# De Moivre's Formula/Rational Index

## 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

- 1964: Murray R. Spiegel:
*Theory and Problems of Complex Variables*... (previous) ... (next): $1$: Supplementary Problems: $94 \ \text{(b)}$