Mathematica has the annoying habit to provide solutions involving incomplete elliptic integrals of the first kind $E(z|m)$, second kind $F(z|m)$ and third kind $\Pi(n;z|m)$ where $z$, $m$ and $n$ are arbitrary complex numbers. However, some numerical interfaces only support classical definitions: $0<m<1$, $-\frac{\pi}{2}<z<\frac{\pi}{2}$ and $n\in \mathbb{R}$. Therefore, what is the simplest transformation formula to reach these easier-to-handle parameter and argument (or at least real values)?
I have consulted traditional formula lists such as P. F. Byrd et al., Handbook of Elliptic Integrals for Engineers and Scientists but it has been surprisingly difficult to find a way to deal with complex $m$ (while the case of negative $m$ is widely covered). As such, this question differs from this question or this other demand.
