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

Abstract

We derive integral representations in terms of the Macdonald functions for the square modulus $s\mapsto |\Gamma (a+is){|}^2$ of the Gamma function and its Fourier transform when $a<0$ and $a\ne -1,-2,\dots$, 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.