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MathGroup Archive 2007

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Re: Question on PrincipalValue in Integrate

  • To: mathgroup at smc.vnet.net
  • Subject: [mg80689] Re: Question on PrincipalValue in Integrate
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Wed, 29 Aug 2007 04:12:44 -0400 (EDT)
  • Organization: The Open University, Milton Keynes, UK
  • References: <fb0u04$j8r$1@smc.vnet.net>

Jung-Tsung Shen wrote:

> A question on the "PrincipalValue" in the option of the command, Integr=
ate:
>
> 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 =CB=9C y || y =CB=9C 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?

For comparison, here is what I get on my Wintel system with version 5.2
and 6.0.1.

In[1]:=
$Version

Out[1]=
5.2 for Microsoft Windows (June 20, 2005)

In[2]:=
Integrate[1/(y-x),{x,-d,d},PrincipalValue\[Rule]True]

Out[2]=
       y             y             y
If[Re[-] >= 1 || Re[-] <= 0 || Im[-] != 0,
       d             d             d

            d
   -Log[1 - -] - Log[y] + Log[d + y],
            y

               1
   Integrate[------, {x, -d, d},
             -x + y

                      y                 y
    Assumptions -> Im[-] == 0 && 0 < Re[-] < 1,
                      d                 d

    PrincipalValue -> True]]

In[3]:=
Integrate[1/(y - x), {x, -d, d}, PrincipalValue ->
    True, Assumptions -> 0 < y < d]

Out[3]=
-Log[d - y] + Log[d + y]

Same expressions, but this time evaluated with version 6.0.1.

In[1]:= $Version

Out[1]= 6.0 for Microsoft Windows (32-bit) (June 19, 2007)

In[2]:= Integrate[1/(y - x), {x, -d, d}, PrincipalValue -> True]

During evaluation of In[2]:= Limit::ldir: Value of Direction -> Sign[d]=

should be a number or Automatic.

During evaluation of In[2]:= Limit::ldir: Value of Direction -> -Sign[d=
]
should be a number or Automatic.

Out[2]=
       y             y              y
If[Re[-] >= 1 || Re[-] <= -1 || Im[-] != 0,
       d             d              d

   -Log[-d + y] + Log[d + y] + Log[-Sign[d]] - Log[Sign[d]],

               1
   Integrate[------, {x, -d, d},
             -x + y

                      y                  y
    Assumptions -> Im[-] == 0 && -1 < Re[-] < 1, PrincipalValue -> Tr=
ue]]
                      d                  d

In[3]:= Integrate[1/(y - x), {x, -d, d}, PrincipalValue -> True,
  Assumptions -> 0 < y < d]

Out[3]=
     d + y
Log[-----]
     d - y

--
Jean-Marc




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