Category:Definitions/Complex Square Roots

This category contains definitions related to Complex Square Roots.
Related results can be found in Category:Complex Square Roots.

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 $r$.

Pages in category "Definitions/Complex Square Roots"

The following 8 pages are in this category, out of 8 total.