Open main menu

Wikipedia β

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 .

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 Eα,1 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

from which the Poincaré asymptotic expansion

follows, which is true for .


Special casesEdit

For   we find

The sum of a geometric progression:


Exponential function:


Error function:


Hyperbolic cosine:


For  , the integral


gives, respectively


Mittag-Leffler's integral representationEdit


where the contour C starts and ends at −∞ and circles around the singularities and branch points of the integrand.

Related to the Laplace transform and Mittag-Leffler summation is the expression




on the negative axis.

See alsoEdit


External linksEdit