Radon transform

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1: 1.14 Integral Transforms

§1.14 Integral Transforms

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§1.14(i) Fourier Transform

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§1.14(iii) Laplace Transform

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Fourier Transform

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Laplace Transform

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2: 18.38 Mathematical Applications

… ►It has elegant structures, including $N$-soliton solutions, Lax pairs, and Bäcklund transformations. While the Toda equation is an important model of nonlinear systems, the special functions of mathematical physics are usually regarded as solutions to linear equations. … ►

Radon Transform

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3: Bibliography D

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Integral Transforms and their Applications. 2nd edition, Applied Mathematical Sciences, Vol. 25, Springer-Verlag, New York.

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External Links:

ISBN 0-387-96080-5, MathReview, ZentralBlatt

Referenced by:

§1.14(vii).

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The Radon Transform and Some of Its Applications. A Wiley-Interscience Publication, John Wiley & Sons Inc., New York.

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Notes:

Reprinted by Dover in 2007.

External Links:

ISBN 0-471-89804-X, MathReview (W. R. Madych), ZentralBlatt

Referenced by:

§18.38(ii).

Encodings:

BibTeX

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Integral transforms and their applications. Third edition, CRC Press, Boca Raton, FL.

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External Links:

ISBN 978-1-4822-2357-6, MathReview Entry, ZentralBlatt

Referenced by:

§1.16(viii).

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BibTeX

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Handbuch der Laplace-Transformation. Bd. II. Anwendungen der Laplace-Transformation. 1. Abteilung. Birkhäuser Verlag, Basel und Stuttgart (German).

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External Links:

MathReview (I. I. Hirschman), ZentralBlatt

Referenced by:

§2.5(i).

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BibTeX

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4: 15.14 Integrals

§15.14 Integrals

►The Mellin transform of the hypergeometric function of negative argument is given by … ►Fourier transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §§1.14 and 2.14). Laplace transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §4.21), Oberhettinger and Badii (1973, §1.19), and Prudnikov et al. (1992a, §3.37). …Hankel transforms of hypergeometric functions are given in Oberhettinger (1972, §1.17) and Erdélyi et al. (1954b, §8.17). …

5: 12.16 Mathematical Applications

… ►PCFs are also used in integral transforms with respect to the parameter, and inversion formulas exist for kernels containing PCFs. …Integral transforms and sampling expansions are considered in Jerri (1982).

6: 2.5 Mellin Transform Methods

§2.5 Mellin Transform Methods

… ►The Mellin transform of $f(t)$ is defined by …The inversion formula is given by … ► … ►

§2.5(iii) Laplace Transforms with Small Parameters

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7: 35.2 Laplace Transform

§35.2 Laplace Transform

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Definition

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Inversion Formula

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Convolution Theorem

►If $g_j$ is the Laplace transform of $f_j$, $j=1,2$, then $g_1{g}_2$ is the Laplace transform of the convolution $f_1\ast f_2$, where …

8: 15.17 Mathematical Applications

… ►The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. … ►Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform. … ►Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)). … ►By considering, as a group, all analytic transformations of a basis of solutions under analytic continuation around all paths on the Riemann sheet, we obtain the monodromy group. …

9: 19.15 Advantages of Symmetry

… ►Symmetry in $x,y,z$ of $R_F(x,y,z)$, $R_G(x,y,z)$, and $R_J(x,y,z,p)$ replaces the five transformations (19.7.2), (19.7.4)–(19.7.7) of Legendre’s integrals; compare (19.25.17). Symmetry unifies the Landen transformations of §19.8(ii) with the Gauss transformations of §19.8(iii), as indicated following (19.22.22) and (19.36.9). (19.21.12) unifies the three transformations in §19.7(iii) that change the parameter of Legendre’s third integral. …

10: 14.31 Other Applications

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§14.31(ii) Conical Functions

… ►These functions are also used in the Mehler–Fock integral transform (§14.20(vi)) for problems in potential and heat theory, and in elementary particle physics (Sneddon (1972, Chapter 7) and Braaksma and Meulenbeld (1967)). The conical functions and Mehler–Fock transform generalize to Jacobi functions and the Jacobi transform; see Koornwinder (1984a) and references therein. …