Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Radius of convergence problem

  1. Nov 6, 2012 #1
    consider the rational function :

    [tex]f(x,z)=\frac{z}{x^{z}-1}[/tex]
    [tex]x\in \mathbb{R}^{+}[/tex]
    [tex]z\in \mathbb{C}[/tex]

    We wish to find an expansion in z that is valid for all x and z. a Bernoulli-type expansion is only valid for :
    [tex] \left | z\ln x \right |<2\pi[/tex]
    Therefore, we consider an expansion around z=1 of the form :
    [tex] \frac{z}{x^{z}-1}=\sum_{n=0}^{\infty}f_{n}(x)(z-1)^{n}[/tex]
    Where [itex] f_{n}(x)[/itex] are suitable functions in x that make the expansion converge. the first two are given by :
    [tex] f_{0}(x)=\frac{1}{x-1}[/tex]

    [tex] f_{1}(x)=\frac{x-x\ln x -1}{(x-1)^{2}}[/tex]
    now i have two questions :
    1-in the literature, is there a similar treatment to this specific problem !? and under what name !?
    2- how can we find the radius of convergence for such an expansion !?
     
    Last edited: Nov 6, 2012
  2. jcsd
  3. Nov 9, 2012 #2
    it's not so hard to prove that the functions [itex]f_{n}(x) [/itex] have the general form :
    [tex]f_{n}(x)=\frac{(-\ln x)^{n-1}}{n!}\left(n\text{Li}_{1-n}(x^{-1})-\ln x\;\text{Li}_{-n}(x^{-1})\right) [/tex]
     
    Last edited: Nov 9, 2012
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook