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author | Justin Gassner <justin.gassner@mailbox.org> | 2024-02-14 07:24:38 +0100 |
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committer | Justin Gassner <justin.gassner@mailbox.org> | 2024-02-14 07:24:38 +0100 |
commit | 28407333ffceca9b99fae721c30e8ae146a863da (patch) | |
tree | 67fa2b79d5c48b50d4e394858af79c88c1447e51 /pages/complex-analysis/one-complex-variable/cauchys-theorem.md | |
parent | 777f9d3fd8caf56e6bc6999a4b05379307d0733f (diff) | |
download | site-28407333ffceca9b99fae721c30e8ae146a863da.tar.zst |
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-rw-r--r-- | pages/complex-analysis/one-complex-variable/cauchys-theorem.md | 45 |
1 files changed, 21 insertions, 24 deletions
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 %} |