I am following Stein and Shakarchi on Complex analysis. In a previous thread, I learnt that if $U$ is a simply connected region with $1 \in U$ and $0 \not \in U$, then there is a branch of the logarithm on $U$.
I have the following two questions:
- I have learnt that on the principal branch of the logarithm, we have the formula $\log z = \log |z| + i \mathrm{arg}(z)$. But from the derivation, it seems to me it should be valid on $U$. Is this correct? If it is false, what would a good alternative formula be?
- If $1 \not\in U$, what would the formula become then?