De Moivre's Formula/Proof 1

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Let $z \in \C$ be a complex number expressed in complex form:

$z = r \paren {\cos x + i \sin x}$


$\forall \omega \in \C: \paren {r \paren {\cos x + i \sin x} }^\omega = r^\omega \map \cos {\omega x} + i \map \sin {\omega x}$


\(\ds \paren {r \paren {\cos x + i \sin x} }^\omega\) \(=\) \(\ds \paren {r e^{i x} }^\omega\) Euler's Formula
\(\ds \) \(=\) \(\ds r^\omega e^{i \omega x}\) Power of Power
\(\ds \) \(=\) \(\ds r^\omega \paren {\map \cos {\omega x} + i \map \sin {\omega x} }\) Euler's Formula


Source of Name

This entry was named for Abraham de Moivre.