Definition:Principal Square Root/Also presented as

Principal Square Root: Also presented as
Let $z \in \C$ be a complex number.

Let $z = x + i y$ where $x, y \in \R$ are real numbers.

The principal square root of $z$ can also be seen presented in the form:
 * $z^{1/2} = \paren {\dfrac 1 2 \paren {r + x} }^{1/2} \pm i \paren {\dfrac 1 2 \paren {r - x} }^{1/2}$

where:
 * $r$ is the modulus of $z$: $r = \sqrt {x^2 + y^2}$
 * the $\pm$ sign is taken to be the same as the sign of $y$.