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