# Mittag-Leffler function

An entire function $E _ \rho ( z)$ of a complex variable $z$, introduced by G. Mittag-Leffler  as a generalization of the exponential function:

$$E _ \rho ( z) = \sum _ { k= } 0 ^ \infty \frac{z ^ {k} }{\Gamma ( 1 + k / \rho ) } ,\ \ 1 \leq \rho < \infty .$$

Since the Mittag-Leffler function and the more general functions of Mittag-Leffler type

$$E _ \rho ( z ; \mu ) = \sum _ { k= } 0 ^ \infty \frac{z ^ {k} }{\Gamma ( \mu + k / \rho ) } ,\ \ \mu , \rho \in \mathbf C ,$$

are widely used in integral representations and transforms of analytic functions, their properties, in particular asymptotic properties, have been studied in great detail (see , ).

How to Cite This Entry:
Mittag-Leffler function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mittag-Leffler_function&oldid=47857
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article