Generalization of the Glasser - Manna - Oloa integral and some new integrals of similar type

TL;DR

This paper generalizes the Glasser-Manna-Oloa integral, introduces a parametric family of similar integrals, and derives new integral representations related to the Hurwitz zeta function and gamma function logarithms.

Contribution

It provides a one-parameter generalization of a known integral and develops methods to find new integrals of similar type, expanding the analytical tools available.

Findings

Derived a parametric generalization of the Glasser-Manna-Oloa integral.

Obtained new integral representations of the Hurwitz zeta function.

Presented formulas involving the logarithm of the gamma function.

Abstract

As was shown in the previous works by other authors, Glasser - Manna - Oloa integral arise in the study of the Laplace transform of the dilogarithm function and can be evaluated in a closed form. In this article, we give a one parametric generalization of the Glasser - Manna - Oloa integral. The method employed in the course of derivation allows to obtain some new integrals of similar type. These include also integral representation of the Hurwitz zeta function and some beautiful formulae involving the logarithm of the gamma function of the argument -ix+ln(2cosx).