:PROPERTIES:
:ID:       b243a8c0-ca7c-40e6-95b4-0f725a1a361f
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#+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