When $a,b,c,d$ are complex numbers^{} and $z$ is a complex number
such that $$ and $C$ is a contour in the
complex $s$plane which goes from $i\mathrm{\infty}$ to $+i\mathrm{\infty}$
chosen such that the poles of $\mathrm{\Gamma}(a+s)\mathrm{\Gamma}(b+s)$ lie
to the left of $C$ and the poles of $\mathrm{\Gamma}(s)$ lie to the
right of $C$, then

$${\int}_{C}\frac{\mathrm{\Gamma}(a+s)\mathrm{\Gamma}(b+s)}{\mathrm{\Gamma}(c+s)}\mathrm{\Gamma}(s){(z)}^{s}\mathit{d}s=2\pi i\frac{\mathrm{\Gamma}(a)\mathrm{\Gamma}(b)}{\mathrm{\Gamma}(c)}F(a,b;c;z)$$ 
