The question is as follow:
Let $H\subset \mathbb{C}^n$ be an simply-connected region. If $f$ is a nowhere vanishing analytic function on $H$, with $f(z)>0$ for all $z\in H\cap\mathbb{R}^n$, does there exist an anlytic function $g$ defined on H such that $g(z)=\sqrt{f(z)}$, for all $z\in H\cap\mathbb{R}^n$?
For the one variable case, the answer is yes. But can we extend this fact to the general case?