and denote by = the relation (congruence) between two functions which
differ by a (constant) linear combination of their periods. Then
(5) F(y) = F(<f>) + F(t)
(6) E(y) = E(cf>) + E(ip) - k2 sincf) sin 0 sin y
are the addition theorems of E (0, &) , 7^(0, k).
The interchange theorem mentioned in sec. 13 A is most conveniently
expressed in terms of the elliptic integral of the third kind
/•* k2 cost!/ sin 0 (1 -A2 sin20)** sin2*
y (1- kz sm20 sin2*)(l- kz sin2*)
= ctn0(l-&2 sin20)^ |TI(0,-&2sin20, &)-F(0, A)]
when it reads
(8) n* (0, 0) - II*(0, $)~F (0) E(xjj)-F (0) £ (0) + ti ni.
Here & has been omitted from all symbols of elliptic integrals and n is an
integer.
Both the addition theorems and the interchange theorem depend on the
connection between elliptic integrals and elliptic functions.
In sec. 13*5 it has been mentioned that a regrouping of the zeros of
G (x) results in changing the modulus. If k was the original modulus, such
a regrouping will lead to one of the following values
(9) &, —-, &', , , .
Elliptic integrals belonging to any two of these moduli are connected by
rational relations (linear transformations)* To the expressions enumerated
in (9) we add
1-4'

TABLE 3. TRANSFORMATIONS OF ELLIPTIC INTEGRALS
k
1
k
k'
1
ik
k'
ik
\-k'
\+k'
sin0
k sin0
— i tan0
— ik tan0
k ysin 0
A((f>,k)
ik sin 0
~ A(0,&)
(1+k ')sin</> cos</>
A(0,&)
COS(j>
A(0,&)
sec<£
A(0,&)
cos<£
COS(f>
A(0,&)
1
A(0,&)
cos 0 —/c sin 0
A(0,&)
F(<M)
kF((f>,k)
-iF(<j>,k)
-ik'F(<j>,k)
k'F(<j>,k)
-ikF(cf>,k)
(l+k')F(<j>,k)
E(<j>,ic)
~[E(<j>,k)-k'2F(<j>,kj\
k
i[£(0, k)-F(<j>, &)-tan0 A(0, A;)]
-^[£(0,A;)~A;'2F(0,A;)-tan0A(0,A;)]
1 r _ sin(i cos(£~l
*' L A(<£,A) J
i 1 _ sin</> cos<2>~|
— £(0,A)-F^,A)-A2———T\
k L M<f>fk) J
2 sin<£ cos<£ j
[E(<j>,k)+k'F(<j>,kj]-(l-k')— |
1+k M<f>,k)
CO
C/l
M
n
r
a
n
3
c/i
CO

13.8
ELLIPTIC FUNCTIONS AND INTEGRALS
317
Elliptic integrals of moduli k and (10) are also connected by rational
relations (Landen's transformation).
Table 3 (p. 316) gives for any of the moduli (9) or (10), denoted by
k, the transformed values cf> in terms of <j> and k, F(0, k) and E(<ft>, k) in
terms of F(<£, k), E (0, k\ <f>9 and k. We continue to use the notation (2)
and introduce the abbreviation
(11) A(0, k) = (I-k2 sin2 <f>YA, AOSir.ANK
The quantity <£ in the table is determined up to multiples of 277 by giving
both sin<^> and cos<£.
We also note the differentiation formulas
dF 1 V E-k'2F sin0 cos01
(12) Tk^I^L k A (0,4) J
_dE E-F
lk= k
13.8. Complete elliptic integrals
We use the following notations for the complete elliptic integrals ,of
the first, second, and third kind.
dx
[(l-%2)(l-£2*2)]*
/*" /*,/l-Az*2\
A {<f>, k)d4>=l ( 2 I
(3) nI = nl^,i)= / -—
! / (1 + v
sin2cS) A(c6, k)
o
dx
r
(l+„*2)[(l-*2)(l-£2*2)]
From 13.6(8),
(4) sta(x»)-rJtor.in*^rf^-i?5s.4-
zn o n\ 2

318
SPECIAL FUNCTIONS
13B
and using this in 13„6(5), (6), and (9),
(5) K(k)=V2n /,(«, V2; l;k2) \k\ < 1
(6) E (k) = Y2 n 2F, (- %, y2; 1; k 2) |A | .< 1
(7) n,(*i)- f ^(-.VB^f—)
n = 0 > '
|A|.<1, |H<1-
In (5) and (6)
(a)J£)
(c)
/>■"•"'- Suffer*"
is Gauss' hypergeometric series, see chapter 2«
Tricomi (1935, 1936) also gave the expansion
(8) K(sina) = 77 > I^M sin[(4n + 1) a] 0<a<Kn
-i m
and the inequality
(9) log4< K+logA'<%7r.
From (5) it is seen that K(&) is a monotonic increasing function of k for
0 < k < L K(0)'= J4 77, and from (9) it is seen that K becomes
logarithmically infinite as k -> 1. More precisely,
(10) K = log(4/A0 + 0(*'Mog*/) A'-O.
On the other hand, (6) shows that E is decreasing for 0 < k < 1, and from
(2)
(11) 1<E<Mtt 0<A<L
Expansions valid near k = 1 have been given by several authors; see, for
instance Radon (1950). We also mention a formula for the integrals of the
third kind developed by Hamel (1932)%
For the computation of complete elliptic integrals of the first and
second kinds by means of theta functions see sec. 13„20„
Corresponding to the transformations of Table 3, there are
transformations of complete elliptic integrals. These are listed in Table 4
(p. 319).
The transformation

TABLE 4. TRANSFORMATIONS OF COMPLETE ELLIPTIC INTEGRALS
k
1 1
k
k'
1
ik
k'
ik
| 1-*'
l + k'
1 2kA
l+k
K(k)
Uk + ;k')
K'
k'(K'+iK)
A'K
*K'
1+*'
K
2
(1 + A)K
K'U)
kK'
K
k'K
A'(K'-*K)
A(K+i'K')
(1 + A')K'
1 + &
K'
2
eU)
E'
(E'+iE-kzK'-ik'2K)
k'
1
E
A'
1
— E'
A
1 E+A'K
1 + A'
2E-fc'2K
' 1 +A
E'U)
IE'
A-
E
1
E
(E'+*E-&2K'-M'2K)
—(e-iE'-&'2k+m2k')
A-
• 2E'-*2K'
l + k'
E'+kK'
l + k
r
r
i—i
H
n
n
H
O
>
a
M
H
O
CO
»—I
VO

320
SPECIAL FUNCTIONS
13.8
is especially important since it may be used to compute K numerically.
The first equation in (12) may also be written
. 2A'X
k '= .
1 + A'
Here k' < k'< 1 if 0 < h '< 1, and if the transformation is repeated, h '
tends rapidly to unity. The corresponding K(0) is Y/w% Now define
as) a0'-*', *;+i=3l1 „ = 0,1,2
n
Then by repeated application of (12),
oo
^■KW-i-Ili^r.
For the four complete elliptic integrals belonging to complementary
moduli we have Legendre's relation
(15) KE'+IfE-KK^ %n.
For particular values of h we list the following relations.
(16) K(2-^=K'(2^)=i^^
,17)K-(,i„^) -•»«(•!.£)
(18) K'(2*-1)-2*K(2*-1)
(19)
(20) K '(e in/3) = e i7T/e K (e i7T/3) = ""' „ e_i7T/6.
2-3*T(2/3)
The first of these relations corresponds to the lemniscate functions which
arise from the inversion of the integral
J(l-xA)-y'dx,
and the last relation corresponds to the equianharmonic case of elliptic
integrals.

13.8
ELLIPTIC FUNCTIONS AND INTEGRALS
321
The complete elliptic integrals of the third kind II ■(*/, k) may be
expressed in terms of incomplete elliptic integrals of the first and second
kind* For v > — 1 this was observed by Legendre, for i/ < — 1 (when the
Cauchy principal value of the integral must be taken) it was proved by
Tricomi* The parameter v is expressed in terms of an auxiliary quantity
69 different expressions being valid in the intervals (— oo, —1), (— 1, —k2)
(-k2, 0) and (0, oo). The results are
(21) ctn 6 A(<9, k) II, ( -esc2 6, k) = E(k) F(6, k) - K(k) E (6, k)
sin 6 cos 6
(22) k' mi(-A2W,A/),*)-K(A)]
Me, w) '
= Vzn - [E(A) - K(k)] F(6, k')-K{k)E{$, k)
(23) ctn e Aid, k) [II, (-k2 sin2 0, k) - K{k)]
= ~EF(6, k) + KE{$, k)
sin 6 cos 6
(24) —TT-rTT Dn,(£2tan20,£)-K(A)cos20)]
Aw, k )
= [E«0 - K(k)] Fid, k')+ K(k)E(0, k'\
Beside K, E, I11 , it is sometimes convenient to introduce
% r*" sin2<^» /•*"■ cos2<i
(25) D(A)-/ yrf<ft, B(A)=/ ; ^
A A(^i) 70 A (<£,£)
, .. / (sin<£ cos <£)2
C(A)=| ?—dd>.
J0 [A(<f>, k)]3 9
With k = k2 we have the differentiation and integration formulas, and
connections between various integrals
K-E E-&'2K
(26) D=——, B=K-D= —
D-B 1
C=—— = — [(2-A2)K-2E].

322
SPECIAL FUNCTIONS
13.8
dK B dE dD D-C
dK 1 — K dK rf/C 1 — K
dB dC B
2 = C, 2k—= 4C
</k dK 1- k
2
(28) Jk</k = 2kB, /e</k = —k(E + B)
o
_[D<f* = ~2E, Jb</* = 2(E + kB), fCdK = 2B.
For series expansions and other formulas for these integrals and for
short numerical tables see Jahnke-Emde (1938, p« 73-84)«
PART TWO: ELLIPTIC FUNCTIONS
13.9. Inversion of elliptic integrals
Historically,^elliptic functions were introduced by inverting elliptic
integrals* To obtain Jacobian elliptic functions consider the relation
(1) u = J*{l-k* sin*t)-*dt = F{<f>,k)
0
between the complex variables u and <f>* We already know that u is a
many-valued function of x = sin<f>; conversely, equation (1) also defines
<f>, or sin <^>, as a (possibly many-valued) function of u. Jacobi puts
(2) <f> = am u = am (u, k)
and adopts as basic functions
(3) sn u = sn(u, k) = sin (am u)
en u = en (u9 k) - cos (amu)
dn u = dn (u, &) = 4(amu, &) = [1 — k2 sin2 (amu)] .
Beside these, the following nine functions are often used
(4) nsiz = l/snu, nciz = l/cniz, nd u = 1/dniz,
cs u = cna/sna, scu= snw/cnw, sd u - snu/dnu,
ds u = dnu/snit, dc u = dnif/cnu, cdu= cna/dna,
the notation being due to Glaisher.
At u = 0, we may put
(5) sn 0 = 0, en 0 = dn 0 = 1,

13.10 ELLIPTIC FUNCTIONS AND INTEGRALS 323
and this clearly defines the three basic functions, and hence also the
nine functions (4), as single-valued analytic functions in some
neighborhood of the origin (except for ns u9 cs u9 ds u which have simple poles
at u - 0 and are analytic in a punctured neighborhood of this point)• The
crucial fact of the theory of elliptic functions is the circumstance that
the functions obtained by analytic continuation of the twelve functions
thus defined in a neighborhood of u = 0 are all single-valued functions of
u, analytic except for an infinity of (simple) poles. This result may be
established by a discussion of the inversion problem for the integral
(1), see Hancock (1917), Neville (1944).
Weierstrass9 elliptic functions present a similar problem. The relation
(6) z = J2^3-s2t-g3r1Adt
between the two complex variables z and w may be inverted to yield
Weierstrass9 p - function
(7) w = #>(*)= p(z; g2,g3),
and p(z) turns out to be single-valued, and analytic except for an infinity
of poles (of the second order).
In either case the inversion problem is a formidable one (except in
the case of the integral (1) in the real field and for 0 < k < 1), and it is
of interest to note that an alternative approach exists and has many
advantages. Weierstrass has shown that a study of doubly periodic
analytic functions leads quite naturally to elliptic functions. Since then it
has become customary to approach elliptic functions from the general
theory of analytic functions. We shall do so in this chapter and establish
the connection with elliptic integrals later, see sec. 13.14*
13.10. Doubly-periodic functions
Let f(z) be a single-valued function which is analytic save for
isolated singularities. A period of this function is a complex number, p, such
that
(1) f(z) = f(z + p)
for all z for which f is analytic. A function which has one (non-zero)
period has an infinity of periods (for instance np for all integers rc). Let
ft be the set of all points in the complex plane which correspond to
periods of a fixed function f(z)% If f(z) happens to be a constant, then ft
is the whole plane. This case excepted it may be proved [see for instance
Tricomi (1937 Chap. I, sec. 2)] that ft is either a system of equidistant
points on a straight line through the origin, or else a point-lattice formed

324
SPECIAL FUNCTIONS
13.10
by the intersections of two families of equidistant parallel lines (line-
lattice)* In the former case f(z) is simply-periodic, in the latter case
doubly-periodic.
We now consider a doubly-periodic function f(z) and the corresponding
point-lattice fi. The point-lattice may be generated (in many ways) as the
points of intersection of two families of equidistant parallel lines, that
is to say by the repetition of congruent parallelograms. Take one such
parallelogram with one of its vertices at 0, and let the other three vertices
be 2co, 2co , 2co + 2co ' Then 2co and 2co' are called a pair of primitive
periods of f(z)9 and all periods are of the form
(2) 2co = 2mco + 2nco' m9 n integers.
Clearly, co'/co is not real, and we may choose the primitive periods in
such a manner that
(3) Jm(o)7o)>0.
This convention will be adhered to throughout this chapter.
A point-lattice may be generated in infinitely many ways from a line-
lattice, that is to say it possesses infinitely many pairs of primitive
periods. Let co9 co ' be primitive half-periods of fi, and let a, jS, y, S be
any integers. Then
(4) co = aco + jSo>', co '= yco + dco '
is certainly a pair of half-periods. If
(5) ad-&y= 1,
then we have, from (4)
• • • ♦
(6) co = 8co — /So), co '- — yco + aco
so that co9 co\ and hence any half-period of f(z), is a linear combination,
with integer coefficients, of co, <2>'and (4) gives another pair of primitive
half-periods. Equivalent pairs of primitive half-periods are connected by
unimodular transformations
CO
1— co _
=
"a j8
CO
_co _
9
a j8
It can be shown [see, for instance, Tricomi (1937, Chap. I, sec. 2)]
that a pair of primitive periods may be chosen in such a manner that
(8) M<|<*>'| and fin(e>'/<*>)> H • 3*
but such a choice will not be assumed in this chapter.

13.11
ELLIPTIC FUNCTIONS AND INTEGRALS
325
Two points of the z-plane are said to be congruent if they differ by a
period. A connected set of points is called a fundamental region if every
point of the plane is congruent to exactly one point of the set. We shall
always choose the fundamental region as a parallelogram, with two sides
and the vertex at which they intersect being counted as part of the
parallelogram, the other two sides and three vertices not forming part of it.
Fixing a z , the points
(9) z = z0 + 2<^ + 277*/ 0<f<l, 0 <7/ < 1
form the fundamental period-parallelogram* Any parallelogram obtained
from this by a translation by a period, that is every set of points
(10) * = *0 + 2(/w + £)fi) + 20i + J7)fi)' 0<f<l, 0<77<l
with a fixed pair of integers (m9 n) is a period-parallelogram, or, shortly,
a mesh*
Since a doubly-periodic function assumes the same value at
congruent points, it is sufficient to describe the behavior of such a function
in any one mesh. Since f(z) has only isolated singularities and isolated
zeros, it is possible to choose the fundamental period parallelogram
(i.e., z ) so that no singularities or zero of f(z) lies on the boundary of
a mesh. This will be assumed in the general theorems of sec.l3911, and
such a mesh will be called a cell*
13.11. General properties of elliptic functions
A doubly-periodic meromorphic function is called an elliptic function,
that is to say, an elliptic function is defined to be a single-valued
doubly-periodic analytic function whose only possible singularities in
the finite part of the plane are poles. In this section f(z) will be such a
function, co9 co' a pair of primitive half-periods of f(z\ and Q. the point-
lattice associated with f(z)*
It may be mentioned here that often Weierstrass' sigma- and zeta-
functions, theta functions, and other functions associated with elliptic
functions are also referred to as elliptic functions (in the wider sense),
but in the present chapter the term "elliptic function" will be used in
the sense of the definition given above.
Every non-constant elliptic function has poles* For if f(z) has no
poles in a mesh then it is bounded there, and hence in the entire plane.
By Liouville's theorem it is then a constant.

