# De Moivre's Formula/Proof 1

Jump to navigation
Jump to search

## Theorem

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

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

Then:

- $\forall \omega \in \C: \paren {r \paren {\cos x + i \sin x} }^\omega = r^\omega \map \cos {\omega x} + i \map \sin {\omega x}$

## Proof

\(\ds \paren {r \paren {\cos x + i \sin x} }^\omega\) | \(=\) | \(\ds \paren {r e^{i x} }^\omega\) | 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 |

$\blacksquare$

The term Power of Power as used here has been identified as being ambiguous.In particular: The link to Power of Power currrently only covers real exponentsIf you are familiar with this area of mathematics, you may be able to help improve $\mathsf{Pr} \infty \mathsf{fWiki}$ by determining the precise term which is to be used.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Disambiguate}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Source of Name

This entry was named for Abraham de Moivre.