\[\mathbf{F}\left(a,b;c;z\right)=\frac{1}{2\pi\mathrm{i}\Gamma\left(a\right)%
\Gamma\left(b\right)\Gamma\left(c-a\right)\Gamma\left(c-b\right)}\int_{-%
\mathrm{i}\infty}^{\mathrm{i}\infty}\Gamma\left(a+t\right)\Gamma\left(b+t%
\right)\Gamma\left(c-a-b-t\right)\Gamma\left(-t\right)(1-z)^{t}\,\mathrm{d}t,\]