Uniform Convergence

**kinkong** · September 1st 2011, 08:04 PM

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.
I need some help please

**kinkong** · September 4th 2011, 04:05 PM

i need some help guys...please help me!

**girdav** · September 5th 2011, 07:24 AM

What are $z_1$ and $z_2$ ?

Thread: Uniform Convergence

LinkBack

Thread Tools

Display

Uniform Convergence

Re: Uniform Convergence

Re: Uniform Convergence

Similar Math Help Forum Discussions

Uniform convergence vs pointwise convergence

Show uniform convergence implies equicontinuity and uniform boundedness

Pointwise convergence to uniform convergence

Pointwise Convergence vs. Uniform Convergence

Uniform Continuous and Uniform Convergence

Search Tags