Past exam of the Brownian motion in complex analysis course of the University of Oxford / 2011 / 2 / a / Solution

Let $\beta (\theta )=w-e^{i\theta }=|\beta (\theta )|e^{ig(\theta )}$. It is never zero, since $|w|<1$. On the unit circle,Dividing these identities givesAny other continuous lift of the argument differs by an integer multiple of $2\pi$, which leaves the exponential unchanged.