# Definition:Square Root/Complex Number

## Definition

### Definition 1

Let $z \in \C$ be a complex number expressed in polar form as $\polar {r, \theta} = r \paren {\cos \theta + i \sin \theta}$.

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

 $\ds z^{1/2}$ $=$ $\ds \set {\sqrt r \paren {\map \cos {\frac {\theta + 2 k \pi} 2} + i \map \sin {\frac {\theta + 2 k \pi} 2} }: k \in \set {0, 1} }$ $\ds$ $=$ $\ds \set {\sqrt r \paren {\map \cos {\frac \theta 2 + k \pi} + i \map \sin {\frac \theta 2 + k \pi} }: k \in \set {0, 1} }$

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 $\polar {r, \theta} = r \paren {\cos \theta + i \sin \theta}$.

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

$z^{1/2} = \set {\pm \sqrt r \paren {\map \cos {\dfrac \theta 2} + i \map \sin {\dfrac \theta 2} } }$

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:

$z^{1/2} = \set {\sqrt {\cmod z} \, e^{\paren {i / 2} \map \arg z} }$

where:

$\sqrt {\cmod z}$ denotes the positive square root of the complex modulus of $z$
$\map \arg z$ 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:

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

## 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 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 see

• Results about complex square roots can be found here.