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We derive Mellin integral representations in terms of Macdonald functions for the squared absolute values  ↦    |   Γ  ⁢   (   a  +   i  ⁢  s    )    |   2{s\mapsto|\Gamma(a+is)|^{2}}    of the gamma function and its Fourier transform whena  &lt;  0{a&lt;0}    is non-integer, generalizing known results in the casea  &gt;  0{a&gt;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.

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