19 - Casorati-Weierstrass or Weierstrass Theorem

Casorati-Weierstrass or Weierstrass Theorem

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

Suppose $f:U\setminus \{a\}\to \mathbb{C}$ is a holomorphic function with isolated essential singularity at $a$ then

For all $\rho >0$ with $B(a,\rho )\subseteq U$ then

$$f(B(a,\rho )\setminus \{a\})\text{ is dense in }\mathbb{C}$$

In other words the closure of $f(B(a,\rho )\setminus \{a\})$ is all of $\mathbb{C}$

Proof

Suppose for contradiction that there is some $\rho >0$ such that

$$z_0\in \mathbb{C}\text{ is not a limit point of }f(B(a,\rho )\setminus \{a\})$$

Then function

$$g(z)=\frac{1}{f(z)-z_0}$$

is bounded and non-vanishing on $B(a,\rho )\setminus \{a\}$

Hence by Riemann’s Removable Singularity Theorem then
$g$ can be extended to a holomorphic function on all of $B(a,\rho )$

But then

$$f(z)=z_0+\frac{1}{g(z)}$$

has at most a pole at $a$ which is a contradiction