Laurent Series

A Laurent series generalizes the Taylor Series to functions that may have isolated singularities, by allowing negative powers of $(z-z_0)$.

Let $f$ be holomorphic on an annulus $\{z\in \mathbb{C}:r<|z-z_0|<R\}$ where $0\le r<R\le \infty$. The Laurent series of $f$ centered at $z_0$ is

$f(z)={\sum }_{n=-\infty }^{\infty }{a}_n(z-z_0{)}^n$

convergent throughout the annulus. The series splits into two parts:

• the analytic part: ${\sum }_{n=0}^{\infty }{a}_n(z-z_0{)}^n$
• the principal part: ${\sum }_{n=1}^{\infty }{a}_{-n}(z-z_0{)}^{-n}$

Coefficient Formula

The coefficients are given by

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

where $\gamma$ is any positively oriented simple closed contour lying in the annulus and encircling $z_0$. This follows directly from the Cauchy Integral Formula. The coefficient $a_{-1}$ is the Residue of $f$ at $z_0$.

Distinction from Taylor Series

A Taylor series requires $f$ to be holomorphic in a full disk around $z_0$, so all negative-power coefficients vanish. A Laurent series is needed when $z_0$ is a singularity or is excluded from the domain, and the principal part captures the singular behavior.

Classification of Isolated Singularities

The structure of the principal part classifies the isolated singularity at $z_0$:

• Removable singularity: the principal part is identically zero (all $a_n=0$ for $n<0$).
• Pole of order $m$: the principal part has finitely many terms, with $a_{-m}\ne 0$ and $a_n=0$ for all $n<-m$.
• Essential singularity: the principal part has infinitely many nonzero terms.

Uniqueness

The Laurent expansion in a given annulus is unique. If two Laurent series agree on the annulus, their coefficients are identical term by term.

Convergence

The analytic part converges for $|z-z_0|<R$ and the principal part converges for $|z-z_0|>r$. Both converge absolutely and uniformly on compact subannuli $r^{\prime }\le |z-z_0|\le R^{\prime }$ with $r<r^{\prime }<R^{\prime }<R$.