De Moivre's Formula/Rational 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:
- $\forall p \in \Q: \paren {r \paren {\cos x + i \sin x} }^p = r^p \paren {\map \cos {p x} + i \, \map \sin {p x} }$
Proof
Write $p = \dfrac a b$, where $a, b \in \Z$, $b \ne 0$.
Then:
\(\ds r^p \paren {\map \cos {p x} + i \, \map \sin {p x} }\) | \(=\) | \(\ds \paren {r^p \paren {\map \cos {p x} + i \, \map \sin {p x} } }^{\frac b b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {r^{b p} \paren {\map \cos {b p x} + i \, \map \sin {b p x} } }^{\frac 1 b}\) | De Moivre's Formula/Integer Index | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {r^a \paren {\map \cos {a x} + i \, \map \sin {a x} } }^{\frac 1 b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {r \paren {\cos x + i \, \sin x } }^{\frac a b}\) | De Moivre's Formula/Integer Index | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {r \paren {\cos x + i \, \sin x } }^p\) |
$\blacksquare$
Also defined as
This result is also often presented in the simpler form:
- $\forall p \in \Q: \paren {\cos x + i \sin x}^p = \map \cos {p x} + i \, \map \sin {p x}$
Also known as
De Moivre's Theorem.
Source of Name
This entry was named for Abraham de Moivre.
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: De Moivre's Theorem: $94 \ \text{(b)}$