\begin{definition}[Definition:Radius of Convergence/Complex Domain]
Let $\xi \in \C$ be a complex number.
For $z \in \C$, let:
:$\ds \map f z = \sum_{n \mathop = 0}^\infty a_n \paren {z - \xi}^n$
be a power series about $\xi$.
The '''radius of convergence''' is the extended real number $R \in \overline \R$ defined by:
:$R = \ds \inf \set {\cmod {z - \xi}: z \in \C, \sum_{n \mathop = 0}^\infty a_n \paren {z - \xi}^n \text{ is divergent} }$
where a divergent series is a series that is not convergent.
As usual, $\inf \O = +\infty$.
\end{definition}