Lösung 3.3:4a

Aus Online Mathematik Brückenkurs 2

This is a typical binomial equation which we solve in polar form.

We write

zi=r(cos+isin)=1cos2+isin2

and, on using de Moivre's formula, the equation becomes

r2(cos2+isin2)=1cos2+isin2.

Both sides are equal when

r22=1=2+2n(n is an arbitrary integer),

which gives that

r=1=4+n(n is an arbitrary integer).

When n=0 and n=1, we get two different arguments for , whilst different values of n only give these arguments plus/minus a multiple of 2.

The solutions to the equation are

z=1cos4+isin41cos43+isin43=21+i−21+i.