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Topic: cauchy principal value double integral
Replies: 5   Last Post: Dec 6, 2012 4:56 AM

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Roland Franzius

Posts: 458
Registered: 12/7/04
Re: cauchy principal value double integral
Posted: Dec 6, 2012 4:56 AM
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Am 04.12.2012 10:08, schrieb daniel.lichtblau0@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




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