De Moivre's Formula/Integer Index

Theorem

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

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

Then:

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

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

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

Then:

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

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

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

Then:

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

Then:

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

De Moivre's Theorem.

This entry was named for Abraham de Moivre.

1968: Ian D. Macdonald: The Theory of Groups ... (previous): Appendix: Elementary set and number theory
1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: De Moivre's Theorem: $20$