On the Laplace transform of the Fréchet distribution

Abstract

We calculate exactly the Laplace transform of the Fréchet distribution in the form $γx^{-(1+γ)} \exp(-x^{-γ})$, $γ> 0$, $0 \leq x < \infty$, for arbitrary rational values of the shape parameter $γ$, i.e. for $γ= l/k$ with $l, k = 1,2, \ldots$. 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.