326
SPECIAL FUNCTIONS
13.11
An elliptic function has only a finite number of poles in any mesh,
and, if it does not vanish identically, only a finite number of zeros
there* For an infinity of poles in a mesh implies the existence of a
limiting point of these poles, and hence an essential singularity. Similarly,
an infinity of zeros of an elliptic function which does not vanish
identically implies the existence of an essential singularity.
The number of poles in a cell, each pole counted according to its
multiplicity, is called the order of the elliptic function. The set of poles
or zeros in a given cell is called an irreducible set.
The sum of the residues of an elliptic function at its poles in any
cell is zero. Let C be the boundary of the cell. The sum of the residues
is
- J f(z)dz
2?ri C
and this is zero, since the integrals along opposite sides cancel.
There is no elliptic function of order one. For such a function has
exactly one simple pole in each cell, and the residue is zero by the
preceding theorem.
An elliptic function of order r assumes, in any mesh, every value
exactly r times (counting multiplicity). To show that f(z) - c has exactly
r zeros, take the mesh so that f'(z)/[f(z) - c] is regular on its boundary C.
The difference between the number of poles and the number of zeros of
f(z) - c is
-Lf
f'(z)
dz
f(z)-c
and in this integral the contributions of opposite sides cancel.
The sum of an irreducible set of zeros is congruent to the sum of an
irreducible set of poles (each zero and pole being repeated according to
its multiplicity). Let C be the boundary of a cell, and let at, ... , arbe
the zeros and /3 , ... , /? the poles of f(z) within C The function
f'(z)/f(z) has a simple pole with residue k at a zero of order ky and a
simple pole with residue -A: at a pole of order k.
(i)
ini Jc f(z) fc-i r H/

13.11
ELLIPTIC FUNCTIONS AND INTEGRALS
327
If the vertices of the cell are z q9 zq + 2co, zQ + 2co + 2co'9 zQ + 2co\ the
integral in (1) is
J_ f2" r(z0 + t)f\z0 + t) (za + 2<o'+t)fXz0+2<o'+t)
27riJ0 |_ f(z0+t) f(z0+2a'+t)
1_ f2"'r(z0 + t)fXz0+t) (z0 + 2a> + t)f'(z0 + 2co + t)
2niJo |_ /"U0 + 0 f(z0 + 2a'+t)
~ \2co[losf(z0 + t)]l»' -2o>'[log/'(2o + f)]^i.
2771
Since f(z) has periods 2co, 2<o\ we see that log/"(zQ), log/(z0 + 2<a'),
and log/(z0 + 2co) differ from each other by integer multiples of 2ni,
and hence the integral in (1) has the value 2m co + 2n<x>\
From these fundamental theorems some corollaries follow immediately.
We mention only two of these.
Two elliptic functions which have the same periods, the same poles,
and the same principal parts at each pole differ by a constant.
The quotient of two elliptic functions whose periods, poles9 and zeros
(and multiplicities of poles and zeros) are the same, is a constant.
All elliptic functions with the same periods (2o), 2gO form a field, ft,
that is the sum, difference, product, or quotient of two such functions
has the same periods. Clearly, any rational function (with constant
coefficients) of such functions belongs to ft. Moreover, the derivative
of any function of ft belongs also to ft, so that ft is a differential field*
An integral of a function of ft does not necessarily belong to ft. Although
(2co9 2co ') is a pair of primitive periods for some functions in ft, and a
pair of periods for all functions of ft, it is not necessarily a pair of
primitive periods for all functions of ft.
From the representation of elliptic functions in terms of certain
standard functions (see sec. 13*14) some additional results easily
follow.
Any two functions, f and g, of ft are connected by an algebraic
equation P(/, g) = 0, where P(x, y) is a polynomial with constant
coefficients, and the algebraic curve P(x, y) = 0 is unicursaL
Any elliptic function satisfies an algebraic differential equationof
the first order, P(f, f) « 0. Here again P (x, y) is a polynomial with
constant coefficients and of genus zero.

328
SPECIAL FUNCTIONS
13. ]1
Any elliptic function, f(z), satisfies an algebraic addition theorem
(2) A [f(u\ f(v\ f(u + i;)] = 0
where A (x9 y, z) is a polynomial whose coefficients are independent of
u, v9 and (2) is satisfied identically in u, v.
Conversely, it may be shown that a single-valued analytic function
of z which satisfies an algebraic addition theorem of the form (2)is either
a rational function of z, or a rational function of e for some A, or else
an elliptic function.
The simplest (non-trivial) elliptic functions are functions of order
two. Among these one may select as standard functions either a function
which has one double pole (with residue zero) in each cell, or else a
function which has two simple poles (with residues equal in magnitude
but opposite in sign) in each cell. The former possibility is chosen in
Weierstrass' theory, the latter in Jacobi's.
13.12. Weierstrass'functions
Let 2<d, 2a)'be a fixed pair of primitive periods,
(1) r = o)7o), Im r > 0
(2) w = w = 2m co + 2n a)'.
2 and II will indicate infinite sums and products taken over all integers
77i, n, and X'and II'sums and products taken over all integers m9 n with
the exception of m = n = 0.
Weierstrass' function p(z) - p(z\oj9 co') in an elliptic function of
periods 2co> 2<d' which is of order two, has a double pole at z = 0, the
principal part of the function at this pole being z~29 and for which
$p(z) — z"2 is analytic in a neighborhood of, and vanishes at, z = 0.These
conditions define $(z) uniquely. To obtain an analytic expression we
first construct a meromorphic function which has double poles, with
principal parts, (z - w)~2, at all points w = wm% The partial fraction
expansion of such a function is
(3) /U) = z-2 + S'[U-u;r2-u;-2].
Moreover, f(z) - z~~2 vanishes at z = 0. We prove that f(z + 2o>) = f(z)
= f(z + 2a>') by rearranging the series and then conclude that f(z)= $>(z)
or
(4V pU) = jp(z|g>, o>') = — +\ -— —T*""?^—1^—^
zc *-J \{z-2mco- 2nco ) {2m(o + 2nco )J

13.12 ELLIPTIC FUNCTIONS AND IN TEGRALS 329
The function &>(z) is an even function of z% Also
(5) &\z) = - 2z~3 - 22'U - w)'3 = - 22U - to)'3.
Integrating term by term we obtain Weierstrass9 zeta function which is
a meromorphic function with simple poles.
(6) £GrWU|<», co") = z-} +2'[(*-ti;r1 +w-' + zw'z]
(7) #>U) = -£'(*).
The function £(z) is an odd function of z% It is not doubly- periodic and
hence not an elliptic function• It is usual to put
(8) £(z + 2g>) «£(*) + 2r/, £(z + 2co ') - £(z) + 2r/'.
Since £(z) is an odd function of z,
(9) 77 = £(o>), r; '= £(a> 0.
By integrating £(z) around a cell one obtains Legendre's relation
(10) T/G)'- 7/ 0)= %7Tl.
Weierstrass9 sigma function is an entire function whose logarithmic
derivative is the zeta function
(ii)aw-a(,k«%.n'{(i-:)-p[:4(i),J}
>/x a'(2) . x a'2(2)-a(z)a"U)
(12)^)=——, *>(*) = j-r .
cKz) a U)
With the abbreviations
(13) g2 = 60l'w-\ g3 = 140l'w-6,
the power series expansion of o(z), and the Laurent series expansions of
<£(z\ p(z\ p\z\ in a neighborhood of the origin are
(14) a(*)«z--7
*2*' *,*' ***"
24 • 3 . 5 23 . 3 • 5 . 7 29 • 32 • 5 • 7
g2g3z"
27 • 32-52.7- 11
as) tfo-i.--^ £al! £l£ +...
z 22• 3 • 5 22 . 5 • 7 24 • 3 • 52. 7

330
SPECIAL FUNCTIONS
13.12
' & z2 22 -5 22. 7 24- 3 . 52
(17) gP (2) = +-£-- + -7— + * r2 + ••• .
z3 2.5 7 23.52
The radius of convergence of these series is equal to the smallest
distance of two points of the point-lattice Q, i.e., the smallest of the four
numbers \2co\9 |2a>'|, |2a> ±2a>'|.
Formulas with Weierstrass' functions may be expressed more
symmetrically when the notation
(18) a>T = a>, co2 = - co - co'9 co3 = co'
<19) VW a- 1,2, 3
is used% We then have
(20) £{z + 2wa) = C(z) + 2rja a = 1, 2,3
(21) a(z + 2<ya) = - a(z) exp Vfy^z + coa)] a = 1, 2, 3,
It is convenient to introduce the three functions
a{z + a) )
(22) C7a(z) = -*- expi-zrja) a = 1,2,3.
<rU>a)
For these we have
(23) aa(z + 2wa) = - aa(z) exp[2rja(z + coa)] a = 1, 2, 3
CTa(z + 2^) = aaU) exp [2qfi(z + co^)] a, )3 = 1, 2, 3, a ^ £.
The function gp'(z) is an odd elliptic function of order three with
periods 2coa, a - 1, 2, 3: it has three zeros in every cell. Now, p'(— a>a)
= p'(co^) since gp'has period 2a>a, and #? '(~<^a) = ~^P '^a^ s*nce fP ' (*}
is an odd function of 2. Thus we see that z = a>a, a = 1, 2, 3 is an
irreducible set of zeros for p'(z). It is customary to put
(24) ea-|p(G)a) a- 1,2,3.
The function gp(z) — #?(a> ) is an elliptic function of order two. It has
double poles at points congruent to 0, and double zeros at points
congruent to co * Since it is of order two, these are the only poles and zeros,
and hence the function [#?(z) - e^ may be defined as a single-valued
function (but it need not have periods 2a>, 2co'9 see sec. 13a 139 13*16).

13*. 13
ELLIPTIC FUNCTIONS AND INTEGRALS
331
13.13. Further properties of Weierstrass* functions
The dependence of p{z) on the half-periods a>, co', is indicated
by writing £?(z|a>, co '), the dependence on the invariants g , g by
fp(z> 82' £3)' anc^ similarly f°r the other functions of Weierstrass.
From the definitions we have the homogeneity relations for arbitrary
(1) %> '(tz\tco9 tco') = t~3 p'{z\co9 co')
p {tz \tco9 tco') = £ ~2 #> (z |<i>, <i>')
£(tz\tco, tco') = *"' £(z|g>, a>')
cr(tz\tco9 tco') = t a(z\co9 co')
(2) p'U*; r4 g2, r6 g3) = r3 ^'(*; g2, g3)
£(**; '~4 g2> r* g3) = t"1 £(*; gt, g3)
ctUz; *~4g2, *~6g3) = ta(z; g 2, g3).
Thus it is seen that Weierstrass' functions depend essentially on two
parameters which may be chosen, for instance, as the ratios of z9 co9 co\
The expressions of the invariants in terms of the periods are given in
13,12(13), Conversely, from 13a9(6) and 13,12(24),
coa=Jeattt*-g2t-g3r*dt.
The functions
p'2(z) and \&(z)-e1]y(z)-e2][&(z)-e3]
are both elliptic functions of order six with periods 2co , a = 1, 2, 3*
They both have an irreducible set of double zeros at coa, a = 1, 2, 3, and
a pole of order six at 0* By the general theorems of sec. 13.11, their
quotient is constant. The value of this constant may be computed from
the expansions 13«12(4)and (5)« Thus we obtain the algebraic differential
equation of Weierstrass' #>-function,
(3) &'Hz) = 4![p(z)~e}][p(z)-e2\[p(z)-e3\.
An alternative form of this differential equation may be obtained from the
remark that
p'2(z)-4,p3(z) + g2p(z)

332
SPECIAL FUNCTIONS
13.13
is an elliptic function of order six at most, and that all possible poles
of this function are congruent to 0. From the expansions 13,12 (16) and
(17) it follows that this function is regular at z = 0, and hence a constant
by sec* 13g 1U The value of this constant, obtained by means of 13,12(16)
and (17), is — g3, and hence the alternative differential equation
(4) #>'2(z) = 4fr>3U)-g2£>U)-g3.
A comparison of the right-hand sides of (3) and (4) shows that e ^
a ~ 1, 2, 3 are the roots of the algebraic equation 4>t3 - g^t - g = 0,
and the formulas for symmetric functions of the roots of algebraic
equations lead to the following formulas
(5) eT+e2 + e3=0,
(6)
2 2 2
1
7
e? + e!+e3=jSS
e,+e2+e3- g2
(7) 16(e2 - e3)2 (e3 - e ,)* (e , - e2f = g\ - 27g* = A.
The last of these expressions is the discriminant of the cubic equation•
The differential equation (4), together with the remark that $>(z) has
a pole, and hence becomes infinite, at z = 0, establishes the relations
13*9(6) and (7), and the connection between Weierstrass' canonical form
of elliptic integrals of the first kind, and Weierstrass' £?- function. From
(4) we also have
(8) tp"(z) = 6tp*(z)-y2g2, p",(z)^12p(z)p'(z)
and, by induction,
fp(2n-2\z) and p<2n+'>(2)/p'(2)
are polynomials of degree n in %>(z)*
The addition theorem of the #>-function may be written in several
forms •
(9) #>U + v)=-- — —- -^(u)-fp(v)
4 |_ p(u)- &>(v) J
(10)
1 p(u) p'iu)
l p(v) p'M
= 0

13.13 ELLIPTIC FUNCTIONS AND INTEGRALS 333
1 d
(11) fp{u + v)= fp(u)
1 d \ p 'U) - p '(v)
2 du L &(u)- fpM J
2 * L pU)-p(t;) J
= p(v)--~ —
<?2
(12) p(a + v) +p(a-v) = 2sp(a)--— {log [#> (a) - p(v)]\.
du
These addition theorems may be obtained in several ways. They may be
proved by observing that the functions on the two sides of the equation
are elliptic functions with the same periods, poles, and principal parts,
and have the same value at some specified point.
From the addition theorems many formulas for Weierstrass' function
follow. We note
(13) pU + G>a)=ea+-* f » 7 a =1,2, 3
a
(14) gj(2z) = -2gj(z) +
[ &" (z) T
\_2g>'(z) J
(15) p(M*)=pU)+[p(*)-e2]*[p(*)-e,]*
+ [pU)-e/[?U)-e,]i( + [pU)-e/[pU)-e/.
In the first of these, a, jS, y is any permutation of 1, 2, 3* Equation
(14) is the duplication formula* The square roots in (15) are to be taken
in accordance with (22)«
There are also corresponding formulas for Weierstrass' zeta and sigma
functions.
(16) C(u + v) = C(u)+CM+-P'U)~P'{v)
2 p(u)-p(v)
(17) a(u + v) a(u - v) = - az(u) az (v)[p (u) - @(v)\
These formulas are sometimes called the addition theorems of the zeta
and thesigma function, although they are not addition theorems as defined
in 13*11 (2). Since £(u) and a(u) are not elliptic functions, they cannot
possess addition theorems. The following formulas may be deduced from
(16) and (17).

