De Moivre's Formula/Negative 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:
- $\forall n \in \Z_{\le 0}: \paren {r \paren {\cos x + i \sin x} }^n = r^n \paren {\map \cos {n x} + i \map \sin {n x} }$
Proof
Let $n = 0$.
Then:
\(\ds r^0 \paren {\map \cos {0 x} + i \map \sin {0 x} }\) | \(=\) | \(\ds 1 \times \paren {\cos 0 + i \sin 0}\) | Definition of Zeroth Power | |||||||||||
\(\ds \) | \(=\) | \(\ds 1 \paren {1 + i 0}\) | Cosine of Zero is One and Sine of Zero is Zero | |||||||||||
\(\ds \) | \(=\) | \(\ds 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {r \paren {\cos x + i \sin x} }^0\) | Definition of Zeroth Power |
Now let $n \in \Z_{<0}$.
Let $n = -m$ where $m > 0$.
Thus:
\(\ds \paren {r \paren {\cos x + i \sin x} }^{-m}\) | \(=\) | \(\ds \frac 1 {\paren {r \paren {\cos x + i \sin x} }^m}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {r^m \paren {\map \cos {m x} + i \map \sin {m x} } }\) | De Moivre's Formula: Positive Integer Index | |||||||||||
\(\ds \) | \(=\) | \(\ds r^{-m} \paren {\map \cos {-m x} + i \map \sin {-m x} }\) | Definition of Complex Division | |||||||||||
\(\ds \) | \(=\) | \(\ds r^n \paren {\map \cos {n x} + i \map \sin {n x} }\) | as $n = -m$ |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: De Moivre's Theorem: $94 \ \text{(a)}$