Category:Definitions/Complex Square Roots

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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 $r$.