Definition:Locally Uniform Convergence/Complex Functions

Definition

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 if and only if:

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:

$\map {D_r} z \subseteq U$