334
SPECIAL FUNCTIONS
13.13
1 ipXz)
(18) £(*±*>aW(*)±!>a+-r^pr a- 1,2, 3
2 p(z)-ea
(19) £(z + 2m co + 2n co') = £(z) + 2m 7} + 2nr\ ' m, n integers
(20) o(z+ 2mco + 2nco') = (-l)m+n+mna(z)
x exp[(z + mco + nco')(2m7i + 2nr}')] m, n integers.
Equations (16) to (18) may be proved by expressing the elliptic functions,
[p\u)- $>'(v)]/[$)(ii) - fp(v)] in terms of zeta functions, @(u) - p(v) in
terms of sigma functions, and fp'(z)/[p(z) - ej in terms of zeta functions
(see the following sections).
It has been mentioned in sec. 13*12, in the lines following 13.12(24),
that [p{z) - ca] may be defined as a single-valued function of z% This
may be done by taking that square root which will make z = 0 a simple
pole with residue unity for this function. Since the principal part near
the origin of p'(z) is — 2z~3, this definition implies that
(21) p'(z) = -2[fp{z)-e,]* [*>(*)-e2]* [p(*)-e,]*.
To obtain an explicit formula for [#> (z-) - cj^, put a = z, 2; = toa, in (17)
and use (20) and 13*12(21).
, v <rU + g> )<t(z - co)
>(*>)-e = * ^~
a2U)a2(fi) J
er2U + &> )
exp[-27ya(z + g> )] =
^1
a(z) J
a2U)a2(«a)
Extracting the square root according to the definition made above,
(22) [p(z)-eJA = aa{z)/a(z).
In particular, putting z = co „,
(23) {ep - eaYA = o^/cK*^).
In relations involving square roots, such as (15), we shall always assume
that the square roots are determined as in (22) and (23)* From (23) and
13*12(22) we have
(24) ie -ea)*- « P expC-^o, ),
p a ff(.&)a)a(&)/3) u p

13.14
ELLIPTIC FUNCTIONS AND INTEGRALS
335
and this equation in combination with Legendre's relation 13« 12 (10) shows
that
(25) (e 1 - e J* = i (e 3 - e })\ (e , - e ,)* = i (e 2 - e ,)*
(e2-e3)* = i'(e3-e2)K.
13.14. The expression of elliptic functions and elliptic integrals in
terms of Weierstrass' functions
We shall now consider the problem of expressing any elliptic function
in terms of standard functions, either as a rational combination of p and
#?'(linear in %> ')> or as a linear combination of zeta functions and their
derivatives, or else as a quotient of two products of sigma functions. Let
f(z) be an elliptic function with periods 2co9 2<o% and let tf>(z), £(z),
a(z) be Weierstrass' functions constructed with primitive periods2co9 2<o\
Expression in terms of p(z) and #?'(z). First, let f(z) be an even
function of z% If f(z) has a zero or pole at z = 0, this zero or pole must
be of even order, and hence f(z) [@(z)]s will be analytic and / 0 at z = 0,
for some integer 5. The zeros and poles of the even function f(z)[p(z)]s
are situated symmetrically to the origin* Let a]9 ... , ah9 -a}9 ... , -ah
be an irreducible set of zeros, and /S 1, ... , jS^, -j8,f ... , -j8^ an
irreducible set of poles, each zero and pole repeated according to its
multiplicity. Then
h
f(z)[p{z)]s [[ —— -C-
is an elliptic function without zeros or poles and hence a constant. An
even elliptic function may be expressed as a rational function of g>(z).
Let f(z) be any elliptic function
f(z)^[f(z) + f{-z)]+fP'{z) f{Zl~i\Z) .
Here f(z) + f(-z) and [f(z) - f(-z)]/p'(z) are even elliptic functions and
hence rational functions of p(z). Thus, any elliptic function may be
expressed in the form
(i) f(z)=R,[fp(zy} + R2[p{z)]jp'(z)
where R (w) and R (w) are rational functions of u>.
From this, in conjunction with the differential equation and addition
theorem of the #>-function it follows easily that any elliptic function has

336
SPECIAL FUNCTIONS
13.14
an algebraic differential equation and an algebraic addition theorem, and
that any two elliptic functions with the same periods are algebraically
connected (see sec. 13«11).
Expression in terms of zeta functions. The function £(z) is not an
elliptic function, but it is easy to see by means of 13„ 13 (19) that
i cr<ru-yr)
r= 1
is an elliptic function if and only if
h
S c-0.
r= 1
Moreover, £'(z) = -#?(z) so that all derivatives of £(z) are elliptic
functions.
Let /S 1 , ... , /3 h bean irreducible set of distinct poles of f(z\ and let
m
s=\ r*S r
be the principal part (the sum of the negative powers in the Laurent
expansion) of f(z) for the neighborhood of z = (3 which is a pole of order
m, • Consider
m
•u>-/<«>- I I ±^-br^{-^-0r\
Now,
£*r ,£(*-j8r)
is an elliptic function since 2 6 t , being the sum of residues at an
irreducible set of poles, is zero (see sec. 13„11)» Also C Kz - j8 ) is
an elliptic function for s = 2, 3 and hence 0(z) is an elliptic
function. Since the principal part of £(z - /3 r) at z = /3^ is (2 -j3 )"!f it
follows that $(z) has no poles at z = /8 1, ... , j3h, hence no poles at all,
and thus is constant. Any elliptic function may be expressed as
l m
r- 1 s= 1

13.14 ELLIPTIC FUNCTIONS AND INTEGRALS 337
Such an expression is especially useful when integrating elliptic
functions. From (2), 13.12(7), and 13.12(12),
(3) Jf(u)du = bQu + c + ^ | bTtX log[aU-i8r)]-6ri2^U-i8r)
m
S = 3 S
+
S = 3
The expansion (2) may be used to establish 13.13(16) and (18).
Expression in terms of sigma functions. Although a(z) itself is not an
elliptic function, it is easy to see by means of 13.13 (20) that
h
is an elliptic function if and only if 2 (a - /3 ) = 0. Now let a., ..., a, ;
r= i r r
jS , ... , jS^ be an irreducible set of zeros and poles of f(z\ each repeated
h
according to its multiplicity. We know (sec. 13«11) that 2 (a — jS ) is
r~ 1 r r
a period, and replacing some of the zeros and poles by congruent ones,
we may assume that 2 (a — /3 ) = 0. We then form *P(z) according to
r- I r r
(4), and see that f(z)/m{z) is an elliptic function without zeros and poles
and hence a constant. Any elliptic function may be expressed as
h
T\ o(z- a)
(5) n,)-\},^0y
where ay , ... , ah is an irreducible set of zeros, and /3t , ... , fi h an
irreducible set of poles, of f(z), each zero and pole repeated according to its
multiplicity, and the sets are so chosen that
h h
(6) S a = 2 fl.
r= I r= I
The representation (5) may be used to prove 13.13(17).
Elliptic integrals. Given an elliptic integral in Weierstrass' canonical
form
(7) 1= J'R (x, y) dx, y2 = 4x3 - g2x - g

338
SPECIAL FUNCTIONS
13.14
we may put
(8) * = p(z;gt,g3), y= fp'(z;g2,g3)
to reduce (7) to
(9) I=jR[p(z), $>'(z)]p'(z)dz.
The integrand is a rational function of $>(z) and $>'(z) and hence an
elliptic function, say f(z)\ it has an expansion (2), and the integral itself
may be evaluated in the form (3)*
The substitution (8) represents points on the algebraic curve
(10) y2 = 4*3-g2*-g3.
The coordinates, x and y, appear as single-valued functions of a parameter
2, which is a uniformizing variable for (10) (see also sec. 13.2).
Any elliptic integral
(11) I=fR(x,y)dx
(12) y2 = G(x) = aQx4 + 4a,*3 + 6o^2+ 4a3« + a4
may be reduced to Weierstrass' functions. We first reduce (12) to Weier-
strass* canonical form as in sec. 13„5, and then proceed as above. To
some extent the computations indicated in sec. 13«5 may be avoided by
using the expressions 13„5(8) for the invariants with which to form
Weierstrass* functions. See, for instance, Bianchi (1916, 371-374) where
the computation of an elliptic integral of the first kind involving (12) is
carried out.
13.15. Descriptive properties and 'degenerate cases of Weierstrass'
functions
In many applications the coefficients of G (x) are real. In this case
13.5(8) shows that also the invariants g2, g3 are real. We shall describe
briefly the behavior of @(z) for real g2 and g3, distinguishing two cases
according as the discriminant A = g\ - 27g* is positive or negative.
First let A > 0. In this case there exists a pair of primitive periods
2(0, 2co' so that (0 is real and (o ' is imaginary. The point-lattice of all
periods may be generated by a rectangular line lattice. The function
p(z) is real on the lines of the lattice,
Re z = 2m (o m integer
t Ira2 = 2n6)' n integer
and also on the half-way lines

13.15
ELLIPTIC FUNCTIONS AND INTEGRALS
339
Re z = (2m + 1) a m integer
i Im z - (2n + 1) o ' n integer*
We have the following symmetry relations in which z^ and zz are real*
&(z}+iz2) = pfa i ~iz2) = p(-z t -iz2)= &(-z x + iz t\
the bars denoting conjugate complex quantities* In this case e , e , e
are real, e t > e 2 > e 3, e, >0, e 3 < 0» As 2 describes the boundary of the
rectangle 0, o, <o + &)', 6)' 0, the function ga(z) decreases from <*> to
e 1 = fp (&)), to e 2 = #? (&) + o)'), to e 3 = p (0) '), to - 00*
Now let A < 0. This case is very different from the first one. There is
again a pair of periods, the first of which is real and the second imaginary,
but they are not primitive periods. There exists, however, a pair of
conjugate complex primitive periods giving a rhombic fundamental
parallelogram* If 2&), 2&)'are a pair of conjugate complex primitive periods, the
diagonals of the period parallelograms are the lines
Rez^mU + w') m integer
i \mz = n(co — &)') n integer
and these are the only lines on which jp(z) is real. Only e2 is real in
this case: e and e are conjugate complex. As z varies along diagonals
of period parallelograms from 0 to cj + cj' to 2co (or 2co') %>(z) decreases
from + 00 to e to — 00,
Degenerate cases of Weierstrass' functions occur when one or both of
the periods become infinite, or, what is the same, two or all three of e ,
e , e coincide. We list the following three cases,
(i) Real period infinite*
(1) e,=e2 = o, e3 = - 2a
(2) g2 = 12a2, g3 = -8a3, v = ~, &'= (12a)-* ni
(3) fpiz; 12a2,-8a3) = a + 3aisinh[(3a)* z]\~z
(4) £U; 12a2, -8a3) = -au + (3a)* ctnh[(3a)*z]
(5) a(z; 12a2,-8a3)=(3ar* sinh[(3a)* z] exp^az2).
(ii) Imaginary period infinite.
(6) et = 2a, <?2=e3 = -a
(7) g =12a2, g=8a3, v = (12a)-% n, u'=ioo.

340
SPECIAL FUNCTIONS
13.15
(8) &(z; 12a2, 8a3)=-a + 3aism[(3aYA z]\~2
(9) £(z; I2a2,8a3) = az + (3a)% ctn[(3a)*z]
(10) a(z; 12a2, 8a3) = (3a)^ sin [(3a)* z] exp(1/2az2).
(iii) Both periods infinite.
(11) e1 = e2 = e3 = 0, g2 = g3 = 0, a) = -Ja)'=oo
(12) £>(*; 0, 0)=z"2, CSz\ 0, 0)=z~\ a(z;0, 0)=*.
In all three cases A = 0.
13.16. Jacobian elliptic functions
Jacobi's function
(1) w = sn u = sn (iz, &)
may be defined as in sec. 13*9 by the integral
(2) u = fj [Q-*2)(l-*2*2)]-*rf*
in which the square root has the value 1 at x = 0, Also sn (0, h) = 1. The
integral may be evaluated in terms of Weierstrass' functions (see sec.
13,14). It turns out that
(3) e , ': e 2 : e 3 = (2 - £ 2) : (2£ 2 - 1) : - (1 + £ 2), z « (e | - e f* u
and
(e -e )1/4
(4) anfot)- ) 3 .
[#>(*)-e3]/2
For the other two basic functions of Jacobi's we have
(5) en (u, k) =—— -rr-
[p(*)-e,r
(6) dn (u, A;) =7—-; hrr*
In (4), (5), (6),
(7) a-le,--)**, a2-'2"*3
e,-e,

13.16
ELLIPTIC FUNCTIONS AND INTEGRALS
341
and all square roots occurring here are uniquely defined by 13.13(22)
and (23), Using these latter relations we may rewrite (4) to (6) as
i/ cr(z) aAz)
1 3 a3(z) <j3(z)
oAz)
dn (u, k) = —.
o3(z)
The nine subsidiary functions 13*9(4) may similarly be expressed in
terms of sigma functions. In what follows these nine functions will be
omitted in general, since the formulas relating to them may easily be
obtained from the formulas for the three basic functions (8).
In sec. 13.9, the Jacobian functions have been established in a
neighborhood of the origin by the inversion of an elliptic integral. Equation
(8) shows that an analytic continuation of these functions leads to single-
valued analytic functions with poles at the zeros of a 3(z)% Moreover, it
is easy to see from (8) and 13a 12 (23) that the Jacobian functions are
doubly-periodic. We put
(9) u = {e%-e3)*z, K=(e1-e3)^ e>, *K'= (e, - e3)* e>'f
call K the real quarter-period and K' the imaginary quarter-period, and
verify that (9) is in accordance with the definition of K and K ' as
complete elliptic integrals in 13,7(1) and (2). The primitive periods of sn,
en, dn may now be found by means of 13,12(23). The zeros of a(z) are
all simple and may be read off 13.12(11), those of aa(z) follow from
13,12(22). This gives the (simple) zeros and poles of Jacobi's functions.
Lastly, 13.12(14) in conjunction with 13.13(23) enables us to determine
the residues of the three functions (8)« The results are shown in Table 5»
TABLE 5. PERIODS, ZEROS, POLES, AND RESIDUES OF JACOBI'S
ELLIPTIC FUNCTIONS
m and n are integers
Function
sn (u,/c)
en (u, k)
dn (u, A;)
Primitive
Periods
4K
2iK'
4K
2K+2»K'
2K
4»K'
Zeros
2ttiK+ 2niK'
(2/7i+ 1)K+ 2niVL'
(2/7i+ 1)K+ (2ti+ 1)»K'
Poles
2ttiK
+ (2»+ 1)»K'
Residues
k
(- l)*+n
ik \
(-1)B+I*

342
SPECIAL FUNCTIONS
13.16
v = iK
v=K + i'K'
v = 0
v = K
If 0 < k2 < 1, then K and K 'are real, and taking also eareal, we see
from (3) that we may take e > e > e , when co becomes real and co'
imaginary. This is the case A > 0 of sec. 13« 15*
For any kz {£ 0, 1) we take
the parallelogram which is one
eighth of the fundamental
parallelogram for sn or dn and denote
its vertices by the letters S, C,
D, N as in the figure. With this
notation, the first letter in the
symbol of the twelve Jacobian
functions shows the position of
a zero, and the second, the
position of a pole. Zeros and
poles are repeated at half-periods.
From Table 5 it is easy to verify that any cell contains two simple
poles (with zero residue-sum) and two simple zeros of any Jacobian
elliptic function. Thus Jacobi's functions sn u, en u9 dn u are elliptic
functions of order 2. Given the modulus, k9 the quarter-periods K, K ' are
determined by 13.7(1) and (2), uniquely if the i-plane is cut from -«* to
-1 and from 1 to °°, Thereupon the data given in Table 5 determine
Jacobi's functions uniquely. We have expressed these in tenns of sigma
functions, but an independent construction in the manner of the
construction of sec, 13,12 is possible. See Neville (1944) where the construction
of all twelve Jacobian functions is carried out in a symmetric manner,
(The reader should note, however, that Neville's notation differs
somewhat from the customary notation adopted in this book,)
Legendre's complete elliptic integrals of the second kind are also
expressible in terms of values of Weierstrass' functions
e . co + 7)
(10) E=-2 ^
(e^ezY
E'= £—2
(e,-e3)>
The modulus, k, and the complementary modulus, k', are determined
uniquely as
an *=■
(e
",>*
h'-
<«.-«>*
K^' ""<«,-«,>*'
Given any modulus, k, equation (3) determines the *?a(up to an
irrelevant factor), and hence the invariants according to 13.13(5). The
Weierstrass functions constructed with these invariants then fully define
the Jacobian functions, their periods, the complete elliptic integrals.

13.17 ELLIPTIC FUNCTIONS AND INTEGRALS 343
Conversely, the Weierstrass functions formed with any invariants
determine Jacobian functions whose modulus is given by (11),
In sec. 13„7 it has been pointed out that the (incomplete) elliptic
integral of the second kind is a single-valued function of u% This defines
Jacobi's function E (u)% Putting cf> = ara(w, k)9 sin<£ = sn(it, k\ and
sin£ = sn (x9 k) in 13.6(2) we find
(12) £U) = JQ"dn2(x, k)dx.
Jacobi's function E {u) is not periodic since
(13) £(u + 2K) = £U)+2E,
E{u + 2iK') = E(u)+2i(K'-E').
Sometimes it is convenient to use the function
E
(14) Z(u) = E(u) u
K
which is singly-periodic, since
(15) ZU + 2K)=Z(u), Z{u + 2iK')=Z{u)-in/K.
Although the functions E (iz), Z (u) are not elliptic functions, they have
many properties similar to those of elliptic functions. See, for instance,
Whittaker and Watson (1927, p. 517-520).
13.17, Further properties of Jacobian elliptic functions
We shall often use the abbreviations
(1) 5 = sn (iz, k), c = en (iz, k), d = dn (u9 A;).
The following basic formulas are consequences of the definitions of
Jacobian functions and of the properties of Weierstrass' p -function.
Differentiation with respect to u will be indicated by a prime. Thus,
(s)'=ds/du {s)" = d2s/du2, etc.
(2) s2 + c2 = l, k2s2 + d2 = l, d2-k2c2 = k'2
(3) {s)'=cd, (c)'=-sd, (d)'=-k2sc
(4) {*)" =-s(d2 + k2c2), (c)"=-cU2-t2s2),
{d)'f =-kzd(c2-s2)
(5) {s)'2 = a-s2)(l-k2s2)
(6) (c)'2 = (l-c2)a2c2+A'2)

344
SPECIAL FUNCTIONS
13.17
(7) y)'* = (l-i2)«2-A'2)
(8) sn (-u) = - sn iz, cn(-iz) = cniz, dn(~iz)=dniz
(9) sn (2K - u) = sniz, en (2K - u) = -en u, dn (2K - u) = dniz
(10) sn(2iK'- u) = -sn«, en (2iK '- iz) = -enu,
dn(2iK'- u) = -dnw.
The power series expansions
u3 u5
(11) sn(it, ft) = u-{l + k2)— + Q + 14ft2 + ft4)
3! 5!
cn(it, ft)- 1-— + (l + 4ft2)- (l + 44ft2 + 16ft4)— + ...
2!
4!
6!
u2 u4 u«
dn(u, ft)=l-ft2 — + ft2(4 + ft2)-^-ft2Q6 + 44ft2 + ft4)— +...
2! 4! 6!
have a radius of convergence
(12) min(|K'|, |2K + iK'|, |2K-£K'|).
The addition theorems may be obtained from the addition theorems of
the #>-function in combination with the transformation (see Table 11, sec.
13,22)
(13) sn (iu9 ft) = / sc (u, ft 0* en (iu, ft) = nc (u, ft ')
dn (iu, ft) = dc (iz, ft ').
In the addition theorems we shall use the abbreviations
(14) s, = snU, , ft), s2 = sn(u2,ft), s£ = sn(i*2, ft')
with similar abbreviations for en, dn. We then have
(15) sn(ttf + u v k) = (s xc 2d 2 + c xd ys 2)/ {I - k1 s2 s\)
cn(u
dn(iz
(16) snU
cn(u
dn(iz
+ u2,k) = (cicz-s]d,s2d2)/a-kzs*sl)
+ u2,A) = W yd2-kzs ,c ,s 2c 2)/{\-kzs\s\)
+ iu2, A)- (s,i2'+ JClrf,s2'c2')/(c2'2 + A2s2 s2'2)
+ »b2, A) = (c,c2'- «,rf, s2'rf2')/(c2'2 + A2s2 s2'2)
+ iu2, A) = «, c2'd2' - iA2s , c, s 2')/(c2'2+ A2s\ s 2'2)

