An integral's journey over the real line

TL;DR

This paper revisits Cauchy's 1826 integral over the real line, providing a new derivation and an analytic continuation to extend its applicability.

Contribution

It introduces a novel derivation and analytic continuation of Cauchy's integral over the real line, enhancing understanding of its properties.

Findings

New derivation of Cauchy's integral

Analytic continuation of the integral

Extended applicability of the integral

Abstract

In 1826 Cauchy presented an Integral over the real line. Al and I thought a derivation would be mighty fine. So we packed our contour integral bags that day, and we now present an analytic continuation this time.