De Moivre's Formula/Exponential Form

Theorem

$\forall \omega \in \C: \paren {r e^{i \theta} }^\omega = r^\omega e^{i \omega \theta}$

$\ds \paren {r e^{i \theta} }^\omega$

$=$

$\ds \paren {r \paren {\cos \theta + i \sin \theta} }^\omega$

Definition of Exponential Form of Complex Number

$\ds $

$=$

$\ds r^\omega \paren {\cos \omega \theta + i \sin \omega \theta}$

$\ds $

$=$

$\ds r^\omega e^{i \omega \theta}$

Definition of Exponential Form of Complex Number

$\blacksquare$

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