The residue is the local coefficient that records the first-order obstruction to making a meromorphic differential holomorphic at an isolated singularity. It is small enough to be read from one Laurent coefficient, but strong enough to control contour integrals around the singularity. This chapter develops the residue from Laurent expansions to the residue theorem.

The main setting is one complex variable. The notation $D(a,r)$ denotes the open disc with centre $a$ and radius $r$, and the punctured disc is $D(a,r)\setminus\{a\}$. A positively oriented circle around $a$ will be denoted by

\begin{align*} \gamma_{a,\rho}(t)&=a+\rho e^{it}, & 0\leq t\leq 2\pi, \end{align*}

where the parameter $\rho$ is chosen so that the circle lies in the domain under discussion.

The differential notation $f(z)\,dz$ is used when the coordinate-invariant flavour matters. For functions on planar domains, the same information is contained in the integral of $f(z)\,dz$, and the residue of the function $f$ at $a$ is the residue of the differential $f(z)\,dz$ in the coordinate $z$. Residues sit between local singularity theory and global contour integration: they turn many integral computations into coefficient extraction, and they expose how poles, removable singularities, and essential singularities contribute to a contour integral.

## Definition

### Laurent Coefficient

The first source of residues is the [Laurent expansion](/page/Laurent%20Series) on a punctured disc. If $f$ is holomorphic on $0<|z-a|<r$, then it has a Laurent series

\begin{align*} \sum_{n\in\mathbb Z} c_n(z-a)^n \end{align*}

converging normally on compact annuli. The coefficient $c_{-1}$ is singled out by integration.

[example: Basic Residue] For \begin{align*} f(z)=\frac{1}{z-a}, \end{align*} we can write the Laurent expansion about $a$ as the single Laurent monomial \begin{align*} f(z)=1\cdot (z-a)^{-1}. \end{align*} Thus the coefficient multiplying $(z-a)^{-1}$ is $1$, so by the definition of residue, \begin{align*} \operatorname{Res}(f,a)=1. \end{align*} For \begin{align*} g(z)=\frac{1}{(z-a)^2}, \end{align*} the Laurent expansion about $a$ is \begin{align*} g(z)=1\cdot (z-a)^{-2}. \end{align*} This expansion has coefficient $1$ on $(z-a)^{-2}$ and coefficient $0$ on $(z-a)^{-1}$. Therefore \begin{align*} \operatorname{Res}(g,a)=0. \end{align*} The residue records only the coefficient of $(z-a)^{-1}$, not the coefficient of the most singular term. [/example]

The example shows that the relevant coefficient is not the most singular term. The obstruction is that the principal part of a Laurent expansion may contain many negative powers, but only one of them contributes to a small contour integral around the singularity. The residue isolates exactly that coefficient, separating the integral contribution from the rest of the local singular behaviour.

### Formal Local Definition

To turn the preceding contour intuition into a reusable local invariant, we need a definition that depends only on the Laurent expansion near the point. The formal point is to ignore all coefficients except the one attached to $(z-a)^{-1}$, because that is the unique term whose integral around a sufficiently small circle can survive.

[definition: Residue] Fix $a\in\mathbb C$. Let $\mathcal{M}_a$ be the set of germs at $a$ of functions holomorphic on some punctured disc $D(a,r)\setminus\{a\}$. The residue at $a$ is the map \begin{align*} \operatorname{Res}_a: \mathcal{M}_a \to \mathbb C, \end{align*} defined by $\operatorname{Res}_a(f)=c_{-1}$, where $c_{-1}$ is the coefficient of $(z-a)^{-1}$ in the Laurent expansion \begin{align*} f(z)=\sum_{n\in\mathbb Z}c_n(z-a)^n \end{align*} about $a$. For a meromorphic differential $f(z)\,dz$ in the coordinate $z$, its residue at $a$ is denoted $\operatorname{Res}_a(f(z)\,dz)$ and is defined to be $\operatorname{Res}_a(f)$. [/definition]

This definition is local: it depends only on the germ of $f$ near $a$, not on the behaviour of $f$ elsewhere in the domain.

We will also use the two-variable notation $\operatorname{Res}(f,a)$ for the same number as $\operatorname{Res}_a(f)$. The subscript form emphasizes the local functional at the point $a$; the two-variable form is convenient in formulas where several singularities are being summed.

## Laurent Expansion and the Residue

Changing $r$ does not change the coefficient, because Laurent expansions agree on overlaps of annuli. The coefficient can also be described by a small contour integral, which is often the most useful formulation. It explains why a local Laurent coefficient can be recovered without knowing the whole Laurent series, and it prepares the passage from small circles to larger contours.

Concretely, if $\gamma_r(t)=a+re^{it}$ is a sufficiently small positively oriented circle around $a$ containing no other singularities, then the Laurent coefficient of $(z-a)^{-1}$ is recovered by