13.17
ELLIPTIC FUNCTIONS AND INTEGRALS
345
(17) sn(2a, A) = 2scd/(l - k2s 4)
en(2a, k)={c2-s2d2)/(l-k2s*)
dn(2u, A) = U2-A2s2c2)/(l-A2s4)
(18) sn$u, ft)=(l-c)x(l + rf)-*
cnW«, ft)=« + c)X (l + rf)"x
dn^u, A) = (rf + A2c + A '*)* (1 + «*)"*.
In (17) and (18) we reverted to the notation (1)8 Equations (16) show
that the values of Jacobi's elliptic functions for any complex u may be
computed if the values of these functions, and also of the functions with
the complementary modulus, are known on the real axis.
We also note the following Fourier expansions
277 v qn~l/2 , , ™
(19) snu = ) — _- sin(2n-l)
AK M4< l-q2"-' 2K
<7
zn u = ) —- cos(2n - 1)
AK a^1l + 72n"1 2K
2n fl Y qn nil 1
in u = — +7 —- cos n
K |_4 £, 1+92" K J
in which
(20) <7 = ei77T=exp(-77K7K).
The expansions (19) are valid in the strip of the complex plane bounded
by the lines ±i'K'+ AK, -<*> < A < «>.
The values of sn, en, dn at the points mK + niK'(m9 n integers) may
be found by means of 13 „ 12 (24): from these the values at the points
(21) VimK + y2niK' m, n integers
may be found by means of (18)* The results for 0 <m9 n < 3 are shown in
Table 6. The points chosen in Table 6 range in each case over one-half
of a cell. The values at the points (21) in the other half of the cell may
be found by means of Table 7, at other points (21) by the periodic
properties of sn, en, dn. All square roots in this table are to be taken as
positive when 0 < A < 1, and are defined by analytic continuation
otherwise.

TABLE 6. SPECIAL VALUES OF JACOBIAN ELLIPTIC FUNCTIONS
sn(i/2/nK+ %nilO
CO
~2 niK x.
0
1 If'
"2 fcK
iK'
fiK'
0
0
»-K
oo
-i&"X
]k
(i + *T*
(2*)"^ [(l + A^ + id-A)*]
U-fcT54
(2*fXt(l+ *)*-* (!-*)*]
K
1
*"*
fc-'
*-*
fK
<1 + *V*
(2ArH[(n-A)H-i(i-/b)^]
(i - *TX
(2A)"^ [(H-A)^ + i(l-/b)^]
PI
n
P
n
n
H
5

TABLE 6 continued
en
2 niK \.
0
1 V
*K'
T *K'
0
1
AT* (1 + *)5*
oo
-k~* (1 + *)*
i"
k'*a + k'r*
k'% (2kr% u-o
-**'* u-*T*
-A;'1'* (2*)""* (1+0
K
0
-AT* (l-*)5*
— ifc~ A;"
-ik~Yz (l-k)/2
fK
-i'K(l+t')",i
-*'* (2k)~% (1+i)
-i4'^(i-n-M
*'x(2*rx(i-«)
M
r
r
H
n
a
n
H
5
en
>
22
a
o
en
CO

TABLE 6 continued
dnforoK + fcniK')
CO
00
"2 ni It ^x^
0
"2 l'K
iK'
3 .wr /
0 j
1
(l + kf
oo
-(1 + A)X
JK
*'*
(K*')X[(1+ *')"-id-*')54]
-*'*
-C/,tf lU+if+ i(l-4f]
K
k'
a-k)'A
0
-a-*)x
fK
A'*
«ft')X[(l+ *')*+id-*')*]
,V*
-«*')" [(l + *')x-i(l-Ox]
to
W
n
>
r
G
n
o
(7)

13.18
ELLIPTIC FUNCTIONS AND INTEGRALS
349
From the addition theorems and Table 6, we may obtain the values of
Jacobi's functions at the point %/n K + l/2ni K '+ uin terms of their values
at u. Table 7 shows the results for the points mK + niK ± u. The table
covers more than a cell in order to exhibit the symmetry around the points
S, C, Z), N of Jacobi's functions. In the table the abbreviations (1) have
been used and when double signs appear, the upper signs refer to mK +
ni K '+ u9 the lower to m K + ni K '— u •
Jacobian elliptic functions may be used for the computation of Weier-
strass* functions when e^, e2, e are given. The modulus of the Jacobian
functions, and the variable of the Jacobian functions are given by 13,16
(7). The periods of Weierstrass' functions follow from 13.16(9), the
quantities 77 and 77'from 13.16(10). Weierstrass* basic function is
(22) p(z) = e3 + e\e* .
sn (a, k)
The three eamay always be numbered in such a fashion that \k\ < L
13.18. Descriptive properties and degenerate cases of Jacobi's elliptic
functions
In many applications we have 0 < k < 1. In this case also 0 < k '< 1,
and 13.8(1) shows that K and K'are real. The point-lattice mK + niK'
may then be generated by a rectangular line-lattice (although the latter
need not correspond to primitive periods). We shall indicate the behavior
of sn u, en u, dn u in this case by diagrams (see below). The notations
outside the figure indicate the position of the lattice-points wK + niK',
the notations inside the figure give the value of the function in question
at the lattice points. Along fully drawn lines the function is real and
between any two consecutive lattice-points it is monotonic. Along the
broken lines the function is imaginary and between any two consecutive
lattice-points it is monotonic. Along lines joining a zero and a pole of a
function the sign of the imaginary part is not at once obvious from the
figure and will be indicated by a — or + symbol.
From these diagrams we see that all three functions are real and
periodic on the lines Im u = 2nK\ The functions sn and en have periods
4K and oscillate between 1 and -1, the function dn has period 2K and
oscillates between 1 and k' on lines corresponding to even n, and
between - 1 and — & 'on lines corresponding to an odd value of n.

350 SPECIAL FUNCTIONS 13.18
TABLE 7. CHANGE OF THE VARIABLE BY QUARTER-AND HALF-PERIODS.
SYMMETRY.
sn(mK + 7ii'K ±ix)
-iK'
0
iK'
2iK'
-K
-d/(kc)
-c/d
-d/(kc)
-c/d
0
±l/Ucs)
±s
± l/(ks)
±s
K
d/Ucc)
c/d
d/(kc)
c/d
2K
+ 1/fts)
+ s
+ l/(ks)
+ s
3K
-d/(kc)
-c/d
-d/{kc)
-c/d
cn(mK + niK ± u)
^\mK
-iK'
0
iK'
2iK'
-K
-ik'/(kc)
±k's/d
ik'/kc
+ k's/d
0
±id/ihs)
c
+ id/(ks)
— c
K
ik'/(kc)
Tk's/d
-ik'/Qcc)
tk's/d
2K
Tid/Qcs)
— c
±id/(ks)
c
3K
-ik'/(kc)
±k's/d
ik'/(kc)
Tk's/d
dn(mK + niK'±u)
-iK'
0
iK'
2iK'
3*K'
-K
+ ik s/c
k'/d
±ik's/c
-k'/d
+ ik s/c
0
±ic/s
d
+ ic/s
-d
± ic/s
K
+ t A; 's /c
k'/d
±ik's/c
-k'/d
+ ik's/c
2K
±ic/s
d
+ tc/s
-J
± ic/s

13.18
ELLIPTIC FUNCTIONS AND INTEGRALS
351
K'jo
1-
Kl°°
+
0 i_0
1
l/k
1
0 "^
+
Oa
—
0
^1
-l/k
-I
o1
-1
oo |
+ 1
OJ
K
2K
3K
4K
sn u for 0 < Re u < 4K, 0 < Im u < 2K'
2iK'|
K
hi
-ik'/k
K
■+■
1-1
2K
~r
ifc'A
3K
-1
4K
en u for 0 < Re u < 4K, 0 < Im u < 2K'

352
SPECIAL FUNCTIONS
13.18
4iK'
3iK'
2iKd
iK'
1
+
—1
oo
1
k' 1
0 oo
-A' -1
0 °°
+
k' 1
0 K 2K
dnu for 0 < Re u< 2K, 0<lmu < 4K'
These diagrams may also be used to determine the signs of the real
and imaginary parts of Jacobi's functions in any of the rectangles. Take,
for instance, the rectangle whose vertices are K, 2K, 2K + iK ', K + i K \
From the diagrams we have on the boundary of this rectangle
Re sn u > 0, Im snu< 0;
Re cnu < 0, Im en u < 0;
Re dn u > 0, Im dn u > 0;
and by the tlieory of conformal mappings these inequalities hold also in
the interior.

13.18
ELLIPTIC FUNCTIONS AND INTEGRALS
/ (u) = 0 or oo
X
/(u)^0,«
SYMMETRIES OF JACOBIAN ELLIPTIC FUNCTIONS
The symmetries of Jacobi's functions may also be read off the
diagrams. Let u and uz lie symmetrically with respect to a zero or pole of
one of Jacobi's functions, f(u\ say, let u 1 and u3 be symmetric with
respect to a lattice point which is neither a zero nor a pole, u and u4
symmetric with respect to a line on which f(u) is real, and u^ and u5
symmetric with respect to a line on which f[u) is imaginary* Then
f(uf) = -f(u2) = f(u3) = f(uA) = -f(u5).
We also note that
(1) |snit| = AT* lmu = (n + y2)K'
(2) |dntt|-A'* Rett-(n + Ji)K.
A rotation by a right angle carries the diagram of sn u essentially
into the diagram of dn u; a rotation by a right angle does not change the
diagram of en u essentially*
A more complete description of the Jacobian elliptic functions for
0 <k < 1 is contained in the relief diagrams given in Jahnke-Emde (1938,
p. 92, 93).

354
SPECIAL FUNCTIONS
13.18
The Jacobian elliptic functions degenerate if one or both of the periods
become infinite, that is, if &2 is 0, 1, or indefinite (the last case being
trivial). As in the case of Weierstrass' functions (see. 13q15)> we list
three cases.
(i) Real period infinite*
(3) 4=1, £'=0, K=~, K'=%77
(4) sn (u, 1) = tanh u, en (u, 1) = dn (u, 1) = sech u.
(ii) Imaginary period infinite.
(5) 4 = 0, £'=1, K=y2n, K'=*c
(6) sn(iz, 0) = sin w, cn(u, 0) = cosu, dn (u, 0) = 1«
(iii) Both periods infinite.
(7) K = K '= oo, sn u = 0, en u = dn u = 1.
13.19. Theta functions
Although functions closely related to theta functions were encountered
by Euler, Jakob Bernoulli, and Fourier, their systematic study and their
exploitation for the theory of elliptic functions is due to Jacobi. Jacobi's
theta functions correspond to the sigma functions of Weie^st^ass, theory.
Like the sigma functions, theta functions are entire functions and hence
certainly not doubly-periodic, yet such that they show a simple behavior
under a translation by a period. Theta functions are more highly
standardized than sigma functions. They are simply periodic, can be represented
by series whose convergence is extraordinarily rapid, and they are the
best means for the numerical computation of elliptic functions.
For Weierstrass' functions we had the -variable 2, the half-periods
a), co ', we put 7=co '/co, and assumed Im r> 0, Jacobi's functions were
represented in terms of u, and the quarter-periods K, K', where
(1) tt-(ef-e3)*zf K = (c, - e 3)* co, iK'*(et - e 3YA eo'.
Theta functions will be expressed in terms of the variable
z u
(2) t/« = ,
2co 2K
the parameter being either
g>' K'
(3) r=—=i Imr>0
co K
or

. 13.19 ELLIPTIC FUNCTIONS AND INTEGRALS 355
(4) q = eirrr = eimu'/a=exp(-irK7K) |?|<1.
The half-periods are l,r. Making use of 13.10(8), we may always achieve
(5) \q\ < exV{-V2n • 3*),
hut such a choice of the primitive periods will not he assumed in what
follows.
The definition of the four theta functions is
(6) 0 {v)=d]{v,q)=di{v\r)=2qX 1 (-l)n?n(n + ,)sin[(2ra+l)ffV]
n= 0
(7) 62{v)=d (t>, ?)=02(t>|r)=2?* 1 qn(n + i)COs[{2n+l)7Tv]
n= 0
(8) OAv)=0 Av,q)=d3(v\r)=l + 2. I qn* cos (2nnv)
n= 0
(9) d(v)-d(v,q)=d4(v\r)-l + 2 1 (-l)"?"2 cosiZnnu).
n= 0
The last of these functions is sometimes denoted by 6Av) or 6(v) simply.
These series converge for all (complex) v and all q satisfying (4)* On
account of the factor qnZ we have excellent convergence. The four series
may be rewritten in the form
(10) 0 (v)-i 1 (_l)nq(n-V2)2 ei<n(2n-\)v
n = — oo
(11) 0Av)= I ?0»-*>2e *"«*»-'>.
n = —oo
(12) 6(v)= 1 qn\
2
(13) d (v)= 1 (-1)"?" ei7r2n",
n=—oo
when they appear as Laurent expansions in the variable exp(inv), and
are convergent for all finite non-zero values of this variable.
All four theta functions are entire functions of v% All four are periodic,
the period of 6 and #2 being 2, and that of d3 and #4 being 1. Their
behavior under the addition of half- and quarter-periods may be seen from
Table 8 in which the abbreviations

TABLE 8. CHANGE OF VARIABLE BY QUARTER- AND HALF-PERIODS. SYMMETRY.
e(v)
0,(v)
e2{v)
<?,w
eAM
e(-v)
-*,(*)
e2(v)
e3M
eAM
0(v + 1)
-6t{v)
-e2M
*,<«>
0AM
eiv + t)
-AMd^iv)
A(v) (92(v)
A(v) 63{v)
-a (v) eiv)
4
eiv + i + r)
A{v)Ot(v)
-A(v)ez{v)
am e3(v)
-a M eA(v)
eiv + k)
e2M
-0,(t»)
eAM
e3M
e(v + Vi r)
iB(v) eA{v)
b{v) e3{v)
bM ez{v)
IBM et(v)
ei.v + vi + viT)
bM e3M
-tBiv) eA(v)
iBM et(v)
b{v) ez(v)

13.19
ELLIPTIC FUNCTIONS AND INTEGRALS
357
(14) A{v) = e-i7T{2v+r\ B{v)=e"i7T{v^r)
have been used. Table 8 also shows the parity of the four theta functions.
Table 8 shows that all four theta functions may be generated by any
one of them by the addition of quarter-periods. From the table, 0f has a
zero at v = 0, and hence zeros at m + nr, where m9n are integers. It can
be proved (by integrating 6'j/f6\ over the boundary of a parallelogram with
vertices ± % ± % t) that these are the only zeros of #:; and Table 8 then
may be used to determine the zeros of the other three theta functions. In
Table 9, m and n are integers.
TABLE 9. ZEROS OF THETA FUNCTIONS
\e(v)
zeros
etiv)
m+ nr
6z(v)
m + l/2 + nr
63{v)
m-ir1A + (n + lA)T
<9>)
m+ (n+ %)r
From the knowledge of the zeros it is possible to obtain infinite
products representing the theta functions, and from these products the partial
fraction expansions of log 6(v) and d'(v)/d(v) follow. From (17) we also
have (19). In the products we use the notation
(15) ?0= 5 (l-qZn)
n= 1
and have
(16) 0y{v)= 2qQq% sin nv H (1 - 2qZn cos 2 nv + qAn)
6 (v)=2q q% cosnv fl (1 + 2q2n cos 2*v + q*")
03(v) = qo n (l+2g2n-' cos2ttv + q*"-2)
n= 1
6n(v) = q0 S (l-2g2n-1 cos 2nv + qAn~2).
1= 1

358
SPECIAL FUNCTIONS
13.1
(17)
f ep) "I v ?2"
°2 2m • 2
" " sin mnv
q " m
2* „:-2,
<7 771
Sin /7177V
gmo) = 1°s<c°s^+4 £ (_1) it
L. 2 J m= "
lojAlll _4 y ?" si"2w™
Lm»J ~ .4, w -
(18)
6'Av)
v.w) y 9™
= nctnnv + 4n- / ;— sin2m7rt;
0»
—- = -77 tan77V + 477 ) (-l)m — sin 2m nv
(W) £, i-92»
e»
,(v) Lt 1-.
sin 2/n 77v
= 4;
,2m
Sin 277177 V
(19) — log
+ 2
(9, (v + u;)
1 sin77(v + w)
= Tlo& ~—? ^~
2 Lsln ^^ ~ ^' J
00
2--
.2m
,2m
sin277i77V sin 2mnw9
1 r^2(v+w;)n 1 | cos77(v + w)
T °g[eyv-u;)J =~10gLcos77(v-M;)J
+2 f hil^:
m= 1
771 1 — q2
s\n2mnv sm2mnw.

