When $a,b,c,d$ are complex numbers and $z$ is a complex number such that $-\pi < \arg (-z) < +\pi$ and $C$ is a contour in the complex $s$ plane which goes from $-i \infty$ to $+ i \infty$ chosen such that the poles of $\Gamma (a+s) \Gamma (b+s)$ lie to the left of $C$ and the poles of $\Gamma (-s)$ lie to the right of $C$ then $$ \int_C {\Gamma (a+s) \Gamma (b+s) \over \Gamma (c+s)} \Gamma (-s) (-z)^s \, ds = 2 \pi i {\Gamma (a) \Gamma (b) \over \Gamma (c)} F (a, b; c; z) $$