Can I write $(ix+\sqrt{1-x^2})=\sqrt{ix+\sqrt{1-x^2}}*\sqrt{ix+\sqrt{1-x^2}}$ when $x \in (-1,1)$?
If not, then what is the rule for breaking up complex functions in square roots?
Sorry if this is elementary, but it confuses me a lot at the moment.
The principal square root of a complex number $z_1$ is defined as being the number $z_2$ such $z_2^2 = z_1$ and $Re(z_2) \geq 0$. So, by definition, $\sqrt{z_1}\sqrt{z_1}=z_1$. The danger is not in splitting a number up into square roots, but in the temptation to recombine them. Consider $\sqrt{-1}\sqrt{-1}=-1$. This is an entirely true statement; $\sqrt{-1}=i$ and $i^2=-1$. However, if we use the identity $\sqrt a \sqrt b = \sqrt{ab}$, which is true when $a$ and $b$ are positive real numbers, we get $\sqrt{-1}\sqrt{-1}=\sqrt{1}$, which is not true.