Expression of Farhi's integral in terms of known mathematical constants

TL;DR

This paper derives a closed-form expression for a specific integral involving the gamma function and sine, expressing it in terms of well-known constants, and provides two independent proofs of this formula.

Contribution

It presents a new explicit formula for Farhi's integral in terms of fundamental constants and offers two different proofs, enriching the understanding of gamma function integrals.

Findings

The integral equals (γ + log(2π))/π, where γ is Euler-Mascheroni constant.

Two proofs are provided: one using the Glaisher-Kinkelin constant, another via Malmstén's integral.

The formula can be derived from the Fourier series expansion of log Gamma.

Abstract

In an interesting article entitled "A curious formula related to the Euler Gamma function", Bakir Farhi posed the open question of whether it was possible to obtain an expression of $$ \boldsymbol{\eta}=2\int_0^1\log\Gamma(x)\,\cdot\sin(2\pi x)\,\mathrm{d}x=0.7687478924\dots $$ in terms of the known mathematical constants as $\pi$, $\log\pi$, $\log 2$, $\Gamma(1/4)$, $e$, etc. In the present work, we show that $$ \boldsymbol{\eta}=\frac{1}{\pi}\left[\gamma+\log(2\pi)\right], $$ where $\gamma$ is the usual Euler-Mascheroni constant, and provide two different proofs, the first one involving the Glaisher-Kinkelin constant, and the second one based on the Malmst\'en integral representation of $\log\Gamma(x)$. The resulting formula can also be obtained directly from the knowledge of the Fourier series expansion of $\log\Gamma(x)$.