Cauchy's residue theorem for a class of real valued functions

TL;DR

This paper extends Cauchy's residue theorem to certain real-valued functions with discontinuities, using Kurzweil-Henstock integrals, and demonstrates the theory with illustrative examples.

Contribution

It introduces a residue theorem for real functions with discontinuities, utilizing Kurzweil-Henstock integrals, which is a novel extension of classical complex analysis concepts.

Findings

Established a residue theorem for real functions with discontinuities.

Connected the Kurzweil-Henstock integral to the difference of endpoint values.

Provided examples illustrating the application of the theorem.

Abstract

Let $[a,b] $ be an interval in $\mathbb{R}$ and let $F$ be a real valued function defined at the endpoints of $[a,b]$ and with a certain number of discontinuities within $[a,b] $. Having assumed $F$ to be differentiable on a set $[a,b] \backslash E$ to the derivative $f$, where $E$ is a subset of $[a,b] $ at whose points $F$ can take values $\pm \infty $ or not be defined at all, we adopt the convention that $F$ and $f$ are equal to 0 at all points of $E$ and show that $\mathcal{KH-}vt\int_{a}^{b}f=F(b) -F(a)$%, where $\mathcal{KH-}$ $vt$ denotes the total value of the \textit{% Kurzweil-Henstock} integral. The paper ends with a few examples that illustrate the theory.