Book:Arthur Erdélyi/Higher Transcendental Functions Volume III

Harry Bateman: Higher Transcendental Functions, Volume III

Published $1953$, Dover Publications

ISBN 0-486-44614-X.

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hosted by Caltech

Contents

CONTENTS

FOREWORD

CHAPTER XIV AUTOMORPHIC FUNCTIONS

14.1. Discontinuous groups and homographic transformations

14.1.1. Homographic transformations

14.1.2. Fixed points. Classification of transformations

14.1.3. Discontinuous groups

14.1.4. Fundamental region

14.2. Definition of automorphic functions

14.3. The icosahedral group

14.4. Parabolic substitutions

14.5. Infinite cyclic group with two fixed points

14.6. Elliptic modular functions

14.6.1. The modular group

14.6.2. The modular function $\mathcal{J}(z)$

14.6.3. Subgroups of the modular group

14.6.4. Modular equations

14.6.5. Applications to number theory

14.7. General theory of automorphic functions

14.7.1. Classification of the groups

14.7.2. General theorems on automorphic functions

14.8. Existence and construction of automorphic functions

14.8.1. General remarks

14.8.2. Riemann surfaces

14.8.3. Automorphic forms, Poincaré's theta series

14.9. Uniformization

14.10. Special automorphic functions

14.10.1. The Riemann-Schwarz triangle functions

14.10.2. Burnside's automorphic functions

14.11. Hilbert's modular groups

14.12. Siegel's functions

References

CHAPTER XV LAMÉ FUNCTIONS

15.1. Introduction

15.1.1. Coordinates of confocal quadrics

15.1.2. Coordinates of confocal cones

15.1.3. Coordinates of confocal cyclides of revolution

15.2. Lamé-Wangerin functions

15.3. Heun's equation

15.4. Solutions of the general Lamé equation

15.5. Lamé functions

15.5.1. Lamé functions of real periods

15.5.2. Lamé functions of imaginary periods. Transformation formulas

15.5.3. Integral equations for Lamé functions

15.5.4. Degenerate cases

15.6. Lamé-Wangerin functions

15.7. Ellipsoidal and sphero-conal harmonics

15.8. Harmonics associated with cyclides of revolution

References

CHAPTER XVI MATHIEU FUNCTIONS, SPHEROIDAL AND ELLIPSOIDAL WAVE FUNCTIONS

16.1. Introduction

16.2. The general Mathieu equation and its solutions

16.3. Approximations, integral relations, and integral equations for solutions of the general Mathieu equation

16.4. Periodic Mathieu functions

16.5. Expansions of Mathieu functions and functions of the second kind

16.6. Modified Mathieu functions

16.7. Approximations and asymptotic forms

16.8. Series, integrals, and expansion problems

SPHEROIDAL WAVE FUNCTIONS

16.9. The differential equation of spheroidal wave functions and its solution

16.10. Further expansions, approximations, integral relations

16.11. Spheroidal wave functions

16.12. Approximations and asymptotic forms for spheroidal wave functions

16.13. Series and integrals involving spheroidal wave functions

ELLIPSOIDAL WAVE FUNCTIONS

16.14. Lamé's wave equation

References

CHAPTER XVII INTRODUCTION TO THE FUNCTIONS OF NUMBER THEORY

17.1. Elementary functions of number theory generated by Riemann's zeta function

17.1.1. Notations and definitions

17.1.2. Explicit expressions and generating functions

17.1.3. Relations and properties

17.2. Partitions

17.2.1. Notations and definitions

17.2.2. Partitions and generating functions

17.2.3. Congruence properties

17.2.4. Asymptotic formulas and related topics

17.3. Representations as a sum of squares

17.3.1. Definitions and notations

17.3.2. Formulas for $r_k(n)$

17.4. Ramanujan's function

17.5. The Legendre-Jacobi symbol

17.6. Trigonometric sums and related topics

17.7. Riemann's zeta function and the distribution of prime numbers

17.8. Characters and $L$-series

17.9. Epstein's zeta function

17.10. Lattice points

17.11. Bessel function identities

References

CHAPTER XVIII MISCELLANEOUS FUNCTIONS

18.1. Mittag-Leffler's function $E_{\alpha}(z)$ and related functions

18.2. Trigonometric and hyperbolic functions of order $n$

18.3. The functions $\nu(x)$ and related functions

CHAPTER XIX GENERATING FUNCTIONS

FIRST PART: GENERAL SURVEY

19.1. Introduction

19.2. Typical examples for the application of generating functions

19.3. General theorems

19.4. Symbolic relations

19.5. Asymptotic representations

19.6. Rational and algebraic functions. General powers

19.7. Exponential functions

19.8. Logarithms, trigonometric and inverse trigonometric functions. Other elementary functions and their integrals

19.9. Bessel functions. Confluent hypergeometric functions (including special cases such as functions of the parabolic cylinder)

19.10. Gamma functions. Legendre functions and Gauss' hypergeometric function. Generalized hypergeometric functions

19.11. Generated functions of several variables

19.12. Some generating functions connected with orthogonal polynomials

19.13. Generating functions of certain continuous orthgonal systems

References

SUBJECT INDEX

INDEX OF NOTATIONS

See also

Book:Harry Bateman/Higher Transcendental Functions Volume I
Book:Harry Bateman/Higher Transcendental Functions Volume II