On a continued fraction expansion of the special function and an explicit expression of the continued fraction convergents

TL;DR

This paper introduces a continued fraction expansion for the exponential integral at infinity, demonstrating its equivalence to the regular continued fraction and providing explicit forms for the truncated rational approximations.

Contribution

It defines a new continued fraction expansion for the exponential integral and proves its equivalence to the standard expansion, with explicit formulas for convergents.

Findings

The continued fraction expansion for $E_1(x)$ at infinity is equivalent to the regular continued fraction.

Explicit representations of truncated rational functions are provided.

The expansion offers a new perspective on approximating the exponential integral.

Abstract

In this paper we define "a continued fraction expansion of the exponential integral $E_{1}(x)$ at infinity", which is analogous to the regular continued fraction expansion of real numbers, and prove that this expansion gives the same continued fraction. Moreover, we give concrete representations of rational functions which are obtained by truncating the continued fraction.