Two integral representations for the logarithm 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.