Fourier Transforms of Distributions and Their Inverses

Fourier Transforms of Distributions and Their Inverses

A Collection of Tables

1st Edition - September 28, 1973

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  • Author: Fritz Oberhettinger
  • eBook ISBN: 9781483219028

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Description

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.

Table of Contents


  • Preface

    Introduction

    Characteristic Functions

    Description and Use of the Tables

    Tables of the Appendix

    References

    Part 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 Classes

    Part 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 Classes

    Appendix: 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 Functions

    List of Abbreviations, Symbols, and Notations

Product details

  • No. of pages: 180
  • Language: English
  • Copyright: © Academic Press 1973
  • Published: September 28, 1973
  • Imprint: Academic Press
  • eBook ISBN: 9781483219028

About the Author

Fritz Oberhettinger

About the Editors

Z. W. Birnbaum

E. Lukacs

Affiliations and Expertise

Bowling Green State University

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