# Difference between revisions of "Book:Arthur Erdélyi/Higher Transcendental Functions Volume III"

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__NOTOC__ | __NOTOC__ | ||

− | {{Book|Higher Transcendental Functions, Volume III|1953|Dover Publications|0-486-44614-X| | + | {{Book|Higher Transcendental Functions, Volume III|1953|Dover Publications|0-486-44614-X|Arthur Erdélyi|author2=Wilhelm Magnus|author3=Fritz Oberhettinger|author4=Francesco G. Tricomi}} |

===Online mirrors=== | ===Online mirrors=== | ||

[http://authors.library.caltech.edu/43491/ hosted by Caltech]<br /> | [http://authors.library.caltech.edu/43491/ hosted by Caltech]<br /> | ||

+ | |||

+ | ===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=== | ===See also=== | ||

− | [[Book: | + | [[Book:Arthur Erdélyi/Higher Transcendental Functions Volume I]]<br /> |

− | [[Book: | + | [[Book:Arthur Erdélyi/Higher Transcendental Functions Volume II]]<br /> |

− | [[Category: | + | [[Category:Book]] |

## Latest revision as of 05:44, 4 March 2018

## Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi: *Higher Transcendental Functions, Volume III*

Published $1953$, **Dover Publications**

- ISBN 0-486-44614-X.

### Online mirrors

### 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

- 14.1. Discontinuous groups and homographic transformations

- 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

- 15.1. Introduction

- 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

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

- 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

- FIRST PART: GENERAL SURVEY
- SUBJECT INDEX
- INDEX OF NOTATIONS

### See also

Book:Arthur Erdélyi/Higher Transcendental Functions Volume I

Book:Arthur Erdélyi/Higher Transcendental Functions Volume II