The following identity is a formula found by Abraham De Moivre (1667 - 1754).

Theorem: De Moivre's Identity, Complex Powers

For all real numbers $\phi\in\mathbb R$ and all natural numbers $n\in\mathbb N,$ the formula


holds, where $\cos$ and $\sin$ denote the cosine and sine functions.

In particular, the power $z^n$ of a complex number written in polar coordinates $z=r\exp(i\phi)$ can be calculated by the formula $$z^n=r^n\exp(in\phi).$$

Proofs: 1

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  1. Modler, F.; Kreh, M.: "Tutorium Analysis 1 und Lineare Algebra 1", Springer Spektrum, 2018, 4th Edition