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?