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.

```

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