I need to numerically compute the hypergeometric function
$$ _2F_1(k,1,c,z) $$
where $k$ is an integer, $c>2$ is a real number and $|z|<1$, using the integral representation
$$ {}_2F_1[a,b;c;z] = {i \, \Gamma(c) \, e^{i\pi (b-c)} \over \Gamma(b) \Gamma(c-b) 2 \sin (\pi(c-b))} \int^{(1+)}_{0} t^{b-1} (1-t)^{c-b-1}(1-t z)^{-a} dt\,, $$
where the contour goes from $t=0$ close to $t=1$ (call it C1), circles $t=1$ counter-clockwise with radius $\rho$ (call it C2) and goes back to $t=0$ (call it C3). We have then
$$ {}_2F_1[k,1;c;z] = {(1-c) e^{-i\pi c} \over 2 i \sin (\pi c)} \int^{(1+)}_{0} (1-t)^{c-2}(1-t z)^{-k} dt\,. $$
The integral in C3 can be shown to be 0, so what's left is C1 and C2. We have $t_- = t-i \delta$ and $t_+ +i \delta$ in C1 and C2, respectively, so the result should be
$$ {}_2F_1[k,1;c;z] = {(1-c) e^{-i\pi c} \over 2 i \sin (\pi c)} \int^{1-\rho}_{0} \left[(1-t_-)^{c-2}(1-t_- z)^{-k} - (1-t_+)^{c-2}(1-t_+ z)^{-k}\right] dt\,. $$
An NIntegrate with small $\delta$ and $\rho$ should do the trick, but it only gives something very close to 0 as $\delta$ gets smaller, and I can check using Hypergeometric2F1 that the result is quite finite. What is wrong in my logic?
