Ta'lim innovatsiyasi va integratsiyasi

THE CLASSICAL MITTAG-LEFFLER FUNCTION AND PROPERTIES

Nematjonov Furqatjon

Keywords: Keywords: analytic, asymptotics, convergence, Cauchy inequality, Taylor coefficients, error function, complementary error function, Hankel path, Wright function, Euler transforms.

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Abstract

Abstract: In this article, we will present the basic properties of the classical
Mittag-Leffler function   E z

. The material can demonstrate some information
starting from the basic definition of the Mittag-Leffler function in terms of a power
series, we discover that for parameter
 with positive real part the function   E z

is
an entire function of the complex variable z. Therefore, we discuss in the first part the
(analytic) properties of the Mittag-Leffler function as an entire function. As well as,
some relations to elementary and special functions before we will show integral
representations and differential relations. After that, we calculate integral transforms;
lastly, relation to fractional calculus is the last part of the article.

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References

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