Why the Mittag-Leffler Function Can Be Considered the Queen Function of the Fractional Calculus?

Abstract

In this survey we stress the importance of the higher transcendental Mittag-Leffler function in the framework of the Fractional Calculus. We first start with the analytical properties of the classical Mittag-Leffler function as derived from being the solution of the simplest fractional differential equation governing relaxation processes. Through the sections of the text we plan to address the reader in this pathway towards the main applications of the Mittag-Leffler function that has induced us in the past to define it as the Queen Function of the Fractional Calculus. These applications concern some noteworthy stochastic processes and the time fractional diffusion-wave equation We expect that in the next future this function will gain more credit in the science of complex systems. Finally, in an appendix we sketch some historical aspects related to the author's acquaintance with this function.

Keywords: Laplace and Fourier transform; Mittag-Lefflller functions; Wright functions; diffusion-wave equation; fractional Poisson process complex systems; fractional calculus; fractional relaxation.

Figure A1

F. Mainardi between R. Gorenflo (left) and M. Caputo (right).

Figure 1

The spectral function $K_{\alpha }\left(r\right)$ for $\alpha =0.25,0.50,0.75,0.90$ in the frequency range $0\le r\le 2$.

Figure 2

The Mittag-Leffler function $e_{\alpha }\left(t\right)$ for $\alpha =0.25,0.50,0.75,0.90,1.$ in the time range $0\le t\le 15$.

Figure 3

Approximations $e_{\alpha }^0\left(t\right)$ (dashed line) and $e_{\alpha }^{\infty }\left(t\right)$ (dotted line) to $e_{\alpha }\left(t\right)$ in ${10}^{-5}\le t\le {10}^{+5}$ for $\alpha =0.25$ (LEFT) and for $\alpha =0/50$ (RIGHT).

Figure 4

Approximations $e_{\alpha }^0\left(t\right)$ (dashed line) and $e_{\alpha }^{\infty }\left(t\right)$ (dotted line) to $e_{\alpha }\left(t\right)$ (LEFT) and the corresponding relative errors (RIGHT) in ${10}^{-5}\le t\le {10}^{+5}$ for $\alpha =0/75$ (LEFT) and for $\alpha =0.90$ (RIGHT).

Figure 5

Plots of ${\varphi }_{\alpha }\left(t\right)$ with $\alpha =1/4,1/2,3/4,1$ versus t; for $0\le t\le 5$.

Figure 6

Plot of the symmetric $M-$Wright function $M_{\nu }\left(\right|x\left|\right)$ for $0\le \nu \le 1/2$. Note that the $M-$Wright function becomes a Gaussian density for $\nu =1/2$.

Figure 7

Plot of the symmetric $M-$Wright type function $M_{\nu }\left(\right|x\left|\right)$ for $1/2\le \nu \le 1$. Note that the MWright function becomes a a sum of two delta functions centered in $x=\pm 1$ for $\nu =1$.