A Functional Identity involving Elliptic Integrals

M. L. Glasser, Yajun Zhou

We show that the following double integral \[\int_{0}^π{\rm d}x \int_0^x {\rm d}y \frac{1}{\sqrt{1-\smash[b]{p}\cos x}\sqrt{1+\smash[b]{q\cos y}}}\]remains invariant as one trades…