Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Henstock-Kurzweil integral"
m (→‎Illustration: typo)
 
(2 intermediate revisions by 2 users not shown)
Line 1: Line 1:
−
The '''Henstock-Kurzweil integral''' (also known as the '''Generalized Reimann integral''') is one of the most widely applicable generalizations of the [[Integral|Reimann integral]], but it also uses a strikingly simple and elegant idea. It was developed independantly by [[Ralph Henstock]] and [[Jaroslav Kurzweil]]
+
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==
Line 6: Line 6:
 
Let <math>L\in\mathbb{R}</math>
 
Let <math>L\in\mathbb{R}</math>
  
−
We say that <math>f</math> is ''Generalised Reimann 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,
+
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{\dot{P}})</math> is the [[Reimann sum]] of <math>f</math> on <math>[a,b]</math> with respect to <math>\mathcal{\dot{P}}</math>
+
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 [[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==
 
==Illustration==
−
The utility of the Henstock -Kurzweil integral is demonstrated by this function, which is not Reimann integrable but is Geeralized Reimann Integrable.
+
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>
  
−
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\left( \frac{1}{n}\right) =n\forall n\in\mathbb{N}</math>
  
 
<math>f(x)=0</math> everywhere else.
 
<math>f(x)=0</math> everywhere else.
  
−
It can be shown that <math>f</math> is not Reimann integrable on <math>[0,1]</math>
+
It can be shown that <math>f</math> is not Riemann integrable on <math>[0,1]</math>
  
 
Let <math>\varepsilon>0</math> be given.
 
Let <math>\varepsilon>0</math> be given.
 +
 
Consider [[gauge]] <math>\delta:[0,1]\rightarrow\mathbb{R}^+</math>
 
Consider [[gauge]] <math>\delta:[0,1]\rightarrow\mathbb{R}^+</math>
  
Line 34: Line 36:
 
Let <math>\mathcal{\dot{P}}</math> be a <math>\delta</math>-fine [[Partition of an interval|partition]] on <math>[0,1]</math>
 
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.  
+
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>
 
<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>
+
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==
 
==References==
Line 45: Line 46:
  
 
==See Also==
 
==See Also==
−
*[[Integral|Reimann Integral]]
+
*[[Integral|Riemann Integral]]
−
*[[Gauge]]
 
  
 
[[Category:Calculus]]
 
[[Category:Calculus]]
  
 
{{stub}}
 
{{stub}}

Latest revision as of 08:46, 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.

Definition

Let $f:[a,b]\rightarrow\mathbb{R}$

Let $L\in\mathbb{R}$

We say that $f$ is Generalized Riemann Integrable on $[a,b]$ if and only if, $\forall\epsilon>0$, there exists a gauge $\delta:[a,b]\rightarrow\mathbb{R}^+$ such that,

if $\mathcal{\dot{P}}$ is a $\delta$-fine tagged partition on $[a,b]$, then $|L-S(f,\mathcal{\dot{P}})|<\epsilon$

Here, $S(f,\mathcal{\dot{P}})$ is the Riemann sum of $f$ on $[a,b]$ with respect to $\mathcal{\dot{P}}$


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 $f:[0,1]\rightarrow\mathbb{R}$

$f\left( \frac{1}{n}\right) =n\forall n\in\mathbb{N}$

$f(x)=0$ everywhere else.

It can be shown that $f$ is not Riemann integrable on $[0,1]$

Let $\varepsilon>0$ be given.

Consider gauge $\delta:[0,1]\rightarrow\mathbb{R}^+$

$\delta\left( \frac{1}{n}\right) =\frac{\varepsilon}{k2^{k+1}}$

$\delta(x)=1$ everywhere else.

Let $\mathcal{\dot{P}}$ be a $\delta$-fine partition on $[0,1]$

The Riemann sum will have maximum value only when the tags are of the form $t_i=\frac{1}{n}$, $n\in \mathbb{N}$. Also, each tag can be shared by at most two divisions.

$S(f,\mathcal{\dot{P}})\leq\sum_{k=1}^{\infty}\frac{\varepsilon}{2^k}<\varepsilon$

But as $\varepsilon>0$ is arbitrary, we have that $f$ is Generalized Riemann integrable or, $\int_0^1 f(x)dx=0$

References

R.G. Bartle, D.R. Sherbert, Introduction to Real Analysis, John Wiley & sons

See Also

This article is a stub. Help us out by expanding it.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Henstock-Kurzweil_integral&oldid=90105"