 Complex square root

For a nonzero complex number $z=r{e}^{i\theta }$ with $r>0$, its two square roots are $\pm \sqrt{r}{e}^{i\theta /2}$. They are distinct and differ by a sign. At $z=0$ the only square root is zero. A principal value convention chooses $\theta =\mathrm{Arg}z\in (-\pi ,\pi ]$ and the plus sign; continuity across the negative real axis then fails. A holomorphic square root is a different, domain-dependent notion: it requires a holomorphic function whose square is $z$ on the specified domain. It cannot exist on every punctured neighborhood surrounding zero.