       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]
>
>
>
> 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:=
\$Version

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

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

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

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

Same expressions, but this time evaluated with version 6.0.1.

In:= \$Version

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

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

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

should be a number or Automatic.

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

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

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

--
Jean-Marc

```

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