Section 2.11 Powers and Roots of Complex Numbers
If a complex number \(z\) is written in exponential form:
\begin{equation}
z=re^{i\phi}\text{,}\tag{2.11.1}
\end{equation}
then the \(n\)th power of \(z\) is:
\begin{align}
z^n\amp = r^n\, (e^{i\phi})^n\notag\\
\amp = r^n\, e^{in\phi}\tag{2.11.2}
\end{align}
and we see that the distance of the point \(z\) from the origin in the complex plane has been raised to the \(n\)th power, but the angle has been multiplied by \(n\text{.}\) Similarly, an \(n\)th root of \(z\) is:
\begin{align}
z^{\frac{1}{n}}\amp = r^{\frac{1}{n}}e^{i\frac{\phi}{n}}\text{.}\tag{2.11.3}
\end{align}
For example, a square root of \(-4=4\exp(i\pi)\) is given by:
\begin{align}
z^{\frac{1}{2}}\amp = (4e^{i\pi})^{\frac{1}{2}}\notag\\
\amp = 2e^{\frac{i\pi}{2}}\notag\\
\amp = 2i\tag{2.11.4}
\end{align}
This is one of the square roots of \(-4\text{;}\) what about the other root?
It turns out that we can find the other root by including in our original expression for \(z\) the multiplicity of angles, all of which give the same point in the complex plane, i.e.
\begin{equation}
z=r\, e^{i(\pi + 2\pi m)}\tag{2.11.5}
\end{equation}
where \(m\) is any positive or negative integer. Now, when we take the root, we get an infinite number of different factors of the form \(\exp(i\frac{2\pi m}{n})\text{.}\) How many of these correspond to different geometric angles in the complex plane? For \(m=\{0, 1, \dots, n-1\}\text{,}\) we will get different angles in the complex plane, but as soon as \(m=n\) the angles will repeat. Therefore, we find \(n\) distinct \(n\)th roots of \(z\text{.}\) Notice that the roots are all at radius \(r^{1/n}\) and they are equally spaced with angle \(\frac{2\pi m}{n}\) between them. If \(z\) is real and positive, then one of these roots will be the positive, real \(n\)th root that you learned about in high school.
Example: Cube Roots of 8.
For example, let’s find the cube roots of 8.
\begin{align}
8\amp =8e^{i(0+2\pi m)}\tag{2.11.6}\\
(8)^{\frac{1}{3}}\amp =8^{\frac{1}{3}}e^{i\frac{2\pi m}{3}} \tag{2.11.7}
\end{align}
For \(m=\{0, 1, 2\}\) we get distinct roots, as expected, one on the real axis and the other two at angles \(\frac{2\pi}{3}\) and \(\frac{4\pi}{3}\text{,}\) in the complex plane. These angles are equally space on a circle of radius \((8)^{\frac{1}{3}}\text{.}\) Other values of \(m\) just repeat these same three roots.
UNDER CONSTRUCTION. Add a figure for the three roots of 8.
