Laurent Series

Two-sided expansion on an annulus, and what the principal part says about a singularity.

The idea

A Laurent series expands a function in powers of $(z - z_{0})$ of both signs, and it is the expansion to use at a point where the function fails to be holomorphic. A Taylor series exists only when the function is holomorphic at $z_{0}$ itself: $\dfrac{e^{z}}{z^{2}}$ is holomorphic at every point of $\mathbb{C}$ except $0$, and near $0$ it has no Taylor expansion.

Theorem (Laurent's theorem).

Let $f$ be holomorphic on an annulus $r < |z - z_{0}| < R$, where $r = 0$ is allowed. Then throughout the annulus $f(z) = \sum_{n = -\infty}^{\infty} a_{n} (z - z_{0})^{n}.$

The annulus is the ring of points, shown in the figure, lying farther than $r$ but nearer than $R$ from the center $z_{0}$; with $r = 0$ it is a punctured disk. In the expansion, the terms with $n \ge 0$ form an ordinary power series, the regular part; the terms with $n < 0$ form the principal part.

Two consequences follow. The expansion belongs to the annulus, not to the point alone, so one function can have different Laurent series about the same $z_{0}$ on different annuli. And when the annulus is a punctured disk, the principal part determines the type of the singularity at $z_{0}$.

Ways to work on it

• Walkthrough. Build a Laurent series, read off its principal part, and classify the singularity.
• Practice. Classify an isolated singularity from the function's expansion.
• Hardest. Expand one function on two different annuli to see the series depend on the region.

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