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:

$\forall p \in \Q: \left({r \left({\cos x + i \sin x}\right)}\right)^p = r^p \left({\cos \left({p x}\right) + i \sin \left({p x}\right)}\right)$


Proof


Also defined as

This result is also often presented in the simpler form:

$\forall p \in \Q: \left({\cos x + i \sin x}\right)^p = \cos \left({p x}\right) + i \sin \left({p x}\right)$


Also known as

De Moivre's Theorem.


Source of Name

This entry was named for Abraham de Moivre.


Sources