Definition:Radius of Convergence/Complex Domain

Definition
Let $\xi \in \C$ be a complex number.

For $z \in \C$, let:
 * $\displaystyle \map f z = \sum_{n \mathop = 0}^\infty a_n \paren {z - \xi}^n$

be a power series about $\xi$.

The radius of convergence is the extended real number $R \in \overline \R$ defined by:


 * $R = \displaystyle \inf \set {\cmod {z - \xi}: z \in \C, \sum_{n \mathop = 0}^\infty a_n \paren {z - \xi}^n \text{ is divergent} }$

where a divergent series is a series that is not convergent.

As usual, $\inf \O = +\infty$.

Also see

 * Existence of Radius of Convergence of Complex Power Series, which shows that:


 * If $\cmod {z - \xi} < R$, then the power series defining $f \paren z$ is absolutely convergent
 * If $\cmod {z - \xi} > R$, then the power series defining $f \paren z$ is divergent.