Classifying Singularities

Every Laurent series carries a principal part — the terms in negative powers of (z - z_0). Remarkably, the shape of that principal part sorts every isolated singularity into exactly one of three kinds. Just count the negative-power terms.

Let f have an isolated singularity at z_0, with Laurent series \sum a_n (z - z_0)^n on a punctured disk. Then:

Removable — no principal part (a_n = 0 for all n < 0). The limit exists; the singularity is an illusion.
Pole of order m — finitely many negative terms, the lowest being a_{-m}/(z - z_0)^m with a_{-m} \ne 0.
Essential — infinitely many negative terms. Behaviour near z_0 is wild.

One worked example of each

Removable: \dfrac{\sin z}{z} at z = 0. Divide the sine series by z:

\frac{\sin z}{z} = \frac{1}{z}\left(z - \frac{z^3}{3!} + \cdots\right) = 1 - \frac{z^2}{3!} + \frac{z^4}{5!} - \cdots.

There are no negative powers, so the singularity is removable: \lim_{z\to 0}\frac{\sin z}{z} = 1. Define the value to be 1 and the function is holomorphic.

Pole of order 3: \dfrac{1}{(z - 1)^3} at z = 1. Its Laurent series is the single term

\frac{1}{(z - 1)^3} = a_{-3}(z - 1)^{-3}, \qquad a_{-3} = 1.

The most negative power is -3, so this is a pole of order m = 3. (Order 1 poles are called simple.)

Essential: e^{1/z} at z = 0. We computed its Laurent series; it has every negative power:

e^{1/z} = 1 + \frac{1}{z} + \frac{1}{2!\,z^2} + \frac{1}{3!\,z^3} + \cdots.

Infinitely many negative terms — an essential singularity.

Near a pole, |f(z)| simply marches to infinity, no matter the direction of approach. An essential singularity is far stranger. Picard's great theorem says that in any punctured neighbourhood of an essential singularity, f takes every complex value — with at most a single exception — infinitely often. Watch e^{1/z} as z \to 0: along the positive real axis it explodes, along the negative real axis it vanishes, and along the imaginary axis it whirls forever around the unit circle. All values, all at once, packed into an arbitrarily small disk.

Pole versus essential, as you approach

Pick the singularity type with the control and move the test point toward z_0 = 0 along the positive real axis with the t slider (smaller t = closer). For the pole 1/z, |f| = 1/t climbs smoothly to infinity. For the essential e^{1/z}, |f| = e^{1/t} on the positive axis but would collapse to 0 from the negative side — the readout shows just how violently the magnitude depends on the approach.