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

3.9 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