The Classical Mittag-Leffler Function

Abstract

In this chapter we present the basic properties of the classical Mittag-Leffler function $E_\alpha (z)$ (see (1.0.1)). The material can be formally divided into two parts. Starting from the basic definition of the Mittag-Leffler function in terms of a power series, we discover that for parameter $\alpha$ with positive real part the function $E_\alpha (z)$ 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. Namely, we calculate its order and type, present a number of formulas relating it to elementary and special functions as well as recurrence relations and differential formulas, introduce some useful integral representations and discuss the asymptotics and distribution of zeros of the classical Mittag-Leffler function.