Mittag-Leffler function

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 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.

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 and \mu real such that \frac{\pi\alpha}{2} 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: :|E_{\alpha,\beta}(z)|\le C\exp\left(\sigma|z|^{1/\alpha}\right) for any positive C and any \sigma1.

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.

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.

Notes

• R Package 'MittagLeffleR' by Gurtek Gill, Peter Straka. Implements the Mittag-Leffler function, distribution, random variate generation, and estimation.

References

• Gorenflo R., Kilbas A.A., Mainardi F., Rogosin S.V., Mittag-Leffler Functions, Related Topics and Applications (Springer, New York, 2014) 443 pages

References

1. 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.
2. 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]
3. Anders Wiman, Über den Fundamentalsatz in der Teorie [sic] der Funktionen $E\_a(x)$, Acta Math 29, 191-201 (1905).
4. Weisstein, Eric W.. "Mittag-Leffler Function".
5. (1962). "Integral Functions". Cambridge Univ. Press.
6. (2014). "Mittag-Leffler Functions, Related Topics and Applications: Theory and Applications". Springer Berlin Heidelberg.
7. Pritz, T. (2003). Five-parameter fractional derivative model for polymeric damping materials. Journal of Sound and Vibration, 265(5), 935-952.
8. Nonnenmacher, T. F., & Glöckle, W. G. (1991). A fractional model for mechanical stress relaxation. Philosophical magazine letters, 64(2), 89-93.