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