Complex square root
= Complex square root
{title2=$w^2=z$}
For a nonzero <complex number> $z=re^{i\theta}$ 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 $\theta=\operatorname{Arg}z\in(-\pi,\pi]$ 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.