Roots of Complex Number/Exponential Form

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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

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