| Title:
|
Cauchy's residue theorem for a class of real valued functions (English) |
| Author:
|
Sarić, Branko |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
60 |
| Issue:
|
4 |
| Year:
|
2010 |
| Pages:
|
1043-1048 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Kurzweil-Henstock integral |
| Keyword:
|
Cauchy's residue theorem |
| MSC:
|
26A24 |
| MSC:
|
26A39 |
| idZBL:
|
Zbl 1224.26029 |
| idMR:
|
MR2738965 |
| . |
| Date available:
|
2010-11-20T13:57:39Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140802 |
| . |
| Reference:
|
[1] Bartle, R. G.: A Modern Theory of Integration.Graduate Studies in Math. Vol. 32, AMS, Providence (2001). Zbl 0968.26001, MR 1817647, 10.1090/gsm/032/07 |
| Reference:
|
[2] Garces, I. J. L., Lee, P. Y.: Convergence theorem for the $H_1$-integral.Taiw. J. Math. 4 (2000), 439-445. MR 1779108, 10.11650/twjm/1500407260 |
| Reference:
|
[3] Gordon, R. A.: The Integrals of Lebesgue, Denjoy, Perron and Henstock.Graduate Studies in Math., Vol. 4, AMS, Providence (1994). Zbl 0807.26004, MR 1288751, 10.1090/gsm/004/09 |
| Reference:
|
[4] Macdonald, A.: Stokes' theorem.Real Analysis Exchange 27 (2002), 739-747. Zbl 1059.26008, MR 1923163, 10.14321/realanalexch.27.2.0739 |
| Reference:
|
[5] Sinha, V., Rana, I. K.: On the continuity of associated interval functions.Real Analysis Exchange 29 (2003/2004), 979-981. MR 2083833, 10.14321/realanalexch.29.2.0979 |
| . |
