For $\Re(b) > 0$, the Gauss hypergeometric function ${}_2F_1[a,b;c;z]$ can be represented by the integral $$ {}_2F_1[a,b;c;z] = {i \, \Gamma(c) \, e^{i\pi (b-c)} \over \Gamma(b) \Gamma(c-b) 2 \sin (\pi(c-b))} \int^{(1+)}_{0} t^{b-1} (1-t)^{c-b-1}(1-t z)^{-a} dt\,. $$ In all references I find, the only specifications about the integration path are that the point $1/z$ lies outside the integration contour, $t^{b-1}$ and $(1-t)^{c-b-1}$ assume their principal values where the contour cuts the interval $(1,\infty)$, and $(1-t z)^{-a} = 1$ at $t = 0$.
I'm having trouble understanding what the contour looks like. Is it supposed to be a closed one? I need to be able to transport this into a code to calculate numerically. Thanks!