The incomplete beta function $B_x(a,b)$ is defined for $x\in [0,1]$ by the integral $$B_x(a,b)=\int_0^x dt t^{a-1}(1-t)^{b-1}.\tag{1}$$
I'm interested in two aspects associated to that. First is the analytic continuation of that for $z\in \mathbb{C}$. As far as I know, this is answered by a relation to the hypergeometric function $$B_z(a,b) = \dfrac{z^a}{a}{_2F_1}(a,1-b;a+1;z)\tag{2}.$$
but where does this come from and for which $z\in\mathbb{C}$ is it valid? The second aspect is that after analytically continuing I'm interested in relating $B_{-z}$ to $B_z$. I expect there is some phase involved because (2) has branch cuts, first for defining $z^a=\exp a\log z$ and secondly as far as I recall the hypergeometric has a branch cut as well.
What is the correct way to take these branch cuts into consideration and evaluating $B_{-z}$ in terms of $B_z$? For example, if $x\in \mathbb{R}$ and we evaluate $B_{-x}$ using (2) we could write $-x=e^{\pm i\pi}|x|$ and depending on the sign the result seems different.