# 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.”

Title radius of convergence of a complex function RadiusOfConvergenceOfAComplexFunction 2013-03-22 14:40:33 2013-03-22 14:40:33 rspuzio (6075) rspuzio (6075) 6 rspuzio (6075) Theorem msc 30B10