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18.4 De Moivre's Theorem

The variable z , z = x + i y can be represented by its length and angle as can any two dimensional vector, and the relation as usual is

r 2 = x 2 + y 2
and
x = r cos ⁡ θ , y = r sin ⁡ θ

De Moivre's Theorem is the statement that e θ = cos ⁡ θ + i sin ⁡ θ . We can therefore write

z = x + i y = r e i θ

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