$$ e^{-i\frac43\pi n} - e^{-i\frac23\pi n}, n\in \mathbb{N} $$ I am trying to simplify this but cant. Any ideas appreciated.
2026-04-07 08:42:52.1775551372
Can this be simplified?
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We have
$$ e^{-i\frac{4}{3}\pi n} - e^{-i\frac{2}{3}\pi n}=e^{i\frac{2}{3}\pi n} - e^{-i\frac{2}{3}\pi n}=2i\sin\left(\frac{2\pi n}3\right)=\left\{\begin{array}\\0&\text{if}&n\equiv0\mod3\\ \pm i\sqrt3&\text{if}&n\equiv\pm1\mod3 \end{array}\right. $$