Mittag-Leffler function

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The Mittag-Leffler function can be used to interpolate continuously between a Gaussian and a Lorentzian 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:[1][2]

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 (Theorem 5.1 of [1])

from which the Poincaré asymptotic expansion

follows, which is true for .

Special cases[edit]

For we find: (Section 2 of [1])

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[edit]

The integral representation of the Mittag-Leffler function is (Section 6 of [1])

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 (Eq (7.5) of [1], with m=0)


See also[edit]

Notes[edit]

  • R Package 'MittagLeffleR' by Gurtek Gill, Peter Straka. Implements the Mittag-Leffler function, distribution, random variate generation, and estimation.

References[edit]

  1. ^ a b c d e Saxena, R. K.; Mathai, A. M.; Haubold, H. J. (2009-09-01). "Mittag-Leffler Functions and Their Applications". Cite journal requires |journal= (help)
  2. ^ Weisstein, Eric W. "Mittag-Leffler Function". mathworld.wolfram.com. Retrieved 2019-09-11.

External links[edit]

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