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Representations of the Glaisher-Kinkelin constant deduced from integrals due to Féaux and Kummer

Jean-Christophe Pain

Abstract

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

Representations of the Glaisher-Kinkelin constant deduced from integrals due to Féaux and Kummer

Abstract

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

Paper Structure

This paper contains 4 sections, 28 equations.

Table of Contents

  1. Introduction
  2. Integral representation resulting from the Féaux formula
  3. Integral representation resulting from the Kummer formula
  4. Conclusion