Question

Let f(z)=arg 1/(1 – z), then find the range of f(z) for |z|=1, z≠1.

(a) (-∞, π/2)

(b) (-π/2, π/2)

(c) (-∞, ∞)

(d) [0, π/2)

This question was addressed to me during an interview.

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

Answer 1

Correct answer is (b) (-π/2, π/2)

Easy explanation: Let y=1/(1-z) ⇒ z=1-1/y

|z|=1 ⇒ |1-1/y|=1 ⇒ |y-1|=|y| ⇒ locus of y is the perpendicular bisector of line segment joining 0 and 1 ⇒ arg y ∈(-π/2, π/2).