Question

Find the range of the function defined by f(z)=Re[2iz/(1-z^2)].

(a) (−∞, 0) ⋃ (0, ∞)

(b) [2, ∞)

(c) (−∞, −1] ⋃ [1, ∞)

(d) (−∞, 0] ⋃ [2, ∞)

I got this question in my homework.

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

Answer 1

Correct option is (c) (−∞, −1] ⋃ [1, ∞)

Explanation: z=2i(x+iy)/(1-(x+iy)^2)=2i(x+iy)/(1-(x^2-y^2+2ixy))

Using 1-x^2=y^2, z=(2ix-2y)/(2y^2-2ixy)=-1/y

∵ –1≤y≤1 ⇒ –1/y≤-1 or -1/y≥1.