13.19 ELLIPTIC FUNCTIONS AND INTEGRALS 359
(-1)" ?"
(19)Tloghn—t\ = 2 *
2 L0(v-w)J
■ sin 2m 77V sin 2m Trie,
,= 1 m l-q2'
1 \ dAv + w)~\ V 1 q'
—loe = 2 /, ;— sin 2m 77i> sin 2m rnv,
2 8|_ «>-.«) J ~, m 1-92"
Equations (16) are valid in the entire r-plane* Of equations (17) and
(18) those relating to 6y and $2 are valid in the strip |Im v\ < Im r, those
relating to #3 and $A in the strip |Im v\ < %Im u Of equations (19), the
first two are valid when |lm v\ + |Im w\ < Im % the last two are valid when
|lm v\ + |Im w\ < /2lm r« From (18) we have
0'(v Tin+nr) 61v)
(20) —— -= - 2n77i a = 1, 2, 3, 4; w, n integers.
0a(i; + 77i + n r) 0a(i;)
Between the squares of theta functions of the same variable there are
the following relations
(2i) e]{v) **«))= b\{v) e\{Q)-e\{v)e\®)
e]M e*(o) = e\(v) e\(.o)- e\M ^(o)
e'M 042(o) = 02d>) 6>2(o)- e\(v) e\io)
Each of these relations may be proved by remarking that the ratio of its
two sides is a doubly periodic function (with periods 1 and r) without
zeros or poles and hence a constant, and evaluating this constant by
using special values of v (half-periods)*
Equations (21) are special cases of the so-called addition formulas of
the theta functions which express
0a{v + w)da(v-w)dl(0)
in terms of squares of theta functions of v and w (see Whittaker and
Watson, 1927, p. 487).
The "theta functions of zero argument"
0'x(0), d2{0), 0,(0), 04(O)
are of especial importance (see see* 13*20). They satisfy several
identities among which the most important are
(22) 0,'(O)=7702(O)03(O)04(O)

360
SPECIAL FUNCTIONS
13.19
(23) 0*(O)+0<(O)=0<(O).
For graphs illustrating the behavior of the theta functions of argument
zero, and for a description and graph of the behavior of 6 (i;|0,l) for real
v see Tricomi's book (1937, p. 137- 140).
Theta functions arise, independently of the theory of elliptic functions,
in the theory of heat conduction and similar boundary value problems.
As is seen from (10)-(13), the functions 0jMx\int\ a = 1, 2, 3, 4,
satisfies the partial differential equation
dx2 dt
In this connection it is worth noting that theta functions have remarkably
simple Laplace transforms.
There are also non-linear differential equations of the first order (the
variable is v) satisfied by quotients of theta functions. These can be
derived very easily from the connection between elliptic functions and
theta quotients (see sec, 13„20)«
Hermite has studied the function
(25) © Av\t)= 2 exp[i77r(rc + Viy)2 + 2i7rv(n + %/x)+ innv]
V-tV n = — oo
(see Hurwitz and Courant, 1925, p« 198-201), Jacobi's four theta functions
are particular cases of Hermite's function,
13.20. The expression of elliptic functions and elliptic integrals in
terms of theta functions. The problem of inversion
Theta functions are very closely related to Weierstrass* sigma
functions: hence the expression of Weierstrass' functions in terms of theta
functions, Jacobi's functions have already been expressed in terms of
Weierstrass' functions and may now be expressed in terms of theta
functions. Lastly, theta functions may also be used to writedown expressions
for complete and incomplete elliptic integrals of the third kind. We shall
use the variable z for Weierstrass' functions, u for Jacobian elliptic
functions and v for theta functions. These are connected by 13,19(2%
The connection between the various notations of periods and other
quantities is given by equations 13,19(1) to (4)*
Weierstrass' functions
(1) a(2)=26)exp^— J -^

13.20 ELUPTIC FUNCTIONS AND INTEGRALS 361
\2co J 0a+,(O)
(2) aaU) = exp[^- J-^ii^l a=l,2,3
7/1 d'Av)
(3) £(*) = -*+ —
(4) FU)-ea+^- f-fe -^-1 a =1,2, 3
^2 L^+,(°) *,<»> J
(5) ,u i «.(«)«>>«»«;'«»
' fe3 6AO)6A0)6A0)6*{v)
(6) 12a>2et=,72[<?<(O)+0*(O)]
12a>2e = *2[<?*(0)-<?*(0)]
Z Z 4
12 6>fe3--ira[0*(O)+0j(O)]
(7) (e2-e/=»(e3-e^=-!.^(0)
(8) g2=lul-J[^(0) + ^3(0)+^(0)]
.Q'WJ(0)+^
" " ' "■ • "(O)][0j(O)+0;«»][02«»-0««»]
27
.3
(9) A* =-IT 9\\0) =^_ [0,(0) 0(0) 04(O)]2
4&>3 ' 4.6J3 2 3 4
i g,-(0) , „i r *r«»
(10) >/ = -- fl/.„. > T) = -- '
12(u 0,'(O) ' 2a> 12a, 6A0)

362
SPECIAL FUNCTIONS
13.20
Equation (1) may be proved by remarking that the quotient of the
functions on its two sides is a doubly-periodic function without poles or
zeros, and approaches 1 as v and z tend to 0. Equation (2) follows by
13,12 (22) and Table 8 of sec. 13,19. Equation (3) follows by logarithmic
differentiation of (1), (4) from (2) and 13,13(22), (5) by (4) and 13,13(21),
(6) and (7) from 13,13(23), (8) from 12,13(5) and (6), (9) from 13,13(7),
(10) from (1) and (3)* All of Weierstrass' functions are formed with periods
2o>, 2<a', and variable z. The variables v and q in the theta functions are
given by 13,19(2) and (4).
Jacobian elliptic functions. The following relations are obtained from
the formulas of sec, 13,16 by means of equations (1) to (10)«
(ID k* = e2{o)/e3(o\ k'* = 04(o)/03(o)
(12) K* = (%n)*e3{0\ K'*«(-KrO*03(O)
0A0)0Av) OMOAv)
(13) sn«=-2 ! , cn«= 4 2
e2(o)e4(v) e2(o)dA(v)
dn u =—=—-—2—r , Z\u) = E\u) u = , x .
d3(0)dA(v) K 2K 6A{v)
Given r, equation (11) determines the modulus of the Jacobian elliptic
functions, (12) the quarter-periods, and (13) the functions themselves. In
applications of elliptic functions, usually k is given and the question
arises whether there always exists a q such that |^| < 1 and
This is known as the problem of inversion % In many practical applications
0<&2<l.ln this case consider
*<4(o,?) n i-q
K®'^ „=, 1 + i
n—
11 i +
2n-t
2n + l
by 13, 19 (16)* As q increases from 0 through real values to 1, the infinite
product decreases monotonically from 1 to 0 and hence (14) has exactly
one solution q for which 0 < q < 1. For other values of A;2 the discussion

13.20
ELLIPTIC FUNCTIONS AND INTEGRALS
363
is much more difficult (see for instance Whittaker and Watson, 1927, p%
480-483) and involves complex values of q. The proof of a unique system
of Jacobian elliptic functions for any given ft2 A 0, 1 may be based on the
theory of elliptic modular functions.
Elliptic integrals. The basic elliptic integrals in Legendre's normal
form, 13»6(l)-(3)» may be computed by means of theta functions. We form
Jacobian elliptic functions with modulus ft, determine the quarter-periods
K and K ', and put
FU, ft)
(15) v=— , <7 = exp(-7rK',/K)
A K.
for the parameter and variable of the theta functions. We then have from
(13)
1 0\v)
(16) E(<f>,k) = £_ + 2Et;.
2K 6A(v)
The computation of elliptic integrals of the third kind is more difficult.
We shall give the results for real cf>, v, and 0 < ft < 1, shall express v in
terms of an auxiliary real parameter y, different expressions being valid
in the intervals (-<*>,-!), (-1, -ft2), (-ft2, 0), (0, oo), use (15), and put
We then have (see Tricomi, 1937, p« 153-158)
cn(y, ft)dn(y, ft) V 1 1
sn(y, ft) I sn2(y, ft) J
p,(» + ff) 1
- log —L. Z- L7~v 0<y<K, v >/3
2 *\j (v-$) J 0(0) y ». i i p
1 \dA$ + v)\ 6'A(0)
= — log—'■ - 4 v 0<y<K, \v\<B
. sn (y, ft ') en (y, ft ')
(19> * I , , 7 II [<£,-dn2(y, *'),*]
dnly, A )
i , [d2(v + ip)~\ e:w
log — - i —* v
2* L^^-^'^J 9«t*/3)
0 <y<K'

364
SPECIAL FUNCTIONS
13.20
cn(y, k)dn(y, h) _
(20) , ,N 7' n [<£, -k2 sn2(y, A), A]
sn(y, A;)
1 fe4(t; + j3)"] (9'(jS)
= Los: —- + —! v
2 *Le>-ie)J e.dg)
dn(y, A') „ T
(21) —TTT-r-PT n **'
siUy, k ) ca(y, k ) |_
0 <y <K
(y,*')
(y, k')
0
i r0> + *j8)1 e;up)
= log —2— +r— 2;
2i *L0>-*'0)J ^ttjS)
0 < y < K '.
In all these formulas logarithms have their principal values. In (18) and
(20) these are real, in (19) and (21), —n < Im log[--«] < 77. The right-hand
sides of (19) and (21) are real. From 13.19(18) and (19) we have
e;ap) " -Zn
(22) i
dt(ifi)
= 77 Ctnh 77)8 — 477 7 ■
^. 1-
sinh 2n nfi
(23) f 3, = 4w > (-DV , sinh2n^/3
(24) — log
2 s
+ 2
6>2(t> +i/3)
L02(»-*/3)
= — tan ' (tanh tt/3 • tamrt;)
2
(-1)"
l-?2"
1 , r 6*.(v + J/S)"| V 1 qn
sin 2n nv • sinh 2nnj3
— sin 2 n 77V • sinh 2 7177)6.
The convergence of the infinite series in (23) and (25) is not always
as rapid as one would wish. When q is not small, the expansions
(26) dA{v± i)8)= 1 + 2 f (~l)n q71* cos (2nnv) cosh(2nn@)
±i 2 (-l)""1 qn sin(27177*;) sinh(2n77j3)
n= 1

13.21
ELLIPTIC FUNCTIONS AND INTEGRALS
365
(27) 0.U/8)=l + 2 1 qn* coshVnnP)
(28) id'(ip)=4n 1 nqn2 sinh(2nnp)
may be used for the computation of the right-hand sides of (19) and (21).
These expansions, and some others which are useful in these
computations, follow from 13,19(6) to (9).
Complete elliptic integrals of the first kind have already been expressed
in terms of theta functions, see (12). For complete elliptic integrals of the
second kind, we have from (6), (7), (10), and 13.16(10),
(29) E 1 —i K ! .
3#43(o) 12k e;(0)
Complete elliptic integrals of the third kind have already been reduced,
in 13„8(21)-(24), to elliptic integrals of the first and second kinds and
hence may be computed by means of theta functions.
Finally we mention that in applying theta functions to the computation
of Jacobian elliptic functions or of elliptic integrals with a given modulus
k, 0 < h < 1, the parameter q of the theta functions may be computed from
(30) q = e + 2€* + 15€9 + 150€13 + ... 2€ = (1 - k '*)/(l + A '*).
13.21. The transformation theory of elliptic functions
The transformation theory of elliptic functions deals with the relations
between elliptic functions belonging to different pairs of primitive periods.
Since any elliptic functions of periods 2a>, 2a>' may be expressed
algebraically in terms of fp(z\co, co 0, it is sufficient to discuss relations
between #?-functions. We shall always assume
(1) Imico 7co) > 0, fin(<5>7&) > 0,
and will summarize briefly the results of the general transformation theory,
referring for proofs and fuller details to the books listed at the end of
this chapter.
We shall say that two functions f(z) and g(z) are algebraically
connected if there is a polynomial in two variables, P (x, y), such that
P [f(z\ g(z)] = 0 identically in z.
A necessary and sufficient condition for %>(u\a), co') and fp(u\co9 co')
to be algebraically connected is the existence of integers a, jS, y, 8, p
such that

366
SPECIAL FUNCTIONS
13.21
(2) pco = aco + /3eo ', po> '= yeo + 8co % D = ad- (3y> 0>
Given (2), clearly both $>(u\co9 co') and p(u\cb, cb') are even elliptic
functions of periods pa>, pco'9 and hence they are rational functions of
$>(u\pco, pco')* Thus, it is sufficient to envisage substitutions (2) with
p = 1, and these we shall write in matrix notation as
■Cl-CXl-
Then the relation between
8
>0.
(4) x = $>(z\co9 co'), y = p{z\co9 co')
is of the form
(5) P(x, y)=0
where P is a polynomial in x and y, linear in x, and of degree D in y%
(The degree in y is elucidated by counting poles.) We call D the degree
or order of the transformation
(6) r =
[;:]■
and shall multiply transformations as matrices,
~aa + /3c a6 + /Sc?H
[a #T T° 6~| Taa +
y 8J Lc d] Lya+
Sc yb + 3d J
The transformations (3) may also be envisaged as Moebius
transformations of the upper half of the complex plane onto itself,
(7)
y + St
a + fir
All transformations of the first order form a group (the modular group).
A necessary and sufficient condition for $p(u\co, co') = p(u\co9 co') is that
co9 co' and co9 co' be connected by a transformation of the first order
(unimodular transformation).
The modular group is generated by the transformations
"1 0'f
(8) A
Ll
c, :>

13.22
ELLIPTIC FUNCTIONS AND INTEGRALS
367
that is, any unimodular transformation is a product of powers of A and B.
Thus, the study of transformations of the first order may be limited to the
study of A and B.
Similarly, the study of transformations of the second order may be
limited to Landen*s transformation
m L-[o i]>
since any transformation of the second order, S, may be decomposed as
S = HLK9 where H and K are unimodular transformations.
13.22. Transformations of the first order
A transformation of the first order leaves the point-lattice 12, of all
periods (sec. 13«10) unchanged. Since Weierstrass* functions o(z), £(z),
$>(z\ and the invariants g2 , g3 , A = g*~ 27 g* depend only on Q, they do
not change. The eamay undergo a permutation. From 13„12(19) and 13%
13(19)
77 = £(g3|g3, co ') = £(d> \co9 a>') = £(aa> + jS co*\co9 co') = arj + f3rj'
ri'=yri + 8r)'
so that 77, 77' undergo the same transformation as co9 co\ The functions
<7a(z) may undergo a permutation. A straightforward computation shows
that A of equation 13*21(8) interchanges the indices 2 and 3, and B the
indices 1 and 3, in e, , e2, e 3 and o x (z), o 2(z) <r3(z).
The behavior of Jacobian elliptic functions under unimodular
transformations is more involved. If a and 8 are odd integers, and /3 and y even
integers, in 13„21(6), we call T a A-transformation. It is easy to verify
that all X - transformations form a subgroup of the modular group, and this
subgroup is called the X- group. For a A-transformation,
e] = p(cb\co9 co ') = p (aco + /3 co '\co9 <w') = p(co) = e 1,
since /3a>'is a period for even /3, and aco differs from co by a period for
odd a. Similarly 42 = e2 and e3 = e 3. From 13 „16 (4)-(6) it is seen that
Jacobi9s functions sn, en, dn are invariant under X-transformations. Any
other unimodular transformation affects Jacobi's elliptic functions.
We shall consider the five transformations
4:13. -l\ -J

368
SPECIAL FUNCTIONS
13.22
The last three may be expressed in terms of A and B.
(2) C = ABA, D = ABAB, E - BAB A.
In Table 10 the six transformations U (identity), A9 ... , E are listed
together with the permutations of e aeffected by them. Every permutation
of e} , e2, e3 occurs. Since the permutation of the eacompletely
determines the transformations of Jacobian elliptic functions, it is sufficient
to consider the transformations (1) in order to obtain all possible
transformations of the first order of Jacobi's elliptic functions.
TABLE 10. PERMUTATIONS OF THE ea
Transformation
U
A
B
C
D
E
CO
CO
CO
0
CO
co+ co'
-co+ co*
0
CO
CO
CO
CO+ CO*
-co
CO
-co
-co — co*
e*.
e1
e.
S
e2
e2
S
«2
e2
e3
e2
e1
S
e1
U
e3
e2
e,
e3
ex
62
This table in combination with 13,16(4), (5), (6), (9), and (11) at once
leads to the transformation formulas recorded in Table 11.
For the transformation of elliptic integrals see Table 3, sec. 13a7, and
Table 4, sec. 13«8.
The transformations of the four theta functions may be derived from the
expression
w1/2Al/8 / z\ z ^
(3) 0,(1/10= FTi exp( -J_ } <j(z) "=-^-> *~ —
77 \ ICO / 2C0 CO
which follows from 13.20(1), (9), and 13*19(2), (3). We know already
how the right-hand side behaves under a transformation 13*21(6) of the
first order and note in particular that
7/ 7} 7} ar)-\-fir)' (3(7)co'-7)'co) (3ni
co co co aw + jSw' coco 2coco

