Cauchy's Theorem

1. Introduction
2. Theorem

1. 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 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.

2. 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 $\oint_{D}f(z)dz = 0$ .

The closed differentials in the complex plane under contour integration are exactly those which satisfy the Cauchy-Riemann equations.

Cauchy's Theorem

Table of Contents

1. Introduction

2. Theorem