Question

For the function f(z)=z^i, what is the value of |f(ω)|+Arg f(ω),  ω being the cube root of unity with Im(ω)>0?

(a) e^-2π/3

(b) e^2π/3

(c) e^-2π/3+2π/3

(d) e^-2π/3-2π/3

I have been asked this question in an interview for internship.

I'd like to ask this question from Functions of a Complex Variable in division Complex Function Theory of Engineering Mathematics

Answer 1

Right answer is (a) e^-2π/3

The best explanation: Let y=z^i⇒ ln y=iln z=i(ln |z|+iarg z)=iln |z|-arg z

⇒ y=e^iln |z|/e^arg z ⇒ |y|=e^arg z and Arg y=ln |z| ⇒ |f(ω)|+Arg f(ω)=e^-2π/3+0=e^-2π/3.