Computing Integer Roots of Complex Numbers

Polar coordinates gives us a really nice way to find a principal root for any complex number $$ Z = x + iy = Re^{i\theta}. $$

One (principal) solution can be found as $z_1 = re^{i\phi}$, where $r=\sqrt[n]{R}$ and $\phi = \frac{\theta}{n}$. This is a solution because $$ (re^{i\phi})^n = r^n e^{i(n\phi)} = R e^{i\theta}. $$

Furthermore, any other root $z_2$ must be $z_1$ times an $n$th root of unity, since $$ \left( \frac{z_2}{z_1} \right)^n = \frac{(z_2)^n}{(z_1)^n}=\frac{Z}{Z}=1. $$ Thus, $z_2 = z_1 e^i\psi$, where $\psi$ is a multiple of $\frac{2\pi}{n}$.

In the diagram, you can see the principal $n$th root of a complez number $Z$. The argument is taken between $0$ and $2\pi$ for $Z$.

Check the box to see the other $n$th roots.

Figure 1: Complex Roots

Drag $Z$ across the positive real axis and you will see why this function is not continuous.