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MathGroup Archive 2004

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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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