Definition:Square Root/Complex Number/Principal Square Root

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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:

$-\dfrac \pi 2 < \arg w \le \dfrac \pi 2$


Also defined as

The principal square root of $z$ can sometimes also be found defined as the element $w$ of $z^{1/2}$ such that:

$0 \le \arg w < \pi$


Also presented as

The principal square root of $z = x + i y$ 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 $y$.


Sources