Closed-form expressions for Farhi's constant and related integrals and its generalization

Abstract

In a recent work, Farhi developed a Fourier series expansion for the function $\,\ln{Γ(x)}\,$ on the interval $(0,1)$, which allowed him to derive a nice formula for the constant $\,η:= 2 \int_0^1{\ln{Γ(x)} \, \sin{(2 πx)} \, dx}$. At the end of that paper, he asks whether $η$ could be written in terms of other known mathematical constants. Here in this work, after deriving a simple closed-form expression for $η$, I show how it can be used for evaluating other related integrals, as well as certain logarithmic series, which allows for a generalization in the form of a continuous function $η(x)$, $x \in [0,1]$. Finally, from the Fourier series expansion of $\,\ln{Γ(x)}$, $x \in (0,1)$, I make use of Parseval's theorem to derive a closed-form expression for $\,\int_0^1{\ln^2{Γ(x)}~dx}$.