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

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## 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
- 3.9 Addition theorems
- 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
- 6.4 Addition and multiplication theorems
- 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
- 7.3 Addition and multiplication theorems
- 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