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… 

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References

SHOWING 1-10 OF 46 REFERENCES

Essentials of Padé approximants

Fractional Derivatives for Physicists and Engineers

On solution of integral equation of Abel-Volterra type

Velocity and displacement correlation functions for fractional generalized Langevin equations

We study analytically a generalized fractional Langevin equation. General formulas for calculation of variances and the mean square displacement are derived. Cases with a three parameter

Mittag-Leffler Functions

This chapter is devoted to a brief summary of the most important properties of Mittag-Leffler functions. These functions play a fundamental role in many questions related to fractional differential

The Multi-index Mittag-Leffler Functions and Their Applications for Solving Fractional Order Problems in Applied Analysis

During the last few decades, differential equations and systems of fractional order (that is arbitrary one, not necessarily integer) begun to play an important role in modeling of various phenomena

On Creep and Relaxation

The creep and the relaxation function of linear systems, for which the principle of superposition is valid, are mutually connected in a simple way. This makes it possible to calculate the