Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 1 / 12G / a / Solution

Termwise differentiation inside the disc givesConsequentlySince $f(1)=0$, the constant is one, so $e^{f(z)}=z$. Thus $f$ is an analytic branch of the logarithm on $D(1,1)$ with the required value.

Given $a\in D$, write $a=|a|e^{i{\theta }_a}$ with $0<{\theta }_a<2\pi$. On $|z/a-1|<1$, defineThen $e^{{\ell }_a(z)}=z$ and $\mathrm{Im}{\ell }_a(a)={\theta }_a$. After shrinking the neighbourhood of $a$, continuity keeps its imaginary part in $(0,2\pi )$.