# Notes - Complex Analysis MT23, Casorati-Weierstrass theorem

> Source: https://ollybritton.com/notes/uni/part-a/mt23/complex-analysis/notes/casorati-weierstrass-theorem/ · Updated: 2024-03-22 · Tags: uni, notes

- [Course - Complex Analysis MT23](https://ollybritton.com/notes/uni/part-a/mt23/complex-analysis/)
	- [Notes - Complex Analysis MT23, Zeroes, singularities and poles](https://ollybritton.com/notes/uni/part-a/mt23/complex-analysis/notes/zeroes-singularities-and-poles/)

### Flashcards
@State the Casorati-Weierstrass theorem (a result about the closure of the image of a function with an isolated essential singularity).::
Suppose:

- $f : U \setminus \{a\} \to \mathbb C$, holomorphic
- $f$ has an isolated essential singularity at $a$.
- $U \subseteq \mathbb C$
- $a \in U$

Then:

- $\forall r > 0$ with $B(a, r) \subseteq U$, $f(B(a, r) \setminus \{a\})$ is dense in $\mathbb C$.

@Prove the Casorati-Weierstrass theorem, i.e. that if 

- $f : U \setminus \{a\} \to \mathbb C$, holomorphic
- $f$ has an isolated essential singularity at $a$.
- $U \subseteq \mathbb C$
- $a \in U$

then:

- $\forall r > 0$ with $B(a, r) \subseteq U$, $f(B(a, r) \setminus \{a\})$ is dense in $\mathbb C$.

::
Suppose, for a contradiction that $\exists \rho > 0$ s.t. $\exists z_0$ that is not a limit point in $f(B(a, \rho) \setminus \{a\})$. Then the function
$$
g(z) = \frac{1}{f(z) - z_0}
$$
is bounded and non-vanishing on $B(a, \rho)\setminus\{a\}$, so by Riemann's removable singularity theorem, it extends to a holomorphic function on $B(a, \rho)$. But then
$$
f(z) = \frac{1}{g(z)} + z_0
$$
which has at most a pole at $z = a$, a contradiction.

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