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

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