Isolated singularity

In the context of Complex Analysis, given the function $f$ we would say that $z_{0} \in C$ is an isolated singularity if there exists $r > 0$ such that $f$ is holomorphic in

B (z_{0}, r) - {z_{0}} .

We can define therefore the Laurent series for $f$ around $z_{0}$

f (z) = \sum_{n = 0}^{\infty} a_{n} (z - z_{0})^{n} + \sum_{n = 1}^{\infty} \frac{b_{n}}{(z - z_{0})^{n}}

It is said that the singularity is removable if $b_{n} = 0$ for $n \geq 1$ . Another special case: pole.

Classification

The principal part $\sum_{n \geq 1} b_{n} (z - z_{0})^{- n}$ determines the type:

Removable: $b_{n} = 0$ for all $n \geq 1$ . Equivalently, $lim_{z \to z_{0}} f (z)$ exists and is finite; assigning $f (z_{0}) := a_{0}$ makes $f$ holomorphic at $z_{0}$ .
Pole of order $k$ : $b_{k} \neq 0$ and $b_{n} = 0$ for $n > k$ — finitely many principal terms. Equivalently, $| f (z) | \to \infty$ as $z \to z_{0}$ . See pole.
Essential: infinitely many $b_{n} \neq 0$ . The function behaves erratically: $f (B (z_{0}, ε) ∖ {z_{0}})$ is dense in $C$ for every $ε > 0$ (Casorati–Weierstrass); by Picard's theorem it omits at most one value, attained infinitely often in every punctured neighborhood. Example: $e^{1 / z} = \sum_{n \geq 0} \frac{1}{n! z^{n}}$ at $0$ .

The trichotomy is exhaustive: every isolated singularity is exactly one of these three types.

Note: singularities need not be isolated — e.g. $z \mapsto 1 / \sin (1 / z)$ has poles accumulating at $0$ , so no Laurent expansion exists on a full punctured disk around it. At $z_{0} = \infty$ , apply the same theory to $g (z) = f (1 / z)$ .