De Moivre's Formula/Proof 1

Theorem

Let $z \in \C$ be a complex number expressed in complex form:

$z = r \paren {\cos x + i \sin x}$

Then:

$\forall \omega \in \C: \paren {r \paren {\cos x + i \sin x} }^\omega = r^\omega \map \cos {\omega x} + i \map \sin {\omega x}$

Proof

\(\ds \paren {r \paren {\cos x + i \sin x} }^\omega\)

\(=\)

\(\ds \paren {r e^{i x} }^\omega\)

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

$\blacksquare$

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Source of Name

This entry was named for Abraham de Moivre.