Two-dimensional Fourier transformations and double Mordell integrals

TL;DR

This paper evaluates Fourier transforms of functions in one and two variables to derive integral identities, reduces double Mordell integrals to products of single integrals, and uncovers new connections to elliptic functions and lattice sums.

Contribution

It introduces a method to reduce double Mordell integrals to products of one-dimensional integrals and finds new identities linking these integrals with quadratic polynomials.

Findings

Double Mordell integrals can be expressed as sums of products of one-dimensional Mordell integrals.

A new quadratic polynomial identity connecting Mordell integrals is established.

An integral similar to Ramanujan's and Glasser's is explicitly calculated.

Abstract

Fourier transformations of several functions of one and two variables are evaluated and then used to derive some integral and series identities. It is shown that certain double Mordell integrals can be reduced to a sum of products of one-dimensional Mordell integrals. As a consequence of this reduction, a quadratic polynomial identity is found connecting products of certain one-dimensional Mordell integrals. An integral that depends on one real valued parameter is calculated reminiscent of an integral previously calculated by Ramanujan and Glasser. Some connections to elliptic functions and lattice sums are discussed.