# Mittag-Leffler function

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:

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

where ${\displaystyle \Gamma }$is the Gamma function .

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

${\displaystyle E_{\alpha ,\beta }(z)={\frac {1}{z}}E_{\alpha ,\beta -\alpha }(z)-{\frac {1}{z\Gamma (\beta -\alpha ),}}}$

from which the Poincaré asymptotic expansion

${\displaystyle E_{\alpha ,\beta }(z)\sim -\sum _{k=1}{\frac {1}{z^{k}\Gamma (\beta -k\alpha )}}}$

follows, which is true for ${\displaystyle z\to -\infty }$.

## Special cases

For ${\displaystyle \alpha =0,1/2,1,2}$  we find

The sum of a geometric progression:

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

For ${\displaystyle \alpha =0,1,2}$ , the integral

${\displaystyle \int _{0}^{z}E_{\alpha ,1}(-s^{2})\,{\mathrm {d} }s}$

gives, respectively

${\displaystyle \arctan(z),}$
${\displaystyle {\tfrac {\sqrt {\pi }}{2}}\operatorname {erf} (z),}$
${\displaystyle \sin(z).}$

## Mittag-Leffler's integral representation

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

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

Related to the Laplace transform and Mittag-Leffler summation is the expression

${\displaystyle \int _{0}^{\infty }e^{-tz}t^{\beta -1}E_{\alpha ,\beta }(t^{\alpha })dt={\frac {z^{-\beta }}{1-z^{-\alpha }}}}$

and

${\displaystyle \int _{0}^{\infty }e^{-tz}t^{\beta -1}E_{\alpha ,\beta }(-t^{\alpha })\,dt={\frac {z^{\alpha -\beta }}{1+z^{\alpha }}}}$

on the negative axis.