13 - Convergence

Uniform Convergence on Compacts

Let $U$ be an open subset of $\mathbb{C}$

If $(f_n)$ is a sequence of functions defined on $U$ then

$f_n\to f$ uniformly on compacts if
For every compact subset $K$ of $U$ then sequence $(f_{n\mid K})$ converges uniformly to $f_{\mid K}$

Note that $f$ is continuous if $f_n$ are also continuous

Holomorphic of Uniform Convergence on Compacts

Suppose that $U$ is a domain

Let sequence of holomorphic functions $f_n:U\to \mathbb{C}$
Suppose $f_n$ converges to $f:U\to \mathbb{C}$ uniformly on compacts in $U$ then

$$f\text{ is holomorphic}$$

Proof

Since $U$ is open, for any $w\in U$ there exists $r>0$ such that $B(w,r)\subseteq U$
As $B(w,r)$ is convex then by Cauchy’s Theorem for Simply Connected Domains then
For every closed path $\gamma :[a,b]\to B(w,r)$ whose image lies in $B(w,r)$ then

$${\int }_{\gamma }{f}_n(z)dz=0\text{ for all }n\in N$$

But as ${\gamma }^{\ast }=\gamma ([a,b])$ is a compact subset of $U$ then $f_n\to f$ uniformly on ${\gamma }^{\ast }$ so

$$0={\int }_{\gamma }{f}_n(z)dz\to {\int }_{\gamma }f(z)dz$$

Hence integral of $f$ around any closed path in $B(w,r)$ is zero
So by Morera’s Theorem then $f$ is holomorphic