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Homeradius of convergence of a complex function

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# radius of convergence of a complex function

Let $f$ be an analytic function defined in a disk of radius $R$ about a point $z_{0}\in\mathbb{C}$. Then the radius of convergence of the Taylor series of $f$ about $z_{0}$ is at least $R$.

For example, the function $a(z)=1/(1-z)^{2}$ is analytic inside the disk $|z|<1$. Hence its the radius of covergence of its Taylor series about $0$ is at least $1$. By direct examination of the Taylor series we can see that its radius of convergence is, in fact, equal to $1$.

Colloquially, this theorem is stated in the sometimes imprecise but memorable form “The radius of convergence of the Taylor series is the distance to the nearest singularity.”

Type of Math Object:

Theorem

Major Section:

Reference

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## Mathematics Subject Classification

30B10*no label found*

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