De Moivre's Formula

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

Then:
 * $\forall \omega \in \C: \left({r \left({\cos x + i \sin x}\right)}\right)^\omega = r^\omega \left({\cos \left({\omega x}\right) + i \sin \left({\omega x}\right)}\right)$

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

Integral Index
This result is often given for integral index only:

Rational Index
Some sources give it for rational index:

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


 * $\forall \omega \in \C:\left({\cos x + i \sin x}\right)^\omega = \cos \left({\omega x}\right) + i \sin \left({\omega x}\right)$

Also known as
De Moivre's Theorem.