Complex Powers (De Moivre)

Powers and roots of complex numbers

Operation
z^(1/n) = r^(1/n)·cis((θ + 2πk)/n), k = 0…n−1
z in polar form
1 ∠ 0°
Modulus |z|
1
The 3 nth roots
  1. w₀ =1(1 ∠ 0°)
  2. w₁ =-0,5 + 0,866i(1 ∠ 120°)
  3. w₂ =-0,5 − 0,866i(1 ∠ -120°)
Complex Powers (De Moivre)Argand diagram showing the 3 nth roots of z = 1, spaced evenly around a circle and joined into a regular polygon.ReIm-1-111w₀w₁w₂