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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 ?

Thanks in advance

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What is the center of the disk in which the power series for $f$ converges? – Hans Engler Dec 11 '12 at 18:41
the conter is the origin – MAK Dec 11 '12 at 18:45
up vote 2 down vote accepted

(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 $f(x)$.

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