De Moivre's Formula/Integer Index
Jump to navigation
Jump to search
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
- 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: $20$