What is $E(\phi \mid m)$ for $|\phi| \ge \pi/2$?

Question

According to details about the elliptic integral of the second kind given by the Wolfram Language & System Documentation Center:

$$\mathrm{For}\;-\pi/2<\phi<\pi/2,\,E(\phi\mid m)=\int_0^\phi{\left(1-m\sin^2(\theta)\right)^{1/2}}d\theta$$

My question is simple. How is $E(\phi\mid m)$ defined for $|\phi|\ge\pi/2$?

Answer 1

For other $\phi$ you can keep the same definition. But of course your integrand may involve the square root of a negative number, so the result may be complex, not real. And, of course, you need to choose a branch somehow.

And I would say the switch-over value is not $\phi = \pi/2$ but $\phi = \arcsin(1/\sqrt{m})$