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

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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: Question about radius of convergence of fractional power series
Loading...