---
title: "The Residue Theorem"
description: "Reduce an integral around a closed contour to a sum of residues, one number per singularity enclosed."
canonical: https://lightmysky.com/learn/mathematics/the-residue-theorem-mt_BN2biZ_zos
source: https://lightmysky.com/learn/mathematics/the-residue-theorem-mt_BN2biZ_zos.md
retrieved: 2026-09-12
---

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# The Residue Theorem

Reduce an integral around a closed contour to a sum of residues, one number per singularity enclosed.

Subject: Mathematics · Area: Complex Analysis · Ages 21 to 22
Page: https://lightmysky.com/learn/mathematics/the-residue-theorem-mt_BN2biZ_zos

## Ready when they can

- Compute a residue at a simple pole and at a pole of higher order
- State the residue theorem with the winding number and apply it to a contour enclosing several poles
- Explain why only the coefficient of the minus-one power survives the integration

## Lesson: One coefficient rules each loop

Residues turn loop integrals into arithmetic. List the enclosed singularities, read one coefficient at each, and add them with weights. This works because every Laurent power except minus one integrates to zero around a loop. All the rest own primitives, so only the minus one coefficient, the residue, survives.

At a simple pole the residue is a one step limit: multiply by z minus the point and evaluate. For z plus three over z minus one, cover up gives three plus one, which is 4. Higher order poles need derivatives one below the pole order. Each case extracts the same minus one coefficient. Watch the trap: 1 over z squared holds no such term, so its residue is 0.

The residue theorem weights every residue by its winding number. A loop circling a pole twice collects double, while a missed pole gives nothing. Clockwise traversal flips every sign through negative winding. Always check direction and enclosure before you apply the formula.

**Tip.** Shrink first, compute second. Picture the contour tightened onto tiny circles around each enclosed hole, then read residues off those circles. Never parametrize the big loop directly when residues will do.

**Recap.** Read one coefficient per hole, weight by winding, and scale by 2 pi i.

## Practice

17 questions on this page, each with its working shown.

## Needs first

- [Laurent Series and Classifying Isolated Singularities](https://lightmysky.com/learn/mathematics/laurent-series-and-classifying-isolated-singularities-mt_WoQV0gD1O-)

## Opens up

- [The Argument Principle and Rouche's Theorem](https://lightmysky.com/learn/mathematics/the-argument-principle-and-rouches-theorem-mt_J1dC428M84)
- [Evaluating Real Integrals by Residues](https://lightmysky.com/learn/mathematics/evaluating-real-integrals-by-residues-mt_Lj_HXtjzAh)