I am currently aware of the following two versions of the global Cauchy Theorem. Which one is stronger?
1.)If the region $U$ is simply connected, then for every closed curve contained therein, the integral of the holomorphic function $f$ defined on $U$ over the curve is zero.
2.) If the domain $D$ is an arbitrary open set, then for every closed curve contained therein with the property that the index is zero for points outside $D$, the integral of the holomorphic function $f$ defined on $D$ is zero. Thanks in advance.