Mittag-Leffler function

An entire function $ E _ \rho ( z) $ of a complex variable $ z $, introduced by G. Mittag-Leffler [1] 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 [2], [3]).

References

[1]	G. Mittag-Leffler, "Sur la représentation analytique d'une branche uniforme d'une fonction monogène" Acta Math. , 29 (1905) pp. 101–181
[2]	M.M. Dzhrbashyan, "Integral transforms and representation of functions in the complex domain" , Moscow (1966) (In Russian)
[3]	A.A. Gol'dberg, I.V. Ostrovskii, "Value distribution of meromorphic functions" , Moscow (1970) (In Russian)
[a1]	M.L. Cartwright, "Integral functions" , Cambridge Univ. Press (1962)