Difference between revisions of "Henstock-Kurzweil integral"
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if <math>\mathcal{\dot{P}}</math> is a <math>\delta</math>-fine [[Partition of an interval|tagged partition]] on <math>[a,b]</math>, then <math>|L-S(f,\mathcal{\dot{P}})|<\epsilon</math> | if <math>\mathcal{\dot{P}}</math> is a <math>\delta</math>-fine [[Partition of an interval|tagged partition]] on <math>[a,b]</math>, then <math>|L-S(f,\mathcal{\dot{P}})|<\epsilon</math> | ||
| − | Here, <math>S(f,\mathcal{\ | + | Here, <math>S(f,\mathcal{\dot{P}})</math> is the [[Reimann sum]] of <math>f</math> on <math>[a,b]</math> with respect to <math>\mathcal{\dot{P}}</math> |
Revision as of 08:25, 16 February 2008
The Henstock-Kurzweil integral (also known as the Generalized Reimann integral) is one of the most widely applicable generalizations of the Reimann integral, but it also uses a strikingly simple and elegant idea. It was developed independantly by Ralph Henstock and Jaroslav Kurzweil
Definition
Let ![$f:[a,b]\rightarrow\mathbb{R}$](http://latex.artofproblemsolving.com/6/7/c/67c320754a36aaec178e61f04bfcbac41c0bf675.png)
Let 
We say that 
is Generalised Reimann Integrable on ![$[a,b]$](http://latex.artofproblemsolving.com/8/e/c/8ecbd1ba3da8f2adef66a63f2ab32c47e63fa734.png)
if and only if, 
, there exists a gauge ![$\delta:[a,b]\rightarrow\mathbb{R}^+$](http://latex.artofproblemsolving.com/c/c/b/ccb9dc6961d05a2a0261ddcddd063dd32e28fcd3.png)
such that,
if 
is a 
-fine tagged partition on , then 
Here, 
is the Reimann sum of on with respect to
The elegance of this integral lies in in the ability of a gauge to 'measure' a partition more accurately than its norm.
See Also
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