# On modifications of the exponential integral with the Mittag-Leffler function

## TL;DR

This paper surveys the Schelkunoff modification of the exponential integral, generalizes it using the Mittag-Leffler function, and introduces a new special function potentially useful in linear viscoelasticity due to its monotonicity properties.

## Contribution

It introduces a novel generalization of the exponential integral with the Mittag-Leffler function, expanding the class of special functions relevant to viscoelasticity.

## Key findings

- The new function exhibits complete monotonicity in the time domain.
- Generalized sine and cosine integral functions are also considered.
- Potential applications in linear viscoelasticity are discussed.

## Abstract

In this paper we survey the properties of the Schelkunoff modification of the Exponential integral and we generalize it with the Mittag-Leffler function. So doing we get a new special function (as far as we know) that may be relevant in linear viscoelasticity because of its complete monotonicity properties in the time domain. We also consider the generalized sine and cosine integral functions