Transformation
A
B
C
D
E
TABLE 11.
CO
• *
CO
CO
co + co'
*
CO
-co
co + co'
*
CO
-co + co'
-co
*
CO
-~(co + co')
TRANSFORMATIONS OF THE FIRST ORDER OF JACOBI'S ELLIPTIC FUNCTIONS
u
k'u
— IU
hi
—ik'u
—iku
k
ik
17
k'
1
T
i
k'
Ik
k'
i
F
k
k'
ik
k
ik'
1
T
•
K
k'K
K'
/Kk + i'kO
kXK'+iK)
kK'
K'
k\K'-iK)
K
kK'
k'K
k(K + iK')
sn (u, k)
sn (u, k)
k'
dn (ut k)
sn (u, k)
—i
en(u, k)
k sn (a, k)
sn (ut k)
— ik'
en (u, k)
sn(a, k)
—ik
dn (u9 k)
9 •
en (ut k)
en (ut k)
dn (ut k)
1
en {u, k)
dn (u, k)
dn (ut k)
en (ut k)
1
dn (u, k)
dnfe, k)
1 J
dn {u, k)
dn (u, k)
en (u, k)
en (u, k)
1
en (u, k)
en (u, k)
dn (u, k)
w
r
r
I—C
n
a
si
n
o
C/)
>
a
3
OS
VO

370
SPECIAL FUNCTIONS
13.22
by 13,12(10), and also that
z z v y + 8t
2co 2(aco + pco') a+j3r ' a+j3r'
Then we have, from (3), the general transformation formula of 6 (v\r) for
transformations of the first order,
(4) 0,
a + /3r
where €B = 1. The factor € accounts for the ambiguity in the fractional
powers in (3) and may be determined by dividing (4) by v, making v -> 0,
and then comparing both sides* The transformations of the other three
theta functions then follow from Table 8, sec. 13„19.
The explicit formulas for the transformations A and B of (1), which
generate the modular group, are as follows.
Transformation A.
(5) v = v, t = 1 + r, q = - q
(6) 0f(t/|r + \)-~e%7Ti 6^v\t\ 62(v\t + 1) = e*^02G;|r)
6^v\t+\)-6a(v\t\ dA(v\T+I) = d3{v\r).
Transformation B.
(7) v = v/t, f = - 1/r, log 4 = 772/log q
(8)
0 f-JLl j =-i(-ir)lA exp(inv2/r)dy(v\T)
d2 (— -—) «(-*»* ex?(i7TV2/r)dA(v\r)
63 (~\-~) -(-*»* exp(irrv2/r) d3(v\r)
In these formulas (-ir) has its principal value (lies in the right half-
plane). Transformation B is known as Jacobi's imaginary transformation*

1«,23
ELLIPTIC FUNCTIONS AND INTEGRALS
371
Transformation B may be used for the numerical computation of theta
functions when q is near 1, or r is very small, when the series for 6x(v\f)
converge somewhat slowly, but those for 6}(v/r\- 1/r) converge very
rapidly. In particular, the asymptotic behavior as q -* 1 may be
investigated in this way, and one obtains
(9) 0,(0, q) - 0S (0, q) - (-ir/log ?)* q -> 1.
13.23. Transformations of the second order
There is essentially only one transformation of the second order, in
the sense that any transformation of the second order is a combination
of Landen's transformation
and two unimodular transformations. In writing down the transformation
formulas we shall observe the following convention. All Weierstrassian
functions whose periods are not indicated are formed with primitive
periods co, co', and all e a, -qa(without dots) are derived from such functions.
Landen's transformation* Weierstrass* functions*
(2) co = Vico, co0 - co*
(3) e1=e1+2(eI-e2)^(e1-e3)^
*2-e1-2(cI-c2)»(c|-c8)»
(4) rJi=r)] + 1Aei co^, rj 2 = 7/2 — 77 z + lAe ,(^2- coz)
^ 3=2r/3+eiW3-
(5) o(z\l/2co, cjO- exp(Hc1 z2)o(z)ol(z)
o,(z\V2co, 6)0=exp(1/2e1z2)[a|(z)-(eT-e2)^(e?-e3)^a2(z)]
o2U\V2co, 6)0=exp(1/2e1z2)[a2(z) + (e1-e2)}4(e1-e3)^a2(z)]
o3(z\y2co9 co') = expO^e, z2)a2(z) oz(z)
(6) £U|% a>, a>') = £(z) + £(z + tj) + e z - ryt

372
SPECIAL FUNCTIONS
13.23
(7) jp(z\lA<oy o>') = p(z) + %>(z - o>t)- e x
= #?U) + —
Since Landen's transformation of Weierstrass* functions involves
ea9 tj , which are not invariant under unimodular transformations, we
record the basic formulas for two other transformations of the second
order.
Gauss9 transformation. Weierstrass' #?-function.
to :]■-
(8) G = I n = - BLB
(9) &(z\co, 1Ag>')= p(r-)+ #>U - a) )- e
3
.P(«)>'-<8')(<8'-<8')
pU)-e3
TAe irrational transformation. Weierstrass' ft»-function
(10) -Z"=[_1 „] --ABLABAB
(11) #>(z|gj, Via + y2co')= p(z) + p{z - co2)- e 2
{e,-et){et-e3)
= kHz)--
p(z)- e
2
Landen's transformation. Jacobian elliptic and theta functions. When
the parameter in a theta function is not indicated, it is understood to be r.
(12) u «(1 + A')u, A = (1-A')/U + A'), A*'-2A'V(1 + A')
.,„. [",.. . ,. l-A'l sn(u, A) cn(u, A)
(13) sn (1 + A ') u, = (1 + k ') ——
L 1 + ft J dn (u, A)
I\, ,« 1-*'1 l-Cl + AOsn2^, A)
en (1 + A ) u, =
L 1 + A ' J dn (u, A)
, Tm ,n 1-*'1 l-d-^')sn2(U,A)
dn (1 + A ) u, =
L 1 + A ' J dn («, A)
(14) v =2v, f =2r, 4 =<?2

13.23 ELLIPTIC FUNCTIONS AND INTEGRALS 373
(iB«;e)i»,).x»'w;«))«;wrf'-7 u'W); ,
3 4
<?2(0|2r)=2-*[<?23(0)-^(0)]*
<?3(0|2r)=2-*[<?2(0)+<?2(0)]*
<?4(O|2r) = [03(O)<?4(O)]*
0,(t>)0,(*>)
(16) 0,(2t>|2r)=-
<?4(0|2r)
<?2M-<?2G>) e2Av)-e2Av)
2<?3(0|2r) 202(O|2r)
0,(2t;|2r)=-* ' 3
0,(2t>|2r)-
fl4(2tt|2r)-
202(O|2r) 203(O|2r)
'3 *-' "4 '
04(O|2r)
Gauss' transformation. Jacobian elliptic functions
(17) d = (1 + t) u, k=2k1A/(l + k\ k '= (1 - i)/(l + it)
(i8)snra+.)u,i£!l ,a+t)i s;(M^ n
L 1 +A; J 1 + J snz(",i)
r,, 7x 24*1 cn(»,A)dn(it,ft)
en ll + «)u, = 1
L 1 + 4 J l + £sn2(u ,k)
r 2k%~] 1-
dn (l+"A)ttf = ■
L l+k J 1 + 1
. — A; sn (iz, A)
A; sn (u, k)
Transformations of higher orders are more involved. We mention here
only the transformation (LB) which is of the fourth order and leads to
the following duplication formulas for the theta functions. All theta
functions have the same parameter r.

374
SPECIAL FUNCTIONS
13.23
, N d.MdAv)9Av)dAv)
(19) eA2v)~2-* z- 3 4
0.(2»)-
dA2v) =
dAo)e3(o)dA{o)
d\(v)d*Av)-d2Av)e2Av)
6A0)d2A0)
e2Av) e2Av) + e2Av) e2Av)
d2A0)$A0)
eAAv)-e4Av) e*Av)-d\
eHo) eHo)
4 4
M
13.24. Elliptic modular functions
An elliptic modular function, /(r), is a function which is regular save
for poles, when Im r > 0, and has the property that fir) and fir) are
algebraically connected whenever r and f are connected by a transformation
of the modular group
ar+ /S _ n
(1) f = a, j3, y, § integers, ad - py = 1.
y r+ 8
[Note that a, ... , y have been renamed as against 13,21(7)J If fir) =
fir) for any transformation of the modular group, then fir) is called an
automorphic function of the modular group*
A first example of such a modular function is the square of the
modulus of the Jacobian elliptic functions. From 13,16(7) and 13,20(14)
e,-e3 0j(O|r)
is an analytic function of r for Im r > 0, with the real r-axis as a natural
boundary. From the invariance of e 9 e 2, eg under X-transformations
(a, 8 odd, jS, y even, see sec. 13,22) it follows that X(r) is an
automorphic function of the \-group. In general, a transformation of the
modular group will permute the e and hence change X(r) into one of the six
values
1 1 A(r) , 1
(3) A(r), l-A(r), ——, —, —j-r—-, ^TTT*
A(r) l-A(r) A(r)-1 AW

13.24
ELLIPTIC FUNCTIONS AND INTEGRALS
375
Since all these are algebraically connected with X(r), this function is
an elliptic modular function*
From 13,12(13), g2, g3, and A = g\ - 27 g32 are homogeneous functions
of degree -4, -6, -12 respectively in o and <a'and the absolute invariant
(4) 4-= ,*' '» =J(r)
A g23~27g^
is a function of r alone: it is analytic in the upper half-plane. A
transformation of the modular group leaves g2 and A unchanged (see sec.
13*22), showing that J(t) is an automorphic function of the modular
group. From 13*13 (6), (7) and 13*16(3), J may be expressed in terms of
X, and from 13*20(8) , (9) in terms of theta functions
4 (1_X + X2)3 1 [dl(0\r)+d6A0\T) + dB(0\T)]3
(5) J(r) =
27 X2(l-X)2 54 08(O|r)083(O|r)08(O|r)
We call two points r, f in the upper half of the complex r-plane
equivalent if they are connected by a transformation (1) of the modular group.
The fundamental region of the modular group is defined by
|r|>l, |r+l|>|r|, |r-l|>|r|.
The upper r half-plane may be subdivided into an infinity of regions,
each bounded by three circular arcs (one or two of which may degenerate
into segments of straight lines), and each equivalent to the fundamental
region. In fact every point in the upper half-plane is equivalent to exactly
one point of the fundamental region.
Given an automorphic function of the modular group, it is sufficient
to investigate the behavior of this function in the fundamental region.
For instance, it may be proved that J(r) assumes every finite value
exactly once in the fundamental region, and this shows that to every
(finite) value of J there is exactly one system of Weierstrassian
functions.
The fundamental region of the X-group is bounded by the straight
lines Re r = + 1 and the circles |2r ± 1| = 1; the boundary points in
Re t > 0 belong to the region, the boundary points for which Re r <0 do
not. It may be proved that X(r) assumes every finite value different from
zero and unity exactly once in the fundamental region of the X-group, and
this is the key to the problem of inversion (sec. 13*20)' it may be used
to prove that the Jacobian elliptic functions are uniquely determined when
the square of the modulus is assigned as any number ^ 0, 1.

376
SPECIAL FUNCTIONS
13.25
13.25. Conformal mappings
Elliptic integrals, elliptic functions and related functions occur in
many important conformal mappings. Many examples of such conformal
mappings, and some further references, are to be found in H. Kober's
"Dictionary of conformal representations" (1952, p. 170-200). In this
section we shall describe some of the most important mappings briefly.
Throughout the section we assume the "real" case,
0 < k < 1, 0 < q < 1, co real, co 'imaginary, K, K 'real,
and put e f > e 2 > e 3. We put Re z = z |f Im z = z and similarly for other
complex variables. In diagrams illustrating conformal mappings from the
plane of one complex variable to the plane of another such variable,
corresponding points will be indicated by the same letter.
CO
,2 = o
0
zi =
J9
Q
= 0
I
II
C
13
C
co -\
CO

fill
I >
e, «,2 = 0
a
J9 C B
I
The mapping w = ${z)
co- co
The function w = fp(z). As z describes the boundary of the rectangle
with vertices 0, co, co + co ', co \ the variable w is real and decreases from
<» to e,, e2, e3, — oo (see sec. 13» 15)» The function maps the interior of
the rectangle on the lower w half-plane. By Schwarz'sreflection principle,
the rectangle with vertices -co\ co — co', co + co\ co' in the 2-plane is
mapped on the whole w plane cut from — <*> to e .
In the lemniscatic case, g3 = 0, g2 > 0, we have e 2 = 0, e 3 = —e t. The
rectangle in the 2-plane becomes a square, the diagonal (iC joining 0 and
co + co ' is mapped on the negative imaginary axis in the ft/-plane, and the
diagonal B<B joining co and a/is mapped on the lower half of the circle
with center at e - 0 and radius e ^ in the u;-plane. The interior of the
rectangular isosceles triangle with vertices xAco + 1Aco,9 co9 co'+ co in the
2-plane is mapped on the fourth quadrant of the circle with radius e^n
the w-plane.

13.25
ELLIPTIC FUNCTIONS AND INTEGRALS
377
The function w = sn (u, k). From sec. 13* 18 it is seen that the interior
of the rectangle with vertices 0, K, K + i K', i°K 'in the a-plane is mapped
on the first quadrant of the w-plane, the rectangle -K, K, K + i'K',
-K+iK'
u2=0
3
in
.&
IV
u( =0
iK'
8
n
i
Q
K + iK'
C
B
Q
tt Q -B
The mapping u; = sn (u, k)
C J9
— K + z'K'is mapped on the upper half of the w-plane, and the rectangle
with vertices ± K ±£K'is mapped on the whole w-plane cut from -« to
-1 and from 1 to <». It can be proved (see for instance Dixon, 1894,
Appendix A) that the lines uy = const., u2 - const., are mapped on the
doubly orthogonal system of confocal bicircular quartics in the if-plane
whose foci are ±1, ±k~\ These quartics are symmetric with respect to
both the w and w axes. The quartics corresponding to u = const, have
two branches, one, encircling KJ9, corresponding to u > 0, the other,
encircling cU9 to u y < 0. The quartics corresponding to u2 « 0 are ovals
encircling &$• In particular, for u 2 = in + Y2) K', we have a circle, see
13« 18(1). See the figure for further details.

378
SPECIAL FUNCTIONS
13.25
The function w = cn(u, k). The interior of the rectangle with vertices
0, K, K + iK', i'K'in the iz-plane is mapped on the fourth quadrant of the
w-plane, the rectangle -K, K, K + iK\ -K + iK'is mapped on the right
w half-plane cut from 0 to 1, the rectangle -iK', K - i K ', K + j'K', j'K'
is mapped on the right half-plane cut from 1 to ©©, and the rectangle with
vertices ± iK', 2K ± i'K' is mapped on the whole u>-plane cut from — ©o
to —1, from 1 to oo, from — joo to — ik '/k, and from ih 'Ik to ioo. The lines
a t = const, u2 = const are mapped on the doubly orthogonal system of
confocal bicircular quartics in the w-plane whose foci are ± 1, ± ih'/k%
Both families are ovals, those corresponding to u f = const around CS,
those corresponding to u2 = const around U3. Both families are symmetric
with respect to the axes w f = 0, w = 0,
iK'
o
„,-0
J9
I
Q
II
J9
K-)
c
B
8
iK'
IV
ffl
2K+iK'
J9
3
J9
Uj = 0
2K
-/K
K-/K' 2K-iK'
The mapping iu = en (u, &)
m
£\ ik'/k II
l «v
IV
3
«,,=()
B Q
c j - ik'/k i
TAe function w = dn («, A). Since it follows from tables 7 (sec. 13,17)
and 11 (sec. 13.22) that
dn («, k) = k'sn{K'-iK + iu, k '),
the mapping w = dn u may be derived from w = sn «.

13.25
ELLIPTIC FUNCTIONS AND INTEGRALS
379
2*K
iK'
l»,-°
3
III
J9
n
a
IV
I
e
c
K+2iK'
K + iK'
u2 = 0
K
The mapping w — dn (ut A;).
In particular, the rectangle with vertices 0, K, K + 2iK\ 2tK'is mapped
on the lower w half-plane in the manner indicated in the figure, and the
rectangle with vertices 0, 2K, 2K + 2t'K', 2tK'is mapped on the whole
w-plane cut from -°o to -1 and from 1 to ©c. The lines u, = const., u =
1 2
const* are mapped on the doubly orthogonal system of confocal bicircular
quartics with foci ± 1, ± h ', and the lines u t = (m + Y2)K in particular
are mapped on the circle with center at w = 0 and radius h ' ^.
The functions w= £z) + e az. Clearly ^(2^ is real, C(iz±) is imaginary,
and since we have from 13*13(18) that
£(co + iz ) - £(<o) = £Uz ) +-
6> Vz )
>(izz)-e i
£(*'+*,)-£(<»')-£(*,)■
2 p\zf)-e.

