Re: cauchy principal value double integral
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- Subject: [mg128966] Re: cauchy principal value double integral
- From: Roland Franzius <roland.franzius at uos.de>
- Date: Thu, 6 Dec 2012 04:56:53 -0500 (EST)
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Am 04.12.2012 10:08, schrieb daniel.lichtblau0 at gmail.com:
> On Monday, December 3, 2012 2:18:53 AM UTC-6, Alex Krasnov wrote:
>> I am unclear how to interpret the following result (Mathematica 8.0.4):
>>
>>
>>
>> In: Integrate[1/(x^2+y^2), {x, -1, 1}, {y, -1, 1}, PrincipalValue -> True]
>>
>> Out: -4*Catalan
>>
>>
>>
>> How is Cauchy principal value defined in this case? Since the integrand is
>>
>> circularly symmetric around (0,0), excluding a shrinking neighborhood
>>
>> around (0,0) is not useful. Perhaps this result is merely an issue with
>>
>> option handling for multiple integrals?
>>
>>
>>
>> Alex
>
> That does indeed look like a bug.
>
The integral doesnt exist and what a Cauchy pincipal value option in two
dimensions may mean to the Integrate subsystem remains obscure.
In polar coordinates the integrand is
dx /\ dy /(x^2+y^2) = dr /\ dphi /r
Any integral containing the origin diverges logarithmically.
--
Roland Franzius