河海大学毕业证办理【somewhat微KAA2238】burchnal

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1: Software Index

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These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

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2: 16.15 Integral Representations and Integrals

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16.15.4 $$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x(1-y),y(1-x))=\frac{\mathrm{\Gamma }\left(\gamma \right)\mathrm{\Gamma }\left({\gamma }^{\prime }\right)}{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(\beta \right)\mathrm{\Gamma }\left(\gamma -\alpha \right)\mathrm{\Gamma }\left({\gamma }^{\prime }-\beta \right)}{\int }_0^1{\int }_0^1\frac{u^{\alpha -1}{v}^{\beta -1}{(1-u)}^{\gamma -\alpha -1}{(1-v)}^{{\gamma }^{\prime }-\beta -1}}{{(1-ux)}^{\gamma +{\gamma }^{\prime }-\alpha -1}{(1-vy)}^{\gamma +{\gamma }^{\prime }-\beta -1}{(1-ux-vy)}^{\alpha +\beta -\gamma -{\gamma }^{\prime }+1}}dudv,$$ $\mathrm{\Re }\gamma >\mathrm{\Re }\alpha >0$, $\mathrm{\Re }{\gamma }^{\prime }>\mathrm{\Re }\beta >0$.

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Symbols:

$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$: fourth Appell function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Source:

Burchnall and Chaundy (1940, (68))

Referenced by:

§16.15

Permalink:

http://dlmf.nist.gov/16.15.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §16.15 and Ch.16

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3: 15.16 Products

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4: Bibliography B

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Expansions of Appell’s double hypergeometric functions. Quart. J. Math., Oxford Ser. 11, pp. 249–270.

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External Links:

Document, MathReview (G. Szegő), ZentralBlatt

Referenced by:

§15.16, (16.15.4), §16.15, §16.16(i).

Encodings:

BibTeX

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Expansions of Appell’s double hypergeometric functions. II. Quart. J. Math., Oxford Ser. 12, pp. 112–128.

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External Links:

Document, MathReview (G. Szegő), ZentralBlatt

Referenced by:

§16.16(i).

Encodings:

BibTeX

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The hypergeometric identities of Cayley, Orr, and Bailey. Proc. London Math. Soc. (2) 50, pp. 56–74.

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External Links:

Document, MathReview (N. A. Hall), ZentralBlatt

Referenced by:

§15.16.

Encodings:

BibTeX

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5: 16.16 Transformations of Variables

… ►see Burchnall and Chaundy (1940, 1941). …