De Moivre's Formula

Theorem
Let $z \in \C$ be a complex number expressed in complex form:
 * $z = r \paren {\cos x + i \sin x}$

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

Exponential Form
De Moivre's Formula can also be expressed thus in exponential form:

Integer Index
This result is often given for integer index only:

Rational Index
Some sources give it for rational index:

Also defined as
This result is also often presented in the simpler form:


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

Also known as
De Moivre's Theorem.