§16.15 Integral Representations and Integrals

16.15.1		$$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)=\frac{\mathrm{\Gamma }\left(\gamma \right)}{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(\gamma -\alpha \right)}{\int }_0^1\frac{u^{\alpha -1}{(1-u)}^{\gamma -\alpha -1}}{{(1-ux)}^{\beta }{(1-uy)}^{{\beta }^{\prime }}}du,$$
$\mathrm{\Re }\alpha >0$, $\mathrm{\Re }\left(\gamma -\alpha \right)>0$,
ⓘ 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: TeX, pMML, png See also: Annotations for §16.15 and Ch.16

16.15.2		$$\begin{array}{l}{F}_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)\\ =\begin{array}{l}\frac{\mathrm{\Gamma }\left(\gamma \right)\mathrm{\Gamma }\left({\gamma }^{\prime }\right)}{\mathrm{\Gamma }\left(\beta \right)\mathrm{\Gamma }\left({\beta }^{\prime }\right)\mathrm{\Gamma }\left(\gamma -\beta \right)\mathrm{\Gamma }\left({\gamma }^{\prime }-{\beta }^{\prime }\right)}\\ \times {\int }_0^1{\int }_0^1\frac{u^{\beta -1}{v}^{{\beta }^{\prime }-1}{(1-u)}^{\gamma -\beta -1}{(1-v)}^{{\gamma }^{\prime }-{\beta }^{\prime }-1}}{{(1-ux-vy)}^{\alpha }}dudv,\end{array}\end{array}$$
$\mathrm{\Re }\gamma >\mathrm{\Re }\beta >0$, $\mathrm{\Re }{\gamma }^{\prime }>\mathrm{\Re }{\beta }^{\prime }>0$,
ⓘ 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: TeX, pMML, png See also: Annotations for §16.15 and Ch.16

16.15.3		$$\begin{array}{l}{F}_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)\\ =\frac{\mathrm{\Gamma }\left(\gamma \right)}{\mathrm{\Gamma }\left(\beta \right)\mathrm{\Gamma }\left({\beta }^{\prime }\right)\mathrm{\Gamma }\left(\gamma -\beta -{\beta }^{\prime }\right)}{\iint }_{\mathrm{\Delta }}\frac{u^{\beta -1}{v}^{{\beta }^{\prime }-1}{(1-u-v)}^{\gamma -\beta -{\beta }^{\prime }-1}}{{(1-ux)}^{\alpha }{(1-vy)}^{{\alpha }^{\prime }}}dudv,\end{array}$$
$\mathrm{\Re }\left(\gamma -\beta -{\beta }^{\prime }\right)>0$, $\mathrm{\Re }\beta >0$, $\mathrm{\Re }{\beta }^{\prime }>0$,
ⓘ 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: TeX, pMML, png 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

where $\mathrm{\Delta }$ is the triangle defined by $u\ge 0$, $v\ge 0$, $u+v\le 1$.

16.15.4		$$\begin{array}{l}{F}_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x(1-y),y(1-x))\\ =\begin{array}{l}\frac{\mathrm{\Gamma }\left(\gamma \right)\mathrm{\Gamma }\left({\gamma }^{\prime }\right)}{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(\beta \right)\mathrm{\Gamma }\left(\gamma -\alpha \right)\mathrm{\Gamma }\left({\gamma }^{\prime }-\beta \right)}\\ \times {\int }_0^1{\int }_0^1\frac{u^{\alpha -1}{v}^{\beta -1}{(1-u)}^{\gamma -\alpha -1}{(1-v)}^{{\gamma }^{\prime }-\beta -1}}{{(1-ux)}^{\gamma +{\gamma }^{\prime }-\alpha -1}{(1-vy)}^{\gamma +{\gamma }^{\prime }-\beta -1}{(1-ux-vy)}^{\alpha +\beta -\gamma -{\gamma }^{\prime }+1}}dudv,\end{array}\end{array}$$
$\mathrm{\Re }\gamma >\mathrm{\Re }\alpha >0$, $\mathrm{\Re }{\gamma }^{\prime }>\mathrm{\Re }\beta >0$.
ⓘ 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: TeX, pMML, png See also: Annotations for §16.15 and Ch.16

For these and other formulas, including double Mellin–Barnes integrals, see Erdélyi et al. (1953a, §5.8). These representations can be used to derive analytic continuations of the Appell functions, including convergent series expansions for large $x$, large $y$, or both. For inverse Laplace transforms of Appell functions see Prudnikov et al. (1992b, §3.40).