On the Laplace transform of the Fréchet distribution

We calculate exactly the Laplace transform of the Fréchet distribution in the form γx−(1 + γ)exp (−x−γ), γ > 0, 0 ≤ x < ∞, for arbitrary rational values of the shape parameter γ, i.e. for γ = l/k with l, k = 1, 2, …. 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.