KU Math 800: Lecture 2/6/2026
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[definition=%counter%] Let $f:\Omega\setminus z_0 \to \CC$ be holomorphic. We say that $z_0$ is a removable singularity of $f$ if we can define $f(z_0)$ such that $f:\Omega \to \CC$ is holomorphic. [/definition]
[example=%counter%] Let $\Omega = \CC^*$ and $f(z) = z$. Then $0$ is removable. [/example]
[theorem=%counter% (Riemann)] Suppose $f:\Omega\setminus z_0 \to \CC$ is holomorphic. If $f$ is bounded on $\Omega\setminus z_0$, then $z_0$ is removable. [/theorem]
[proof] Without loss of generality, assume that $\Omega$ be a small disc centered at $z_0$. We claim that for $z \in \Omega \setminus z_0$, we have
\[f(z) = \frac{1}{2\pi i}\int_{C=\partial D} \frac{f(\zeta)}{\zeta - z} \dd\zeta.\]Granted the claim, we set
\[f(z_0) = \frac{1}{2\pi i} \int_{C} \frac{f(\zeta)}{\zeta - z_0} \dd\zeta.\]By Theorem 5.4, the function
\[f(z) = \frac{1}{2\pi i} \int_{C} \frac{f(\zeta)}{\zeta - z} \dd{\zeta}\]is holomorphic in $z$. To prove the claim above, use the usual keyhold contour argument. [/proof]
[corollary=%counter%] Suppose that $f$ has an isolated singularity at the point of $z_0$. Then $z_0$ is a pole of $f$ if and only if $|f(z)| \to \infty$ as $z \to z_0$. [/corollary]
[definition=%counter%] Let $f:\Omega \setminus z_0 \to \CC$ be holomorphic. We say that $z_0$ is an essential singularity if $z_0$ is not removable and not a pole. [/definition]
[theorem=%counter% (Casorati-Weierstrass)] Suppose $f$ is holomorphic in the punctured disc $D_r(z_0)\setminus\{z_0\}$ and has an essential singularity at $z_0$. Then, the image of $D_r(z_0) \setminus \{z_0\}$ under $f$ is dense in the complex plane. [/theorem]