On some properties of the  Mittag-Leffler function $\mathbf{E_\alpha(-t^\alpha)}$,      completely monotone for $\mathbf{t> 0}$ with  $\mathbf{0<\alpha<1}$

Francesco Mainardi

doi:10.3934/dcdsb.2014.19.2267

Article Contents

2014, Volume 19, Issue 7: 2267-2278. Doi: 10.3934/dcdsb.2014.19.2267

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On some properties of the Mittag-Leffler function $\mathbf{E_\alpha(-t^\alpha)}$, completely monotone for $\mathbf{t> 0}$ with $\mathbf{0<\alpha<1}$

Francesco Mainardi^1,

1.
Department of Physics and Astronomy, University of Bologna, and INFN, Via Irnerio 46, Bologna, I-40126

Received: March 31, 2013

Revised: June 30, 2013

Published: August 2014

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Abstract

We analyse some peculiar properties of the function of the Mittag-Leffler (M-L) type, $e_\alpha(t) := E_\alpha(-t^\alpha)$ for $0<\alpha<1$ and $t>0$, which is known to be completely monotone (CM) with a non-negative spectrum of frequencies and times, suitable to model fractional relaxation processes. We first note that (surprisingly) these two spectra coincide so providing a universal scaling property of this function, not well pointed out in the literature. Furthermore, we consider the problem of approximating our M-L function with simpler CM functions for small and large times. We provide two different sets of elementary CM functions that are asymptotically equivalent to $e_\alpha(t)$ as $t\to 0$ and $t\to +\infty$. The first set is given by the stretched exponential for small times and the power law for large times, following a standard approach. For the second set we chose two rational CM functions in $t^\alpha$, obtained as the Pad\`e Approximants (PA) $[0/1]$ to the convergent series in positive powers (as $t\to 0$) and to the asymptotic series in negative powers (as $t\to \infty$), respectively. From numerical computations we are allowed to the conjecture that the second set provides upper and lower bounds to the Mittag-Leffler function.

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Mathematics Subject Classification: Primary: 26A33, 33E12; Secondary: 35S10, 45K05.

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