Definition:Locally Uniform Convergence/Complex Functions

Definition
Let $U \subseteq \C$ and $f_n : U \to \C$ be a sequence of functions.

For $z \in U$ let $D_r(z)$ be the disk of radius $r$ about $z$.

Then $f_n$ converges to $f$ locally uniformly if for each $z \in U$, there is an $r > 0$ such that $f_n$ converges uniformly to $f$ on $D_r(z)$ (and $D_r(z)$ is contained in $U$).