---
title: "Cauchy's Theorem and Deforming a Contour"
description: "Prove that an analytic function integrates to zero around a closed loop it is analytic inside, and use that to slide contours freely."
canonical: https://lightmysky.com/learn/mathematics/cauchys-theorem-and-deforming-a-contour-mt_aacYFVN-HH
source: https://lightmysky.com/learn/mathematics/cauchys-theorem-and-deforming-a-contour-mt_aacYFVN-HH.md
retrieved: 2026-09-12
---

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# Cauchy's Theorem and Deforming a Contour

Prove that an analytic function integrates to zero around a closed loop it is analytic inside, and use that to slide contours freely.

Subject: Mathematics · Area: Complex Analysis · Ages 20 to 21
Page: https://lightmysky.com/learn/mathematics/cauchys-theorem-and-deforming-a-contour-mt_aacYFVN-HH

## Ready when they can

- Derive Cauchy's theorem from Green's theorem plus the Cauchy-Riemann equations
- Deform a contour past a region of analyticity without changing the integral
- Explain what fails when a singularity lies inside the loop

## Lesson: Why closed loops often integrate to zero

Cauchy's theorem says an analytic function integrates to zero around a closed loop it is analytic inside. Split f into u plus iv and the loop integral becomes two real circulations. Green plus Cauchy-Riemann zeroes both curls, and multiplying out shows the real part is u dx minus v dy.

**Example.** z squared is analytic everywhere, so its integral around any closed loop is 0. But 1 divided by z blows up at the origin, so its unit circle integral is 2 pi i, not zero. The missing analyticity at one interior point is the whole story.

You may slide a contour across any region where f stays analytic without changing the integral. The old and new paths differ by loops with zero integral. This freedom replaces awkward contours with friendly circles: a preserved value like 3 plus 4 stays 7 on the deformed loop.

**Tip.** A single interior singularity breaks the spell. Never apply the theorem after spotting a blowup inside, and never deform across one. The unit circle with 1 divided by z is the classic nonzero counterexample to keep in mind.

**Recap.** Analytic inside means zero on the loop and free sliding nearby, while one singularity inside changes everything.

## Practice

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

## Needs first

- [Conservative Fields and Path Independence](https://lightmysky.com/learn/mathematics/conservative-fields-and-path-independence-mt_5djKpI5F33)
- [Contour Integrals Along Parametrised Paths](https://lightmysky.com/learn/mathematics/contour-integrals-along-parametrised-paths-mt_CaJ7ruAv8P)
- [Green's Theorem in the Plane](https://lightmysky.com/learn/mathematics/greens-theorem-in-the-plane-mt_UDgivmPVEJ)

## Opens up

- [Cauchy's Integral Formula and Derivatives of Every Order](https://lightmysky.com/learn/mathematics/cauchys-integral-formula-and-derivatives-of-every-order-mt_mViFwrFNWQ)