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

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 these two spectra coincide so providing a universal scaling property of this function. 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 $t\to \infty$.

17 pages, 12 figures. This paper is based on an invited talk delivered by the Author at the First Workshop on Fractional Calculus and Its Applications, UAE University at Al Ain, UAE, 25-26 April 2013

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• Time-fractional derivatives in relaxation processes: a tutorial survey
• The Mittag-Leffler function
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