Simple evaluation of one of Malmst\'en's integrals

TL;DR

This paper presents a simple real analysis method to evaluate Malmstén's logarithmic integral, utilizing geometric series and Kummer's Fourier expansion, offering an accessible alternative to complex contour techniques.

Contribution

It introduces a straightforward real analysis approach to evaluate a classical logarithmic integral, simplifying previous complex contour methods.

Findings

The integral is evaluated explicitly using real analysis techniques.

Geometric series and Kummer's Fourier expansion are key tools.

The method provides an accessible alternative to contour integration.

Abstract

The logarithmic integral no. 4.325.7 from Gradshteyn and Ryzhik's tables of integrals was first evaluated by Malmst\'en. Recently, Blagouchine used contour integration methods to evaluate a family of logarithmic integrals that contains this integral. We evaluate the integral in a simple, straightforward manner mainly by means of real analysis. The main ingredients of the evaluation are the use of geometric series and Kummer's Fourier series expansion for the logarithm of the gamma function.