On the Laplace transform of the Fréchet distribution

Abstract

We calculate exactly the Laplace transform of the Fréchet distribution in the form $\gamma x^{-(1+\gamma )}\exp (-x^{-\gamma })$, $\gamma >0$, $0\le x<\infty$, for arbitrary rational values of the shape parameter $\gamma$, i.e. for $\gamma =l/k$ with $l,k=1,2,\dots$. The method employs the inverse Mellin transform. The closed form expressions are obtained in terms of Meijer G functions and their graphical illustrations are provided. A rescaled Fréchet distribution serves as a kernel of Fréchet integral transform. It turns out that the Fréchet transform of one-sided Lévy law reproduces the Fréchet distribution.

10 pages, 4 figures; one reference added

References in corpus (1)

• Exact and explicit probability densities for one-sided Levy stable distributions