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