1. Dec 4, 2011

### jackmell

Suppose I have the Laurent series with region of convergence given below:

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

Can I conclude the Laurent-Puiseux series:

$$f(\sqrt{z})=\sum_{n=-\infty}^{\infty} a_n \left(\sqrt{z}\right)^n$$

has a region of convergence $3<|z|<5$? 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