In mathematics, the Mittag-Leffler function Eα,β is a special function, a complex function which depends on two complex parameters α and β. It may be defined by the following series when the real part of α is strictly positive:
where is the gamma function. When , it is abbreviated as . For , the series above equals the Taylor expansion of the geometric series and consequently .
In the case α and β are real and positive, the series converges for all values of the argument z, so the Mittag-Leffler function is an entire function. This function is named after Gösta Mittag-Leffler. This class of functions are important in the theory of the fractional calculus.
For α > 0, the Mittag-Leffler function is an entire function of order 1/α, and is in some sense the simplest entire function of its order.
The Mittag-Leffler function satisfies the recurrence property (Theorem 5.1 of )
from which the Poincaré asymptotic expansion
follows, which is true for .
For we find: (Section 2 of )
The sum of a geometric progression:
For , we have
For , the integral
gives, respectively: , , .
Mittag-Leffler's integral representation
The integral representation of the Mittag-Leffler function is (Section 6 of )
where the contour C starts and ends at −∞ and circles around the singularities and branch points of the integrand.
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. It can even take a long time before a constant asymptotic value is reached. Therefore, a lot of Maxwell elements are required to describe relaxation behavior with sufficient accuracy. This ends in a difficult optimization problem in order to identify a large number of material parameters. On the other hand, over the years, the concept of fractional derivatives has been introduced to the theory of viscoelasticity. Among these models, the fractional Zener model was found to be very effective to predict the dynamic nature of rubber-like materials with 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. 
- R Package 'MittagLeffleR' by Gurtek Gill, Peter Straka. Implements the Mittag-Leffler function, distribution, random variate generation, and estimation.
- Saxena, R. K.; Mathai, A. M.; Haubold, H. J. (2009-09-01). "Mittag-Leffler Functions and Their Applications". arXiv:0909.0230v2. Cite journal requires
- Weisstein, Eric W. "Mittag-Leffler Function". mathworld.wolfram.com. Retrieved 2019-09-11.
- 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.
This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. (September 2015) (Learn how and when to remove this template message)
- Mittag-Leffler, M.G.: Sur la nouvelle fonction E(x). C. R. Acad. Sci. Paris 137, 554–558 (1903)
- Mittag-Leffler, M.G.: Sopra la funzione E˛.x/. Rend. R. Acc. Lincei, (Ser. 5) 13, 3–5 (1904)
- 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
- Igor Podlubny (1998). "chapter 1". Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications. Mathematics in Science and Engineering. Academic Press. ISBN 0-12-558840-2.
- Kai Diethelm (2010). "chapter 4". The analysis of fractional differential equations: an application-oriented exposition using differential operators of Caputo type. Lecture Notes in Mathematics. Heidelberg and New York: Springer-Verlag. ISBN 978-3-642-14573-5.
- Mittag-Leffler function math handbook mathHandbook.com
- Mittag-Leffler function: MATLAB code
- Mittag-Leffler and stable random numbers: Continuous-time random walks and stochastic solution of space-time fractional diffusion equations