I am trying to solve this problem:
Does there exist a function $f(z)$, that is analytic at $E=\{x+iy :x>y\}$ and provides $f^2(z)=z$ for every $z \in \mathbb C$.
I have seen a solution that assumes $f(z)=\sqrt z$, but i have a bad feeling about this way.
Do you have a good solution (or convince me that the square root solution is good).
Thanks.