Mittag-Leffler function


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
from which the Poincaré asymptotic expansion
follows, which is true for.

Special cases

For we find:
Error function:
The sum of a geometric progression:
Exponential function:
Hyperbolic cosine:
For, we have
For, the integral
gives, respectively:,, .

Mittag-Leffler's integral representation

The integral representation of the Mittag-Leffler function is
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