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

The relation between H(k), inverse Laplace transform of a relaxation function I(t), and H-beta(k), inverse Laplace transform of I(t(beta)), is obtained. It is shown that for beta < 1 the function H-beta(k) can be expressed in terms of H(k) and of the Levy one-sided distribution L-beta(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 L-1/4(k) is also obtained.