![]() Fourier Transforms of Distributions (Tables IV and V).- List of Functions. Exponential Fourier Transforms (Tables III).- IV. 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. 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. Table of Fourier Transform Pairs Function, f(t) Fourier Transform, F(w) Definition of Inverse Fourier Transform Definition of Fourier Transform 1 F (w ). 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. This means that g(y) for the remaining part of y cannot be given in a reasonably simple form. In some cases the result function g(y) is given over a partial range of y only. A possible analytic continuation to complex parameters y* should present no difficulties. ![]() The transform parameter y in (A) and (B) is assumed to be positive, while in (C) negative values are also included. 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. Fourier Transform Table Time Signal Fourier Transform 1, t < < 2 ( ) u t 0.5 ( ) 1/ j u t ( ) ( ) 1/ j t( ) 1, << t c c ( ), real j c e c, real bt e u t b >( ), 0, 0 1 > b j b jto, real e o 2 ( ), real o o p t ( ) sinc /2 t sinc / 2.![]() Known errors have been correc- ted, apart from the addition of a considerable number of new results, which involve almost exclusively higher functions. I also checked the book "Table of Integrals, Series, and Products" by Gradshteyn and Ryzhik, and couldn't find the thing I'm looking for.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). ![]() The function $f(\xi)=(i-\xi)^a\cdot(\log(i-\xi))^b$ can be defined for all real $\xi$ (by choosing appropriate branches of $\log(i-\xi)$ and $\log(\log(i-\xi))$), and the inverse Fourier transform of $f(\xi)$ makes sense as a distribution on $\mathbb$ As was commented below, the Erdelyi book "Tables of Integral Transforms" is the same as the one I referred to above when I mentioned the Bateman project. Note that I failed to find the answer not only online, but also in the standard books with Fourier transform tables (such as "Tables of Integral Transforms" from the Bateman project). My question is in fact motivated by one concrete example (so if you know the answer to this one, please let me know!). I was wondering if someone can suggest a website (or some online document) containing an $extensive$ table of Fourier transforms? When I try obvious Google searches, like "table of Fourier transforms", the several dozen top results give extremely short tables.
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