S80
SPECIAL FUNCTIONS
13.25
the first of these two
expressions is imaginary, the second ^
real. Investigating the mapping
of the rectangle with vertices
0, co, co + co ', co in the e-plane,
we find that QB and CJ9 are
mapped on horizontal lines, and
BC and GJ9 on vertical lines in
the if-plane (a = 1, 2, 3),
Moreover,
w((l)~oo9 w(\£) = rj + e aco,
w(C)= 77 -h 77' + eJ,co + co'\ w(J9)
We have to discuss the signs oi -q + eaco and of (77' + eaco')/i* From
13.16(9), (10), (11) and 13.8(25), (26), we have
(e, ~e3YlA (r/ + e} co) = E >0
(e1 ~e3)~1A (r) + e2 <y) = E - (e , - e3)~Vz (e f -e2)<y
= E-A'2K = £2B >0
(e^e 3)"K (*? + e 3 co) = E - (e , - e z)% co
= E-K = -&2D<0
-i(et-e3rlA (ri'+e3co') = -E'<0
-Ke1-e3r^(77/+e26)0 = -E/-(e1-.e3rM(e2-e3)^/
= -E'+&2K'=-A'2B'<0
-i(e}-e3r*{ri'+exo') = -E'-(el-ejXio>'
= K'-E'=&'2D'>0.
In the figures illustrating the mapping w = £(z) + eaz of the rectangle
GBCJQ, the abbreviations
Tf + eaco = Ha9 -q'+ea co'=H^i
were used. From our discussion,
//1>//2>o>//3, h;>o>h;>h;.
CO+ CO
z-plane
= 77'+e */.

13. 25 ELLIPTIC FUNCTIONS AND INTEGRALS 381
In each case that portion of the plane which is to the left of GBCJ8G (in
this order) is the map of the rectangle. By reflection on the sides of the
rectangle we find the following results. The function w - £(z) + e %z maps
lU) =U
1 W2 = H1
J9 C
",=»,
w2 = 0
13
w=£(z) +
elz
fl Ct
13
C
wz = K
w2 = 0
J9
u> = £(z) + e3z
w}=0
a
the interior of the rectangle with vertices ± co, ± co + 2co'in the z-plane
on the region exterior to two semi-infinite horizontal strips with corners
± #t , ± //j + 2iH]'in the «;-p]ane. The function w = £(z) + ezz maps the
interior of the rectangle with vertices ± co ± co' in the z-plane on the
exterior of the rectangle with vertices ± Hz ± iH'z in the w-plane. The
function w = £(z) + e2z maps the interior of the rectangle with comers
± co\ 2 co ± ca'in the £-plane on the region exterior to two semi-infinite
vertical strips with corners ± iH3 , 2ff 3 ± iff 3 in the w-plane.
The mapping w = £(z) + e 2z maybe combined with one of the preceding
mappings to map the exterior of a rectangle on a half-plane.
"■1 - "
w2 = 0
|«,2«//;
B
C
r
w = £(z) + e 2 z

382
SPECIAL FUNCTIONS
REFERENCES
Appell, Paul and Emile Lacour, 1922: Fonctions elliptiques et applications,
Gauthier-Villars, Paris.
Bianchi, Luigi, 1916: Lezioni sulla teoria delle funzioni di variabile complessa
e delle funzioni ellittiche, 2nd edition, Spoerri, Pisa.
Burkhardt, Heinrich and Georg Faber, 1920: Elliptische Funktionen 3rd edition,
Gruyter, Berlin.
Burnside, W.S. and A.W. Panton, 1892: Theory of equations, 3rd edition,
Longmans, Roberts and Green, London*
Clebsch, Alfred, 1865: /. /• Math. 64, 210-270.
Dixon, A.C., 1894: Elliptic functions with examples, Macmillan, and Co. Ltd.,
London.
Enriques, Federigo and Oscar Chisini, 1934: Funzioni ellittiche ed abeliane
Vol. IV Delia Teoria geometrica delle equazioni, ecc. Zanichelli, Bologna.
Fletcher, Allan, J.C.P. Miller, and Louis Rosenhead, 1946: An index of
mathematical tables, Scientific Computing Service, London.
Fricke, Robert, 1913: Elliptische Funktionen, Encyklopadie, der mathematischen
Wissenschaften, v. % pt. 2, p. 181-348, B.G. Teubner, Leipzig.
Fricke, Robert, 1916-1922: Die elliptischen Funktionen und ihre Anwendungen
I, II, B.G. Teubner, Leipzig.
Grobner, Wolfgang and Nikolaus Hofreiter, 1949: lntegraltafel, Erster Teil,
Unbestimmte Integrale, Springer-Verlag Wien.
Grobner, Wolfgang and Nikolaus Hofreiter, 1950: lntegraltafel, Zweiter Teil,
Bestimmte Integrale, Springer-Verlag Wien.
Hamel, Georg, 1932: S.-B. Berlin Math. Ges. 31, 17-22.
Hancock, Harris, 1917: Elliptic integrals, Wiley.
Humbert, Pierre, 1922: Introduction a I*etude des fonctions elliptiques, Hermann
and Cie., Paris.
Hurwitz, Adolph and Richard Courant, 1925: Funktionentheorie, 2nd edition,
Springer, Berlin.
Jahnke, Eugen and Fritz Emde, 1938: Tables of functions with formulae and
curves, B.G. Teubner, Leipzig and Berlin.
Kober, Hermann, 1952: Dictionary of conformal representations, Dover.
Konig, Robert and Maximilian Krafft, 1928: Elliptische Funktionen, Gruyter,
Berlin.
Low, A.R., 1950: Normal elliptic functions, University of Toronto press.

ELLIPTIC FUNCTIONS AND INTEGRALS
383
REFERENCES
Magnus, Wilhelm and Fritz Oberhettinger, 1949: Formeln und Satze fur die
speziellen Funktionen der mathematischen Physik, Springer-Verlag, Berlin,
Gottingen, Heidelberg.
Meyer zur Capellen, Walther, 1950: Integraltafeln, Springer-Verlag, Berlin,
Gottingen, Heidelberg.
Milne-Thomson, LJVL, 1950: Jacobian elliptic function tables, Dover, New York.
Neville, E.H., 1944: Jacobian Elliptic Functions, Oxford, Clarendon Press.
Oberhettinger, Fritz and Wilhelm Magnus, 1949: Anwendung der elliptischen
Funktionen in Physik und Technik, Springer, Berlin.
Prasad, Ganesh, 1948: An introduction to the theory of elliptic functions and
higher trans c en dentals, University of Calcutta.
Radon, Brigitte, 1950: Atti Accad. Naz. Lincei, Mem., CL Sci. Fis. Mat. Nat.
(8) 2, 69-109.
Roberts, W.R. Westropp, 1938: Elliptic and hyperelliptic integrals and allied
theory, Cambridge University Press.
Spenceley, G.W. and RJVf. Spenceley, 1947: Smithsonian Elliptic Functions
Tables, The Smithsonian Institute Washington.
Tricomi, F.G., 1935: Boll. Un. Mat. Ital. 14, 213-218 and 277-282.
Tricomi, F.G., 1936: Boll. Un. Mat. Ital. 15, 102-195.
Tricomi, F.G., 1937: Funzioni ellittichef Bologna, Zanichelli, 1951 (German
edition, Leipzig, Akad. Verlagsgesellschaft, 1948: second Italian edition
1951).
Whittaker, E.T. and G.N. Watson, 1927: Modern Analysis, 4th edition, Cambridge
University press.

SUBJECT INDEX
All numbers refer to pages. Numbers in italics refer to the definitions.
Abelian integrals, 295 ff.
Absolute invariant, 375
Airy's integrals, 22
Anger's function J (z), 35 ff., 84, 99, 103
Appell series, 280 ff.
Approximation of quadratically
integrable functions, 156
Associated polynomials, 162 ff.
Automorphic functions, 296, 374
B
Barnes' integral representations of
Bessel functions, 21 ff.
Basset's function,
(see modified Bessel function of
the third kind)
Bessel coefficients, 6
generating function for, 7
integral representations for, 13 ff., 81
Bessel function of the first kind, 4
derivative with respect to the order, 7
duplication formula for, 45
inequalities for, 14, 66
series involving, 63 ff.
zeros of, 59 ft.
Bessel function of the second kind, 4
of integer order, 7
of order zero, 8
zeros of, 61 ff.
Bessel function of the third kind, 4
zeros of, 62
Bessel functions,
addition theorems for, 43 ff., 101 ff.
analytic continuation of, 12, 80
and wave motion, 2ff.
as limits of Jacobi polynomials, 173
as limits of Laguerre polynomials,
191
asymptotic expansions for, 22 ff.,
85 ff.
differential equations for, 13
differentiation formulas for, 11
integrals involving, 45 ff., 57,
90 ff.
integral representations for,
14 ff., 57 ff., 81 ff.
notations for, 3
of imaginary order, 87 ff.
of order ± V2t 10, 79
of order n + V2 (see spherical
Bessel functions)
power series for products of, 10 ff.
recurrence relations for, 12
relations with Legendre functions,
55 ff.
series involving, 58, 63 ff., 98 ff.
Wronskians of, 12, 79 ff.
zeros of, 58 ff.
Bessel polynomials, 10
Bessel's differential equation, 3ff.
Bessel 5s inequality, 157
Bilinear forms, 284
Biorthogonal system, 265
Birational invariant, 295
Birational transformation, 295
Cell, 325
Christoffel-Darboux formula, 159,
269
Christoffel numbers, 161
Classical orthogonal polynomials,
163 ff.
(see also Legendre, Cegenbauer
384

SUBJECT INDEX
385
Jacobi, Hermite, Laguerre
polynomials).
characterization of, 164
differential equation for, 166 ff.
differentiation formula for, 167
properties of, 164, 166 ff.
Complete elliptic integrals,
(see elliptic integrals)
Confluent hypergeometric functions,
expansion in terms of parabolic
cylinder functions, 124
Conformal mappings,
involving elliptic functions and
integrals, 376 ff.
Convergence in mean of generalized
Fourier expansions, 157
Convolution, 45
Cornu's spiral, 151
Cosine integral, 145 ff*
Cut Bessel functions, 22
D
Didon series, 280 ff.
Dini series of Bessel functions, 70 ff*
Doubly-periodic functions, 323 ff.
(see also elliptic functions)
E
Elliptic functions, 294 ff., 322 ff., 325
addition theorems satisfied by, 328
differential equations satisfied by,
327
expression of, in terms of p(z),
p'(z), 335
expression of, in terms of sigma
functions, 337
expression of, in terms of zeta
functions, 336 ff.
general properties of, 325 ff.
integrals of, expressed in terms of
Weierstrass' functions, 337
Jacobian, 322 ff., 340 ff.
Jacobian, addition theorems for,
344
Jacobian, degenerate cases of, 354
Jacobian, expressed in terms of
theta functions, 362
Jacobian, expressed in terms of
Weierstrass* functions, 340 ff.
Jacobian, Landen's
transformation of, 372
Jacobian, linear transformation
of, 367 ff.
Jacobian, periods, zeros, poles
and residues of, 341
Jacobian, quadratic
transformations of, 372 ff.
Jacobian, special values of,
346 ff.
Jacobian, with 0 < k < 1, 349 ff.
Neville's notation for, 294, 342
order of, 326
residues of, 326
transformations of, 365 ff.
Weierstrass', 323, 328 ff.
Weierstrass', addition theorem
tor, 332 ff.
Weierstrass', differential
equation for, 331 ff.
Weierstrass', degenerate cases
of 339 ff.
Weierstrass,' duplication
formula for, 333
Weierstrass', expressed in terms
of theta functions, 360 ff.
Weierstrass', Landen's
transformation of, 37Iff.
Weierstrass', linear
transformations of, 367 ff.
Weierstrass', quadratic
transformations of, 37Iff.
Weierstrass', with real
invariants, 338 ff.
Elliptic integrals, 294 ff.
addition theorems for, 315
complete, expressed in terms
of hypergeometric series,
318
complete, integration
formulas for, 322
complete, Legendre's relation
for, 320
complete, of the first, second,
and third kind, 314, 317ff.

386
SPECIAL FUNCTIONS
complete, particular cases
of, 320
complete, transformations
of, 318ff.
differentiation formulas for,
317, 321
expressed in terms of theta
functions, 363 ff.
expressed in terms of
Weierstrass' functions,
337 ff.
interchange theorem for,
303 ff., 315
inversion of, 322 ff.
Landen's transformation of,
317
Legendre's, evaluation of,
308 ff.
Legendre's form of, 300ff.,
314ff.
linear transformations of,
315ff.
Low's form of, 301
moduli of periodicity of, 303
of the first, second, and third
kinds, 299 ff., 313 ff.
periods of, 303, 314
reduction of, 296ff., 304ff.
reduction to Legendre's normal
form, 305 ff.
reduction to Weierstrass'
normal form, 304 ff.
singularities of, 303, 314
Weierstrass' form of, 299 ff.
Elliptic modular functions, 374 ff.
Equianharmonic elliptic functions
and integrals, 306, 320
Error functions, 147 ff.
connection with parabolic
cylinder functions, 119
expansions in terms of Bessel
functions, 148
power series expansions of, 147
repeated integrals of, 149
Exponential integrals, 143 ff.
expressed in terms of confluent
hypergeometric functions, 143
expressed in terms of incomplete
gamma functions, 143
generalizations of, 145
F
Field
differential 327
of elliptic functions, 327
Fourier-Bessel series, 70 ff., 104
Fourier coefficients (generalized),
156
Fourier series (generalized) 156
Fresnel integrals, 149 ff.
connection with error functions,
149
connection with incomplete
gamma functions, 149
Functions
of the paraboloid of revolution,
126 ff.
of the parabolic cylinder
(see parabolic cylinder
functions)
Fundamental period-parallelogram,
325
Fundamental region,
of the modular group, 375
of the A-group, 375
Funk-Hecke theorem, 247
G
Gauss transforms, 194
multi-dimensional, 289 ff.
Gegenbauer polynomials, 164,
174 ff., 235 ff.
asymptotic behavior as
n -> oo, 198
connection with Legendre
functions, 177
expressed as hypergeometric
functions, 175 ff.
generating functions of, 177
inequalities for, 206 ff.
integral representations for,
177
monotonic properties of, 208
recurrence formula for, 175
Rodrigues' formula for, 175
series of, 177 ff., 213 ff.
zeros of, 203 ff.

SUBJECT INDEX
387
Gegenbauer's addition theorem
for Bessel functions, 43 fF*
Gegenbauer's polynomials
A (z), B (z), 34
Generalized Dirichlet series, 72ff.
Genus of algebraic curves, 295
Glaisher's notation,
of Jacobian elliptic functions,
322
Graf's addition theorem for Bessel
functions, 44 ff.
Gram's determinant, 155
Gubler's integral representations,
of Bessel functions, 17 ff.
H
Hankel's function,
(see Bessel function of the
third kind)
Hankel's infinite integral
involving Bessel functions, 49
Hankel's integral representations of
Bessel functions, 15 ff.
Hankel's inversion theorem, 73
- Hardy's generalization of, 73 ff.
Hankel's symbol (v, m), 10
Harmonic polynomials, 237 ft,
complete set of, 239 ff.
Hermite polynomials, 164, 192 ff.
addition theorems for, 196
asymptotic behavior as n -> oo,
20 Iff.
connection with Laguerre
polynomials, 193, 195.
differential equations for, 193
differentiation formula for, 193
expansion of analytic functions
in terms of, 211
expansions in series of spherical
Bessel functions, 201 ff.
expressed as confluent hypergeo-
metric functions, 194
generating functions for, 194
inequalities for, 208
integral representation for, 194
integrals involving, 194 ff.
mean convergence of series of, 210
monotonic properties of, 208
recurrence relation for, 193
relation to parabolic cylinder
functions, 117
Rodrigues' formula for, 193
series of, 193 ff., 216
zeros of, 204 ff.
Hermite polynomials of several
variables, 283 ff.
Hurwitz's theorem on zeros of
Bessel functions, 59
generalizations of, 59
Hyperelliptic integrals, 296
Hypergeometric polynomials,
(see Jacobi polynomials)
Hypergeometric series,
of Lauricella, 275
Hyperspherical harmonics, 232 ff.
I
Incomplete gamma functions, 133 ff.
asymptotic expansions for
large x, 135
asymptotic representations for
large a, 140
connection with confluent
hypergeometric functions, 133f
136
continued fraction expansion, 136
differentiation formulas, 135
expansions in inverse factorials,
139
expansions in series of Bessel
functions, 139
expansions in series of Laguerre
polynomials, 139
integral representations of, 137f
138
loop integral for, 137
Nielsen's expansion for, 139
power series expansions of, 135
recurrence relations for, 134
zeros of, 141 ff.
Integral equations involving Bessel
functions, 76 ff.
Integral representations of arbitrary
functions in terms of Bessel
functions, 73 ff.

