De Moivre's Formula

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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:

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


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:

$z = r \left({\cos x + i \sin x}\right)$

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

\(\displaystyle \left({r \left({\cos x + i \sin x}\right)}\right)^\omega\) \(=\) \(\displaystyle \left({r e^{ix} }\right)^\omega\) $\quad$ Euler's Formula $\quad$
\(\displaystyle \) \(=\) \(\displaystyle r^\omega e^{i \omega x}\) $\quad$ Exponent Combination Laws: Power of Power $\quad$
\(\displaystyle \) \(=\) \(\displaystyle r^\omega \left({\cos \left({\omega x}\right) + i \sin \left({\omega x}\right)}\right)\) $\quad$ Euler's Formula $\quad$

$\blacksquare$


Also defined as

This result is also often presented in the simpler form:

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


Also known as

De Moivre's Theorem.


Source of Name

This entry was named for Abraham de Moivre.


Sources