From the Schaar and L\"osch-Schoblick integrals to representations of the Glaisher-Kinkelin constant

TL;DR

This paper introduces two novel integral representations of the Glaisher-Kinkelin constant's logarithm based on different formulations of the Binet function, providing new insights into its mathematical structure.

Contribution

It presents two new integral formulas for the Glaisher-Kinkelin constant, expanding the understanding of its representations beyond known formulas.

Findings

Derived two new integral representations of the Glaisher-Kinkelin constant.

Connected the representations to different formulations of the Binet function.

Showed that these new formulas are not easily deducible from existing ones.

Abstract

In this article, we present two integral representations of the logarithm of the Glaisher-Kinkelin constant, relying on two different integral formulations of the so-called Binet function $\mu(x)$. The first one is attributed to Schaar (and also often referred to as ``the second Binet formula''), and the second one is due to L\"osch and Schoblik. It seems that the two new expressions (formulas (28) and (33) of the present article) of the Glaisher-Kinkelin constant, can not be easily deduced from know ones.