12. Remark 2.3.2 - Characterization of Isolated Singularities

We now have almost enough information to characterize the different types of Isolated Singularities. We list them here:

Removable Singularities

(Analytic) Let $f : U \to C$ be holomorphic on some open set $U \subseteq C$ . Suppose $z_{0}$ is an isolated singularity of $f$ , and let $B_{r} (z_{0}) - {z_{0}} \subseteq U$ by definition. If the Laurant series of $f$ 1;;vanishes, then $z_{0}$ is a removable singularity
(Geometric) Let $f : U \to C$ be holomorphic on some open set $U \subseteq C$ . Suppose $z_{0}$ is an isolated singularity of $f$ , and accordingly let $B_{r} (z_{0}) - {z_{0}} \subseteq U$ . Suppose there exists some $R \leq r$ such that 1;; $f$ is bounded on $B_{R} (z_{0}) - {z_{0}}$ , then by 11. Theorem 2.3.2 we have that $z_{0}$ is a removable singularity.

Poles

(Anayltic) Let $f : U \to C$ be holomorphic on some open set $U \subseteq C$ . Suppose $z_{0}$ is an isolated singularity of $f$ , and let $B_{r} (z_{0}) - {z_{0}} \subseteq U$ by definition. If the principal part of the Laurant Series 1;;is a polynomial (in $\frac{1}{x}$ ), then $z_{0}$ is a pole of $f$ .
(Geometric) Let $f : U \to C$ be holomorphic on some open set $U \subseteq C$ . Suppose $z_{0}$ is an isolated singularity of $f$ , and let $B_{r} (z_{0}) - {z_{0}} \subseteq U$ by definition. If for all $M > 0$ , 1;;there exists some $0 < ϵ < r$ such that $| f (z) | > M$ for all $z \in B_{ϵ} (z_{0}) - {z_{0}}$ , then $z_{0}$ is a pole.

Essential Singularities

(Analytic) Let $f : U \to C$ be holomorphic on some open set $U \subseteq C$ . Suppose $z_{0}$ is an isolated singularity of $f$ , and let $B_{r} (z_{0}) - {z_{0}} \subseteq U$ by definition. If the principal part of the Laurant Series is 1;;infinite, then $z_{0}$ is an essential singularity.
Let $f : U \to C$ be holomorphic on some open set $U \subseteq C$ . Suppose $z_{0}$ is an isolated singularity of $f$ , and let $B_{r} (z_{0}) - {z_{0}} \subseteq U$ by definition. Now suppose that there is some $M > 0$ such 1;;that for any $0 < ϵ < r$ there exists a $z \in B_{ϵ} (z_{0}) - {z_{0}}$ such that $| f (z) | \leq M$ . Further suppose that $| f |$ is unbounded in $B_{ϵ} (z_{0}) - {z_{0}}$ . Then $z_{0}$ is an essential singularity of $f$ . This has a more explicit characterization in the following theorem.

Remark

Notice that by the intermediate value theorem, we have that $| f |$ hits every positive real number. What this means is that for every $r > 0$ , we can find a $z \in U$ such that $| f (z) | = r$ . Casorati and Weierstrauss go one step further and say essentially that $f (B_{ϵ}) \cap δ B_{r} (0)$ is dense for every $r$ .