Re: Strange PV results from Integrate
- To: mathgroup at smc.vnet.net
- Subject: [mg51288] Re: Strange PV results from Integrate
- From: carlos at colorado.edu (Carlos Felippa)
- Date: Tue, 12 Oct 2004 01:57:54 -0400 (EDT)
- References: <ckd61e$522$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
carlos at colorado.edu (Carlos Felippa) wrote in message news:<ckd61e$522$1 at smc.vnet.net>...
> 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.
An update: a student with 5.0.1 did the tests and the unevaluated
integral cases return the values 0 and 2*I*Pi^2.
The last value is fine and agrees with Szego theorem. I had the wrong
sign in the real part of his Toeplitz asymptotic expansion. The PV
for Log[ a*(2+2*Cos[x])] in [-Pi,Pi] should be 2*Pi*Log[a],
so the - sign is correct if a=1/2.