Question

Let f(z)=(z^2–z–1)^7. If  α^2+α+1=0 and Im(α)>0, then find f(α).

(a) 128α

(b) -128α

(c) 128α^2

(d) -128α^2

The question was posed to me during an internship interview.

I'm obligated to ask this question of Functions of a Complex Variable in chapter Complex Function Theory of Engineering Mathematics

Answer 1

Right option is (c) 128α^2

To explain: Note that α=ω. Therefore, f(α)=f(ω)=(ω^2–ω–1)^7

=(ω^2+ω^2)^7=(2ω^2)^7=2^7ω^14=128ω^2=128α^2.