Past exam of the mathematics course of the University of Cambridge / 2005 / ib / Paper 4 / 4A (Complex Analysis)

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.

Where both derivatives exist, the ordinary product rule givesNeither denominator vanishes. Integrate over a common piecewise-smooth subdivision and use the contour characterization of the winding number:

On the right half-plane a single-valued holomorphic logarithm is $\mathrm{Log}z=\log |z|+i\mathrm{Arg}z$, with $-\pi /2<\mathrm{Arg}z<\pi /2$, and its derivative is $1/z$. ThusThis proves the conclusion directly from the integral definition of the winding number, without appealing to its homotopy invariance.

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.