De Moivre's Formula/Integer Index

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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:

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


Positive Index

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

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

Then:

$\forall n \in \Z_{>0}: \paren {r \paren {\cos x + i \sin x} }^n = r^n \paren {\map \cos {n x} + i \, \map \sin {n x} }$


Negative Index

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

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

Then:

$\forall n \in \Z_{\le 0}: \paren {r \paren {\cos x + i \sin x} }^n = r^n \paren {\map \cos {n x} + i \, \map \sin {n x} }$


Corollary

Then:

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


Also known as

De Moivre's Theorem.


Source of Name

This entry was named for Abraham de Moivre.


Sources