# The Cauchy integral theorem

## Idea

Cauchy’s integral theorem states that contour integrals of holomorphic functions, in any simply connected subspace of the complex plane, are invariant under homotopy of paths.

## Statement

Let ${\gamma }_{1},{\gamma }_{2}$ be two homotopic loops in a simply connected open subset $D\subseteq ℂ$. Let $f$ be a holomorphism on $D$. Then we have

${\int }_{{\gamma }_{1}}f\left(z\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}z={\int }_{{\gamma }_{2}}f\left(z\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}z$\int_{\gamma_1} f(z) \,\mathrm{d}z = \int_{\gamma_2} f(z) \,\mathrm{d}z

In particular we have

${\int }_{{\gamma }_{1}}f\left(z\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}z=0$\int_{\gamma_1} f(z) \,\mathrm{d}z = 0

for ${\gamma }_{2}≔0$.

category: analysis

Revised on March 20, 2013 19:23:48 by Toby Bartels (64.89.53.9)