Answer to Question #11997 in Complex Analysis for Ahmed Awad

Question

find all complex numbers z such that z^4 =-1. write the answers in both polar from and the form a+ib, and sketch them in the complex plane.

Answer 1

z = (-1)^(1/4) =
= ( cosπ + i·sinπ )^(1/4) =
= |de Moivre's formula| =
= cos((π + 2πk)/4) + i·sin((π + 2πk)/4), k = 0, 1, 2, 3.

z1 = |k = 0| = cos(π/4)& + i·sin(π/4)& = (1/√2)(1+i);
z2 = |k = 1| = cos(3π/4) + i·sin(3π/4) = (1/√2)(-1+i);
z3 = |k = 2| = cos(5π/4) + i·sin(5π/4) = (1/√2)(-1-i);
z4 = |k = 3| = cos(7π/4) + i·sin(7π/4) = (1/√2)(1-i).