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MathGroup Archive 2003

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Re: Improper integral

  • To: mathgroup at smc.vnet.net
  • Subject: [mg44665] Re: Improper integral
  • From: "John Doty" <jpd at w-d.org>
  • Date: Thu, 20 Nov 2003 03:16:31 -0500 (EST)
  • References: <6C03C616-19CE-11D8-864F-00039311C1CC@mimuw.edu.pl> <bpfftf$lvs$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

In article <bpfftf$lvs$1 at smc.vnet.net>, "Andrzej Kozlowski"
<akoz at mimuw.edu.pl> wrote:


> More on this theme: can somebody explain what the concept of a  
> "principal value" of an integral is good for?

The place I've used it is Fourier analysis. In practical applications you
often need the rule that the tranform of an even function is real and the
transform of an odd function is imaginary. I don't know of any practical
situation where this can get you into trouble. If the function you're
tranforming has singularities in interesting places (and they often do!)
you're in "principal value" territory whether or not the author of your
favorite text has mentioned this particular piece of jargon.

Hildebrand in "Advanced Calculus for Applications" devotes four pages to
the Cauchy principal value. He notes "[principal values of improper
integrals] arise frequently in the aerodynamic theory of airfoils". He
doesn't actually spell out the applications motivating his examples, but
they are all Fourier integrals (integrating the product of a sine or
cosine with something else). 

> 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.

What's nontrivial about it is that it gives you a license to ignore
certain non-problems without having to go through elaborate arguments
involving limits.

-- 
| John Doty		"You can't confuse me, that's my job."
| Home: jpd at w-d.org
| Work: jpd at space.mit.edu


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