Let $f$ be analytic in a finite simply-connected region $\mathcal{R}$ described by the inside of a closed curve $C$ (that can contain multiple points) and on its boundary $C$. Then
$$ \oint_C f \overset{?}{=} 0$$
Intuitively the Cauchy's Theoreom is also valid for path with multiple point because a closed path with multiple points can be chopped in different regions (as many as parts of the path there are between a multiple point and the next one) and in each part the Cauchy's Theorem can be applied individually. So, if the theorem applies for each region, then the theorem applies for the entire path. A good example is the Lemniscata, where the theorem is obviously satisfied.
Any help with the mathematical proof?