A two-parameter generalization of the complete elliptic integral of second kind

TL;DR

This paper introduces a two-parameter generalization of the complete elliptic integral of the second kind, expressed via Appell functions and reducible to simpler elliptic integrals, with brief mention of physical applications.

Contribution

It presents a novel two-parameter generalization of the elliptic integral of the second kind using Appell functions and simplifies it to basic elliptic integrals.

Findings

Generalization expressed in terms of Appell function F4

Reduction to bilinear form involving K and E

Potential applications in physics briefly discussed

Abstract

A two-parameter generalization of the complete elliptic integral of second kind is expressed in terms of the Appell function $F_{4}$. This function is further reduced to a quite simple bilinear form in the complete elliptic integrals $K$ and $E$. The physical applications are briefly mentioned.