closed curve theorem

Let $U\subset\mathbb{C}$ be a simply connected domain, and suppose $f:U\longrightarrow\mathbb{C}$ is holomorphic. Then

$\int_{C}f(z)\ dz=0$

for any smooth closed curve $C$ in $U$ .

More generally, if $U$ is any domain, and $C_{1}$ and $C_{2}$ are two homotopic smooth closed curves in $U$ , then

$\int_{C_{1}}f(z)\ dz=\int_{C_{2}}f(z)\ dz.$

for any holomorphic function $f:U\longrightarrow\mathbb{C}$ .