Representations of the Glaisher-Kinkelin constant deduced from integrals due to F\'eaux and Kummer

TL;DR

This paper derives two integral formulas for the logarithm of the Glaisher-Kinkelin constant using classical integral representations of the Gamma function, linking them to series expansions.

Contribution

It introduces new integral representations of the Glaisher-Kinkelin constant based on Féaux and Kummer's formulas, connecting integrals to series expansions.

Findings

Two integral representations of log Glaisher-Kinkelin constant derived

Connection established between integrals and series expansion of the constant

Provides analytical tools for studying the constant's properties

Abstract

We present two integral representations of the logarithm of the Glaisher-Kinkelin constant. The calculations are based on definite integral expressions of $\log\Gamma(x)$, $\Gamma$ being the usual Gamma function, due respectively to F\'eaux and Kummer. The connection between the second formula and an infinite series expansion of the Glaisher-Kinkelin constant is also outlined.