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Henstock-Kurzweil integral

Revision as of 08:25, 16 February 2008 by Shreyas patankar (talk | contribs) (→‎Definition)

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

Let $L\in\mathbb{R}$

We say that $f$ is Generalised Reimann 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 Reimann 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.

See Also

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