Asymptotic expansion of the modified exponential integral involving the Mittag-Leffler function

TL;DR

This paper derives asymptotic expansions for a generalized exponential integral involving the Mittag-Leffler function, extending previous definitions and providing numerical validation for the accuracy of these expansions.

Contribution

It extends the asymptotic analysis of the exponential integral to a broader class involving the two-parameter Mittag-Leffler function, including sine and cosine integrals.

Findings

Derived asymptotic expansions for the generalized exponential integral.

Extended definitions to include two-parameter Mittag-Leffler functions.

Numerical examples confirm the accuracy of the expansions.

Abstract

We consider the asymptotic expansion of the generalised exponential integral involving the Mittag-Leffler function introduced recently by Mainardi and Masina [{\it Fract. Calc. Appl. Anal.} {\bf 21} (2018) 1156--1169]. We extend the definition of this function using the two-parameter Mittag-Leffler function. The expansions of the similarly extended sine and cosine integrals are also discussed. Numerical examples are presented to illustrate the accuracy of each type of expansion obtained.