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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:

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 .

Contents

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

 

and

 

on the negative axis.

See alsoEdit

ReferencesEdit

External linksEdit