THE CLASSICAL MITTAG-LEFFLER FUNCTION AND PROPERTIES - Nematjonov Furqatjon (Ta'lim innovatsiyasi va integratsiyasi, 2024)
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.