Mittag-Leffler function

In mathematics, the Mittag-Leffler function $E_{\alpha \beta }$ is special function, a complex function which depends on two complex parameters $\alpha$ and $\beta$. It may be defined by the following series when the real part of $\alpha$ is strictly positive:

$E_{\alpha \beta }(z)=\sum _{k=0}^{\infty }{z^{k} \over \Gamma (\alpha k+\beta )}$

In this case, the series converges for all values of the argument z, so the Mittag-Leffler function is an entire function.

Relationship to the error function

The error function is a special case of the Mittag-Leffler function:

$w(z)=\exp(-z^{2}){\mbox{erfc}}(-iz)=E_{1/2,1}(iz)$

Mittag-Leffler function at PlanetMath.