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Aus Online Mathematik Brückenkurs 2

If we use w=z−1 as a new unknown and move the term 4 over to the right-hand side, we have a binomial equation,

w4=−4.

We can solve this equation in the usual way by using polar form and de Moivre's formula. We have

w−4=r(cos+isin)=4(cos+isin)

and the equation becomes

r4(cos4+isin4)=4(cos+isin).

The only way that both sides can be equal is if the magnitudes agree and the arguments do not differ by anything other than a multiple of 2,

r44=4=+2n(n is an arbitrary integer),

which gives us that

r=44=2=4+2n(n is an arbitrary integer).

For n=01, 2 and 3, the argument assumes the four different values

4, 43, 45and47

and for other values of n we obtain values of which are equal to those above, apart from multiples of 2. Thus, we have four solutions,

w=2cos4+isin42cos43+isin432cos45+isin452cos47+isin47=1+i−1+i−1−i1−i,

and the original variable z is

z=2+ii−i2−i.