De Moivre's Formula
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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:
- $\forall \omega \in \C: \paren {r \paren {\cos x + i \sin x} }^\omega = r^\omega \paren {\map \cos {\omega x} + i \, \map \sin {\omega x} }$
Exponential Form
De Moivre's Formula can also be expressed thus in exponential form:
- $\forall \omega \in \C: \paren {r e^{i \theta} }^\omega = r^\omega e^{i \omega \theta}$
Integer Index
This result is often given for integer index only:
Let $z \in \C$ be a complex number expressed in complex form:
- $z = r \paren {\cos x + i \sin x}$
Then:
\(\ds \forall n \in \Z: \, \) | \(\ds \paren {r \paren {\cos x + i \sin x} }^n\) | \(=\) | \(\ds r^n \paren {\map \cos {n x} + i \map \sin {n x} }\) | |||||||||||
\(\ds \) | \(=\) | \(\ds r^n \cos n x + i r^n \sin n x\) |
Rational Index
Some sources give it for rational index:
Let $z \in \C$ be a complex number expressed in complex form:
- $z = r \paren {\cos x + i \sin x}$
Then:
- $\forall p \in \Q: \paren {r \paren {\cos x + i \sin x} }^p = r^p \paren {\map \cos {p x} + i \, \map \sin {p x} }$
Proof 1
\(\ds \paren {r \paren {\cos x + i \sin x} }^\omega\) | \(=\) | \(\ds \paren {r e^{i x} }^\omega\) | Euler's Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds r^\omega e^{i \omega x}\) | Power of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds r^\omega \paren {\map \cos {\omega x} + i \map \sin {\omega x} }\) | Euler's Formula |
$\blacksquare$
Also defined as
This result is also often presented in the simpler form:
- $\forall \omega \in \C: \paren {\cos x + i \sin x}^\omega = \map \cos {\omega x} + i \, \map \sin {\omega x}$
Also known as
De Moivre's Theorem.
Source of Name
This entry was named for Abraham de Moivre.
Historical Note
De Moivre's Formula was discovered by Abraham de Moivre around $1707$.
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $6.9$: De Moivre's Theorem
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): de Moivre's theorem
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): de Moivre's theorem
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): De Moivre's Theorem
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $6$: Basic Algebra