De Moivre's Formula/Positive Integer Index/Proof 2

Theorem

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

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

Then:

$\forall n \in \Z_{>0}: \paren {r \paren {\cos x + i \sin x} }^n = r^n \paren {\map \cos {n x} + i \map \sin {n x} }$

Proof

From Product of Complex Numbers in Polar Form: General Result:

$z_1 z_2 \cdots z_n = r_1 r_2 \cdots r_n \paren {\map \cos {\theta_1 + \theta_2 + \cdots + \theta_n} + i \, \map \sin {\theta_1 + \theta_2 + \cdots + \theta_n} }$

Setting $z_1 = z_2 = \cdots = z_n = r \paren {\cos x + i \sin x}$ gives the result.

Source of Name

This entry was named for Abraham de Moivre.

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations: $(2.15)$
• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: De Moivre's Theorem