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.