Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 12 / 4 / a / Solution

For $|w|\le 1/2$, the power series of the principal complex logarithm is absolutely convergent and givesThus $C_1=2$ works. The value $1-w$ stays in the right half-plane, so there is no logarithm branch ambiguity. This is the bounds for the principal logarithm near one estimate used to control product tails.