Find and open connected set $G \subset \mathbb{C}$ and two continuous functions $f$ and $g$ defined on $G$ such that $f(z)^2=g(z)^2=1-z^2$ for all $z \in G$. Can you make $G$ maximal? Are $f$ and $g$ analytic?
I have tried to put $f(z)=\frac{1}{\dfrac{d}{dz}-iLog(iz+(1-z²)^\frac{1}{2})}$ for all complex $z$ such that arg($z$) $\in (-\pi, \ \pi)$. Is it works?