A Quadruple Definite Integral Expressed in Terms of the Lerch Function

Robert Reynolds,Allan Stauffer

doi:10.3390/sym13091638

Abstract

A quadruple integral involving the logarithmic, exponential and polynomial functions is derived in terms of the Lerch function. Special cases of this integral are evaluated in terms of special functions and fundamental constants. Almost all Lerch functions have an asymmetrical zero-distribution. The majority of the results in this work are new.

Highlights

In this paper we derive the quadruple definite integral given by ∞∞∞∞(t + z)−m(x + y)m−1e−p(x+z)−q(t+y) logk a(x + y) t+z dxdydzdt (1)where the parameters k, a, p, q and m are general complex numbers
In this present work we provide a formal derivation for a quadruple integral not present in the current literature
In this work our goal is to expand upon the current literature of definite quadruple integrals by providing a formal derivation in terms of the Lerch function

Summary

Significance Statement

Quadruple definite integrals are widely used in a vast number of areas spanning mathematics and physics, from integrating over a four-dimensional volume, integrating over a Lagrangian density in field theory and four-dimensional Fourier transforms of a function of spacetime (x, y, z, t). After perusing the current literature, the authors found many applications of quadruple integrals In some cases these integrals were separable and in some cases asymptotic expansions were used to attain a solution. To the best of our knowledge the authors were unable to find quadruple definite integrals involving the logarithmic, exponential and polynomial functions derived in terms of a closed form solution. In this present work we provide a formal derivation for a quadruple integral not present in the current literature. This integral features a kernel with the product of the logarithmic, exponential and polynomial functions. In this work our goal is to expand upon the current literature of definite quadruple integrals by providing a formal derivation in terms of the Lerch function

Introduction

Definite Integral of the Contour Integral

The Lerch Function The Lerch function has a series representation given by

Discussion