pages/complex-analysis/one-complex-variable/cauchys-theorem.md


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

# {{ page.title }}

{% theorem Cauchy’s Theorem (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

$$
\int_{\gamma} f(z) \, dz = 0
$$
{% endtheorem %}

{% proof %}
{% endproof %}

{{ page.title }} has a converse:

{% theorem * 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 $G$.
{% endtheorem %}