# Uniform Convergence

• Sep 1st 2011, 08:04 PM
kinkong
Uniform Convergence
I need some help in proving the following lemma

Lemma . if $f_j$ are holomorphic functions on $U$ and $f_j \rightarrow$ uniformly on compact subsets of $U$, then any derivatives
$(\frac{\partial}{\partial z_1})^l (\frac{\partial}{\partial z_2})^k f_j$
converges to
$(\frac{\partial}{\partial z_1})^l (\frac{\partial}{\partial z_2})^k f$
Uniformly on compacts sets.

Proof:
i tried to fix $P \in U$ and choose $r > 0$ such that $\overline{D}^2(P,r) \subseteq U$. i have some problem expressing $f_j on D^2(P,r)$ as a Cauchy integral of $f_j$ on $\partial D^2(P,r)$ and differentiating it under integral sign.
What are $z_1$ and $z_2$?