Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 3 / 12F / iii / Solution

The principal argument is continuous and single-valued on $\mathbb{C}\setminus (-\infty ,0]$. The given loop stays in this slit plane, and its initial and final values have the same principal argument. The definition therefore gives $\boxed{w(\gamma ;0)=0}$.