reciprocal-modulus transformation

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41: 16.16 Transformations of Variables

§16.16 Transformations of Variables

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§16.16(i) Reduction Formulas

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§16.16(ii) Other Transformations

… ►For quadratic transformations of Appell functions see Carlson (1976).

42: Bille C. Carlson

… ►The main theme of Carlson’s mathematical research has been to expose previously hidden permutation symmetries that can eliminate a set of transformations and thereby replace many formulas by a few. …This symmetry led to the development of symmetric elliptic integrals, which are free from the transformations of modulus and amplitude that complicate the Legendre theory. …

43: 31.2 Differential Equations

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$F$-Homotopic Transformations

… ►By composing these three steps, there result $2^3=8$ possible transformations of the dependent variable (including the identity transformation) that preserve the form of (31.2.1). ►

Homographic Transformations

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Composite Transformations

►There are $8\cdot 24=192$ automorphisms of equation (31.2.1) by compositions of $F$-homotopic and homographic transformations. …

44: Mourad E. H. Ismail

… ►Ismail serves on several editorial boards including the Cambridge University Press book series Encyclopedia of Mathematics and its Applications, and on the editorial boards of 9 journals including Proceedings of the American Mathematical Society (Integrable Systems and Special Functions Editor); Constructive Approximation; Journal of Approximation Theory; and Integral Transforms and Special Functions. …

45: 18.17 Integrals

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§18.17(v) Fourier Transforms

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Jacobi

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Ultraspherical

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§18.17(vi) Laplace Transforms

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§18.17(vii) Mellin Transforms

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46: 20.7 Identities

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§20.7(iv) Reduction Formulas for Products

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§20.7(vi) Landen Transformations

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§20.7(viii) Transformations of Lattice Parameter

… ►These are specific examples of modular transformations as discussed in §23.15; the corresponding results for the general case are given by Rademacher (1973, pp. 181–183). ►

§20.7(ix) Addendum to 20.7(iv) Reduction Formulas for Products

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47: 3.11 Approximation Techniques

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Laplace Transform Inversion

►Numerical inversion of the Laplace transform (§1.14(iii)) … ►

Example. The Discrete Fourier Transform

… ►is called a discrete Fourier transform pair. ►

The Fast Fourier Transform

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48: 16.15 Integral Representations and Integrals

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16.15.1 $$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)=\frac{\mathrm{\Gamma }\left(\gamma \right)}{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(\gamma -\alpha \right)}{\int }_0^1\frac{u^{\alpha -1}{(1-u)}^{\gamma -\alpha -1}}{{(1-ux)}^{\beta }{(1-uy)}^{{\beta }^{\prime }}}du,$$ $\mathrm{\Re }\alpha >0$, $\mathrm{\Re }\left(\gamma -\alpha \right)>0$,

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Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/16.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.15 and Ch.16

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16.15.2 $$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)=\frac{\mathrm{\Gamma }\left(\gamma \right)\mathrm{\Gamma }\left({\gamma }^{\prime }\right)}{\mathrm{\Gamma }\left(\beta \right)\mathrm{\Gamma }\left({\beta }^{\prime }\right)\mathrm{\Gamma }\left(\gamma -\beta \right)\mathrm{\Gamma }\left({\gamma }^{\prime }-{\beta }^{\prime }\right)}{\int }_0^1{\int }_0^1\frac{u^{\beta -1}{v}^{{\beta }^{\prime }-1}{(1-u)}^{\gamma -\beta -1}{(1-v)}^{{\gamma }^{\prime }-{\beta }^{\prime }-1}}{{(1-ux-vy)}^{\alpha }}dudv,$$ $\mathrm{\Re }\gamma >\mathrm{\Re }\beta >0$, $\mathrm{\Re }{\gamma }^{\prime }>\mathrm{\Re }{\beta }^{\prime }>0$,

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Symbols:

$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$: second Appell function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/16.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.15 and Ch.16

