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

Put $\eta =({\gamma }_2-a)/({\gamma }_1-a)$. The strict inequality gives $|\eta -1|<1$, so $\mathrm{Re}\eta >0$. This is a closed piecewise-smooth curve in the right half-plane, and part (b) gives $n(\eta ,0)=0$. Since ${\gamma }_2-a=({\gamma }_1-a)\eta$, part (a) yieldsThis is the dominated perturbation preserves winding number principle, here derived by logarithmic differentiation.