Past exam of the mathematics course of the University of Cambridge / 2005 / ib / Paper 4 / 4A / Solution

For a closed piecewise continuously differentiable path avoiding $a$, the winding number isBreak the parameter interval at the finitely many derivative discontinuities when evaluating this integral. Translating the curve gives $n(\gamma ,a)=n(\gamma -a,0)$, which will also be used below.