Definition:Square Root/Complex Number/Definition 3

Definition

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.

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: $3.7.26$