Is there any way to re-write or simplify this function for $x,y\in\mathbb{R}$, in the limit $x\rightarrow\pm\infty$? Or any laws regarding symmetry with respect to $x\rightarrow -x$ or $y\rightarrow -y$?
The reason for asking this question is that I have a lengthy term that contains a sum of all kinds of combinations like $B(ix,2\pm iy,0)\pm B(\pm ix,2\pm iy,0)$. All relations that I found for the incomplete Beta function are invalid for the last argument being zero.
My hypothesis is that
$\lim_{x\rightarrow\infty}\frac{-B(-ix,2+iy,0)}{e^{\pi y}B(ix,2+iy,0))}=1$
but I'm struggling with the proof.