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MathGroup Archive 2004

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Re: Polylogarithm Integration - Bis

  • To: mathgroup at smc.vnet.net
  • Subject: [mg46135] Re: Polylogarithm Integration - Bis
  • From: Bill Rowe <readnewsciv at earthlink.net>
  • Date: Sat, 7 Feb 2004 04:02:32 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

On 2/6/04 at 4:15 AM, D at D.gov (D) wrote:

>Some more example for the integral:

>Integrate[PolyLog[2, Exp[I*(x - y)]], {y, 0, 2*Pi}]

>with x=5/2, I get 0 as expected. But with x=2.5 I get

>19.634954084936204 + I*1.7763568394002505`*^-15

>which is not close to zero!

When you use x = 5/2 you are asking Mathematica to use exact numbers. So, Integrate returns the correct exact result of 0. But when you use 2.5 you are asking Mathematica to use machine precision numbers. Integrate does not seem to be coded to avoid problems with loss of precision that occurs with machine precision numbers. If you want to use machine precision, use NIntegrate instead of Integrate.

When I use NIntegrate with x = 2.5 on this problem I get

-1.551478671979467*^-6 - 7.615056976939538*^-7*I

and a warning that numerical integration is converging too slowly. Clearly, this result is close to zero. And coupled with the warning strongly indicates there are issues with numerical precision for this particular integral.
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