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

TL;DR

This paper investigates the properties of the Mittag-Leffler function $E_eta(-t^eta)$, demonstrating its complete monotonicity, spectral properties, and providing asymptotic approximations with simpler functions for modeling fractional relaxation.

Contribution

It reveals the universal spectral scaling of the Mittag-Leffler function and offers novel asymptotic approximations using elementary completely monotone functions.

Findings

Spectral spectra of the function coincide, indicating a universal scaling property.

Asymptotic equivalence of the Mittag-Leffler function with simpler CM functions at small and large times.

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 $t \to \infty$.