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authorJustin Gassner <justin.gassner@mailbox.org>2024-02-14 07:24:38 +0100
committerJustin Gassner <justin.gassner@mailbox.org>2024-02-14 07:24:38 +0100
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tree67fa2b79d5c48b50d4e394858af79c88c1447e51 /pages/complex-analysis/one-complex-variable/cauchys-theorem.md
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downloadsite-28407333ffceca9b99fae721c30e8ae146a863da.tar.zst
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diff --git a/pages/complex-analysis/one-complex-variable/cauchys-theorem.md b/pages/complex-analysis/one-complex-variable/cauchys-theorem.md
index 15412bc..2445b8b 100644
--- a/pages/complex-analysis/one-complex-variable/cauchys-theorem.md
+++ b/pages/complex-analysis/one-complex-variable/cauchys-theorem.md
@@ -3,37 +3,34 @@ 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
->
-> $$
-> \int_{\gamma} f(z) \, dz = 0
-> $$
+{% 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-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 $G$.
+{% 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 %}