Cauchy-Goursat Theorem

Theorem
Let $U$ be a simply connected open subset of the complex plane $\C$.

Let $\gamma : [a,b] \to U$ be a closed path in $U$.

Let $f:U \to \C$ be holomorphic in $U$. Then


 * $\displaystyle \oint_\gamma f(z)\ dz = 0$

Note
This is a special case of the Residue Theorem.