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.
Inhalt
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 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
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