Residue

Definition

Let $c \in \mathbb{C}$, let $r > 0$, and let $f$ be holomorphic on the punctured disc $0 < |z - c| < r$, so that $c$ is an isolated singularity of $f$. Let $f(z) = \sum_{n \in \mathbb{Z}} a_n (z - c)^n$ be the Laurent expansion of $f$ on this punctured disc. The residue of $f$ at $c$ is the coefficient $$\operatorname{Res}(f, c) = a_{-1}.$$

Remarks

For each $\rho$ with $0 < \rho < r$, $\operatorname{Res}(f, c) = \frac{1}{2 \pi i} \oint_{|z - c| = \rho} f(z) \, dz$. This follows from the coefficient formula for $n = -1$. The residue is $0$ at a removable singularity. It can be $0$ at a pole too: $\operatorname{Res}(1/z^2, 0) = 0$. For example, $\operatorname{Res}(1/z, 0) = 1$ and $\operatorname{Res}(e^{1/z}, 0) = 1$. Residues at simple poles are easy to compute, see residue at a simple pole.

Explanations

L1 human

When you integrate a Laurent series term by term around a small circle, every term gives $0$ except the term $a_{-1}/(z - c)$, which gives $2 \pi i a_{-1}$. So the residue is the one coefficient that the integral remembers.