Definition:Square Root/Complex Number/Definition 2

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

• Equivalence of Definitions of Square Root of Complex Number

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: $3.7.26$