Barnes’ integral representation of the hypergeometric function

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

$${\int }_C\frac{\mathrm{\Gamma }(a+s)\mathrm{\Gamma }(b+s)}{\mathrm{\Gamma }(c+s)}\mathrm{\Gamma }(-s){(-z)}^s𝑑s=2\pi i\frac{\mathrm{\Gamma }(a)\mathrm{\Gamma }(b)}{\mathrm{\Gamma }(c)}F(a,b;c;z)$$