Double integral of logarithm and exponential function expressed in terms of the Lerch function

TL;DR

This paper derives a new double integral involving logarithm and exponential functions, expressing it via the Lerch function using contour integrals, and provides a useful table of integral pairs for future reference.

Contribution

It introduces a novel contour integral method to relate double integrals to special functions, expanding the set of known integral identities.

Findings

Derived a double integral connected to the Bessel function

Expressed the integral in terms of the Lerch function

Provided a new table of integral pairs

Abstract

In this work the authors use their contour integral method to derive a double integral connected to the modified Bessel function of the second kind and express it in terms of the Lerch function. There are some useful results relating double integrals of certain kinds of functions to ordinary integrals for which we know no general reference. Thus a table of integral pairs is given for interested readers. The majority of the results in this work are new.