\[\mathbf{F}\left(a,b;c;z\right)={\mathrm{e}}^{-b\pi\mathrm{i}}\frac{\Gamma\left%
(1-b\right)}{2\pi\mathrm{i}\Gamma\left(c-b\right)}\int_{1}^{(0+)}\frac{t^{b-1}%
(1-t)^{c-b-1}}{(1-zt)^{a}}\,\mathrm{d}t,\]