Polylogarithm Integration

*To*: mathgroup at smc.vnet.net*Subject*: [mg46084] Polylogarithm Integration*From*: D <D at D.gov>*Date*: Thu, 5 Feb 2004 04:03:03 -0500 (EST)*Organization*: University of Oslo, Norway*Sender*: owner-wri-mathgroup at wolfram.com

Hi, I've been told to try Mathematica, that is supposed to be more efficient for integrations. Here is my very first try with Mathematica 5: Integrate[ PolyLog[2, Exp[I*(x - y)]], {y, 0, 2*Pi} ] Instead of 0 (at least if | Exp[I*x] | <= 1), after a long time, I get an uggly: \!\(If[x >= 2\ \[Pi] || x <= 0, \[Pi]\^3 - \[Pi]\ Log[\(-\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \ x\)\)]\^2 + 2\ \[ImaginaryI]\ \[Pi]\^2\ Log[ 1 - \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ x\)] - 2\ \[ImaginaryI]\ \[Pi]\^2\ Log[ 1 - \[ExponentialE]\^\(\[ImaginaryI]\ x\)], Integrate[ PolyLog[2, \[ExponentialE]\^\(\[ImaginaryI]\ \((x - y)\)\)], {y, 0, 2\ \[Pi]}, Assumptions -> \(! \((x >= 2\ \[Pi] || x <= 0)\)\)]]\) If I replace x by any numerical real value, it works. But if I put an imaginary numerical, for example x=I, I get: I*( 4*Pi^2 - PolyLog[2,1/e] + PolyLog[3,1/e] ) that, I believe, is wrong. Questions: Is it a bug in Mathematica 5? Or, did I misunderstood something about polylogarithms? How can I workaround the problem? Thanks for your help. D.