A functional identity involving elliptic integrals

Abstract

We show that the following double integral

$$\begin{aligned} \int _{0}^\pi \mathrm {d}\, x\int _0^x\mathrm {d}\, y\frac{1}{\sqrt{1-\smash [b]{p}\cos x}\sqrt{1+\smash [b]{q\cos y}}} \end{aligned}$$

remains invariant as one trades the parameters p and q for $p'=\sqrt{1-p^2}$ and $q'=\sqrt{1-q^2}$, respectively. This invariance property is suggested from symmetry considerations in the operating characteristics of a semiconductor Hall effect device.