From 777f9d3fd8caf56e6bc6999a4b05379307d0733f Mon Sep 17 00:00:00 2001
From: Justin Gassner <justin.gassner@mailbox.org>
Date: Tue, 12 Sep 2023 07:36:33 +0200
Subject: Initial commit

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 .../one-complex-variable/cauchys-theorem.md        | 39 ++++++++++++++++++++++
 1 file changed, 39 insertions(+)
 create mode 100644 pages/complex-analysis/one-complex-variable/cauchys-theorem.md

(limited to 'pages/complex-analysis/one-complex-variable/cauchys-theorem.md')

diff --git a/pages/complex-analysis/one-complex-variable/cauchys-theorem.md b/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
+# 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 $G$.
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