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Question about radius of convergence of fractional power series

  1. Dec 4, 2011 #1
    Suppose I have the Laurent series with region of convergence given below:

    [tex]f(z)=\sum_{n=-\infty}^{\infty} a_n z^n,\quad \sqrt{3}<|z|<\sqrt{5}[/tex]

    Can I conclude the Laurent-Puiseux series:

    [tex]f(\sqrt{z})=\sum_{n=-\infty}^{\infty} a_n \left(\sqrt{z}\right)^n[/tex]

    has a region of convergence [itex]3<|z|<5[/itex]? I don't know, maybe it's obvious now that I look at it. But I'd like to know what some of you think also. Is there something maybe that I'm not considering? Also, I wish to go in the reverse direction with that. That is, if I have a Laurent-Puiseux I believe converges in a region, can I consider just the region of convergence of the ordinary power series f(z) to prove the region of convergence of the Laurent-Puiseux series?

    Or are there other ways of determining the radius of convergence of fractional power series?
    Last edited: Dec 4, 2011
  2. jcsd
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