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?