Tables of Fourier Transforms and Fourier Transforms of Distributions

ISBN-13: 9783540506300 / Angielski / Miękka / 1990 / 259 str.

Fritz Oberhettinger

Opis

These tables represent a new, revised and enlarged version of the previously published book by this author, entitled Tabellen zur Fourier Transformation (Springer Verlag 1957). Known errors have been correc- ted, apart from the addition of a considerable number of new results, which involve almost exclusively higher functions. Again, the follow- ing tables contain a collection of integrals of the form J f(x)cos(xy)dx Fourier Cosine Transform (Al o (B) J f(x)sin(xy)dx Fourier Sine Transform o (C) ge(y) = J f(x)eixYdx Exponential Fourier Transform -00 Clearly, (A) and (B) are special cases of (C) if f(x) is respec- tively an even or an odd function. The transform parameter y in (A) and (B) is assumed to be positive, while in (C) negative values are also included. A possible analytic continuation to complex parameters y* should present no difficulties. In some cases the result function g(y) is given over a partial range of y only. This means that g(y) for the remaining part of y cannot be given in a reasonably simple form.Under certain conditions the following inversion formulas for (A), (B), (C) hold: (A' ) f(x) = 2 J g (y)cos(xy)dy 11 0 c 2 J (B') f (x) gs(y)sin(xy)dy 11 0 -1 00 -ix (C' ) f(x) = (211) J ge(y)e Ydy In the following parts I, II, III tables for the transforms (A), (B) and (C) are given.

Spis treści

I. Fourier Cosine Transforms (Tables I).- 1.1 Algebraic Functions.- 1.2 Arbitrary Powers.- 1.3 Exponential Functions.- 1.4 Logarithmic Functions.- 1.5 Trigonometric Functions.- 1.6 Inverse Trigonometric Functions.- 1.7 Hyperbolic Functions.- 1.8 Orthogonal Polynomials.- 1.9 Gamma- and Related Functions.- 1.10 The Error- and the Fresnel Integrals.- 1.11 The Exponential- and Related Integrals.- 1.12 Legendre Functions.- 1.13 Bessel Functions of Arguments x, x2 and 1/x.- 1.14 Bessel Functions of Argument (ax2 + bx + c)1/2.- 1.15 Bessel Functions of Trigonometric and Hyperbolic Arguments.- 1.16 Bessel Functions of Variable Order.- 1.17 Modified Bessel Functions of Arguments x, x2 and 1/x.- 1.18 Modified Bessel Functions of Argument (ax2 + bx + c)1/2.- 1.19 Modified Bessel Functions of Trigonometric and Hyperbolic Arguments.- 1.20 Modified Bessel Functions of Variable Order.- 1.21 Functions Related to Bessel Functions.- 1.22 Parabolic Cylinder- and Whittaker Functions.- 1.23 Elliptic Integrals.- II. Fourier Sine Transforms (Tables II).- 2.1 Algebraic Functions.- 2.2 Arbitrary Powers.- 2.3 Exponential Functions.- 2.4 Logarithmic Functions.- 2.5 Trigonometric Functions.- 2.6 Inverse Trigonometric Functions.- 2.7 Hyperbolic Functions.- 2.8 Orthogonal Polynomials.- 2.9 Gamma- and Related Functions.- 2.10 The Error- and the Fresnel Integrals.- 2.11 The Exponential- and Related Integrals.- 2.12 Legendre Functions.- 2.13 Bessel Functions of Arguments x, x2 and 1/x.- 2.14 Bessel Functions of Argument (ax2 + bx + c)1/2.- 2.15 Bessel Functions of Trigonometric and Hyperbolic Arguments.- 2.16 Bessel Functions of Variable Order.- 2.17 Modified Bessel Functions of Arguments x, x2 and 1/x.- 2.18 Modified Bessel Functions of Argument (ax2 + bx + c)1/2.- 2.19 Modified Bessel Functions of Trigonometric and Hyperbolic Arguments.- 2.20 Modified Bessel Functions of Variable Order.- 2.21 Functions Related to Bessel Functions.- 2.22 Parabolic Cylinder- and Whittaker Functions.- 2.23 Elliptic Integrals.- III. Exponential Fourier Transforms (Tables III).- IV. Fourier Transforms of Distributions (Tables IV and V).- List of Functions.