Question on square roots of complex functions

Question

LeI have a function $f(x)$ where $x\in R$ and $|x|<1$. Now, I can write $f(x)=\sqrt{1-x^2}g(x)$ for some non-divergent $g(x)$ in the domain of definition of $x$.
Can I write the following:
$$\sqrt{f(x)}=\sqrt{\sqrt{1-x^2}g(x)}=\sqrt{\sqrt{1-x^2}}\sqrt{g(x)}=(1-x^2)^{1/4}\sqrt{g(x)}$$ ? If not, why?

Answer 1

The problem is that the square root of a complex map is ambiguous. For example you can’t get a continuous square root map on $\mathbb C\setminus \{0\}$.

The square root of $e^{i\pi}$ can be $e^{i\dfrac{\pi}{2}}$ or $e^{-i\dfrac{\pi}{2}}$