The orginal problem is "Calculating $\oint_{L} \frac{xdy - ydx}{x^2 + y^2}$, where L is a smooth, simple closed, and postively oriented curve that does not pass through the orgin".
But what if I modify the hypothesis and allow non-simple closed curve? I mean is there something like green formula that allows us calculuating "non-simple closed curve integral"?
EDIT:
It seems to me that this integral should also resemble the simple closed one. Because common interior line integral should be canceled only remaining the outter curve. Am I right?