Roots of Complex Number/Exponential Form

Theorem

Let $z := r e^{i \theta}$ be a complex number expressed in exponential form, such that $z \ne 0$.

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Then the $n$th roots of $z$ are given by:

$z^{1 / n} = \set {r^{1 / n} e^{i \paren {\theta + 2 \pi k} / n}: k \in \set {0, 1, 2, \ldots, n - 1} }$

Principal Root

The principal $n$th root of $z$ is the value of $r^{1/n} e^{i \theta / n}$ such that:

$-\pi < \theta \le \pi$

Proof

$\ds z^{1 / n}$

$=$

$\ds \paren {r e^{i \theta} }^{1 / n}$

Definition of Exponential Form of Complex Number

$\ds $

$=$

$\ds \paren {r \paren {\cos x + i \sin x} }^{1 / n}$

Definition of Polar Form of Complex Number

$\ds $

$=$

$\ds \set {r^{1 / n} \paren {\cos \paren {\dfrac {\theta + 2 \pi k} n} + i \sin \paren {\dfrac {\theta + 2 \pi k} n} }: k \in \set {0, 1, 2, \ldots, n - 1} }$

Roots of Complex Number

$\ds $

$=$

$\ds \set {r^{1 / n} e^{i \paren {\theta + 2 \pi k} / n}: k \in \set {0, 1, 2, \ldots, n - 1} }$

Definition of Exponential Form of Complex Number

$\blacksquare$

Sources

1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 3$. Roots of Unity
1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.7$ Complex Numbers and Functions: Roots: $3.7.28$
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 7$: Operations with Complex Numbers in Polar Form: $7.28$