Mittag-Leffler function

File:MittagLeffler GaussToLorentz.gif

In mathematics, the Mittag-Leffler functions are a family of special functions. They are complex-valued functions of a complex argument z, and moreover depend on one or two complex parameters.

The one-parameter Mittag-Leffler function, introduced by Gösta Mittag-Leffler in 1903,Mittag-Leffler, M.G.: Sur la nouvelle fonction E(x). C. R. Acad. Sci. Paris 137, 554–558 (1903), and several more papers in the following years.Haubold,H J and Mathai,A M and Saxena,R K, [https://onlinelibrary.wiley.com/doi/epdf/10.1155/2011/298628 J Appl Math 2011, 298628] can be defined by the Maclaurin series

: $E_{\alpha} (z) = \sum_{k=0}^\infty \frac{z^k}{\Gamma(\alpha k + 1)},$

where $\Gamma(x)$ is the gamma function, and $\alpha$ is a complex parameter with $\operatorname{Re}\left(\alpha \right)> 0$ .

The two-parameter Mittag-Leffler function, introduced by Wiman in 1905,Anders Wiman, Über den Fundamentalsatz in der Teorie [sic] der Funktionen $E_a(x)$ , Acta Math 29, 191-201 (1905). is occasionally called the generalized Mittag-Leffler function. It has an additional complex parameter $\beta$ , and may be defined by the series{{Cite web|url=http://mathworld.wolfram.com/Mittag-LefflerFunction.html|title=Mittag-Leffler Function|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-09-11}}

: $E_{\alpha, \beta} (z) = \sum_{k=0}^\infty \frac{z^k}{\Gamma(\alpha k + \beta)},$

When $\beta=1$ , the one-parameter function $E_\alpha = E_{\alpha,1}$ is recovered.

In the case $\alpha$ and $\beta$ are real and positive, the series converges for all values of the argument $z$ , so the Mittag-Leffler function is an entire function. This class of functions are important in the theory of the fractional calculus.

See below for three-parameter generalizations.

Some basic properties

For $\alpha >0$ , the Mittag-Leffler function $E_{\alpha,\beta}(z)$ is an entire function of order $1/\alpha$ , and type $1$ for any value of $\beta$ . In some sense, the Mittag-Leffler function is the simplest entire function of its order. The indicator function of $E_{\alpha}(z)$ is{{cite book |last1=Cartwright|first1=M. L. |title=Integral Functions |date=1962 |publisher=Cambridge Univ. Press |isbn=052104586X}}{{rp|p=50}}

$h_{E_\alpha}(\theta)=\begin{cases}\cos\left(\frac{\theta}{\alpha}\right),&\text{for }|\theta|\le\frac 1 2 \alpha\pi;\\0,&\text{otherwise}.\end{cases}$

This result actually holds for $\beta\neq1$ as well with some restrictions on $\beta$ when $\alpha=1$ .{{rp|p=67}}

The Mittag-Leffler function satisfies the recurrence property (Theorem 5.1 of )

: $E_{\alpha,\beta}(z)=\frac{1}{z}E_{\alpha,\beta-\alpha}(z)-\frac{1}{z \Gamma(\beta-\alpha)},$

from which the following asymptotic expansion holds : for $0<\alpha<2$ and $\mu$ real such that

$\frac{\pi\alpha}{2} < \mu < \min(\pi, \pi\alpha)$ then for all $N\in\mathbb{N}^*, N\neq 1$ , we can show the following asymptotic expansions (Section 6. of ):

-as $\,|z|\to+\infty, |\text{arg}(z)|\leq \mu$ :

: $E_{\alpha}(z) = \frac{1}{\alpha} \exp(z^{\frac{1}{\alpha}}) - \sum\limits_{k=1}^{N} \frac{1}{z^k\, \Gamma(1-\alpha k)} + O\left(\frac{1}{z^{N+1}}\right)$ ,

-and as $\,|z|\to+\infty, \mu\leq|\text{arg}(z)|\leq\pi$ :

: $E_{\alpha}(z) = - \sum\limits_{k=1}^N \frac{1}{z^k\Gamma(1-\alpha k)} + O\left(\frac{1}{z^{N+1}}\right)$ .

A simpler estimate that can often be useful is given, thanks to the fact that the order and type of $E_{\alpha,\beta}(z)$ is $1/\alpha$ and $1$ , respectively:{{rp|p=62}}

: $|E_{\alpha,\beta}(z)|\le C\exp\left(\sigma|z|^{1/\alpha}\right)$

for any positive $C$ and any $\sigma>1$ .

Special cases

For $\alpha=0$ , the series above equals the Taylor expansion of the geometric series and consequently $E_{0,\beta}(z)=\frac{1}{\Gamma(\beta)}\frac{1}{1-z}$ .

For $\alpha=1/2,1,2$ we find: (Section 2 of )

Error function:

: $E_{\frac{1}{2}}(z) = \exp(z^2)\operatorname{erfc}(-z).$

Exponential function:

: $E_{1}(z) = \sum_{k=0}^\infty \frac{z^k}{\Gamma (k + 1)} = \sum_{k=0}^\infty \frac{z^k}{k!} = \exp(z).$

Hyperbolic cosine:

: $E_{2}(z) = \cosh(\sqrt{z}), \text{ and } E_{2}(-z^2) = \cos(z).$

For $\beta=2$ , we have

: $E_{1,2}(z) = \frac{e^z-1}{z},$

: $E_{2,2}(z) = \frac{\sinh(\sqrt{z})}{\sqrt{z}}.$

For $\alpha=0,1,2$ , the integral

: $\int_0^z E_{\alpha}(-s^2) \, {\mathrm d}s$

gives, respectively: $\arctan(z)$ , $\tfrac{\sqrt{\pi}}{2}\operatorname{erf}(z)$ , $\sin(z)$ .

