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A series expansion in terms of the polylogarithm function

  1. Oct 25, 2012 #1
    we have the complex valued function :
    [tex]f(z)=\sum_{n=0}^{\infty}a_{n}\text{Li}_{-n}(z)\;\;\;\;\;\;\;(\left | z\right |<1)[/tex]
    we wish to recover the coefficients [itex]a_{n}[/itex] . the only thing i though would work is to try and come up with a function [itex]\phi(n,x)[/itex], such that:
    [tex]\int f(z)\phi(n,z)dz=a_{n}[/tex]
    but that's about as far as i've gotten. any help is appreciated.
    The question is motivated by the following:

    suppose that for some analytic function [itex]g(z)[/itex] we have the values of the function at +ive integers, so we can write a Taylor development :
    now suppose that the following summation is convergent in some open disk:
    Using the above Taylor expansion, and the definition of the Polylogarithm function, we have :
    [tex]\sum_{m=1}^{\infty}g(m)z^{m}=:f(z)=\sum_{n=0}^{\infty} a_{n}\text{Li}_{-n}(z)\;\;\left | z\right |<1[/tex]
    The plan is to recover the coefficients [itex]a_{n}[/itex], and the thus the Taylor expansion of [itex]g(z)[/itex]
    Last edited: Oct 26, 2012
  2. jcsd
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