388
SPECIAL FUNCTIONS
Invariants
of a quartic, 305
of Weierstrass* elliptic functions,
and integrals, 305, 329
Inversion,
problem of, 326 ff., 375
Irreducible set,
of zeros or poles of an elliptic
function, 326
J
Jacobi-Anger formula, 7
Jacobi function of the second
kind, 170 ff.
Jacobi polynomials, 164, 168 ff.,
224, 226
asymptotic behavior as
n -* oo of, 198
connection with Besse]
functions, 173
connection with Laguerre
polynomials, 191
convergence of series of, 210 ff.
differential equation for, 169
expansion of analytic functions
in terms of, 211
expressed as hypergeometric
functions, 170
generating function of, 172
inequalities for, 205 ff.
integral representation for, 172
mean convergence of series of,
210
polynomials associated with, 171
recurrence formula for, 169
represented as finite differences,
173
Rodrigues* formula for, 169
series of, 172, 212 ff.
zeros of, 202 ff.
Jacobian elliptic functions,
(see elliptic functions, Jacobian)
K
Kapteyn series,
of Bessel functions, 66 ff., 103
of the second kind, 67 ff.
Kelvin's functions, 6, 101
L
Lagrange an interpolation, 161
Laguerre polynomials, 164, 188 ff.,
226
Abel summability of series
of, 211
asymptotic behavior as n -> o©
of, 199 ff.
connected with Bessel functions,
191
connected with Jacobi
polynomials, 191
differential equations for, 188
differentiation formulas for,
189, 190
expansion of analytic functions,
in terms of, 211
expansions in series of Bessel
functions, 199 ff.
expressed as confluent
hypergeometric functions, 189
generating functions of 189
inequalities for, 205 ff.
integral representations of, 190
integrals involving, 191
mean convergence of series of,
210
monotonic properties of, 207 ff.
recurrence relations for, 188, 190
represented as finite differences,
191
Rodrigues' formula for, 188
series of, 188 ff., 214ff.
zeros of, 204 ff*
Lambda-group, 367
Laplace transform, 45ff.-, 191
Laplace's expansion, 242, 281 ff.
Lattice,
line, 324
point, 323
Legendre function of the second
kind, 180 ff.
Legendre functions, 182 ff.
connection with Gegenbauer
polynomials, 177
relations with Bessel functions,
55 ff.
Legendre polynomials, 164, 178 ff.
addition theorem for, 182 ff.

SUBJECT INDEX
389
asymptotic behavior as
n -* oo of, 197
differential equation for, 179
differentiation formulas for,
179
equi converge nee theorem for
series of, 211
expressed as hypergeometric
functions, 180
generating functions of, 182
Hilb's formula for, 197
inequalities for, 205 ff.
integral representations of, 182
monotonic properties of, 208
recurrence relation for, 179
series of, 182 ff., 214
Legendre's relation, 320, 329
Lemniscate functions 320, 376
Line-lattice, 324
Logarithmic integral, 143
(see also exponential integrals)
Lommel's functions 5 (z),
5 (z)40S., 73 ff., 84 ff.
Special cases of, 4Iff.
Lommel's functions of two. variables
U (w, z), V {w, z), 42
V V
Lommel's polynomials R (z), 34 ff.
M
Macdonald's integral representations
of products of Bessel functions,
53 ff.
Maxwell's theory of poles, 251
Meijer's generalization of Laplace
transforms, 75
Mesh, 325
Modified Bessel function of the first
kind, 5
Modified Bessel function of the third
kind, 5
duplication formula for, 45
of integer order, .9
of order zero, 9
zeros of, 62 ff.
Modified Bessel functions, 5ff.
addition theorems for, 43 ff., 102
analytic continuation of, 80
asymptotic expansions for, 22ff.,
86 ff.
differentiation formulas for, 79
integral representations of,
I8ff„ 82ff.
integrals involving, 45 ff.,
56 ff., 89
of order ± V2t 10, 79
of order n 4- K (see spherical
Bessel functions)
relations with Legendre functions,
56 ff-
recurrence relations for, 79
Wronskians of, 80
Modular functions
(see elliptic modular functions)
Modular group, 366, 374 ff.
fundamental region of, 375
Modulus of elliptic functions and
integrals, 306
Moment problem, 163
Moments, 157
N
Neumann series
of Bessel functions 63 ff., 98ff.
of the second kind, 65
Neumann's function
(see Bessel function of the second
kind)
Neumann's polynomials 0 (z)t 32ff.
n
Neumann's polynomials Q (z), 34
n
Nicholson's formula for Bessel
functions of large order and
variable, 28, 88
Nicholson's integral representations
of products of Bessel functions, 54
0
Orthogonal group, 257 ff.
Cayley's representation of
^e, 257
Orthogonal invariant, 233

390
SPECIAL FUNCTIONS
Orthogonal polynomials, 153 ff.
and continued fractions, 162 ff.
and mechanical quadrature, 160
Christoffel-Darboux formula for,
159
expansion problems relating to,
209 ff.
extremum properties of, 160
in the circle, 273 ff.
in the sphere and hypersphere,
273 ff.
in the triangle, 269 ff.
of a discrete variable, 221 ff.
of several variables, 264ff.
recurrence relations for, 158ff.
zeros of, 158
Orthogonal system, 154
of parabolic cylinder functions,
122
of polynomials (see orthogonal
polynomials)
Orthogonalization, 154 ff.
Orthonorraal system, 154
P
Parabolic cylinder $
coordinates of, 115
Parabolic cylinder functions, 116,
117
addition theorem for, 123, 124
asymptotic expansions of, 122ff.
Cherry's theorem for, 124 ff.
connection with error functions,
119
differential equation for, 116
generating function of, 119
integral representations of, 119,
120
integrals involving, 121, 122
real zeros of, 126
Wronskians of, 117
Paraboloid of revolution,
coordinates of, 115
functions of (see functions of the
paraboloid of revolution)
Parseval's formula, 157
Period parallelogram, 325
fundamental, 325
Periods,
of elliptic functions, 328, 341
of elliptic integrals, 303, 314
of functions, 324
primitive, 324
Point-lattice, 323
Poisson's integral for Bessel
functions, 14
Polar coordinates,
hyperspherical, 233
Polynomials,
(see also orthogonal
polynomials, classical
orthogonal polynomials, Jacobi,
Gegenbauer, Legendre,
Tchebichef, Laguerre, and
Hermite polynomials)
harmonic, 237ff.
of N. Achyeser, 218
of A.C. Aitken and H.T. Gonin,
224
of Appell, 269 ff.
of H. Bateman, 224
of S. Bernstein and G. Szego,
217
of L.V. Charlier, 222, 226ff.
of J.P. Gram and H.E.H. Green-
leaf, 225
of W. Hahn, 165, 222, 224
of E. Heine, 218
of Hermite and Angelescu, 283
of Hermite and Didon 273 ff.
of Hermite-Didon-Appell-Kampe
de Feriet, 259 ff.
of M. Krawtchouk, 222, 224 ff.
of J. Meixner, 222t 225, 221
of F. Pollaczek, 218 ff.
of Tchebichef, 222, 223 ff.
Primitive periods, 324
Products of Bessel functions,
integral representations for, 47 ff.
53ff„ 96 ff.,
power series for, 10ff.
Q
Quadratic forms, 283 ff.
Quaternions, 255

SUBJECT INDEX
391
Raabe's integrals, 144
Rational curves, 295
Rodrigues' formula, 179
finite difference analogue of,
222 ff.
generalized, 164, 169, 175,
193, 276, 279, 285
Scalar product,
of functions, 153, 264
of vectors, 232, 273
Schlafli*s integral representations
of Bessel functions, 17
Schlafli's polynomials 5 (z), 34
n
Schromilch series of Bessel functions,
68ff., 103 ff.
generalized, 68
Separation theorems, 162
Sine integrals, 145 ft*
generalizations of, 147
Sommerfeld's integral representations,
of Bessel functions, 19 ff.
Sommerfeld's notation for spherical
Bessel functions, 10
Sommerfeld's wave, 125
Sonine's expansion, 43, 64
Sonine-Polya theorem, 205
Sonine's integrals involving Bessel
functions, 46
Spherical Bessel functions, 9, 78, 79
expressed in terms of elementary
functions, 10, 78, 79
Sommerfeld's notation for, 10, 78,
79
Spherical harmonics, 232 ff.
Spherical polynomials,
(see Legendre polynomials)
Spherical surface harmonics, 240 ff«
addition theorem of, 242 ff.
completeness of, 241
four-dimensional, 253 ff.
generating function of, 248 ff.
Maxwell's theory of, 251 ff.
orthogonal property of, 240 ff.
three-dimensional, 248 ff.
transformation of, 256 ff.
Stationary phase,
method of, 27 ff.
Steepest descents,
method of, 24
applied to modified Bessel
functions of the third
kind, 24 ff.
Struve's functions, 18, 37R.,
47, 68 ff., 74, 89, 99, 103
integrals involving, 98
Tchebichef polynomials, 183 ff.
differential equations for, 185
differentiation formulas for,
185, 186
expressed as hypergeometric
functions, 186
generating functions, of, 186
recurrence relations for, 185
Rodrigues' formulas for, 185
Theta functions, 354ff.
(see also elliptic functions)
expression of elliptic functions
and integrals in terms of,
360 ff.
Hermite's, 360
of zero argument, 359
transformations of, 368 ff.
Transformation,
Landen's, see elliptic functions,
elliptic integrals
of elliptic functions and integrals,
see elliptic functions,
elliptic integrals
unimodular, 324
Truncated exponential series, 136 ff.
connection with incomplete
gamma functions, 136
U
Ultraspherical polynomials,
(see Gegenbauer polynomials)
Unicursal algebraic curves, 295, 327
Uniformizing variable, 295, 338
Unimodular transformation, 324, 366

392
SPECIAL FUNCTIONS
W
Watson's formulas for Bessel functions
of large order and variable, 29, 89
Weber and Schafheitlin,
discontinuous integral of, 51 ff.
Weber - Hermite function,
(see also parabolic cylinder function)
differential equation for, 116
Weber's function E (z), 35 ff., 84, 103
Weierstrass' elliptic functions,
(see elliptic functions, Weierstrass')
Weierstrass' sigma function, 329 ff*
(see also elliptic functions)
Weierstrass' zeta function, 329 ff.
(see also elliptic functions)
Weight function, 153

INDEX OF NOTATIONS
D complete elliptic integral, 321
am u Jacobi's function, 322
A (t) Airy function, 200
A it), A (t) Airy integrals, 29
A (z) Gegenbauer's
polynomial, 34
B
bei x, bei x Kelvin's functions, 6
v
ber xf ber x Kelvin's functions, 6
B (z) Gegenbauer's polynomial, 34
B complete elliptic integral, 321
C
cd u Glaisher's function, 322
en u Jacobi's elliptic function, 322
cs u Glaisher's function, 322
C (x) Fresnel integral, 149
C (xt a) generalized Fresnel
integral, 149
C (x) Gegenbauer polynomials, 174
n
Chi x modified cosine integral, 146
Ci x cosine integral, 145
C complete elliptic integral, 321
D
dc u Glaisher's function, 322
dn u Jacobi's elliptic function, 322
ds u Glaisher's function, 322
D (z) Parabolic cylinder function, 117
E
e = p(co ), 330
e (x) Truncated exponential
series, 136
E Incomplete elliptic integral of
the second kind, 300, 313
E (u) Jacobi's function, 343
E (*), exponential integral, 143
E (x) function used in astrophysics
and nuclear physics, 134
E* (x) modified exponential integral, 143
E Appell's polynomials, 271
mn
Ei (x) exponential integral, 143
Erf x error function, 147
Erfc x complementary error function, 147
Erfi x modified error function, 147
E, E ' complete elliptic integrals of the
second kind, 314, 317
E (z) Weber's function, 35
F
F Incomplete elliptic integral of the
first kind, 300, 313
F Appell's polynomial, 271
c3 Appell's polynomial, 270
G
g , g invariants of Weierstrass'
elliptic functions, 299, 305
393

394 SPECIAL FUNCTIONS
G (x ,...,# ) Hermite poly-
m , ...,m i n' ^ }
1 n
nomials of several variables, 285
y Gauss transform, 195
(jr^ multi-dimensional Gauss
transform, 289
nV,
multi-dimensional Gauss
^ transform, 290
kei x, ker x modified Kelvin
v v
functions, 6
K (x) function used in astrophysics
n
and nuclear physics, 134
K (z) modified Bessel function of
v
the third kind, 5
K, K' complete elliptic integrals of
the first kind, 314, 317
H
1 1
h = i+ —+... + — ,8
* 2 ro
hei x. her x Kelvin's
v v
functions, 6
H (x) error function, 147
H (x) Hermite polynomials, 193
n
H (z), H (z) Bessel functions
V V
of the third kind, 4
H & , ... , x ) Hermite
m , ..., m 1 7i
I n
polynomial of several variables, 285
H (z) Struve's function, 37
v
I (a) polynomial, 140
n
li (re) logarithmic integral, 143
L, L Laplace transformy 46, 191
L (x) Laguerre polynomial, 188
L (z) modified Struve function, 38
N
nc u Glaisher's function, 322
nd u Glaisher's function, 322
ns u Glaisher's function, 322
0
0 (z) Neumann's polynomial, 32
I
i erfc x repeated integral of error
function, 149
I (z) modified Bessel function of the
first kind, 5
P {x) Legendre polynomial, 178
Pa'P (x) Jacobi polynomial, 168
p(z) Weierstrass' elliptic function,
323t 328
J(r) absolute invariant, 375
J (z) Bessel function of the first
first kind, 4
J (z) cut Bessel function of the
v, m
first kind, 21
J (z) Anger's function, 35
k modulus of Jacohi's elliptic functions
and integrals, 300, 306
177T 0 ._
q = e ,345
q (x) polynomials associated
with Jacobi polynomials, 171
Q (x) Legendre function of the
n
second kind, 180

NOTATIONS
395
Q (x) Legendre function of the
second kind on the cut, 181
Q P (x) Jacobi function of the
n
second kind, 170
R
R (z) Lommel's polynomials, 34
s (z) Lommel's function, 40
sc u Glaisher's function, 322
sd u Glaisher's function, 322
si x sine integral, 145
sn u Jacobi's elliptic function, 322
S (x) Fresnel integral, 149
S (x, a) generalized FTesnel integral,
149
S (z) Schlafli's polynomial, 34
S (z) Lommel's function, 40
Shi x modified sine integral, 146
Si x sine integral, 145
T (x) Tchebichef polynomial, 183
U (x) Tchebichef polynomial, 183
U (w, z) Lommel's function of two
variables, 42
U (x ,...,# ) polynomials
m , ...,m ^1 ' ' n * 7
1 n
of Hermite and Didon, 277
V (w, z) Lommel's function of two
variables, 42
V (fc ,...,# ) polynomials
771 , ... ,771 1 71
1 n
of Hermite and Didon, 274 ff.
W
W Wronskian, 12
Y (z) Bessel function of the second
v
kind, 4
Y {6, <£>) spherical surface harmonic,
250
Z (u) Jacobi's function, 343
Z (z) Bessel function, 2, 48
GREEK LETTERS
a(x) error function, 147
yidf x) incomplete gamma function,
133
y (a, x) modified incomplete gamma
function, 140
y*(at x) modified incomplete gamma
function, 133
r(ez, x) complementary incomplete
gamma function, 133
A Laplace's operator, 2, 115, 234
l\-jg -27g discriminant, 332
A(<& k\ 317
i/=£(fi>\i/'=£UA 329
77 = £{co ), 330
'a ^ a
£(z) Weierstrass* zeta function, 329
Q.(v)9 ... , 6 (v) Theta functions, 355
@ (v) Hermite's theta function, 360
X(r) modular function, 374
II incomplete elliptic integral of the
third kind, 301, 313
II complete elliptic integral of the
third kind, 317
a{z) Weierstrass' sigma function, 329
a (z) sigma functions, 330
r=co'/co, 328

396
SPECIAL FUNCTIONS
CO, a>' periods of Weierstrass*
elliptic functions, 328
(ti periods of Weierstrass'
elliptic functions, 330
fl (z) Neumann's polynomial,
34
MISCELLANEOUS NOTATIONS
arg z argument (or phase) of
complex number z
Im z imaginary part of complex
number z
Re z real part of complex
number z
y Euler-Mascheroni constant
(see vol. I, p. 1)
d
dx
d
dxk
V Bessel's differential operator, 4
v «)
at
m !
(a) =r(o + /i)/T(o)
n
(v, m) Hankel's symbol, 10
(y, b) scalar product of vectors,
232, 273
(0, ift) scalar product of functions,
153, 264
HjPII length of vector £, 232
~ approximate or asymptotic equality,
J Cauchy principal value of an
integral,
r(0+)
J loop integral, 15
+ oo