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Strange PV results from Integrate
*To*: mathgroup at smc.vnet.net
*Subject*: [mg51269] Strange PV results from Integrate
*From*: carlos at colorado.edu (Carlos Felippa)
*Date*: Mon, 11 Oct 2004 01:25:28 -0400 (EDT)
*Sender*: owner-wri-mathgroup at wolfram.com
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.
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