Mathematical Methods for Physics and Engineering - Matematica.NET

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONSIm ze 2iπ/3 2π/32π/31Re ze −2iπ/3Figure 3.10 The solutions of z 3 =1.Not surprisingly, given that |z 3 | = |z| 3 from (3.10), all the roots of unity haveunit modulus, i.e. they all lie on a circle in the Argand diagram of unit radius.The three roots are shown in figure 3.10.The cube roots of unity are often written 1, ω and ω 2 . The properties ω 3 =1and 1 + ω + ω 2 = 0 are easily proved.3.4.3 Solving polynomial equationsA third application of de Moivre’s theorem is to the solution of polynomialequations. Complex equations in the form of a polynomial relationship must firstbe solved for z in a similar fashion to the method for finding the roots of realpolynomial equations. Then the complex roots of z may be found.◮Solve the equation z 6 − z 5 +4z 4 − 6z 3 +2z 2 − 8z +8=0.We first factorise to give(z 3 − 2)(z 2 +4)(z − 1) = 0.Hence z 3 =2orz 2 = −4 orz = 1. The solutions to the quadratic equation are z = ±2i;to find the complex cube roots, we first write the equation in the formz 3 =2=2e 2ikπ ,where k is any integer. If we now take the cube root, we getz =2 1/3 e 2ikπ/3 .98