Singularities and Their Behavior

The behavior of a holomorphic function near an isolated singularity is completely characterized by its Laurent expansion. The three types of singularities — removable, poles, and essential — exhibit dramatically different behavior.

Removable Singularities

Theorem5.5Riemann's removable singularity theorem

Let $f$ be holomorphic on $0 < |z - z_0| < R$ . The following are equivalent:

$z_0$ is a removable singularity (the principal part of the Laurent series vanishes).
$\lim_{z \to z_0} f(z)$ exists and is finite.
$f$ is bounded in some punctured neighborhood of $z_0$ .
$\lim_{z \to z_0} (z-z_0)f(z) = 0$ .

ExampleDetecting removable singularities

The function $f(z) = \frac{z^3 - 1}{z - 1}$ has a removable singularity at $z = 1$ since $\lim_{z \to 1} f(z) = \lim_{z \to 1} (z^2 + z + 1) = 3$ . After removal, $f$ extends to an entire function.

Similarly, $\frac{e^z - 1}{z}$ has a removable singularity at $z = 0$ with limit $1$ .

Poles

Definition5.4Pole

A function $f$ has a pole of order $m$ at $z_0$ if the Laurent series has the form

$f(z) = \frac{a_{-m}}{(z-z_0)^m} + \cdots + \frac{a_{-1}}{z-z_0} + a_0 + a_1(z-z_0) + \cdots$

with $a_{-m} \neq 0$ . A pole of order $1$ is called a simple pole. Equivalently, $f$ has a pole of order $m$ at $z_0$ if and only if $g(z) = (z-z_0)^m f(z)$ has a removable singularity at $z_0$ with $g(z_0) \neq 0$ .

Theorem5.6Characterization of poles

The following are equivalent:

$f$ has a pole at $z_0$ .
$\lim_{z \to z_0} |f(z)| = \infty$ .
$1/f$ has a zero at $z_0$ (after defining $1/f(z_0) = 0$ ).
$f$ has a pole of order $m$ if and only if $1/f$ has a zero of order $m$ .

ExampleResidue at a pole

For a simple pole at $z_0$ : $\text{Res}(f, z_0) = \lim_{z \to z_0}(z-z_0)f(z)$ .

For a pole of order $m$ : $\text{Res}(f, z_0) = \frac{1}{(m-1)!}\lim_{z \to z_0} \frac{d^{m-1}}{dz^{m-1}}[(z-z_0)^m f(z)]$ .

For $f(z) = \frac{1}{(z-1)^2(z+1)}$ : at $z=1$ (pole of order 2), $\text{Res}(f, 1) = \frac{d}{dz}\frac{1}{z+1}\big|_{z=1} = -\frac{1}{4}$ .

Essential Singularities

Theorem5.7Casorati-Weierstrass theorem

If $f$ has an essential singularity at $z_0$ , then for every $\varepsilon > 0$ , the image $f(\{0 < |z-z_0| < \varepsilon\})$ is dense in $\mathbb{C}$ .

Theorem5.8Picard's great theorem

If $f$ has an essential singularity at $z_0$ , then in every punctured neighborhood of $z_0$ , $f$ takes every complex value infinitely often with at most one exception.

ExampleEssential singularity of $e^{1/z}$

The function $e^{1/z}$ has an essential singularity at $z = 0$ . To find $z$ near $0$ with $e^{1/z} = w$ (for any $w \neq 0$ ), solve $1/z = \log w + 2\pi i k$ , giving $z_k = 1/(\log w + 2\pi ik) \to 0$ as $k \to \infty$ . The only omitted value is $w = 0$ (since $e^{1/z} \neq 0$ ), confirming Picard's theorem.

Behavior Summary

RemarkClassification summary

| Type | $\lim_{z \to z_0} f(z)$ | Laurent principal part | Behavior near $z_0$ | |------|-------------------------|----------------------|---------------------| | Removable | Finite limit exists | None (empty) | $f$ extends holomorphically | | Pole of order $m$ | $\infty$ | Finitely many terms | $|f(z)| \sim C/|z-z_0|^m$ | | Essential | No limit exists | Infinitely many terms | $f$ takes almost all values |

The trichotomy is exhaustive: every isolated singularity falls into exactly one of these categories.