# Definition:Square Root/Complex Number/Principal Square Root

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## Definition

Let $z \in \C$ be a complex number.

Let $z^{1/2} = \set {w \in \C: w^2 = z}$ be the square root of $z$.

The **principal square root** of $z$ is the principal branch of the $2$nd power of $w$.

Hence, by the conventional definition of the principal branch of the natural logarithm of $z$, it is the element $w$ of $z^{1/2}$ such that:

- $-\dfrac \pi 2 < \arg w \le \dfrac \pi 2$

## Also defined as

The **principal square root** of $z$ can sometimes also be found defined as the element $w$ of $z^{1/2}$ such that:

- $0 \le \arg w < \pi$

## Also presented as

The principal square root of $z = x + i y$ can also be seen presented in the form:

- $z^{1/2} = \paren {\dfrac 1 2 \paren {r + x} }^{1/2} \pm i \paren {\dfrac 1 2 \paren {r - x} }^{1/2}$

where:

- $r$ is the modulus of $z$: $r = \sqrt {x^2 + y^2}$
- the $\pm$ sign is taken to be the same as the sign of $y$.

## 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: $3.7.26$