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MathGroup Archive 2007

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Re: Question on PrincipalValue in Integrate

  • To: mathgroup at smc.vnet.net
  • Subject: [mg80702] Re: Question on PrincipalValue in Integrate
  • From: dimitris <dimmechan at yahoo.com>
  • Date: Wed, 29 Aug 2007 04:19:29 -0400 (EDT)
  • References: <fb0u04$j8r$1@smc.vnet.net>

On 28 =C1=FD=E3, 13:39, "Jung-Tsung Shen" <jus... at gmail.com> wrote:
> A question on the "PrincipalValue" in the option of the command, Integrat=
e:
>
> Mathematica (v5.0 Mac) gives the following command
>
> Integrate[1/(y-x), {x, -d, d}, PrincipalValue -> True]
>
> the answer
>
> If[y > 0 && y < d, I Pi - Log[d - y] + Log[d + y], Integrate[1/(-x +
> y), {x, -d, d}, Assumptions -> d   y || y   0]]
>
> But shouldn't the first part of the answer by - Log[d - y] + Log[d +
> y], without the I Pi? This can be computed using the very definition
> of the principal value.
>
> Any comments are greatly appreciated.
>
> Thanks.
>
> JT
>
> PS. Recently I have found several verified bugs in v5.0. Maybe it's
> time to upgrade to v6.0?

In[12]:=
$Version
Integrate[1/(y - x), {x, -d, d}, PrincipalValue -> True, Assumptions -
> Element[y,Reals] && d > 0]
PiecewiseExpand[%]

Out[12]=
"5.2 for Microsoft Windows (June 20, 2005)"

Out[13]=
If[d <= y || y <= 0, -Log[-d + y] + Log[d + y], Integrate[1/(-x + y),
{x, -d, d}, Assumptions -> y > 0 && d > y,
   PrincipalValue -> True]]

Out[14]=
Piecewise[{{-Log[d - y] + Log[d + y], d - y > 0 && y > 0}}, -Log[-d +
y] + Log[d + y]]

Cheers
Dimitris

PS
It is high time we both upgrade!



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