# How does the radius of convergence depend on the point about which the series is expanded?

For a given analytic germ $f(z)=\sum_{n=0}^{\infty} a_n (z-z_0)^n$, and a simple curve $\gamma: [0,1]\to \mathbb{C}$ such that $\gamma(0)=z_0$, suppose one may analytically continue $f(z)$ along $\gamma$. So for each point $\tilde{z} =\gamma(t)$ there is a corresponding power series $f(z;\tilde{z})=\sum_{n=0}^{\infty} \tilde{a}_n (z-\tilde{z})^n$ with its own radius of convergence $r(\tilde{z})$. My question is, what can be said about this function $r$? How regular is it? Is it continuous?

What about for more than one variable?

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Is there a similar result for $f$ a function of more than one complex variable? –  AppliedSide Jan 20 '11 at 5:21