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.
from which the Poincaré asymptotic expansion
follows, which is true for .
The sum of a geometric progression:
For , we have
For , the integral
gives, respectively: , , .
Mittag-Leffler's integral representation
where the contour C starts and ends at −∞ and circles around the singularities and branch points of the integrand.
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- 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