# Book:Arthur Erdélyi/Higher Transcendental Functions Volume I

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

Published $1953$, Dover Publications

ISBN 0-486-44614-X.

### Contents

PREFACE
FOREWARD
INTRODUCTION
1.1. Definition of the gamma function
$(1)$ (and $(1)$)
$(2)$ (and $(2)$
$(3)$
$(4)$
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
$(1)$
$(2)$
$(3)$
$(4)$
$(5)$
$(6)$ (and $(6)$)
$(7)$ (and $(7)$)
$(8)$
$(9)$ (and $(9)$)
$(10)$
$(11)$
$(12)$
$(13)$
$(14)$
$(15)$
$(16)$
$(17)$
$(18)$
$(19)$
1.6. The gamma and beta functions expressed as contour integrals
1.7. The $\psi$ function
$(1)$
-----------
$(4)$
-----------
1.7.1. Function equations for $\psi(z)$
$(8)$
$(9)$
$(10)$
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 $\Phi(z,s,v)= \displaystyle\sum_{n=0}^{\infty} (v+n)^{-s}z^n$
$(1)$
--------------
1.11.1 Euler's dilogarithm
$(22)$ (also (22) and (22))
$(23)$
1.12. The zeta function of Riemann
1.13. Bernoulli's numbers and polynomials
$(1)$
1.14. Euler numbers and polynomials
$(1)$
$(2)$
$(3)$
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
$(1)$ (and $(1)$ and $(1)$)
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
2.7.1. Group of the hypergeometric equation
2.7.2. Schwarz's function
2.7.3. Uniformization
2.7.4. Zeros
2.8. The hypergeometric series
$(1)$
$(31)$
$(32)$
$(33)$
$(34)$
$(35)$
$(36)$
2.9. Kummer's series and the relations between them
2.10. Analytic continuation
2.11. Quadratic and higher transformations
2.12. Integrals
References
3.1. Introduction
3.2. The solutions of Legendre's differential eqaution
3.3.1. Relations between Legendre functions
3.3.2. Some further relations with hypergeometric series
3.4. Legendre's functions on the cut
3.5. Trigonometric expansions for $P_{\nu}^{\mu}(\cos \theta)$ and $Q_{\nu}^{\mu}(\cos \theta)$
3.6.1. Special values of $\mu$ and $\nu$
3.6.2. Legendre polynomials
3.7. Integral representations
3.8. Relations between contiguous Legendre functions
3.9.1. Asymptotic expansions
3.9.2. Behavior of the Legendre functions near the singular points
3.10. Expansions in terms of Legendre functions
3.11. The addition theorems
3.12. Integrals involving Legendre functions
3.13. The ring or toroidal functions
3.14. The conical functions
3.15. Gegenbauer functions
3.15.1. Gegenbauer polynomials
3.15.2. Gegenbauer functions
3.16. Some other notations
References
4.1. Introduction
$(1)$
$(2)$
$(2)$
4.2. Differential equations
$(1)$
$(2)$
$(3)$
$(4)$
$(5)$
$(6)$
4.3. Identities and recurrence relations
4.4. Generalized hypergeometric series with unit argument in the case $p=q+1$
4.5. Transformations of ${}_{q+1}F_q$ and values for arguments other than unity
4.6. Integrals
4.7. Various special results
4.8. Basic hypergeometric series
$(1), (2)$
$(3)$
$(4)$
$(5)$
$(6)$
$(7)$
$(8)$
References
5.1. Various generalizations
5.2. Definition of the $E$-function
5.2.1. Recurrence relations
5.2.2. Integrals
5.3. Definition of the $G$-function
5.3.1. Simple identities
5.4. Differential equations
5.4.1. Asymptotic expansions
5.5. Series and integrals
5.5.1. Series of $G$-functions
5.5.2. Integrals with $G$-functions
5.7. Hypergeometric series in two variables
5.7.1. Horn's list
5.7.2. Convergence of the series
5.8. Integral representations
5.8.1. Double integrals of Euler's type
5.8.2. Single integrals of Euler's type
5.8.3. Mellin-Barnes type double integrals
5.9. Systems of partial differential equations
5.9.1. Ince's investigation
5.10. Reduction formulas
5.11. Transformations
5.12. Symbolic forms and expansions
5.13. Special cases
5.14. Further Series
References
6.1. Orientation
6.2. Differential equations
6.3. The general solution of the confluent equation near the origin
6.4. Elementary relations for the $\Phi$ function
6.5. Basic integral representations
6.6. Elementary relations for the $\Psi$ function
6.7. Fundamental systems of solutions of the confluent equation
6.7.1. The logarithmic case
6.8. Further properties of the $\Psi$ function
6.9. Whittaker functions
6.9.1. Bessel functions
6.9.2. Other particular confluent hypergeometric functions
6.10. Laplace transforms and confluent hypegeometric functions
6.11. Integral representations
6.11.1. The $\Phi$ function
6.11.2. The $\Psi$ function
6.12. Expansions in terms of Laguerre polynomials and Bessel functions
6.13. Asymptotic behavior
6.13.1. Behavior for large $|x|$
6.13.2. Large parameters
6.13.3. Variable and parameters large
6.14. Multiplication theorems
6.15. Series and integral formulas
6.15.1. Series
6.15.2. Integrals
6.15.3. Products of confluent hypergeometric functions
6.16. Real zeros for real $a,c$
6.17. Descriptive properties for real $a,c,x$
References
SUBJECT INDEX
INDEX OF NOTATIONS