Mittag-Leffler function

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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:

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 cases[edit]

For we find

The sum of a geometric progression:

Exponential function:

Error function:

Hyperbolic cosine:

For , the integral

gives, respectively

Mittag-Leffler's integral representation[edit]

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

and

on the negative axis.

See also[edit]

References[edit]

External links[edit]

This article incorporates material from Mittag-Leffler function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.