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arXiv · 1203.2322

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

Branko Sari\'
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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.

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Branko Sari\'. 2012-03-11. Cauchy's residue theorem for a class of real valued functions. https://doi.org/10.1007/s10587-010-0068-5

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