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diff --git a/mindmap/Cauchy's Theorem.org b/mindmap/Cauchy's Theorem.org new file mode 100644 index 0000000..38f6c06 --- /dev/null +++ b/mindmap/Cauchy's Theorem.org @@ -0,0 +1,27 @@ +: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 +If $D$ is a bounded domain with piecewise smooth boundary and $f$ is an analytic function which extends smoothly to $D \cup \partial D$, then $\oint_{D}f(z)dz = 0$. +#+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 |