Book:Arthur Erdélyi/Higher Transcendental Functions Volume I

Harry Bateman: Higher Transcendental Functions, Volume I

Published $1953$, Dover Publications

ISBN 0-486-44614-X.

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BiBTeX

@book {MR698779,
    AUTHOR = {Erd{\'e}lyi, Arthur and Magnus, Wilhelm and Oberhettinger,
              Fritz and Tricomi, Francesco G.},
     TITLE = {Higher transcendental functions. {V}ol. {I}},
      NOTE = {Based on notes left by Harry Bateman,
              With a preface by Mina Rees,
              With a foreword by E. C. Watson,
              Reprint of the 1953 original},
 PUBLISHER = {Robert E. Krieger Publishing Co., Inc., Melbourne, Fla.},
      YEAR = {1981},
     PAGES = {xiii+302},
      ISBN = {0-89874-069-X},
   MRCLASS = {33-02 (01A75)},
  MRNUMBER = {698779},
}

Contents

PREFACE

FOREWARD

INTRODUCTION

1.1. Definition of the gamma function

1.2. Functional equations satisfied by $\Gamma(z)$

1.3. Expressions for some infinite products in terms of the gamma function

1.4. Some infinite sums connected with the gamma function

1.5. The beta function

1.6. The gamma and beta functions expressed as contour integrals

1.7. The $\psi$ function

1.7.1. Function equations for $\psi(z)$

1.7.2. Integral representations for $\psi(z)$

1.7.3. The theorem of Gauss

1.7.4. Some infinite series connected with the $\psi$-function

1.8. The function $G(z)$

1.9. Expressions for the function $\log \Gamma(z)$

1.9.1. Kummer's series for $\log \Gamma(z)$

1.10. The generalized zeta function

1.11. The function $\Phi(z,s,v)= \displaystyle\sum_{n=0}^{\infty} (v+n)^{-s}z^n$

1.11.1 Euler's dilogarithm

1.12. The zeta function of Riemann

1.13. Bernoulli's numbers and polynomials

1.14. Euler numbers and polynomials

1.14.1. The Euler polynomials of higher order

1.15. Some integral formulas connected with the Bernoulli and Euler polynomials

1.16. Polygamma functions

1.17. Some expansions for $\log \Gamma(1+z)$, $\psi(1+z)$, $G(1+z)$, and $\Gamma(z)$

1.18. Asymptotic expansions

1.19. Mellin-Barnes integrals

1.20. Power series of some trigonometric functions

1.21. Some other notations and symbols

References

2.1. The hypergeometric series

2.1.1. The hypergeometric equation

2.1.2. Elementary relations

2.1.3. The fundamental integral representation

2.1.4. Analytic continuation of the hypergeometric series

2.1.5. Quadratic and cubic transformations

2.1.6. $F(a,b;c;z)$ as function of the parameters

2.2. The degenerate case of the hypergeometric equation

2.2.1. A particular solution

2.2.2. The full solution and asymptotic expansion in the general case

2.3. The full solution and asymptotic expansion in the general case

2.3.1. Linearly independent solutions of the hypergeometric equation in the non-degenerate case

2.3.2. Asymptotic expansions

2.4. Integrals representing or involving hypergeometric functions

2.5. Miscellaneous results

2.5.1. A generating function

2.5.2. Products of hypergeometric series

2.5.3. Relations involving binomial coefficients and the incomplete beta function

2.5.4. A continued fraction

2.5.5. Special cases of the hypergeometric function

2.6. Riemann's equation

2.6.1. Reduction to the hypergeometric equation

2.6.2. Quadratic and cubic transformations

2.7. Conformation representations