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

Sarić, Branko. "Cauchy's residue theorem for a class of real valued functions." Czechoslovak Mathematical Journal 60.4 (2010): 1043-1048. <http://eudml.org/doc/196565>.

@article{Sarić2010,
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] $. 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 Kurzweil-Henstock integral. The paper ends with a few examples that illustrate the theory.},
author = {Sarić, Branko},
journal = {Czechoslovak Mathematical Journal},
keywords = {Kurzweil-Henstock integral; Cauchy's residue theorem; Kurzweil-Henstock integral; Cauchy's residue theorem},
language = {eng},
number = {4},
pages = {1043-1048},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Cauchy's residue theorem for a class of real valued functions},
url = {http://eudml.org/doc/196565},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Sarić, Branko
TI - Cauchy's residue theorem for a class of real valued functions
JO - Czechoslovak Mathematical Journal
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 4
SP - 1043
EP - 1048
AB - 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 Kurzweil-Henstock integral. The paper ends with a few examples that illustrate the theory.
LA - eng
KW - Kurzweil-Henstock integral; Cauchy's residue theorem; Kurzweil-Henstock integral; Cauchy's residue theorem
UR - http://eudml.org/doc/196565
ER -