De Moivre's Formula/Exponential Form
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Theorem
- $\forall \omega \in \C: \paren {r e^{i \theta} }^\omega = r^\omega e^{i \omega \theta}$
Proof
\(\ds \paren {r e^{i \theta} }^\omega\) | \(=\) | \(\ds \paren {r \paren {\cos \theta + i \sin \theta} }^\omega\) | Definition of Exponential Form of Complex Number | |||||||||||
\(\ds \) | \(=\) | \(\ds r^\omega \paren {\cos \omega \theta + i \sin \omega \theta}\) | De Moivre's Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds r^\omega e^{i \omega \theta}\) | Definition of Exponential Form of Complex Number |
$\blacksquare$
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: $3.7.16$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 7$: Operations with Complex Numbers in Polar Form: $7.27$
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Euler's Formula