Two forms of the integral representations of the Mittag-Leffler function

TL;DR

This paper explores two different contour integral representations of the two-parameter Mittag-Leffler function, analyzing their derivation, properties, and relative advantages through rigorous theorems.

Contribution

It introduces and proves two distinct real-variable integral forms of the Mittag-Leffler function, highlighting their respective benefits and limitations.

Findings

Two integral representations of $E_{ ho, u}(z)$ are established.

Theorems comparing the advantages and disadvantages of each form.

Transition from complex to real variables is systematically analyzed.

Abstract

The integral representation of the two-parameter Mittag-Leffler function $E_{\rho,\mu}(z)$ is considered in the paper that expresses its value in terms of the contour integral. For this integral representation, the transition is made from integration over a complex variable to integration over real variables. It is shown that as a result of such a transition, the integral representation of the function $E_{\rho,\mu}(z)$ has two forms: the representation ``A'' and ``B''. Each of these representations has its advantages and drawbacks. In the paper, the corresponding theorems are formulated and proved, and the advantages and disadvantages of each of the obtained representations are discussed.