Question

Determine the region of the w plane into the first quadrant mapped by the transformation w =z²

Answer 1

Let w = z² , z = re^iθ . Then w = r² e^2iθ . Thus points at (r, θ) are rotated by a further angle θ and their modulus stretched by a factor r.

First quadrant of z-plane

All points in 1st quadrant occupy r > 0, 0 ≤ θ ≤ π/2 . 

Thus all points in w(= ρe^iφ)-plane occupy ρ > 0, 0 ≤ φ ≤ π, i.e., the upper half of the w-plane.