Mittag-Leffler function

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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 répresentation 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)