Question

Determine the region of the w-plane into which the triangle formed by x = 1, y = 1 and x + y = 1 is mapped under 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.

Region bounded by x = 1, y = 1m, x + y = 1.

If w = z² we have

Thus we have:

(z)

Pick a points inside the region ABC to see where it goes and confirm

ABC → A'B'C' shaded region.

Note that w' = f(z) = 2z, so unless z = 0 the local angles should be conserved.

Thus is ∠BAC = ∠ABC = π/4

then ∠B'A'C' (or the tangents’ angle at A' ) is π/4

∠A'B'C' (or the tangents’ angle at B' ) is π/4

Similarly ∠ACB = π/2 and ∠A'C'B' is π/2