Mittag-Leffler function

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In mathematics, the Mittag-Leffler function Eα,β is 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:

E_{\alpha, \beta} (z) = \sum_{k=0}^\infty \frac{z^k}{\Gamma(\alpha k + \beta)}.

In this case, 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.

[edit] Special cases

Exponential function:

E_{1,1}(z) = \sum_{k=0}^\infty \frac{z^k}{\Gamma (k + 1)} = \sum_{k=0}^\infty \frac{z^k}{k!} = \exp(z).

Error function:

E_{1/2,1}(z) = \exp(z^2)\operatorname{erfc}(-z).

Sum of a geometric progression:

E_{0,1}(z) = \frac{1}{1-z}.

Hyperbolic cosine:

E_{2,1}(z) = \cosh(\sqrt{z}).

[edit] References

  • Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications. (Mathematics in Science and Engineering, vol. 198), by Igor Podlubny. Hardcover. Publisher: Academic Press; (October 1998). ISBN 0-12-558840-2 (see Chapter 1).

[edit] External links

This article incorporates material from Mittag-Leffler function on PlanetMath, which is licensed under the GFDL.