
Re: cauchy principal value double integral
Posted:
Dec 6, 2012 4:56 AM


Am 04.12.2012 10:08, schrieb daniel.lichtblau0@gmail.com: > On Monday, December 3, 2012 2:18:53 AM UTC6, 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

