From 55e29f03ac3b537843f85892a1323e1f46321675 Mon Sep 17 00:00:00 2001 From: Preston Pan Date: Sat, 11 Apr 2026 13:14:04 -0700 Subject: new articles and snippets --- mindmap/Cauchy's Theorem.org | 29 +++++++++++++++++++++++++++++ 1 file changed, 29 insertions(+) create mode 100644 mindmap/Cauchy's Theorem.org (limited to 'mindmap/Cauchy's Theorem.org') diff --git a/mindmap/Cauchy's Theorem.org b/mindmap/Cauchy's Theorem.org new file mode 100644 index 0000000..01d507a --- /dev/null +++ b/mindmap/Cauchy's Theorem.org @@ -0,0 +1,29 @@ +:PROPERTIES: +:ID: b243a8c0-ca7c-40e6-95b4-0f725a1a361f +:END: +#+title: Cauchy's Theorem +#+author: Preston Pan +#+description: Spinning around the complex plane. +#+options: broken-links:t + +* Introduction +Cauchy's theorem is the analogue of Green's Theorem for complex variables. It is a part of many equivalent statements made about analytic +functions. For example: +- exact differentials are closed. +- The harmonic conjugates of analytic functions satisfy the Cauchy-Riemann equations. +- Closed differentials describe [[id:6f2aba40-5c9f-406b-a1fa-13018de55648][conservative force]] fields. +- Harmonic functions satisfy Laplace's Equation. +- Under contour integration, the closed differentials are exactly those differentials which also satisfy the Cauchy-Riemann equations. +- A function is analytic iff it satisfies the Cauchy-Riemann equations. +- Analytic functions are conformal mappings except at their zeros. +and many more, are statements about the same set of objects, posed in different ways. +* Theorem +#+begin_theorem +\begin{align} +\oint_{D}f(z)dz = 0 +\end{align} +#+end_theorem + +#+begin_proof +The closed differentials in the complex plane under contour integration are exactly those which satisfy the Cauchy-Riemann equations. +#+end_proof -- cgit v1.3