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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