An introduction to the Barnes double gamma function with an application to an integral involving the cotangent function

TL;DR

This paper introduces the Barnes double gamma function and demonstrates its application to evaluate Kinkelin's integral involving the cotangent function using a direct approach.

Contribution

It provides a novel application of the Barnes double gamma function to evaluate a classical integral involving cotangent.

Findings

Derived a new evaluation of Kinkelin's integral

Connected Barnes double gamma function with cotangent integrals

Presented a direct method for integral evaluation

Abstract

We show a direct approach to Kinkelin's integral involving the cotangent function.