De Moivre's Formula/Rational Index

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Theorem

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

$z = r \left({\cos x + i \sin x}\right)$

Then:

$\forall p \in \Q: \paren {r \paren {\cos x + i \sin x} }^p = r^p \paren {\map \cos {p x} + i \, \map \sin {p x} }$


Proof


Also defined as

This result is also often presented in the simpler form:

$\forall p \in \Q: \paren {\cos x + i \sin x}^p = \map \cos {p x} + i \, \map \sin {p x}$


Also known as

De Moivre's Theorem.


Source of Name

This entry was named for Abraham de Moivre.


Sources