Oberhettinger, Fritz Tables of Fourier transforms and Fourier transforms of distributions. Transl. from the German. Rev. and enlarged ed. (English) Zbl 0694.42017 Berlin etc.: Springer-Verlag. viii, 259 p. DM 64.00 (1990). From the preface: These tables represent a new, revised and enlarged version of the previously published book by this author, entitled “Tabellen zur Fourier-Transformation” (German) Berlin etc.: Springer Verlag (1957; Zbl 0077.12002). Known errors have been corrected, apart from the addition of a considerable number of new results, which involve almost exclusively higher functions. Again, the following tables contain a collection of integrals of the form (A) \(g_ c(y)=\int^{\infty}_{0} f(x)\cos (xy)\, dx\) Fourier Cosine Transform, (B) \(g_s(y)=\int^{\infty}_{0} f(x)\sin(xy)\,dx\) Fourier Sine Transform, (C) \(g_e(y)=\int^{\infty}_{- \infty} f(x)e^{ixy}\,dx\) Exponential Fourier Transform. Clearly, (A) and (B) are special cases of (C) if \(f(x)\) is respectively 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\(^\prime\)) \(f(x)=\frac{2}{\pi}\int^{\infty}_{0} g_c(y)\cos(xy)\,dy\), (B\(^\prime\)) \(f(x)=\frac{2}{\pi}\int^{\infty}_{0}g_s(y)\sin(xy)\,dy\), (C\(^\prime\)) \(f(x)=(2\pi)^{-1}\int^{\infty}_{-\infty}g_e(y)e^{- ixy}\,dy\). In the following parts I, II, III tables for the transforms (A), (B) and (C) are given. The parts I and II are subdivided into 23 sections each involving the same class of functions. The first and the second column (in parenthesis) refers to the location of the correspondent page number for the cosine- and sine-transform respectively. Compared with the before-mentioned previous edition, a new part IV titled “Fourier Transforms of Distributions” has been added. In this, those functions \(f(x)\) occurring in the parts I–III have been singled out which represent so-called probability density (or frequency distribution) functions. The corresponding normalization factors are likewise listed. Cited in 2 ReviewsCited in 43 Documents MSC: 42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 42-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to harmonic analysis on Euclidean spaces 65A05 Tables in numerical analysis Keywords:tables of Fourier transforms; Fourier transforms of distributions; probability density; normalization factors Citations:Zbl 0077.12002 PDFBibTeX XMLCite \textit{F. Oberhettinger}, Tables of Fourier transforms and Fourier transforms of distributions. Transl. from the German. Rev. and enlarged ed. Berlin etc.: Springer-Verlag (1990; Zbl 0694.42017) Digital Library of Mathematical Functions: §10.22(vi) Compendia ‣ §10.22 Integrals ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions §10.43(vi) Compendia ‣ §10.43 Integrals ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions §11.10(x) Integrals and Sums ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions §1.14(viii) Compendia ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods §11.7(v) Compendia ‣ §11.7 Integrals and Sums ‣ Struve and Modified Struve Functions ‣ Chapter 11 Struve and Related Functions §11.9(iv) References ‣ §11.9 Lommel Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions Nicholson-type Integral ‣ §12.12 Integrals ‣ Properties ‣ Chapter 12 Parabolic Cylinder Functions §13.10(iv) Fourier Transforms ‣ §13.10 Integrals ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions §13.23(ii) Fourier Transforms ‣ §13.23 Integrals ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions §18.17(ix) Compendia ‣ §18.17 Integrals ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials §6.14(iii) Compendia ‣ §6.14 Integrals ‣ Properties ‣ Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals §7.14(iii) Compendia ‣ §7.14 Integrals ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals