# §16.15 Integral Representations and Integrals

 16.15.1 $\mathop{{F_{1}}\/}\nolimits\!\left(\alpha;\beta,\beta^{\prime};\gamma;x,y% \right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(\gamma\right)}{\mathop{\Gamma% \/}\nolimits\!\left(\alpha\right)\mathop{\Gamma\/}\nolimits\!\left(\gamma-% \alpha\right)}\int_{0}^{1}\frac{u^{\alpha-1}(1-u)^{\gamma-\alpha-1}}{(1-ux)^{% \beta}(1-uy)^{\beta^{\prime}}}du,$ $\realpart{\alpha}>0$, $\realpart{(\gamma-\alpha)}>0$,
 16.15.2 $\mathop{{F_{2}}\/}\nolimits\!\left(\alpha;\beta,\beta^{\prime};\gamma,\gamma^{% \prime};x,y\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(\gamma\right)% \mathop{\Gamma\/}\nolimits\!\left(\gamma^{\prime}\right)}{\mathop{\Gamma\/}% \nolimits\!\left(\beta\right)\mathop{\Gamma\/}\nolimits\!\left(\beta^{\prime}% \right)\mathop{\Gamma\/}\nolimits\!\left(\gamma-\beta\right)\mathop{\Gamma\/}% \nolimits\!\left(\gamma^{\prime}-\beta^{\prime}\right)}\int_{0}^{1}\!\!\!\int_% {0}^{1}\frac{u^{\beta-1}v^{\beta^{\prime}-1}(1-u)^{\gamma-\beta-1}(1-v)^{% \gamma^{\prime}-\beta^{\prime}-1}}{(1-ux-vy)^{\alpha}}dudv,$ $\realpart{\gamma}>\realpart{\beta}>0$, $\realpart{\gamma^{\prime}}>\realpart{\beta^{\prime}}>0$,
 16.15.3 $\mathop{{F_{3}}\/}\nolimits\!\left(\alpha,\alpha^{\prime};\beta,\beta^{\prime}% ;\gamma;x,y\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(\gamma\right)}{% \mathop{\Gamma\/}\nolimits\!\left(\beta\right)\mathop{\Gamma\/}\nolimits\!% \left(\beta^{\prime}\right)\mathop{\Gamma\/}\nolimits\!\left(\gamma-\beta-% \beta^{\prime}\right)}\iint_{\Delta}\frac{u^{\beta-1}v^{\beta^{\prime}-1}(1-u-% v)^{\gamma-\beta-\beta^{\prime}-1}}{(1-ux)^{\alpha}(1-vy)^{\alpha^{\prime}}}dudv,$ $\realpart{(\gamma-\beta-\beta^{\prime})}>0$, $\realpart{\beta}>0$, $\realpart{\beta^{\prime}}>0$,

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

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

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).