Re: Strange PV results from Integrate

*To*: mathgroup at smc.vnet.net*Subject*: [mg51275] Re: [mg51269] Strange PV results from Integrate*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Tue, 12 Oct 2004 01:57:38 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

On 11 Oct 2004, at 14:25, Carlos Felippa wrote: > > 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. > > Mathematica 5.0 certainly gives the correct values to the first four integrals. As for the last one, well, that's what happens: Integrate[Log[1 + Cos[x]], {x, -Pi, Pi}] (-Pi)*Log[4] N[%] -4.35517 Although you assert this is wrong the result given by Integrate is confirmed by the following numerical check: <<NumericalMath`NLimit` f[t_?NumericQ]:=NIntegrate[Log[1+Cos[x]],{x,-Pi+t,Pi-t}] NLimit[f[t],t->0] -4.35517 However, as I have never heard of the first (or any other) Szego theorem I shall leave to the experts the question who is right and wrong here.