Lösung 3.3:2b

Aus Online Mathematik Brückenkurs 2

The equation z3=−1 is a so-called binomial equation, which we solve by writing both sides in polar form. We have

z−1=r(cos+isin)=1(cos+isin)

and, with the help of de Moivre's formula, the equation becomes

r3(cos3+isin3)=1(cos+isin).

Both sides are equal when their magnitudes are equal and the arguments differ by a multiple of 2,

r33=1=+2n(n is an arbitrary integer),

which gives that

r=1=3+32n(n is an arbitrary integer).

For every third integer n, the solution formula gives in principal the same value for the argument (the difference is a multiple of 2), so the equation has in reality three solutions (for n=0, 1 and 2),

z=1cos3+isin31cos+isin1cos35+isin35=21+i3−121−i3.

We obtain the typical behaviour that the solutions are corner points in a regular polygon (a triangle in this case because the degree of the equation is 3.