From the Schaar and Lösch-Schoblick integrals to representations of the Glaisher-Kinkelin constant

TL;DR

This work derives two novel integral representations for the logarithm of the Glaisher–Kinkelin constant $A$ by exploiting two integral forms of the Binet function $μ(x)$. By applying Schaar's form $μ(x)=2x\int_0^{\infty}\frac{\arctan t}{e^{2\pi x t}-1}\,dt$ and, separately, Lösch–Schoblik's form $μ(x)=-\frac{x}{π}\int_0^{\infty}\frac{\log(1-e^{-2π t})}{t^{2}+x^{2}}\,dt$, and integrating over $x$ on $[0,1/2]$ with Fubini–Tonelli swaps, the authors obtain two explicit integral expressions for $\log A$. The first yields $\log A=\frac{\log 2}{9}+\frac{1}{24}+\frac{2}{3\pi}\int_0^{\infty}\frac{1}{e^{t}-1}\left[\frac{\pi}{4}-\frac{1}{2}\arctan\left(\frac{\pi}{t}\right)+\frac{t}{4\pi}\log\left(1+\frac{\pi^{2}}{t^{2}}\right)\right] dt$, and the second gives $\log A=\frac{1}{24}+\frac{\log 2}{9}-\frac{1}{3\pi}\int_0^{\infty}\log(1-e^{-2π t})\log\left(1+\frac{1}{4t^{2}}\right) dt$. These results connect classical Binet representations with new integral forms for $\log A$, offering alternative avenues for analyzing this constant and suggesting further work via derivatives of the Binet function.

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 $μ(x)$. The first one is attributed to Schaar (and also often referred to as ``the second Binet formula''), and the second one is due to Lösch 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.