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MathGroup Archive 2007

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Re: PolyLog help

  • To: mathgroup at smc.vnet.net
  • Subject: [mg73702] Re: PolyLog help
  • From: "sashap" <pavlyk at gmail.com>
  • Date: Mon, 26 Feb 2007 06:05:37 -0500 (EST)
  • References: <eroqhg$9iu$1@smc.vnet.net>

Hi Jude,

Defining series of dilogarithm is indeed convergent inside the unit
circle
only. Dilogarithm itself, however, is an analytic function in whole
complex
plane, with a branch-cut from 1 to Infinity. It is said, that
dilogarithm
is analytically continued outside the unit circle. I can not reveal
the precise details on how Mathematica computes the dilogarithm
outside
the unit circle, but you could think of your own way to do that using
appropriate relations listed on

http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/PolyLog2/17/01/01/

Another point to note, is the dilogarithm satisfies
a differential equation (see the aforementioned websites for details),
which
could be used for numerical integration to a point outside the unit
circle.

I hope this answered your question.

Oleksandr Pavlyk
Special Functions Developer
Wolfram Research

--------------

On Feb 24, 1:46 am, Jude Bowyer <j.bow... at ucl.ac.uk> wrote:
> Hi all,
>
> Does anyone know how Mathematica evaluates this function?
>
> As far as I understood, the dilogarithm function PolyLog[2,z] is a series like sum^{infinity}_{k=1} (z^k)/(k^2) where k is a positive integer and |z| <= 1.
>
> Mathematica returns to me a function PolyLog[2,-5R] in something I am trying to analyse, where 0 < R <= 1.. but when I plot in this range it produces a convergent output, whereas I would expect it to be divergent (and hence unplottable) beyond R = 0.2
>
> Does anyone know how Mathematica does this?




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