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.
Best Answer
Both definitions agree, so the winding number is homotopy invariant. See Stewart and Tall, "Complex Analysis", section 7.5. You might be also interested in this question and, perhaps, my answer therein.