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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:

Contents

E α , β ( z ) = ∑ k = 0 ∞ z k Γ ( α k + β ) .

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

E α , β ( z ) = 1 z E α , β − α ( z ) − 1 z Γ ( β − α ) ,

from which the Poincaré asymptotic expansion

E α , β ( z ) ∼ − ∑ k = 1 1 z k Γ ( β − k α )

follows, which is true for z → − ∞ .

Special cases

For α = 0 , 1 / 2 , 1 , 2 we find

The sum of a geometric progression:

E 0 , 1 ( z ) = ∑ k = 0 ∞ z k = 1 1 − z .

Exponential function:

E 1 , 1 ( z ) = ∑ k = 0 ∞ z k Γ ( k + 1 ) = ∑ k = 0 ∞ z k k ! = exp ⁡ ( z ) .

Error function:

E 1 / 2 , 1 ( z ) = exp ⁡ ( z 2 ) erfc ⁡ ( − z ) .

Hyperbolic cosine:

E 2 , 1 ( z ) = cosh ⁡ ( z ) .

For α = 0 , 1 , 2 , the integral

∫ 0 z E α , 1 ( − s 2 ) d s

gives, respectively

arctan ⁡ ( z ) , π 2 erf ⁡ ( z ) , sin ⁡ ( z ) .

Mittag-Leffler's integral representation

E α , β ( z ) = 1 2 π i ∫ C t α − β e t t α − z d t

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

∫ 0 ∞ e − t z t β − 1 E α , β ( t α ) d t = z − β 1 − z − α

and

∫ 0 ∞ e − t z t β − 1 E α , β ( − t α ) d t = z α − β 1 + z α

on the negative axis.

References

Mittag-Leffler function Wikipedia
(Text) CC BY-SA


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