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Re: Strange PV results from Integrate


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.


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