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:

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


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:

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

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

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


Also known as

De Moivre's Theorem.


Source of Name

This entry was named for Abraham de Moivre.


Sources