I have been using the imaginary-modulus transformation for elliptic integrals of the second kind $E(\phi,k)$ and I would love to know how it can be derived. This transformation has already been discussed in a number of previous questions (remarkably here), but I have not been able to find a reference where a derivation or proof was provided. Part of the problem might be insufficient mathematical knowledge on my part: I keep finding books where transformations with the same name are explained for elliptic functions, but I do not understand the details and I am not able to determine whether there is an obvious connection to the specific formula I am using:
$E(\phi,ik)=\frac{1}{\kappa'}\Big[E(\theta,\kappa)-\frac{\kappa^2sin(\theta)cos(\theta)}{\sqrt{1-\kappa^2sin^2(\theta)}}\Big]$
I would very much appreciate if anyone could give me any hints on how to derive that formula, even if it is just a rough indication of where it comes from, which might be complex math that I am not familiar with.