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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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

See also

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