Abstract

A closed form expression for a triple integral not previously considered is derived, in terms of the Lerch function. Almost all Lerch functions have an asymmetrical zero-distribution. The kernel of the integral involves the product of the logarithmic, exponential, quotient radical, and polynomial functions. Special cases are derived in terms of fundamental constants; results are summarized in a table. All results in this work are new.

Highlights

  • In this work, a general integral theorem was developed, from which integral and closed form results, in terms of the Lerch function, Hurwitz zeta function, polylogarithm function, and the Riemann zeta function could be expressed

  • Our present work aims to supply a formal derivation and a specific triple integral, and express it in terms of a special function

  • The definite integral obtained in this present work is given by

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Summary

Significance Statement

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Some useful applications of these integrals are in finding the the mass of a ball of radius, say (r) whose density (ρ) is proportional to the squared distance from the center; another application is in finding the moment of inertia of a right circular homogeneous cone about its axis. These integrals are used to find the mass of a planet, where its radius and density are expressed in terms of a formula. These results will provide a new process and formula for current work requiring such results

Introduction
Definite Integral of the Contour Integral
The Lerch Function
Infinite Sum of the Contour Integral
Derivation of Table Entries
Discussion

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