Compactification of log z Riemann Surface

Question

I've been reading the 'Road to Reality' book of Roger Penrose and in the chapter on Riemann Surfaces, there is a note that we can compactify the log z Riemann Surface into a sphere. But I don't see the way to do it and find any clues in literature. Maybe, someone here knows the answer. Thanks!

Answer 1

Because $z\to \log z+2ik\pi$ is locally biholomorphic the Riemann surface of $\log z$ is just $\{ \log z + 2i k\pi, z\in \Bbb{C}^*\}= \Bbb{C}$ its compactification is the Riemann surface $\Bbb{C}\cup \infty$