= Solution
For a closed piecewise continuously differentiable curve avoiding $a$, its <winding number> is
$$
\boxed{I(\gamma,a)=\frac1{2\pi i}\int_\gamma\frac{dz}{z-a}.}
$$
Equivalently, for a parametrization $\gamma:[u,v]\to\mathbb C$, it is $(2\pi i)^{-1}\int_u^v\gamma'(t)/(\gamma(t)-a)\,dt$, summed over the smooth pieces. A continuous choice of the argument of $\gamma(t)-a$ along the parameter interval changes by an integer multiple of $2\pi$ between the endpoints; that integer is the winding number.
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