# Book:Wilhelm Magnus/Formulas and Theorems for the Special Functions of Mathematical Physics/Third Edition

## Wilhelm Magnus, Fritz Oberhettinger and Raj Pal Soni: Formulas and Theorems for the Special Functions of Mathematical Physics

Published $1966$, Springer-Verlag.

### Contents

Chapter I. The gamma function and related functions
1.1 The gamma function
1.2 The function $\psi(z)$
1.3 The Riemann zeta function $\zeta(z)$
1.4 The generalized zeta function $\zeta(z,\alpha)$
1.5 Bernoulli and Euler polynomials
1.6 Lerch's transcendent $\phi(z,s,\alpha)$
1.7 Miscellaneous results
Literature
Chapter II. The hypergeometric function
2.1 Definitions and elementary relations
2.2 The hypergeometric differential equation
2.3 Gauss' contiguous relations
2.4 Linear and higher order transformations
2.5 Integral representations
2.6 Asymptotic expansions
2.7 The Riemann differential equation
2.8 Transformation formulas for Riemann's $P$-function
2.9 The generalized hypergeometric series
2.10 Miscellaneous results
Literature
Chapter III. Bessel functions
3.1 Solutions of the Bessel and the modified Bessel differential equation
3.2 Bessel functions of integer order
3.3 Half odd integer order
3.4 The Airy functions and related functions
3.5 Differential equations and a power series expansion for the product of two Bessel functions
3.6 Integral representations for Bessel, Neumann and Hankel functions
3.7 Integral representations for the modified Bessel functions
3.8 Integrals involving Bessel functions
3.10 Functions related to Bessel functions
3.11 Polynomials related to Bessel functions
3.12 Series of arbitrary functions in terms of Bessel functions
3.13 A list of series involving Bessel functions
3.14 Asymptotic expansions
3.15 Zeros
3.16 Miscellaneous
Literature
Chapter IV. Legendre functions
4.1 Legendre's differential equation
4.2 Relations between Legendre functions
4.3 The functions $P^u_v(x)$ and $Q^u_v(x)$. (Legendre functions on the cut)
4.4 Special values for the parameters
4.5 Series involving Legendre functions
4.6 Integral representations
4.7 Integrals involving Legendre functions
4.8 Asymptotic behavior
4.9 Associated Legendre functions and surface spherical harmonics
4.10 Gegenbauer functions, toroidal functions and conical functions
Literature
Chapter V. Orthogonal polynomials
5.1 Orthogonal systems
5.2 Jacobi polynomials
5.3 Gegenbauer or ultraspherical polynomials
5.4 Legendre Polynomials
5.5 Generalized Laguerre polynomials
5.6 Hermite polynomials
5.7 Chebychev (Tchebichef) polynomials
Literature
Chapter VI. Kummer's function
6.1 Definitions and some elementary results
6.2 Recurrence relations
6.3 The differential equation
6.5 Integral representations
6.6 Integral transforms associated with ${}_1F_1(a;c;z)$, $U(a,c,z)$
6.7 Special cases and its relation to other functions
6.8 Asymptotic expansions
6.9 Products of Kummer's functions
Literature
Chapter VII. Whittaker function
7.1 Whittaker's differential equation
7.2 Some elementary results
7.4 Integral representations
7.5 Integral transforms
7.6 Asymptotic expansions
7.7 Products of Whittaker functions
Literature
Chapter VIII. Parabolic cylinder functions and parabolic functions
8.1 Parabolic cylinder functions
8.2 Parabolic functions
Literature
Chapter IX. The incomplete gamma function and special cases
9.1 The incomplete gamma function
9.2 Special cases
Literature
Chapter X. Elliptic integrals, theta functions and elliptic functions
10.1 Elliptic integrals
10.2 The theta functions
10.3 Definition of the Jacobian elliptic functions by the theta functions
10.4 The Jacobian zeta function
10.5 The elliptic functions of Weierstrass
10.6 Connections between the parameters and special cases
Literature
Chapter XI. Integral transforms
Examples for the Fourier cosine transform
Examples for the Fourier sine transform
Examples for the exponential Fourier transform
Examples for the Laplace transform
Examples for the Mellin transform
Examples for the Hankel transform
Examples for the Lebedev, Mehler and generalized Mehler transform
Example for the Gauss transform
11.1 Several examples of solution of integral equations of the first kind
Literature
Chapter XII. Transformation of systems of coordinates
12.1 General transformation and special cases
12.2 Examples of separation of variables
Literature
List of special symbols
List of functions
Index