Integrals of $K$ and $E$ from Lattice Sums

TL;DR

This paper provides closed-form evaluations of integrals involving elliptic integrals K and E, leveraging connections with theta functions, lattice sums, and Eisenstein series, and introduces new high-dimensional lattice sum results.

Contribution

It introduces novel methods to evaluate integrals with elliptic integrals using advanced mathematical structures and presents new lattice sum evaluations, including in 10 dimensions.

Findings

Closed-form evaluations for integrals involving elliptic integrals K and E.

New lattice sum evaluations, including in 10 dimensions.

Connections established between elliptic integrals, theta functions, and Eisenstein series.

Abstract

We give closed form evaluations for many families of integrals, whose integrands contain algebraic functions of the complete elliptic integrals $K$ and $E$. Our methods exploit the rich structures connecting complete elliptic integrals, Jacobi theta functions, lattice sums, and Eisenstein series. Various examples are given, and along the way new (including 10-dimensional) lattice sum evaluations are produced.