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What is the radius of convergence of a power series in two real variables?

If I were to fix one of the variables (i.e. make it a real constant), then would the radius of convergence simply be related to the nearest singularity?

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A power series in several complex variables doesn't have a radius of convergence. In general, the domain of convergence is a logaritmically convex Reinhardt domain, but not necessarily a ball. Consider for example the series $$\sum_{k=0}^\infty z^k w^k \qquad\text{and}\qquad \sum_{j=0}^\infty \sum_{k=0}^\infty z^j w^k$$ respectively.

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What if the variables were only allowed to take real values? – poirot May 14 '13 at 19:53
That doesn't matter. (Besides, the natural way to study power series is with complex variables.) – mrf May 14 '13 at 19:55
So only power series in a single variable have a radius of convergence in general. Ok, thanks. – poirot May 14 '13 at 19:57
What of I were to fix one of the variables? Then it would effectively become a power series in only one variable, right? – poirot May 14 '13 at 19:58
Right. For each fixed value of $w$ say, you get a power series in the other variable, which has a radius of convergence (depending on $w$). – mrf May 14 '13 at 20:00

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