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

Sarić, Branko

Czechoslovak Mathematical Journal, Tome 60 (2010) no. 4, pp. 1043-1048.

Résumé

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] $. Assuming $F$ 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}\hbox {\rm -vt}\int _a^bf=F( b) -F( a) $, where $\mathcal {KH}\hbox {\rm -vt}$ denotes the total value of the {\it Kurzweil-Henstock} integral. The paper ends with a few examples that illustrate the theory.

Détail

Classification : 26A24, 26A39
Keywords: Kurzweil-Henstock integral; Cauchy's residue theorem