Consider the calculation of roots of an equation z N = where N 1 is

Consider the calculation of roots of an equation z^N = α where N ≥ 1 is an integer and α = ∣α∣e^jφ a nonzero complex number.

(a) First verify that there are exactly N roots for this equation and that they are given by z_k = re^jθk where r = ∣α∣^1/N and θ_k = (φ + 2πk)/N for k = 0, 1, … ,N − 1.

(b) Use the above result to find the roots of the following equations

(i) z² = 1,               (ii) z² = − 1,          (iii) z³ = 1,             (iv) z³ = − 1

and plot them in a polar plane (i.e., indicating their magnitude and phase). Explain how the roots are distributed in the polar plane.