---
tag: 032HKYS329
kind: statement
role: theorem
name: Residue theorem
aliases: [cauchys-residue-theorem]
msc: ["30E20"]
uses: [0JFGKKWWFK, 0X7SBS8E0F, 00FWNP9TXY, 0DG0Z72CQD]
sources:
  - url: https://en.wikipedia.org/wiki/Residue_theorem
    locator: Statement of Cauchy's residue theorem
same_as:
  wikidata: Q830513
  wikipedia: Residue theorem
---

## Statement

Let $U \subseteq \mathbb{C}$ be a simply connected open set. This means that $U$ is connected, and each closed path in $U$ can be continuously deformed in $U$ to a constant path.
Let $a_1, \dots, a_n$ be distinct points of $U$, and let $f$ be [[holomorphic|0DG0Z72CQD]] on $U \setminus \{a_1, \dots, a_n\}$.
Let $\gamma$ be a closed, continuous and piecewise continuously differentiable path in $U \setminus \{a_1, \dots, a_n\}$.
Then
$$\oint_\gamma f(z) \, dz = 2 \pi i \sum_{k=1}^{n} n(\gamma, a_k) \operatorname{Res}(f, a_k),$$
where $n(\gamma, a_k)$ is the [[winding number|0X7SBS8E0F]] of $\gamma$ around $a_k$ and $\operatorname{Res}(f, a_k)$ is the [[residue|0JFGKKWWFK]] of $f$ at $a_k$.

## Remarks

If $\gamma$ is a simple closed curve traveled once counterclockwise, then $\oint_\gamma f(z) \, dz$ is $2 \pi i$ times the sum of the residues at the points $a_k$ inside $\gamma$.
With no points $a_k$, the theorem is [[Cauchy's integral theorem|01MPZXYVYF]].
For example, $\oint_{|z| = 2} \frac{dz}{z^2 + 1} = 2 \pi i \left( \frac{1}{2i} - \frac{1}{2i} \right) = 0$, by the [[residues at the simple poles|01991NCP7H]] $i$ and $-i$.
The theorem is the main tool for the [[evaluation of real integrals|0476QH9846]].

## Explanations

### L1 human

To integrate around a closed curve, you only need to know the residues at the bad points inside the curve.
Each bad point contributes $2 \pi i$ times its residue, once for each turn of the curve around it.