Laurent Series

Laurent series extend Taylor series to include negative powers, enabling the representation of holomorphic functions near isolated singularities. They are essential for classifying singularities and computing residues.

Definition and Convergence

Definition5.2Laurent series

A Laurent series centered at $z_0$ is a doubly infinite series

$\sum_{n=-\infty}^{\infty} a_n(z - z_0)^n = \sum_{n=0}^{\infty} a_n(z-z_0)^n + \sum_{n=1}^{\infty} a_{-n}(z-z_0)^{-n}.$

The first sum is the analytic part (or regular part) and the second is the principal part.

Theorem5.4Laurent series theorem

Let $f$ be holomorphic on the annulus $A = \{z : r < |z - z_0| < R\}$ where $0 \leq r < R \leq \infty$ . Then $f$ has a unique Laurent series representation

$f(z) = \sum_{n=-\infty}^{\infty} a_n(z-z_0)^n$

converging absolutely and uniformly on compact subsets of $A$ , with coefficients

$a_n = \frac{1}{2\pi i}\oint_\gamma \frac{f(z)}{(z-z_0)^{n+1}}\,dz$

where $\gamma$ is any positively oriented simple closed curve in $A$ encircling $z_0$ .

Examples

ExampleLaurent series of $e^{1/z}$

The function $e^{1/z}$ is holomorphic on $\mathbb{C} \setminus \{0\}$ . Substituting $w = 1/z$ into $e^w = \sum w^n/n!$ :

$e^{1/z} = \sum_{n=0}^{\infty} \frac{1}{n! z^n} = 1 + \frac{1}{z} + \frac{1}{2z^2} + \frac{1}{6z^3} + \cdots$

The principal part has infinitely many terms, so $z = 0$ is an essential singularity.

ExampleLaurent series from partial fractions

For $f(z) = \frac{1}{z(z-1)}$ on the annulus $0 < |z| < 1$ , use partial fractions:

$f(z) = \frac{-1}{z} + \frac{1}{z-1} = -\frac{1}{z} - \frac{1}{1-z} = -\frac{1}{z} - \sum_{n=0}^\infty z^n.$

On the annulus $|z| > 1$ :

$f(z) = -\frac{1}{z} + \frac{1}{z}\cdot\frac{1}{1-1/z} = -\frac{1}{z} + \sum_{n=1}^\infty \frac{1}{z^n} = \sum_{n=2}^\infty \frac{1}{z^n}.$

Uniqueness and Coefficient Formulas

RemarkUniqueness of Laurent series

The Laurent series representation in a given annulus is unique. This is because the coefficients $a_n$ are determined by the integral formula, which depends only on the values of $f$ on any circle in the annulus. Different annuli (e.g., $0 < |z| < 1$ vs. $|z| > 1$ ) can yield different Laurent series for the same function.

RemarkResidue from Laurent coefficients

The coefficient $a_{-1}$ in the Laurent series of $f$ around $z_0$ is the residue of $f$ at $z_0$ :

$\text{Res}(f, z_0) = a_{-1} = \frac{1}{2\pi i}\oint_\gamma f(z)\,dz.$

This connects Laurent series directly to the residue theorem.

Classification of Singularities

Definition5.3Isolated singularity classification

Let $f$ be holomorphic on $0 < |z - z_0| < R$ with Laurent series $\sum a_n(z-z_0)^n$ . The singularity at $z_0$ is:

A removable singularity if $a_n = 0$ for all $n < 0$ (the principal part vanishes).
A pole of order $m$ if $a_{-m} \neq 0$ and $a_n = 0$ for all $n < -m$ .
An essential singularity if infinitely many $a_n$ with $n < 0$ are nonzero.

ExampleExamples of singularity types

$\frac{\sin z}{z} = 1 - z^2/6 + \cdots$ has a removable singularity at $z = 0$ .
$\frac{1}{z^3}$ has a pole of order $3$ at $z = 0$ .
$e^{1/z}$ has an essential singularity at $z = 0$ .
$\frac{\cos z - 1}{z^2} = -\frac{1}{2} + \frac{z^2}{24} - \cdots$ has a removable singularity at $z = 0$ .