Difference between revisions of "Henstock-Kurzweil integral"
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− | The '''Henstock-Kurzweil integral''' (also known as the '''Generalized | + | The '''Henstock-Kurzweil integral''' (also known as the '''Generalized Riemann integral''') is one of the most widely applicable generalizations of the [[Integral|Riemann integral]], but it also uses a strikingly simple and elegant idea. It was developed independently by [[Ralph Henstock]] and [[Jaroslav Kurzweil]]. |
==Definition== | ==Definition== | ||
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Let <math>L\in\mathbb{R}</math> | Let <math>L\in\mathbb{R}</math> | ||
− | We say that <math>f</math> is '' | + | We say that <math>f</math> is ''Generalized Riemann Integrable'' on <math>[a,b]</math> if and only if, <math>\forall\epsilon>0</math>, there exists a [[gauge]] <math>\delta:[a,b]\rightarrow\mathbb{R}^+</math> such that, |
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 [[Riemann sum]] of <math>f</math> on <math>[a,b]</math> with respect to <math>\mathcal{\dot{P}}</math> |
− | The elegance of this integral lies in in the ability of a | + | 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 Riemann integrable but is Generalized Riemann Integrable. | ||
+ | |||
+ | Consider the function <math>f:[0,1]\rightarrow\mathbb{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 Riemann 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 [[Riemann 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 Riemann 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== | ||
− | *[[Integral| | + | *[[Integral|Riemann Integral]] |
− | |||
[[Category:Calculus]] | [[Category:Calculus]] | ||
{{stub}} | {{stub}} |
Latest revision as of 09:46, 31 January 2018
The Henstock-Kurzweil integral (also known as the Generalized Riemann integral) is one of the most widely applicable generalizations of the Riemann integral, but it also uses a strikingly simple and elegant idea. It was developed independently by Ralph Henstock and Jaroslav Kurzweil.
Contents
Definition
Let
Let
We say that is Generalized Riemann Integrable on
if and only if,
, there exists a gauge
such that,
if is a
-fine tagged partition on
, then
Here, is the Riemann 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 Riemann integrable but is Generalized Riemann Integrable.
Consider the function
everywhere else.
It can be shown that is not Riemann integrable on
Let be given.
Consider gauge
everywhere else.
Let be a
-fine partition on
The Riemann 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 Riemann integrable or,
References
R.G. Bartle, D.R. Sherbert, Introduction to Real Analysis, John Wiley & sons
See Also
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