Relation between the inverse Laplace transforms of I(tβ) and I(t): Application to the Mittag-Leffler and asymptotic inverse power law relaxation functions

Abstract

The relation between H(k), inverse Laplace transform of a relaxation function I(t), and Hβ(k), inverse Laplace transform of I(tβ), is obtained. It is shown that for β < 1 the function Hβ(k) can be expressed in terms of H(k) and of the Levy one-sided distribution Lβ(k). The obtained results are applied to the Mittag-Leffler and asymptotic inverse power law relaxation functions. A simple integral representation for the Levy one-sided density function L1/4(k) is also obtained.