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:

$\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 $

$=$

$\ds r^n \cos n x + i r^n \sin n x$

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:

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

Then:

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

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

De Moivre's Theorem.

This entry was named for Abraham de Moivre.

1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.7$ Complex Numbers and Functions: Powers: $3.7.17$
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: $20$