Definition:Locally Uniform Convergence/Complex Functions

Definition
Let $U \subseteq \C$ be an open set.

Let $\left\langle{f_n}\right\rangle$ be a be a sequence of functions $f_n : U\to\C$.

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

Then $f_n$ converges to $f$ locally uniformly :
 * for each $z \in U$, there is an $r > 0$ such that $f_n$ converges uniformly to $f$ on $D_r \left({z}\right)$

and:
 * $D_r \left({z}\right) \subseteq U$

Also see

 * Equivalence of Local Uniform Convergence and Compact Convergence