# A Functional Identity involving Elliptic Integrals

## TL;DR

This paper demonstrates a symmetry property of a specific double integral involving elliptic integrals, showing its invariance under a parameter transformation inspired by semiconductor device symmetry.

## Contribution

It introduces a novel invariance property of a double integral involving elliptic integrals, linked to symmetry considerations in semiconductor physics.

## Key findings

- The double integral remains invariant under parameter transformation p to p' and q to q'
- The invariance is motivated by symmetry in Hall-effect device operations
- Provides new insights into elliptic integral identities

## Abstract

We show that the following double integral \[\int_{0}^\pi {\rm d}x \int_0^x {\rm d}y \frac{1}{\sqrt{1-\smash[b]{p}\cos x}\sqrt{1+\smash[b]{q\cos y}}}\]remains invariant as one trades 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 characterstics of a semiconductor Hall-effect device.