Mellin transform

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31: Bibliography S

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Asymptotic expansions of Mellin transforms and analogues of Watson’s lemma. SIAM J. Math. Anal. 16 (4), pp. 896–906.

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External Links:

ISSN 0036-1410, Document, Link, MathReview (John S. Lew), ZentralBlatt

Referenced by:

§2.3(vi).

Encodings:

BibTeX

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Asymptotic expansion of Mellin transforms in the complex plane. Int. J. Pure Appl. Math. 71 (3), pp. 465–480.

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External Links:

ISSN 1311-8080, MathReview Entry, ZentralBlatt

Referenced by:

§2.3(vi).

Encodings:

BibTeX

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32: 9.12 Scorer Functions

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9.12.24 $$\mathrm{Hi}\left(z\right)=\frac{3^{-2/3}}{2{\pi }^2\mathrm{i}}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\mathrm{\Gamma }\left(\frac{1}{3}+\frac{1}{3}t\right)\mathrm{\Gamma }\left(-t\right){(3^{1/3}{\mathrm{e}}^{\pi \mathrm{i}}z)}^{t}dt,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral and $z$: complex variable

Keywords:

Mellintransform

Source:

Exton (1983, (2.1), p. 100)

Permalink:

http://dlmf.nist.gov/9.12.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(vii), §9.12(vii), §9.12 and Ch.9

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33: 3.5 Quadrature

… ►Other contour integrals occur in standard integral transforms or their inverses, for example, Hankel transforms (§10.22(v)), Kontorovich–Lebedev transforms (§10.43(v)), and Mellin transforms (§1.14(iv)). …

34: Bibliography W

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The cubic transformation of the hypergeometric function. Quart. J. Pure and Applied Math. 41, pp. 70–79.

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External Links:

ZentralBlatt

Referenced by:

§15.8(v), §15.8(v).

Encodings:

BibTeX

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Some transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 26 (2), pp. 257–272.

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External Links:

Document, ZentralBlatt

Referenced by:

§16.6.

Encodings:

BibTeX

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The Airy transform. Amer. Math. Monthly 86 (4), pp. 271–277.

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External Links:

ISSN 0002-9890, FullText, Document, MathReview (A. G. Ramm), ZentralBlatt

Referenced by:

(9.10.13), §9.10(ix), §9.10(v).

Encodings:

BibTeX

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The Laplace Transform. Princeton Mathematical Series, v. 6, Princeton University Press, Princeton, NJ.

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External Links:

FullText, MathReview (J. D. Tamarkin), ZentralBlatt

Referenced by:

§1.14(vi), §1.15(viii).

Encodings:

BibTeX

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Explicit error terms for asymptotic expansions of Mellin convolutions. J. Math. Anal. Appl. 72 (2), pp. 740–756.

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External Links:

ISSN 0022-247X, Document, MathReview (John S. Lew), ZentralBlatt

Referenced by:

§2.6(ii), §2.6(iv).

Encodings:

BibTeX

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35: Bibliography P

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Asymptotics and Mellin-Barnes Integrals. Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-79001-8, MathReview, ZentralBlatt

Referenced by:

§1.14(iv), §10.32(iii), §10.32(ii), §10.46, §10.9(iii), §10.9(ii), §16.11(i), §16.11(ii), §2.10(iii), §2.11(iii), §2.11(v), §2.5(ii), §2.5(ii), Ch.2, §24.7(ii), §25.9, §5.11(ii), §5.11(iii), §5.13, §5.19(ii), §5.6(ii), Ch.5, §8.6(ii), Profile Richard B. Paris.

Encodings:

BibTeX

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Smoothing of the Stokes phenomenon using Mellin-Barnes integrals. J. Comput. Appl. Math. 41 (1-2), pp. 117–133.

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External Links:

ISSN 0377-0427, Document, MathReview (Anatoly Kilbas), ZentralBlatt

Referenced by:

§2.11(v).

Encodings:

BibTeX

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A Kummer-type transformation for a ${}_2{}^{}F_2^{}$ hypergeometric function. J. Comput. Appl. Math. 173 (2), pp. 379–382.

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External Links:

ISSN 0377-0427, Document, MathReview (P. Anandani), ZentralBlatt

Referenced by:

§16.6.

Encodings:

BibTeX

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Bounds for ratios of modified Bessel functions. Integral Transform. Spec. Funct. 9 (4), pp. 293–298.

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External Links:

ISSN 1065-2469, Document, MathReview, ZentralBlatt

Referenced by:

§10.37.

Encodings:

BibTeX

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Fourier Series and Integral Transforms. Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-59209-7; 0-521-59771-4, MathReview, ZentralBlatt

Referenced by:

§1.14(vii).

Encodings:

BibTeX

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36: 35.8 Generalized Hypergeometric Functions of Matrix Argument

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Kummer Transformation

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Thomae Transformation

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Laplace Transform

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§35.8(v) Mellin–Barnes Integrals

… ►These multidimensional integrals reduce to the classical Mellin–Barnes integrals (§5.19(ii)) in the special case $m=1$. …