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.