
R: Re: cauchy principal value double integral
Posted:
Dec 5, 2012 3:09 AM


Since the integrand is a positive quantity, obviously the integral has to be positive. Principal value has nothing to do with this integral since there is not the cancellation effect that requires a change in sign of the integer and. Mathematica 5.1 answers (correctly) with the non convergence of integra l. Also Mathematica8 gives the same answer, with the option Principal ValueFalse.
I have divided the integration square in eight triangles by the two diagonals and consider the first one in the first quadrant: (0<x<1}&&(y,0,x) And cut away a small e0 part of x near zero. The integral then becomes
R[a,e0]=8*Integrate[Integrate[1/(x^2 + y^2)^a, {y, 0, x}], {x, e0, 1}]
and performing the integrals (first the internal) is immediate to find that
R[a_,e0_]:=4 F[a](1e0^(2(1a))/(1a)
where
F[a_]:= Hypergeometric2F1[1/2, a, 3/2, 1)
This last term results to be always positive 0<F[a]<1. So convergence/diverence of integral is determined by the other factor, i.e. convergence for e0 >0 requires that a<1; For instance a=1/2 R[1/2,0]=8 ArcSAinh[1]
Cheers, Rob
Messaggio originale Da: daniel.lichtblau0@gmail.com [mailto:daniel.lichtblau0@gmail.com] Inviato: marted=EC 4 dicembre 2012 10.10 A: mathgroup@smc.vnet.net Oggetto: Re: cauchy principal value double integral
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.
Daniel Lichtblau Wolfram Research
RSE SpA ha adottato il Modello Organizzativo ai sensi del D.Lgs.231/2001, in forza del quale l'assunzione di obbligazioni da parte della Societ=E0 avvie ne con firma di un procuratore, munito di idonei poteri. RSE adopts a Compliance Programme under the Italian Law (D.Lgs.231/2001). Ac cording to this RSE Compliance Programme, any commitment of RSE is taken by the signature of one Representative granted by a proper Power of Attorney. Le informazioni contenute in questo messaggio di posta elettronica sono ris ervate e confidenziali e ne e' vietata la diffusione in qualsiasi modo o for ma. Qualora Lei non fosse la persona destinataria del presente messaggio, La invitiamo a non diffonderlo e ad eliminarlo, dandone gentilmente comunicazi one al mittente. The information included in this email and any attachments are confidential and may also be privileged. If you are not the correct rec ipient, you are kindly requested to notify the sender immediately, to cancel it and not to disclose the contents to any other person.

