---
title: "Contour Integrals Along Parametrised Paths"
description: "Integrate a complex function along a path by parametrising it, and bound the result by length times maximum modulus."
canonical: https://lightmysky.com/learn/mathematics/contour-integrals-along-parametrised-paths-mt_CaJ7ruAv8P
source: https://lightmysky.com/learn/mathematics/contour-integrals-along-parametrised-paths-mt_CaJ7ruAv8P.md
retrieved: 2026-09-12
---

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# Contour Integrals Along Parametrised Paths

Integrate a complex function along a path by parametrising it, and bound the result by length times maximum modulus.

Subject: Mathematics · Area: Complex Analysis · Ages 20 to 21
Page: https://lightmysky.com/learn/mathematics/contour-integrals-along-parametrised-paths-mt_CaJ7ruAv8P

## Ready when they can

- Evaluate a contour integral directly from a parametrisation
- Show the value is unchanged by reparametrising and reversed by reversing the path
- Apply the estimation lemma to bound an integral without evaluating it

## Lesson: Walking a function along a path

A contour integral accumulates f along a parametrised path. Put gamma of t into f, multiply by the velocity gamma prime, and integrate in t. On gamma of t equal t over 0 to 3, the integral of z dz becomes the integral of t dt, which is 9 over 2, or 4.5.

**Example.** Circles use gamma of t equal e to the it. Integrating 1 along the upper half from 1 to minus 1 gives end minus start, which is minus 2. Integrating 1 divided by z around the full circle gives 2 pi i, since the velocity cancels the integrand down to i.

Parametrising the same curve twice as fast changes nothing, because the velocity factor compensates exactly. Only the path plus its direction matter. Reversing direction negates the whole integral, so clockwise and anticlockwise runs are negatives of each other.

**Tip.** When exact evaluation is hopeless, bound instead. The estimation lemma says the modulus of the integral is at most the maximum of f times the path length. Length 3 with f at most 4 gives a bound of 12.

**Recap.** Feed the path into the function, weight by velocity, and watch direction while speed stays irrelevant.

## Practice

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

## Needs first

- [Harmonic Functions and What Analyticity Forces](https://lightmysky.com/learn/mathematics/harmonic-functions-and-what-analyticity-forces-mt_jqajP39XI4)
- [Complex Differentiability and the Cauchy-Riemann Equations](https://lightmysky.com/learn/mathematics/complex-differentiability-and-the-cauchy-riemann-equations-mt_nZNmZcl9Ss)
- [Vector Fields and Line Integrals](https://lightmysky.com/learn/mathematics/vector-fields-and-line-integrals-mt_VZxKEmsVgO)

## Opens up

- [Cauchy's Theorem and Deforming a Contour](https://lightmysky.com/learn/mathematics/cauchys-theorem-and-deforming-a-contour-mt_aacYFVN-HH)