Integral representation of a hypergeometric

Question

I need to be able to express the hypergeometric function $$ {}_2F_{1}[k,1;-c;-z] $$ as an integral which can be computed numerically, where $k$ is an integer, $c$ is a real number and $z$ is real with $z>1$. With this restrictions for the arguments, the only option I found was $$ {}_2F_1[a,b;c;z] = {i \, \Gamma(c) \, e^{i\pi (b-c)} \over \Gamma(b) \Gamma(c-b) 2 \sin (\pi(c-b))} \int^{(1+)}_{0} t^{b-1} (1-t)^{c-b-1}(1-t z)^{-a} dt\,, $$ where the contour goes from $t=0$ close to $t=1$, circles $t=1$ counter-clockwise and goes back to $t=0$. I've calculated this integral but is still diverges. Is there any other option? Thanks.