Consider the complex gamma function, denoted by $\Gamma(\sigma+it)$.

Now, let's fix $\sigma$ and let t vary. Then consider the following expression:


For any choice of $\sigma$ such that $\Gamma(\sigma)$ isn't a pole, this will appear to be (almost) a two-sided probability density function, save for that it isn't normalized. It decays around as quickly as $e^{-|t|}$, somewhat resembles the function $e^{-\sqrt{1+t^2}}$, and appears related to the hyperbolic distribution.

  1. Is there a precise closed-form expression for this function in terms of elementary or Louvillian functions?

  2. Is there a name for this probability distribution (assuming it's normalized)?

  • $\begingroup$ Maybe this question is better suited for math overflow? $\endgroup$
    – Ryan
    Sep 30, 2015 at 19:00


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