Definition:Square Root/Complex Number/Definition 2
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Definition
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 $r$.
Also see
Sources
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.7$ Complex Numbers and Functions: Roots: $3.7.26$