• DOI:10.3934/dcdsb.2014.19.2267
• Corpus ID: 119699143

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

@article{Mainardi2013OnSP,
  title={On some properties of the Mittag-Leffler function \$\mathbf\{E\_\alpha(-t^\alpha)\}\$, completely monotone for \$\mathbf\{t> 0\}\$ with \$\mathbf\{0<\alpha<1\}\$},
  author={Francesco Mainardi},
  journal={Discrete and Continuous Dynamical Systems-series B},
  year={2013},
  volume={19},
  pages={2267-2278}
}

• F. Mainardi
• Published 1 May 2013
• Mathematics
• Discrete and Continuous Dynamical Systems-series B

We analyse some peculiar properties of the function
of the Mittag-Leffler (M-L) type,
$e_\alpha(t) := E_\alpha(-t^\alpha)$ for $0 0$, 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.