Notes - Complex Analysis MT23, Casorati-Weierstrass theorem

Created: March 22, 2024

Flashcards

Can you state the Casorati-Weierstrass theorem, which gives 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 $\mathbb C$.

Quickly 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 $z = z _ 0$, a contradiction.



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