De Moivre's Formula/Exponential Form

From ProofWiki
Jump to navigation Jump to search

Theorem

$\forall \omega \in \C: \paren {r e^{i \theta} }^\omega = r^\omega e^{i \omega \theta}$


Proof

\(\displaystyle \paren {r e^{i \theta} }^\omega\) \(=\) \(\displaystyle \paren {r \paren {\cos x + i \sin x} }^\omega\) Definition of Exponential Form of Complex Number
\(\displaystyle \) \(=\) \(\displaystyle r^\omega \paren {\cos \omega x + i \omega \sin x}\) De Moivre's Formula
\(\displaystyle \) \(=\) \(\displaystyle r^\omega e^{i \omega \theta}\) Definition of Exponential Form of Complex Number

$\blacksquare$


Sources