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

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 $\mathcal{K}\mathcal{H}-vt{\int }_a^{b}f=F(b)-F(a)$%, where $\mathcal{K}\mathcal{H}-$ $vt$ denotes the total value of the \textit{% Kurzweil-Henstock} integral. The paper ends with a few examples that illustrate the theory.

6 pages