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Lets start with a simple example: We take the analytic function $$ f:R -> R , f(x)= \frac{1}{1+x^2}$$ if we do a Taylor expansion around 0 we get a power series with convergence radius of 1. How can we see this easily? We know that the radius of convergence is the distance of the expansion point to the first point of singularity in the complex domain for the complex continuation of the function. In this case the complex continuation of f has 2 singularities, i and -i. So for each expansion point we can easily determine the convergence radius of the power series. This is all clear to me.

What about higher dimension analytical functions, like: $$g: R^3 -> R, g(x,y,z)=\frac{1}{1+ x^2 + y^2 + z^2}$$ you can do a higher dimensional taylor series as well. Is there some similiar theorem about convergence of the power series?

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  • $\begingroup$ Nobody? Any suggestions how to improve the question? $\endgroup$ – lalala Aug 22 '17 at 15:42
  • $\begingroup$ The condition for a real function to be analytic is similar, as you can find here. Based on this can you reformulate the question to be more precise? $\endgroup$ – Victor Palea Aug 26 '17 at 16:24

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