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 $f:U\\longrightarrow\\mathbb{C}$ .

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