Mittag-Leffler's integral representation

The integral representation of the Mittag-Leffler function is (Section 6 of )

: $E_{\alpha,\beta}(z)=\frac{1}{2\pi i}\oint_C \frac{t^{\alpha-\beta}e^t}{t^\alpha-z} \, dt,
\Re(\alpha)>0, \Re(\beta)>0,$

where the contour $C$ starts and ends at $-\infty$ and circles around the singularities and branch points of the integrand.

Related to the Laplace transform and Mittag-Leffler summation is the expression (Eq (7.5) of with $m=0$ )

: $\int_0^{\infty}e^{-t z} t^{\beta-1} E_{\alpha,\beta}(\pm r\, t^\alpha) \,dt
= \frac{z^{\alpha-\beta}}{z^{\alpha}\mp r},
\Re(z)>0, \Re(\alpha)>0, \Re(\beta)>0.$

Three-parameter generalizations

One generalization, characterized by three parameters, is

$E_{\alpha,\beta}^\gamma(z)=\left ( \frac{1}{\Gamma(\gamma)} \right )\sum\limits_{k=1}^\infty \frac{\Gamma(\gamma + k)z^k}{k! \Gamma(\alpha k + \beta ) },$

where $\alpha, \beta$ and $\gamma$ are complex parameters and $\Re(\alpha)>0$ .{{Cite book |last1=Gorenflo |first1=Rudolf |url=https://link.springer.com/10.1007/978-3-662-43930-2 |title=Mittag-Leffler Functions, Related Topics and Applications: Theory and Applications |last2=Kilbas |first2=Anatoly A. |last3=Mainardi |first3=Francesco |last4=Rogosin |first4=Sergei V. |date=2014 |publisher=Springer Berlin Heidelberg |isbn=978-3-662-43929-6 |series=Springer Monographs in Mathematics |location=Berlin, Heidelberg |language=en |doi=10.1007/978-3-662-43930-2}}

Another generalization is the Prabhakar function

$E_{\alpha, \beta}^{\gamma}(z) = \sum_{k=0}^{\infty} \frac{(\gamma)_k z^k}{k!\Gamma(\alpha k + \beta)},$

where $(\gamma)_k$ is the Pochhammer symbol.

Applications of Mittag-Leffler function

One of the applications of the Mittag-Leffler function is in modeling fractional order viscoelastic materials. Experimental investigations into the time-dependent relaxation behavior of viscoelastic materials are characterized by a very fast decrease of the stress at the beginning of the relaxation process and an extremely slow decay for large times, i.e. it takes a long time to approach a constant asymptotic value. Therefore, many Maxwell elements are required to describe relaxation behavior to sufficient accuracy. This results in a difficult optimization problem in order to identify the large number of material parameters required. On the other hand, over the years, the concept of fractional derivatives has been introduced into the theory of viscoelasticity. Among these models, the fractional Zener model was found to be very effective for predicting the dynamic nature of rubber-like materials using only a small number of material parameters. The solution of the corresponding constitutive equation leads to a relaxation function of the Mittag-Leffler type. It is defined by the power series with negative arguments. This function represents all essential properties of the relaxation process under the influence of an arbitrary and continuous signal with a jump at the origin.Pritz, T. (2003). Five-parameter fractional derivative model for polymeric damping materials. Journal of Sound and Vibration, 265(5), 935-952.Nonnenmacher, T. F., & Glöckle, W. G. (1991). A fractional model for mechanical stress relaxation. Philosophical magazine letters, 64(2), 89-93.

Notes

R Package [https://CRAN.R-project.org/package=MittagLeffleR 'MittagLeffleR' ] by Gurtek Gill, Peter Straka. Implements the Mittag-Leffler function, distribution, random variate generation, and estimation.

References

[https://www.springer.com/gp/book/9783662439296 Gorenflo R., Kilbas A.A., Mainardi F., Rogosin S.V., Mittag-Leffler Functions, Related Topics and Applications (Springer, New York, 2014)] 443 pages {{isbn|978-3-662-43929-6}}
{{cite book|title=Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications|series=Mathematics in Science and Engineering|author=Igor Podlubny|publisher=Academic Press|year=1998|isbn=0-12-558840-2|chapter=chapter 1}}
{{cite book|author=Kai Diethelm|title=The analysis of fractional differential equations: an application-oriented exposition using differential operators of Caputo type|location=Heidelberg and New York|publisher=Springer-Verlag|year=2010|series=Lecture Notes in Mathematics|isbn=978-3-642-14573-5|chapter=chapter 4}}

External links

[http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=8738&objectType=FILE Mittag-Leffler function: MATLAB code]
[https://arxiv.org/abs/0707.2582 Mittag-Leffler and stable random numbers: Continuous-time random walks and stochastic solution of space-time fractional diffusion equations]

Category:Special functions

Category:Analytic functions