Question about radius of convergence of fractional power series

[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?

 

[fresh_42]

You can conclude this. Just look at it as a variable substitution $u=\sqrt{z}$. Then $f(\sqrt{z})=f(u)=\sum a_nu^n = \sum a_n\sqrt{z}^n$ as you wrote. Now we have $\sqrt{3}<|u|=|\sqrt{z}|<\sqrt{5}$ as region of conversion in $u$ and get $3 <|u|^2=|z|<5\,.$