# Complex Variables

\(\zeta = e^{\pi i / 3} = \cos(\pi i / 3) + i \sin(\pi i / 3)\) is a complex cube root of unity; its complex conjugate, \(\bar{\zeta} = e^{2 \pi i / 3}\), is another root.

\(\alpha = e^{\pi i / p}\) is a complex p-th root of unity, then \(\sum_{q = 0}^{p} e^{\pi i / q} = \alpha + \alpha^2 + \ldots + \alpha^p = 0\).