Extended Mellin integral representations for the absolute value of the gamma function.

We derive Mellin integral representations in terms of Macdonald functions for the squared absolute value s ↦ | Γ ⁢ ( a + i ⁢ s ) | 2 {s\mapsto|\Gamma(a+is)|^{2}} of the gamma function and its Fourier transform when a < 0 {a<0} is non-integer, generalizing known results in the case a > 0 {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 a Fokker–Planck equation. [ABSTRACT FROM AUTHOR]