Euler integral for the hypergeometric function
= Euler integral for the hypergeometric function
{c}
For $\Re c>\Re b>0$ and a fixed branch away from the cut,
$$
{}_2F_1(a,b;c;z)=\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}\int_0^1s^{b-1}(1-s)^{c-b-1}(1-zs)^{-a}\,ds.
$$
Expand $(1-zs)^{-a}$ for $|z|<1$ and integrate termwise using the <beta function> to recover the <Gauss hypergeometric function> series. <Analytic continuation> extends the identity in $z$.