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author | Justin Gassner <justin.gassner@mailbox.org> | 2023-09-12 07:36:33 +0200 |
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committer | Justin Gassner <justin.gassner@mailbox.org> | 2024-01-13 20:41:27 +0100 |
commit | 777f9d3fd8caf56e6bc6999a4b05379307d0733f (patch) | |
tree | dc42d2ae9b4a8e7ee467f59e25c9e122e63f2e04 /pages/complex-analysis/one-complex-variable/cauchys-theorem.md | |
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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 new file mode 100644 index 0000000..15412bc --- /dev/null +++ b/pages/complex-analysis/one-complex-variable/cauchys-theorem.md @@ -0,0 +1,39 @@ +--- +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 $G$. |