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

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{{Book|Formulas and Theorems for the Special Functions of Mathematical Physics |1966|Springer-Verlag||Wilhelm Magnus|author2 = Fritz Oberhettinger|author3=Raj Pal Soni}}

===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