Two integral representations for the logarithm of the Glaisher-Kinkelin constant

TL;DR

This paper introduces two new integral formulas for calculating the logarithm of the Glaisher-Kinkelin constant, simplifying numerical evaluation by leveraging known integral representations of the Gamma function.

Contribution

It provides novel integral representations of the Glaisher-Kinkelin constant's logarithm based on classical Gamma function integrals, enhancing computational efficiency.

Findings

The second integral representation is numerically easier to evaluate.

Both representations are derived from established Gamma function integrals.

The methods facilitate more efficient computation of the Glaisher-Kinkelin constant.

Abstract

We present two integral representations of the logarithm of the Glaisher-Kinkelin constant. Both are based on a definite integral of $\ln[\Gamma(x + 1)]$, $\Gamma$ being the usual Gamma function. The first one relies on an integral representation of $\ln[\Gamma(x + 1)]$ due to Binet, and the second one results from the so-called Malmst\'en formula. The numerical evaluation is easier with the latter expression than with the former.