When is the value of elliptic integral complex?

Question

Are there conditions on $\phi$ and $k$ which determines if the value of elliptic integrals $F(\left.\phi\right| k)$ and $E(\left.\phi\right| k)$ is real or complex?

Answer 1

The integrals are real if $k\le1$ or $|\phi|\le\arcsin\frac1{\sqrt{k}}$, assuming that $k$ and $\phi$ are real. Otherwise the integrals are complex.

This boils down to the fact that $\sqrt{1-k\sin^2\phi}$ is real if and only if $1-k\sin^2\phi\ge0.$