For Hypergeometric function ${}_2F_1(a_1,a_2;b_1;z)$, if $a_1$, $a_2$ are negative integers, it will be terminated and convergence is not a problem. Under this circumstances, does anyone know the integral representation of the Hypergeometric function? In particular, what if $b_1=\frac{1}{2}$.The only integral representation I searched out applies uniquely to $Re(b_1)>Re(a_2)>0$: \begin{equation} {}_2F_1(a_1,a_2;b_1;z)=\frac{\Gamma(b_1)}{\Gamma(a_2)\Gamma(b_1-a_2)}\int_0^1dt\, t^{a_2-1}(1-t)^{b_1-a_2-1}(1-zt)^{-a_1} \end{equation}
But it doesn't apply to the negative integers when $b_1=1/2$.