Multiple integral representations of the Catalan's constant

TL;DR

This paper introduces new integral representations of Catalan's constant, including double, single, and higher-dimensional integrals involving the Lerch function, supported by illustrative examples.

Contribution

It provides novel integral formulas for Catalan's constant, expanding the mathematical tools available for its analysis and computation.

Findings

Derived a double integral representation of Catalan's constant.

Established a general theorem for single integral representation.

Presented a multiple integral representation in higher dimensions using the Lerch function.

Abstract

In this paper, we present several novel integral representations of Catalan's constant. We begin by deriving an initial result expressed as a double integral. Subsequently, as a consequence of this result, we establish a general theorem that enables the representation of Catalan's constant in terms of a single integral. Finally, we provide a multiple integral representation of Catalan's constant in dimensions greater than or equal to two using the Lerch function. The results are accompanied by illustrative examples.