H I G H E R T R A N S C E N D E N T A L F U N C T I O N S

Volume I I

Based, in part, 011 notes left by

Harry Bateman

Late Professor of Mathemalics, 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-241 Task Order XIV with the Office of Naval Research

Project Designation Number: NR 013-015

T t 3̂

NEW YORK TORONTO LONDON

McGRAW-HILL BOOK COMPANY, INC. 1953

CONTENTS

FOREWORD ix

CHAPTER VII

BESSEL FUNCTIONS

FIRST PART: THEORY

7.1. Introduction 1 7.2. Bessel's differential equation 3 7.2.1. Bessel 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 \ l 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

Xl l 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. Sorae other notations and related functions 42 7.6. Addition theorems 43 7 .6 .1 . Gegenbauer 's addition theorem 43 7.6.2. G r a f s addition theorem 44 7.7. Integral formulas 45 7 .7 .1 . Indefinite integrals 45 7.7.2. Fini te 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. Relat ions between Besse l and Legendre functions . . . 55 7.9. Zeros of the Besse l functions 57 7.10. Series and integral representat ions of arbitrary

functions 63 7.10.1. Neumann's ser ies 63 7.10.2. Kapteyn se r ies 66 7.10.3. Schlö'milch se r i es 68 7.10.4. Fourier-Bessel and Dini se r i es 70 7.10.5. Integral representat ions of arbitrary functions 73

SECOND PART: FORMULAS

7.11. Elementary relations and miscel laneous formulas . . . 78 7.12. Integral representat ions 81 7.13. Asymptotic expansions 85 7.13.1. Large variable 85 7.13.2. Large order 86 7.13.3. Transi t ional 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 Besse l functions 98

References 106

CONTENTS xiii

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 inte grals 119 Asymptotic expansions 122 Representation of functions in terms of the D {x). . . 123 Seri«s 123

. 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 c lass ica l polynomials 163 10.7. General properties of the c lass ica l 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. Inequali t ies for the c lass ica l polynomials 205 10.19. Expansion problems *. . . 209 10-20. Examples of expansions 212 10.22. Some c l a s se s 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. Charl ier ' s polynomials 226

References 228

CONTENTS xv

CHAPTER XI

SPHERICAL AND HYPERSPHERICAL HARMONIC POLYNOMIALS

11.1. Preliminaries 232 11.1.1. Vectors 232 11.1.2. Gegenbauer polynomials 235 11.2. Harmonie polynomials 237 11.3 Surface harmonics 240 11.4. The addition theorem 242 11.5. The case p = 1, hin, p) = 2n + 1 248 11.5.1. A generating funetion for surface harmonics in the

three-dimensional case • 248 11.5.2. Maxwell's theory of poles 251 11.6. The case p = 2, hin, p) = in + l ) 2 253 11.7. The transformation formula for spherical harmonics . 256 11.8. The polynomials of Hermite-Kampe, de Feriet . . . . 259

References 262

CHARTER 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. Appell's 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. Bas ic 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. Ell ipt ic integrals 295 13.3. Reduction of ell iptic integrals 296 13.4. Periods and singulari t ies of ell iptic integrals . . . 302 13.5. Reduction of G (x) to normal form 304 13.6. Evaluation of Legendre ' s ell iptic integrals . . . . 308 13.7. Some further properties of Legendre ' s elliptic

normal integrals 314 13.8. Complete ell iptic integrals 317

PART TWO: 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 Weiers t rass ' funct ions . . . . 331 13.14. The expression of elliptic functions and ell iptic

integrals in terms of Weiers t rass ' 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 ell iptic functions • • 343 13.18. Descriptive properties and degenerate cases of

Jacobi ' s ell iptic 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 ell iptic functions . . . 365

!

CONTENTS xvii

13.22. Trans formations 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