Difference between revisions of "Henstock-Kurzweil integral"
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− | The elegance of this integral lies in in the ability of a [[gauge]] to 'measure' a partition more accurately than its [[Partition of an interval|norm]]. | + | The elegance of this integral lies in in the ability of a [[gauge]] to 'measure' a partition more accurately than its [[Partition of an interval|norm]] |
+ | |||
+ | ==Illustration== | ||
+ | The utility of the Henstock -Kurzweil integral is demonstrated by this function, which is not Reimann integrable but is Geeralized Reimann Integrable. | ||
+ | |||
+ | Consider the function <math>f:[0,1]\rightarrow\mathh{R}</math> | ||
+ | <math>f\left( \frac{1}{n}\right) =n\forall n\in\mathbb{N}</math> | ||
+ | |||
+ | <math>f(x)=0</math> everywhere else. | ||
+ | |||
+ | It can be shown that <math>f</math> is not Reimann integrable on <math>[0,1]</math> | ||
+ | |||
+ | Let <math>\varepsilon>0</math> be given. | ||
+ | Consider [[gauge]] <math>\delta:[0,1]\rightarrow\mathbb{R}^+</math> | ||
+ | |||
+ | <math>\delta\left( \frac{1}{n}\right) =\frac{\varepsilon}{k2^{k+1}}</math> | ||
+ | |||
+ | <math>\delta(x)=1</math> everywhere else. | ||
+ | |||
+ | Let <math>\mathcal{\dot{P}}</math> be a <math>\delta</math>-fine [[Partition of an interval|partition]] on <math>[0,1]</math> | ||
+ | |||
+ | The [[Reimann sum]] will have maximum value only when the tags are of the form <math>t_i=\frac{1}{n}</math>, <math>n\in \mathbb{N}</math>. Also, each tag can be shared by at most two divisions. | ||
+ | |||
+ | <math>S(f,\mathcal{\dot{P}})\leq\sum_{k=1}^{\infty}\frac{\varepsilon}{2^k}<\varepsilon</math> | ||
+ | |||
+ | But as <math>\varepsilon>0</math> is arbitrary, we have that <math>f</math> is Generalized Reimann integrable or, <math>\int_0^1 f(x)dx=0</math> | ||
+ | |||
+ | |||
+ | ==References== | ||
+ | R.G. Bartle, D.R. Sherbert, <i>Introduction to Real Analysis</i>, John Wiley & sons | ||
==See Also== | ==See Also== |
Revision as of 00:16, 19 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
Contents
[hide]Definition
Let
Let
We say that is Generalised Reimann Integrable on if and only if, , there exists a gauge 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
Illustration
The utility of the Henstock -Kurzweil integral is demonstrated by this function, which is not Reimann integrable but is Geeralized Reimann Integrable.
Consider the function $f:[0,1]\rightarrow\mathh{R}$ (Error compiling LaTeX. Unknown error_msg)
everywhere else.
It can be shown that is not Reimann integrable on
Let be given. Consider gauge
everywhere else.
Let be a -fine partition on
The Reimann sum will have maximum value only when the tags are of the form , . Also, each tag can be shared by at most two divisions.
But as is arbitrary, we have that is Generalized Reimann integrable or,
References
R.G. Bartle, D.R. Sherbert, Introduction to Real Analysis, John Wiley & sons
See Also
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