Cauchy-Goursat Theorem/Corollary 1

Corollary

Let $f: U \to \C$ be a holomorphic function, where $U \subseteq \C$ is an open set.

Let $C$ be a simple closed contour in $U$.

Let $\Int C \subseteq U$, where $\Int C$ denotes the interior of $C$.

Then:

$\ds \oint_C \map f z \rd z = 0$

Proof

By Interior of Simply Closed Contour Extends to Simply Connected Domain, there exists a simply connected domain $V$ such that $\Int C \subseteq V \subseteq U$, and $C$ is a contour in $V$.

The result now follows from the main Cauchy-Goursat Theorem.

$\blacksquare$

Sources

• 2001: Christian Berg: Kompleks funktionsteori: $\S 3.1$