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

## TL;DR

This paper derives closed-form expressions for Farhi's constant and related integrals involving the gamma function, providing new formulas and a generalization for evaluating complex integrals and series.

## Contribution

It presents a simple closed-form for Farhi's constant and introduces a generalized function for related integrals, expanding the analytical tools for gamma function analysis.

## Key findings

- Closed-form expression for Farhi's constant ta
- Evaluation of related integrals and series using derived formulas
- Generalization of ta(x) for continuous parameterization

## Abstract

In a recent work, Farhi developed a Fourier series expansion for the function $\,\ln{\Gamma(x)}\,$ on the interval $(0,1)$, which allowed him to derive a nice formula for the constant $\,\eta := 2 \int_0^1{\ln{\Gamma(x)} \, \sin{(2 \pi x)} \, dx}$. At the end of that paper, he asks whether $\eta$ could be written in terms of other known mathematical constants. Here in this work, after deriving a simple closed-form expression for $\eta$, 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 $\eta(x)$, $x \in [0,1]$. Finally, from the Fourier series expansion of $\,\ln{\Gamma(x)}$, $x \in (0,1)$, I make use of Parseval's theorem to derive a closed-form expression for $\,\int_0^1{\ln^2{\Gamma(x)}~dx}$.