On the integral representations of $|\Gamma (z)|^2$ and its Fourier transform

TL;DR

This paper derives new integral representations for the squared modulus of the Gamma function and its Fourier transform using Macdonald functions, extending known results to negative parameters and applications to the Fokker-Planck equation.

Contribution

It introduces generalized integral representations for |Γ(a+is)|^2 and its Fourier transform for negative a, expanding the mathematical understanding of these functions.

Findings

Derived integral representations using Macdonald functions for negative a

Extended known results from positive to negative a values

Applied the representations to solutions of the Fokker-Planck equation

Abstract

We derive integral representations in terms of the Macdonald functions for the square modulus $s\mapsto | \Gamma ( a + i s ) |^2$ of the Gamma function and its Fourier transform when $a<0$ and $a\not= -1,-2,\ldots $, generalizing known results in the case $a>0$. This representation is based on a renormalization argument using modified Bessel functions of the second kind, and it applies to the representation of the solutions of the Fokker-Planck equation.