# Homotopy invariance of winding number in complex analysis.

In topology, the winding number is homotopy invariant under the definition $n(\gamma,a)=\frac{\tilde{\theta}(\beta)-\tilde{\theta}(\alpha)}{2\pi}.$

I assume the must be true in the framework of complex analysis. Suppose you take as definition for the winding number $n(C,a)$ of a curve $C$ through $a$ to be $$n(C,a)=\frac{1}{2\pi i}\int_C\frac{dz}{z-a}.$$

Is is still true that $n(C,a)$ is hopotopy invariant under smooth curves $C$ not going through $a$? Thank you.

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It seems intuitively that it would be the case that $n(C,a)$ is homotopy invariant under $C$, but I don't know how to prove it. – Samuel Reid Feb 25 '12 at 7:12