# §16.15 Integral Representations and Integrals

 16.15.1 $\mathop{{F_{1}}\/}\nolimits\!\left(\alpha;\beta,\beta^{\prime};\gamma;x,y% \right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(\gamma\right)}{\mathop{\Gamma% \/}\nolimits\!\left(\alpha\right)\mathop{\Gamma\/}\nolimits\!\left(\gamma-% \alpha\right)}\int_{0}^{1}\frac{u^{\alpha-1}(1-u)^{\gamma-\alpha-1}}{(1-ux)^{% \beta}(1-uy)^{\beta^{\prime}}}\mathrm{d}u,$ $\Re{\alpha}>0$, $\Re{(\gamma-\alpha)}>0$,
 16.15.2 $\mathop{{F_{2}}\/}\nolimits\!\left(\alpha;\beta,\beta^{\prime};\gamma,\gamma^{% \prime};x,y\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(\gamma\right)% \mathop{\Gamma\/}\nolimits\!\left(\gamma^{\prime}\right)}{\mathop{\Gamma\/}% \nolimits\!\left(\beta\right)\mathop{\Gamma\/}\nolimits\!\left(\beta^{\prime}% \right)\mathop{\Gamma\/}\nolimits\!\left(\gamma-\beta\right)\mathop{\Gamma\/}% \nolimits\!\left(\gamma^{\prime}-\beta^{\prime}\right)}\int_{0}^{1}\!\!\!\int_% {0}^{1}\frac{u^{\beta-1}v^{\beta^{\prime}-1}(1-u)^{\gamma-\beta-1}(1-v)^{% \gamma^{\prime}-\beta^{\prime}-1}}{(1-ux-vy)^{\alpha}}\mathrm{d}u\mathrm{d}v,$ $\Re{\gamma}>\Re{\beta}>0$, $\Re{\gamma^{\prime}}>\Re{\beta^{\prime}}>0$,
 16.15.3 $\mathop{{F_{3}}\/}\nolimits\!\left(\alpha,\alpha^{\prime};\beta,\beta^{\prime}% ;\gamma;x,y\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(\gamma\right)}{% \mathop{\Gamma\/}\nolimits\!\left(\beta\right)\mathop{\Gamma\/}\nolimits\!% \left(\beta^{\prime}\right)\mathop{\Gamma\/}\nolimits\!\left(\gamma-\beta-% \beta^{\prime}\right)}\iint_{\Delta}\frac{u^{\beta-1}v^{\beta^{\prime}-1}(1-u-% v)^{\gamma-\beta-\beta^{\prime}-1}}{(1-ux)^{\alpha}(1-vy)^{\alpha^{\prime}}}% \mathrm{d}u\mathrm{d}v,$ $\Re{(\gamma-\beta-\beta^{\prime})}>0$, $\Re{\beta}>0$, $\Re{\beta^{\prime}}>0$, Symbols: $\mathop{{F_{3}}\/}\nolimits\!\left(\NVar{\alpha},\NVar{\alpha^{\prime}};\NVar{% \beta},\NVar{\beta^{\prime}};\NVar{\gamma};\NVar{x},\NVar{y}\right)$: third Appell function, $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\mathrm{d}\NVar{x}$: differential of $x$ and $\Re{}$: real part Notes: Erdélyi et al. (1953a, 5.8.1(3)) contains a typo. Referenced by: Other Changes 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. Reported 2016-10-08 See also: Annotations for 16.15

where $\Delta$ is the triangle defined by $u\geq 0$, $v\geq 0$, $u+v\leq 1$.

 16.15.4 $\mathop{{F_{4}}\/}\nolimits\!\left(\alpha,\beta;\gamma,\gamma^{\prime};x(1-y),% y(1-x)\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(\gamma\right)\mathop{% \Gamma\/}\nolimits\!\left(\gamma^{\prime}\right)}{\mathop{\Gamma\/}\nolimits\!% \left(\alpha\right)\mathop{\Gamma\/}\nolimits\!\left(\beta\right)\mathop{% \Gamma\/}\nolimits\!\left(\gamma-\alpha\right)\mathop{\Gamma\/}\nolimits\!% \left(\gamma^{\prime}-\beta\right)}\int_{0}^{1}\!\!\!\int_{0}^{1}\frac{u^{% \alpha-1}v^{\beta-1}(1-u)^{\gamma-\alpha-1}(1-v)^{\gamma^{\prime}-\beta-1}}{(1% -ux)^{\gamma+\gamma^{\prime}-\alpha-1}(1-vy)^{\gamma+\gamma^{\prime}-\beta-1}(% 1-ux-vy)^{\alpha+\beta-\gamma-\gamma^{\prime}+1}}\mathrm{d}u\mathrm{d}v,$ $\Re{\gamma}>\Re{\alpha}>0$, $\Re{\gamma^{\prime}}>\Re{\beta}>0$.

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).