Definition:Square Root/Complex Number/Principal Square Root

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

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

The principal square root of $z$ is the principal branch of the $2$nd power of $w$.

Hence, by the conventional definition of the principal branch of the natural logarithm of $z$, it is the element $w$ of $z^{1/2}$ such that:
 * $-\dfrac \pi 2 < \arg w \le \dfrac \pi 2$