Book:Arthur Erdélyi/Higher Transcendental Functions Volume III

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Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi: 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:Arthur Erdélyi/Higher Transcendental Functions Volume I
Book:Arthur Erdélyi/Higher Transcendental Functions Volume II

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