A statement of Cauchy's theorem is that:
Let $\gamma$ be a closed, null homotopic contour on an open,bounded and simply connected subset $\Omega$ of $\mathbb{C}$. If $f:\Omega\mapsto\mathbb{C}$ is holomorphic, then $\int_{\gamma}f(z)dz=0$
I get that $\gamma$ would have residues if $\Omega$ wasn't simply connected and the integral wouldn't be equal to zero, but what is the importance of the null homotopic, closed $\gamma$ assumption and the open,bounded $\Omega$ assumption?
Thanks