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16.15.3 $$F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)=\frac{\mathrm{\Gamma }\left(\gamma \right)}{\mathrm{\Gamma }\left(\beta \right)\mathrm{\Gamma }\left({\beta }^{\prime }\right)\mathrm{\Gamma }\left(\gamma -\beta -{\beta }^{\prime }\right)}{\iint }_{\mathrm{\Delta }}\frac{u^{\beta -1}{v}^{{\beta }^{\prime }-1}{(1-u-v)}^{\gamma -\beta -{\beta }^{\prime }-1}}{{(1-ux)}^{\alpha }{(1-vy)}^{{\alpha }^{\prime }}}dudv,$$ $\mathrm{\Re }\left(\gamma -\beta -{\beta }^{\prime }\right)>0$, $\mathrm{\Re }\beta >0$, $\mathrm{\Re }{\beta }^{\prime }>0$,

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Symbols:

$F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)$: third Appell function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Notes:

Erdélyi et al. (1953a, 5.8.1(3)) contains a typo.

Referenced by:

Erratum (V1.0.14) for Equation (16.15.3), Erratum (V1.0.14) for Equation (16.15.3)

Permalink:

http://dlmf.nist.gov/16.15.E3

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.14):

A stray “$f$” was removed; it appeared after the comma at the end of this equation.

See also:

Annotations for §16.15 and Ch.16

… ►For inverse Laplace transforms of Appell functions see Prudnikov et al. (1992b, §3.40).

49: Bibliography O

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Tables of Lebedev, Mehler and Generalized Mehler Transforms. Mathematical Note Technical Report 246, Boeing Scientific Research Lab, Seattle.

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Referenced by:

§1.14(viii).

Encodings:

BibTeX

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Tables of Fourier Transforms and Fourier Transforms of Distributions. Springer-Verlag, Berlin.

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External Links:

ISBN 3-540-50630-6; 0-387-50630-6, MathReview, ZentralBlatt

Referenced by:

§1.14(viii), §10.22(vi), §10.43(vi), §11.10(x), §11.7(v), §11.9(iv), §12.12, §13.10(iv), §13.23(ii), §18.17(ix), §6.14(iii), §7.14(iii).

Encodings:

BibTeX

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Tables of Laplace Transforms. Springer-Verlag, Berlin-New York.

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Notes:

Table errata: Math. Comp. v. 50 (1988), no. 182, pp. 653–654.

External Links:

ISBN 0-387-06350-1, MathReview, ZentralBlatt

Referenced by:

§1.14(viii), §10.22(vi), §10.43(vi), §11.5(iii), §11.7(v), §11.9(iv), §12.12, §12.5(iv), §13.10(ii), §13.23(i), §14.17(v), §15.14, §18.17(ix), §20.10(ii), §5.13, §6.14(iii), §6.7(iv), §7.14(iii), §7.7(iii), §8.6(iii).

Encodings:

BibTeX

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Tables of Bessel Transforms. Springer-Verlag, Berlin-New York.

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Notes:

Table errata: Math. Comp. v. 65 (1996), no. 215, p. 1386.

External Links:

ISBN 0-387-05997-0, MathReview, ZentralBlatt

Referenced by:

§1.14(viii), §10.22(v), §10.43(v), §10.43(vi), §11.10(x), §11.5(iii), §11.7(v), §11.9(iv), §13.10(v), §13.23(iii), §15.14, §18.17(ix), §8.14, §8.6(iii).

Encodings:

BibTeX

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Tables of Mellin Transforms. Springer-Verlag, Berlin-New York.

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External Links:

ISBN 0-387-06942-9, MathReview, ZentralBlatt

Referenced by:

§1.14(viii), §10.22(vi), §10.43(vi), §11.10(x), §11.5(iii), §11.7(v), §11.9(iv), §12.12, §12.5(iv), §13.10(iii), §13.23(i), §14.17(vi), §15.14, §18.17(ix), §20.10(i), §5.13, §6.14(iii), §6.7(iv), §7.14(iii), §7.7(iii).

Encodings:

BibTeX

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50: 15.8 Transformations of Variable

§15.8 Transformations of Variable

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§15.8(i) Linear Transformations

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§15.8(iii) Quadratic Transformations

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§15.8(v) Cubic Transformations

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Ramanujan’s Cubic Transformation

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