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 formula for the constant η := 2∫₀¹ ln Γ(x) sin(2πx) dx. At the end of that paper, he asks whether η could be written in terms of other known mathematical constants. Here, after deriving a simple closed-form expression for η, we 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 ∈ [0, 1]. Finally, from the Fourier series expansion of ln Γ(x), x ∈ (0, 1), we make use of Parseval's theorem to derive a closed-form expression for ∫₀¹ ln²Γ(x) dx.