# De Moivre's Formula

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## 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:

$\forall \omega \in \C: \paren {r e^{i \theta} }^\omega = r^\omega e^{i \omega \theta}$

### Integer Index

This result is often given for integer index only:

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

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

Then:

$\forall n \in \Z: \paren {r \paren {\cos x + i \sin x} }^n = r^n \paren {\map \cos {n x} + i \, \map \sin {n x} }$

### Rational Index

Some sources give it for rational index:

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 1

 $\ds \paren {r \paren {\cos x + i \sin x} }^\omega$ $=$ $\ds \paren {r e^{i x} }^\omega$ Euler's Formula $\ds$ $=$ $\ds r^\omega e^{i \omega x}$ Power of Power $\ds$ $=$ $\ds r^\omega \paren {\map \cos {\omega x} + i \map \sin {\omega x} }$ Euler's Formula

$\blacksquare$

## 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.

## Source of Name

This entry was named for Abraham de Moivre.

## Historical Note

De Moivre's Formula was discovered by Abraham de Moivre around $1707$.