# Residue theorem with winding numbers

In my studies, I've been uniquely using the Residue Theorem with no more than a single winding around each singularity. Actually, my professor has never mentioned winding-numbers in the Residue Theorem.

I've been reading discussions about complex integrals, where some people suggest solutions that take winding numbers into account. Is there a general advantage in doing so?

## migrated from mathoverflow.netJan 31 at 18:02

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Residue Theorem. Let $$M\subset\mathbb{C}$$ be open. Let $$\gamma\subset M$$ be a closed curve freely homotopic to a constant curve within $$M$$. Assume that $$f$$ is holomorphic on $$M\setminus S$$, where $$S\subset M$$ is a discrete subset of isolated singularities of $$f$$ disjoint from $$\gamma$$. Then $$\int_\gamma f=2\pi i\sum_{p\in S} w_p(\gamma)\mathrm{res}_p(f),$$ where the winding number $$w_p(\gamma)$$ vanishes for all but finitely many points $$p\in S$$.