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Re: Re: Improper integral
*To*: mathgroup at smc.vnet.net
*Subject*: [mg44689] Re: [mg44667] Re: Improper integral
*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>
*Date*: Fri, 21 Nov 2003 05:13:12 -0500 (EST)
*References*: <6C03C616-19CE-11D8-864F-00039311C1CC@mimuw.edu.pl> <bpfftf$lvs$1@smc.vnet.net> <200311200816.DAA01569@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
On 20 Nov 2003, at 17:16, Jean-Claude Poujade wrote:
> Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote in message
> news:<bpfftf$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
>
>
In fact my question about the usefulness of concept of PrincipalValue
of an integral was motivated by curiosity (stimulated by your original
posting). I was surprised that I had (as it seemed to me) never come
across any use of it. But that was not actually true; after Daniel
Lichtblau pointed out that it was used in distribution theory I looked
up one of the books that I once had read (more precisely skimmed
through) and it was there, in some detail in fact. The principal value
of an integral is actually itself an integral but not of a function but
of a distribution.
The other question is whether Mathematica should have a full
implementation of this. At this time it seems that it only computes
principal values of integrals with singularities at points in the
complex plane. In fact distribution theory uses precisely the principal
values of integrals of the type that appeared in your example, that is,
with singularities at infinity. The omission seems not to be an
accident. I have looked at other cases at it appears that
PrincipalValue for integrals from -Infinity to Infinity is not
implemented in any case. Considering this more carefully one notices
that the most obvious implementations of both kinds of principal
values ("finite" and "infinite") would conflict with one another. I
don't think it is an insurmountable difficulty but it seems to me that
the current state of affairs is not a bug but a deliberate decision.
One reason may be is that the principal value is so easy to compute
(just take a suitable limit), that it does not appear worth the effort
given that the problem of implementing both types of principal values
is not entirely trivial.
Andrzej Kozlowski
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