De Moivre's Formula/Integer Index/Corollary

Corollary to De Moivre's Formula: Positive Integer Index

$\forall n \in \Z: \paren {\cos x + i \sin x}^n = \map \cos {n x} + i \map \sin {n x}$

Proof

$\cos x + i \sin x$ is a complex number expressed in polar form $\polar {r, \theta}$ whose complex modulus is $1$ and whose argument is $x$.

From De Moivre's Formula: Integer Index:

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

The result follows by setting $r = 1$.

$\blacksquare$

Source of Name

This entry was named for Abraham de Moivre.

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations
• 1990: H.A. Priestley: Introduction to Complex Analysis (revised ed.) ... (previous) ... (next): $1$ The complex plane: Complex numbers $\S 1.2$ The algebraic structure of the complex numbers