Definition:Square Root/Complex Number/Principal Square Root

Definition
Let $z \in \C$ be a complex number.

Let $z^{1/2} = \left\{{w \in \C: w^2 = z}\right\}$ be the square root of $z$.

The principal square root of $z$ is the element $w$ of $z^{1/2}$ such that:
 * $\begin{cases} \operatorname{Im} \left({w}\right) > 0 : & \operatorname{Im} \left({z}\right) \ne 0 \\

\operatorname{Re} \left({w}\right) \ge 0 : & \operatorname{Im} \left({z}\right) = 0 \end{cases}$

Also defined as
Equivalently, the case where $\operatorname{Im} \left({z}\right) = 0$ can be reported as $w \ge 0$, as in this case $w$ is wholly real.