# Radius of convergence of this series

Let $f$ an analytic function of complex variable, with radius of convergence $r$. What about the radius of $$x^pf(x),$$where $p$ is a complex parameter ?

What is the center of the disk in which the power series for $f$ converges? –  Hans Engler Dec 11 '12 at 18:41
(I assume you mean convergence radius around $x=0$.) This function is only single valued around $x = 0$ if $p \in \mathbb{Z}$. It extends over $x=0$ only if $p \in \mathbb{N}$ in which case its radius of converge is the same as that of $f(x)$.