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Formulas and Theorems for the Special Functions of Mathematical Physics (3rd Ediion) by Dr. Wilhelm Magnus, Dr. Fritz Oberhettinger, Dr. Raj Pal Soni

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Formulas and Theorems for the Special Functions of Mathematical Physics (3rd Ediion) written by Dr. Wilhelm Magnus , Professor at the New York University, Courant Institute of Mathematical Sciences and Dr. Fritz Oberhettinger , Professor at the Oregon State University, Department of Mathematics and Dr. Raj Pal Soni , Mathematician, International Business Machines Corporation. This is a new and enlarged English edition of the book which, under the title "Formeln und Satze fur die Speziellen Funktionen der mathematischen Physik" appeared in German in 1946. Much of the material (part of it unpublished) did not appear in the earlier editions. We hope that these additions will be useful and yet not too numerous for the purpose of locating .with ease any particular result. Compared to the first two (German) editions a change has taken place as far as the list of references is concerned. They are generally restricted to books and monographs and accomodated at the end of each individual chapter. Occasional references to papers follow those results to which they apply. The authors felt a certain justification for this change. At the time of the appearance of the previous edition nearly twenty years ago much of the material was scattered over a number of single contributions. Since then most of it has been included in books and monographs with quite exhaustive bibliographies. For information about numerical tables the reader is referred to "Mathematics of Computation", a periodical published by the American Mathematical Society; "Handbook of Mathematical Functions" with formulas, graphs and mathematical tables National Bureau of Standards Applied Mathematics Series, 55, 1964, 1046 pp., Government Printing Office, Washington, D.C., and FLETCHER, MILLER, ROSENHEAD, Index of Mathematical Tables, Addison-Wesley, Reading, Mass.).

The Cambridge Handbook of Physics Formulas written by Dr. Wilhelm Magnus, Dr. Fritz Oberhettinger, Dr. Raj Pal Soni cover the following topics.

  • I. The gamma function and related functions
    1.1 The gamma function .
    1.2 The function P (z)
    1.3. The Riemann zeta function C(z)
    1.4 The generalized zeta function C(z, "')
    1.6 Bernoulli and Euler polynomials
    1.6 Lerch
    s transcendent «P(z, 5, "')
    1.7 Miscellaneous results Literature •••.

  • 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

  • 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
    3.6 Differential equations and a power series expansion for the product of two Bessel functions
    3.6 Integral representatio~s 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
    3.16 Zeros
    3.16 Wscellaneous

  • IV. Legendre functions
    4.1 Legendre
    s differential equation
    4.2 Relations between Legendre functions
    4.3 The functions P~(x) and Q:(x). (Legendre functions on the cut)
    4.4 Special values for the parameters
    4.6 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

  • V. Orthogonal polynomials
    5.1 Orthogonal systems
    5.2 Jacobi polynomials
    5.3 Gegenbauer or ultraspherical polynomials
    5.4 Legendre Polynomials
    5.6 Generalized Laguerre polynomials
    5.7 Hermite polynomials
    5.8 Chebychev (Tchebichef) polynomials

  • 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 tFt(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

  • 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

  • VIII. Parabolic cylinder functions and parabolic
    8.1 Parabolic cylinder functions
    8.2 Parabolic functions .
    Appendix to Chapter VIII.

  • IX. The incomplete gamma function and special cases.
    9.1 The incomplete gamma function
    9.2 Special cases
    Literature .

  • X. Elliptic integrals, theta functions and elliptic
    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.

  • 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 generalised Mehler transform
    Example for the Gauss. transform
    11.1 Several examples of solution of integral equations of the first kind
    Appendix to Chapter XI

  • XII. Transformation of systems of coordinates
    12.1 General transformation and special cases
    12.2 Examples of separation of variables
    List of special symbols
    List of functions

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