These questions come from 2.2 of Ahlfor's famous text. I admit that defining branches of power functions and log functions in $\mathbb C$ has been conceptually difficult for me, and I think having a few more classical examples spelled out could help me in my understanding.
Give a precise definition of a single-valued branch of $\sqrt{1+z}+\sqrt{1-z}$ in a suitable region, and prove that it is analytic.
Same problem for $\log {\log{z}}$.
Unfortunately, I have no work to show here, since I have difficulty gaining a solid, geometric grasp of the detailed analysis in the discussion preceding these problems in Ahlfors' text. A kick in the right direction would be the approach most appreciated here.
I'm doing my qualifier studying from Greene and Krantz (having long abandoned Conway) but Ahlfors has been a good supplement so far. I appreciate your input as I continue to work through these problems.