Re: Strange PV results from Integrate
- To: mathgroup at smc.vnet.net
- Subject: [mg51275] Re: [mg51269] Strange PV results from Integrate
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Tue, 12 Oct 2004 01:57:38 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
On 11 Oct 2004, at 14:25, Carlos Felippa wrote:
>
> Could somebody pls check if these results persist in 5.0? Thx.
>
> $Version
> "4.2 for Power Macintosh (August 27, 2002)"
>
> Integrate[Log[ 2+2*Cos[x] ], {x,-Pi,Pi}]
> 0 (* correct *)
>
> Integrate[Log[ 2*(1+Cos[x]) ], {x,-Pi,Pi}]
> identical integrand returns unevaluated
>
> Integrate[Log[ -2-2*Cos[x] ], {x,-Pi,Pi}]//InputForm;
> (2*I)*Pi^2 (* correct *)
>
> Integrate[Log[ -2*(1+Cos[x]) ], {x,-Pi,Pi}]
> identical integrand returns unevaluated
>
> Integrate[Log[ 1+ Cos[x] ], {x,-Pi,Pi}]
> (I/2)*Pi^2 - 2*Pi*Log[1 + I] - Pi*Log[2] (* wrong *)
>
> FullSimplify[Integrate[Log[ 1+ Cos[x] ], {x,-Pi,Pi}]]
> -(Pi*Log[4]) (* wrong *)
>
> These integrals arise on applying the first Szego theorem
> to some benchmark infinite Toeplitz matrices. Results
> labelled "wrong" contradict the theorem.
>
>
Mathematica 5.0 certainly gives the correct values to the first four
integrals. As for the last one, well, that's what happens:
Integrate[Log[1 + Cos[x]], {x, -Pi, Pi}]
(-Pi)*Log[4]
N[%]
-4.35517
Although you assert this is wrong the result given by Integrate is
confirmed by the following numerical check:
<<NumericalMath`NLimit`
f[t_?NumericQ]:=NIntegrate[Log[1+Cos[x]],{x,-Pi+t,Pi-t}]
NLimit[f[t],t->0]
-4.35517
However, as I have never heard of the first (or any other) Szego
theorem I shall leave to the experts the question who is right and
wrong here.