Definition:Square Root/Complex Number/Principal Square Root

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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 element $w$ of $z^{1/2}$ such that:

$\begin{cases} \map \Im w > 0 : & \map \Im z \ne 0 \\ \map \Re w \ge 0 : & \map \Im z = 0 \end{cases}$

Also defined as

Equivalently, the case where $\map \Im z = 0$ can be reported as $w \ge 0$, as in this case $w$ is wholly real.