§16.15 Integral Representations and Integrals

Keywords:: Appell functions, analytic continuation, integrals
Permalink:: http://dlmf.nist.gov/16.15

16.15.1 $\mathop{{F_{{1}}}\/}\nolimits\!\left(\alpha;\beta,\beta^{{\prime}};\gamma;x,y\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(\gamma\right)}{\mathop{\Gamma\/}\nolimits\!\left(\alpha\right)\mathop{\Gamma\/}\nolimits\!\left(\gamma-\alpha\right)}\int _{0}^{1}\frac{u^{{\alpha-1}}(1-u)^{{\gamma-\alpha-1}}}{(1-ux)^{{\beta}}(1-uy)^{{\beta^{{\prime}}}}}du,$ $\realpart{\alpha}>0$ , $\realpart{(\gamma-\alpha)}>0$ ,

Symbols:: $\mathop{{F_{{1}}}\/}\nolimits\!\left(\alpha;\beta,\beta^{{\prime}};\gamma;x,y\right)$ : Appell function, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral and $\realpart{}$ : real part
Permalink:: http://dlmf.nist.gov/16.15.E1
Encodings:: TeX, pMML, png

16.15.2 $\mathop{{F_{{2}}}\/}\nolimits\!\left(\alpha;\beta,\beta^{{\prime}};\gamma,\gamma^{{\prime}};x,y\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(\gamma\right)\mathop{\Gamma\/}\nolimits\!\left(\gamma^{{\prime}}\right)}{\mathop{\Gamma\/}\nolimits\!\left(\beta\right)\mathop{\Gamma\/}\nolimits\!\left(\beta^{{\prime}}\right)\mathop{\Gamma\/}\nolimits\!\left(\gamma-\beta\right)\mathop{\Gamma\/}\nolimits\!\left(\gamma^{{\prime}}-\beta^{{\prime}}\right)}\int _{0}^{1}\!\!\!\int _{0}^{1}\frac{u^{{\beta-1}}v^{{\beta^{{\prime}}-1}}(1-u)^{{\gamma-\beta-1}}(1-v)^{{\gamma^{{\prime}}-\beta^{{\prime}}-1}}}{(1-ux-vy)^{{\alpha}}}dudv,$ $\realpart{\gamma}>\realpart{\beta}>0$ , $\realpart{\gamma^{{\prime}}}>\realpart{\beta^{{\prime}}}>0$ ,

Symbols:: $\mathop{{F_{{2}}}\/}\nolimits\!\left(\alpha;\beta,\beta^{{\prime}};\gamma,\gamma^{{\prime}};x,y\right)$ : Appell function, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral and $\realpart{}$ : real part
Permalink:: http://dlmf.nist.gov/16.15.E2
Encodings:: TeX, pMML, png

16.15.3 $\mathop{{F_{{3}}}\/}\nolimits\!\left(\alpha,\alpha^{{\prime}};\beta,\beta^{{\prime}};\gamma;x,y\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(\gamma\right)}{\mathop{\Gamma\/}\nolimits\!\left(\beta\right)\mathop{\Gamma\/}\nolimits\!\left(\beta^{{\prime}}\right)\mathop{\Gamma\/}\nolimits\!\left(\gamma-\beta-\beta^{{\prime}}\right)}\iint _{{\Delta}}\frac{u^{{\beta-1}}v^{{\beta^{{\prime}}-1}}(1-u-v)^{{\gamma-\beta-\beta^{{\prime}}-1}}}{(1-ux)^{{\alpha}}(1-vy)^{{\alpha^{{\prime}}}}}dudv,$ $\realpart{(\gamma-\beta-\beta^{{\prime}})}>0$ , $\realpart{\beta}>0$ , $\realpart{\beta^{{\prime}}}>0$ ,

Symbols:: $\mathop{{F_{{3}}}\/}\nolimits\!\left(\alpha,\alpha^{{\prime}};\beta,\beta^{{\prime}};\gamma;x,y\right)$ : Appell function, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ and $\realpart{}$ : real part
Permalink:: http://dlmf.nist.gov/16.15.E3
Encodings:: TeX, pMML, png

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

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

Symbols:: $\mathop{{F_{{4}}}\/}\nolimits\!\left(\alpha;\beta;\gamma,\gamma^{{\prime}};x,y\right)$ : Appell function, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral and $\realpart{}$ : real part
Permalink:: http://dlmf.nist.gov/16.15.E4
Encodings:: TeX, pMML, png

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).