# De Moivre's Formula/Rational Index

## Theorem

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

$z = r \paren {\cos x + i \sin x}$

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

Write $p = \dfrac a b$, where $a, b \in \Z$, $b \ne 0$.

Then:

 $\ds r^p \paren {\map \cos {p x} + i \, \map \sin {p x} }$ $=$ $\ds \paren {r^p \paren {\map \cos {p x} + i \, \map \sin {p x} } }^{\frac b b}$ $\ds$ $=$ $\ds \paren {r^{b p} \paren {\map \cos {b p x} + i \, \map \sin {b p x} } }^{\frac 1 b}$ De Moivre's Formula/Integer Index $\ds$ $=$ $\ds \paren {r^a \paren {\map \cos {a x} + i \, \map \sin {a x} } }^{\frac 1 b}$ $\ds$ $=$ $\ds \paren {r \paren {\cos x + i \, \sin x } }^{\frac a b}$ De Moivre's Formula/Integer Index $\ds$ $=$ $\ds \paren {r \paren {\cos x + i \, \sin x } }^p$

$\blacksquare$

## 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.