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Re: Improper integral
*To*: mathgroup at smc.vnet.net
*Subject*: [mg44667] Re: Improper integral
*From*: poujadej at yahoo.fr (Jean-Claude Poujade)
*Date*: Thu, 20 Nov 2003 03:16:33 -0500 (EST)
*References*: <6C03C616-19CE-11D8-864F-00039311C1CC@mimuw.edu.pl> <bpfftf$lvs$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote in message news:<bpfftf$lvs$1 at smc.vnet.net>...
> More on this theme: can somebody explain what the concept of a
> "principal value" of an integral is good for? I have been a
> professional mathematician for years and have been involved in several
> different areas of research, and yet never came across any use for it.
> I have more then a dozen texts on analysis yet none of them mentions
> it. The only books where I can find it mentioned are books for
> physicists and engineers (one is the well known text by Riley, Hobson
> and Bence, the other a book in Polish) and they both give one line
> definitions without any examples of use (and do not mention poles at
> infinity). At first sight it seems a pretty trivial and useless
> concept, so I would like to know if it really has any serious
> applications.
>
> Andrzej Kozlowski
Andrzej,
I'm sorry my subject was perhaps not enough explicit.
I'm not arguing about the usefulness of the "principal value"
concept : I just want to be allowed to remove as easily as
possible some exceptions (for instance in probabilities)
and I want Mathematica to comply with generally agreed definitions.
Let me quote Borowski & Borwein's dictionary of mathematics :
< Cauchy principal value : the evaluation of an improper integral
< on the interval [-Infinity,Infinity] as the symmetric (two-sided)
< limit of the integral on intervals of the form [-n,n].
< This may well converge even if the sum of the two ordinary improper
< integrals over [-Infinity,a] and [a,Infinity] does not.
Given that definition, it seems to me that Mathematica doesn't return
what it should. That's all I mean.
---
jcp
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