# Definition:Square Root/Complex Number

## Definition

### Definition 1

Let $z \in \C$ be a complex number expressed in polar form as $\left \langle{r, \theta}\right\rangle = r \left({\cos \theta + i \sin \theta}\right)$.

The square root of $z$ is the $2$-valued multifunction:

 $\displaystyle z^{1/2}$ $=$ $\displaystyle \left\{ {\sqrt r \left({\cos \left({\frac {\theta + 2 k \pi} 2}\right) + i \sin \left({\frac {\theta + 2 k \pi} 2}\right) }\right): k \in \left\{ {0, 1}\right\} }\right\}$ $\displaystyle$ $=$ $\displaystyle \left\{ {\sqrt r \left({\cos \left({\frac \theta 2 + k \pi}\right) + i \sin \left({\frac \theta 2 + k \pi}\right) }\right): k \in \left\{ {0, 1}\right\} }\right\}$

where $\sqrt r$ denotes the positive square root of $r$.

### Definition 2

Let $z \in \C$ be a complex number expressed in polar form as $\left \langle{r, \theta}\right\rangle = r \left({\cos \theta + i \sin \theta}\right)$.

The square root of $z$ is the $2$-valued multifunction:

$z^{1/2} = \left\{ {\pm \sqrt r \left({\cos \left({\dfrac \theta 2}\right) + i \sin \left({\dfrac \theta 2}\right) }\right)}\right\}$

where $\pm \sqrt r$ denotes the positive and negative square roots of $r$.

### Definition 3

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

The square root of $z$ is the $2$-valued multifunction:

$z^{1/2} = \left\{ {\sqrt {\left\vert{z}\right\vert} e^{\left({i / 2}\right) \arg \left({z}\right)} }\right\}$

where:

$\sqrt {\left\vert{z}\right\vert}$ denotes the positive square root of the complex modulus of $z$
$\arg \left({z}\right)$ denotes the argument of $z$ considered as a multifunction.

### Definition 4

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

The square root of $z$ is the $2$-valued multifunction:

$z^{1/2} = \left\{ {w \in \C: w^2 = z}\right\}$

## Principal Square Root

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:

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