 Solution

For a closed piecewise continuously differentiable curve avoiding $a$, its winding number isEquivalently, for a parametrization $\gamma :[u,v]\to \mathbb{C}$, it is $(2\pi i{)}^{-1}{\int }_u^v{\gamma }^{\prime }(t)/(\gamma (t)-a)dt$, 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.