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

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 $t \to \infty$.