On the integral representations of |Γ(z)|² and its Fourier transform

2014/10/19 by Nicolas Privault, Privault, Nicolas
Mathematics · #Advanced Mathematical Identities #Fractional Differential Equations Solutions #Mathematical functions and polynomials #math.CA

paper · pdf · doi:10.48550/arxiv.1410.5043

arxiv created 2014/10/19 · arxiv updated 2014/10/21

Abstract

We derive integral representations in terms of the Macdonald functions for the square modulus s↦ | Γ( a + i s ) |² of the Gamma function and its Fourier transform when a<0 and a\not= -1,-2,… , 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.

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