Cauchy's Theorem

Definition

Let $f:D\to \mathbb{C}$ be holomorphic. Suppose ${\gamma }_1$ and ${\gamma }_2$ are two (parametrized) closed curves in $D$ such that ${\gamma }_1\sim {\gamma }_2$. Then we have
${\int }_{{\gamma }_1}f={\int }_{{\gamma }_2}f.$

Proof

The proof can be found here.