Singularities & Poles

Removable, pole, or essential — read it off the Laurent principal part.

The idea

An isolated singularity of a function $f$ is a point $z_{0}$ such that $f$ is holomorphic on a punctured disk $0 < |z - z_{0}| < r$ but not at $z_{0}$ itself. Every isolated singularity is of one of three kinds.

A removable singularity is repaired by a choice of value: $\dfrac{z^{2} - 1}{z - 1}$ equals $z + 1$ away from $1$, and defining its value at $1$ to be $2$ makes it holomorphic there. A pole is a point where $|f(z)|$ grows without bound as $z \to z_{0}$, at the rate of a definite power, as $\dfrac{1}{(z-1)^{2}}$ does at $1$. An essential singularity is neither: near $0$, $\sin(1/z)$ is bounded along the real axis, unbounded along the imaginary axis, and comes arbitrarily close to every complex value in every neighbourhood of $0$.

The Laurent series of $f$ about $z_{0}$ separates the three kinds by its principal part. No negative powers means removable; finitely many, the most negative being $(z - z_{0})^{-m}$, means a pole of order $m$; infinitely many means essential. For a quotient no expansion is needed: if the numerator vanishes to order $j$ at $z_{0}$ and the denominator to order $k$, the series begins at the power $j - k$.

Ways to work on it

• Walkthrough. Classify removable, pole, and essential singularities from the Laurent series.
• Practice. Find the order of a pole at the origin.
• Hardest. Classify a singularity from its Laurent expansion and extract a coefficient.

Not sure where to start? Take the ten-question placement test.