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.