\begin{definition}[Definition:Locally Uniform Convergence/Complex Functions]
Let $U \subseteq \C$ be an open set.
Let $\sequence {f_n}$ be a be a sequence of functions $f_n : U \to \C$.
For $z \in U$, let $\map {D_r} z$ be the open disk of radius $r$ about $z$.
Then $f_n$ '''converges to $f$ locally uniformly''' {{iff}}:
:for each $z \in U$, there is an $r \in \R_{>0}$ such that $f_n$ converges uniformly to $f$ on $\map {D_r} z$
and:
:$\map {D_r} z \subseteq U$
{{Proofread}}
\end{definition}