---
title: Cauchy's Theorem
parent: One Complex Variable
grand_parent: Complex Analysis
nav_order: 2
# cspell:words
---

# {{ page.title }}

{: .theorem-title }
> {{ page.title }} (Homotopy Version)
>
> Let $G$ be a connected open subset of the complex plane.
> Let $f : G \to \CC$ be a holomorphic function.
> If $\gamma_0$, $\gamma_1$ are homotopic closed curves in $G$, then
>
> $$
> \int_{\gamma_0} \! f(z) \, dz =
> \int_{\gamma_1} \! f(z) \, dz 
> $$
>
> If $\gamma$ is a null-homotopic closed curve in $G$, then
>
> $$
> \int_{\gamma} f(z) \, dz = 0
> $$

{% proof %}
{% endproof %}

{{ page.title }} has a converse:

{: .theorem-title }
> Morera's Theorem
>
> Let $G \subset \CC$ be open and let $f : G \to \CC$ be a continuous function.
> If $\int_{\gamma} f(z) \, dz = 0$ for every contour $\gamma$ contained in $G$,
> then $f$ is holomorphic in $G$.