# Mittag-Leffler 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:

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

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.

## Special cases

$E_{1,1}(z) = \sum_{k=0}^\infty \frac{z^k}{\Gamma (k + 1)} = \sum_{k=0}^\infty \frac{z^k}{k!} = \exp(z).$
$E_{1/2,1}(z) = \exp(z^2)\operatorname{erfc}(-z).$

Sum of a geometric progression:

$E_{0,1}(z) = \frac{1}{1-z}.$
$E_{2,1}(z) = \cosh(\sqrt{z}).$

## Mittag-Leffler's integral representation

$E_{\alpha,\beta}(z)=\frac{1}{2\pi i}\int_C \frac{t^{\alpha-\beta}e^t}{t^\alpha-z} \, dt$

where the contour C starts and ends at −∞ and circles around the singularities and branch points of the integrand.