Linear fractional map 2i, -2, -2i to -2, -2i, 2

A conformal mapping w = f(z) is a mapping that preserves angles between any two curves, both in magnitude and in sense (direction), at every point where f′(z) ≠ 0. Every analytic function with f′(z) ≠ 0 is conformal.

Finding the transformation

z₁ = 2i, z₂ = −2, z₃ = −2i map to w₁ = −2, w₂ = −2i, w₃ = 2.

Cross-ratio form of the bilinear transformation:

(w − w₁)(w₂ − w₃) / [(w − w₃)(w₂ − w₁)] = (z − z₁)(z₂ − z₃) / [(z − z₃)(z₂ − z₁)]

Right side

(z − 2i)(−2 + 2i) / [(z + 2i)(−2 − 2i)]. Now (−2 + 2i)/(−2 − 2i) = (−1 + i)/(−1 − i) = −i (multiply top and bottom by −1 + i: (−1 + i)²/2 = −2i/2 = −i).

So the right side = −i (z − 2i)/(z + 2i).

Left side

(w + 2)(−2i − 2) / [(w − 2)(−2i + 2)]. Now (−2 − 2i)/(2 − 2i) = (−1 − i)/(1 − i) = (−1 − i)(1 + i)/2 = −(1 + i)²/2 = −i.

So the left side = −i (w + 2)/(w − 2).

Equate

(w + 2)/(w − 2) = (z − 2i)/(z + 2i)

Cross-multiply: (w + 2)(z + 2i) = (w − 2)(z − 2i)

wz + 2iw + 2z + 4i = wz − 2iw − 2z + 4i → 4iw = −4z → w = −z/i = iz.

Answer: w = iz

Check: 2i → i·2i = −2 ✓. −2 → −2i ✓. −2i → i(−2i) = 2 ✓.