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Fourier Transforms of Distributions and Their Inverses

A Collection of Tables

1st Edition - September 28, 1973

Author: Fritz Oberhettinger

Editors: Z. W. Birnbaum, E. Lukacs

eBook ISBN:

9 7 8 - 1 - 4 8 3 2 - 1 9 0 2 - 8

Fourier Transforms of Distributions and Their Inverses: A Collection of Tables is a collection of tables on the integrals of Fourier transforms of distributions and their inverses… Read more

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Fourier Transforms of Distributions and Their Inverses: A Collection of Tables is a collection of tables on the integrals of Fourier transforms of distributions and their inverses involving the class of functions which are nonnegative and integrable over the interval. The emphasis is on the probability densities, and a number of examples are provided. This book is organized into two parts and begins with an introduction to those properties of characteristic functions which are important in probability theory, followed by a description of the tables and their use. The first three tables contain Fourier transforms of absolutely continuous distribution functions, namely, even functions (including Legendre functions); functions vanishing identically for negative values of the argument (including arbitrary powers); and functions that do not belong to either of the above classes. The transform pairs are numbered consecutively and arranged systematically according to the analytical character of the frequency function. The next two tables give the inverse transforms of the functions listed in the first and third tables, respectively. This monograph will appeal to students and specialists in the fields of probability and mathematical statistics.

PrefaceIntroduction Characteristic Functions Description and Use of the Tables Tables of the Appendix ReferencesPart I. Tables of Fourier Transforms Table I. Even Functions 1. Algebraic Functions 2. Arbitrary Powers 3. Exponential Functions 4. Logarithmic Functions 5. Trigonometric Functions 6. Inverse Trigonometric Functions 7. Hyperbolic Functions 8. Gamma Functions (Including Incomplete Gamma Functions) and Related Functions 9. Elliptic Integrals and Legendre Functions 10. Bessel Functions 11. Modified Bessel Functions 12. Functions Related to Bessel Functions 13. Parabolic Cylindrical Functions Table II. Functions Vanishing Identically for Negative Values of the Argument 1. Algebraic Functions 2. Arbitrary Powers 3. Exponential Functions 4. Logarithmic Functions 5. Trigonometric Functions 6. Inverse Trigonometric Functions 7. Hyperbolic Functions 8. Gamma and Related Functions 9. Elliptic Integrals and Legendre Functions 10. Bessel Functions 11. Modified Bessel Functions 12. Parabolic Cylindrical Functions Table III. Functions Not Belonging to Either of These ClassesPart II. Tables of the Inverse Transforms of Part I Table IΑ· Even Functions 1. Algebraic Functions 2. Arbitrary Powers 3. Exponential Functions 4. Logarithmic Functions 5. Trigonometric Functions 6. Inverse Trigonometric Functions 7. Hyperbolic Functions 7a. Orthogonal Polynomials 8. Gamma Functions (Including Incomplete Gamma Functions) and Related Functions 9. Elliptic Integrals and Legendre Functions 10. Bessel Functions 11. Modified Bessel Functions 12. Functions Related to Bessel Functions 13. Parabolic Cylindrical Functions and Whittaker Functions Table IIA. Functions Vanishing Identically for Negative Values of the Argument Table IIIA. Functions Not Belonging to Either of These ClassesAppendix: Distribution Functions and Their Fourier Transforms Found in the Statistical Literature Table A. Univariate Density Functions Table B. Univariate Discrete Distributions Table C. Multivariate Density FunctionsList of Abbreviations, Symbols, and Notations