Question

Let f(z)=|1–z|, if zk=cos(2kπ/10)+isin(2kπ/10), then find the value of f(z1)×f(z2)×…×f(z9).

(a) 10

(b) 15

(c) 20

(d) 30

I had been asked this question in homework.

Enquiry is from Functions of a Complex Variable in chapter Complex Function Theory of Engineering Mathematics

Answer 1

Right option is (a) 10

Best explanation: z^10–1=(z-1)(z-z1)…(z-z9) ⇒ (z-z1)(z-z2) …(z-z9)=1+z+z^2+…+z^9.

Now, putting  z=1, we get, (z-z1)(z-z2)…(z-z9)=f(z1)×f(z2)×…×f(z9)=10.