Conformal mapping; map 1, i, -1 to 0, 1, ∞

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 = 1, z₂ = i, z₃ = −1 map to w₁ = 0, w₂ = 1, w₃ = ∞.

Since w₁ = 0 corresponds to z = 1 and w₃ = ∞ corresponds to z = −1, the transformation has the form

w = k (z − 1)/(z + 1).

Use z = i, w = 1:

1 = k (i − 1)/(i + 1). Since (i − 1)/(i + 1) = [(i − 1)(1 − i)]/[(1 + i)(1 − i)] = 2i/2 = i, we get 1 = k·i, so k = 1/i = −i.

Answer: w = −i (z − 1)/(z + 1) = i (1 − z)/(1 + z)

(Cross-ratio check: with w₃ → ∞ the formula reduces to (w − w₁)/(w₂ − w₁) = (z − z₁)(z₂ − z₃)/[(z − z₃)(z₂ − z₁)], giving w = (z − 1)(i + 1)/[(z + 1)(i − 1)] = −i(z − 1)/(z + 1) since (i + 1)/(i − 1) = −i.)

Check: z = 1 gives 0 ✓. z = i gives −i·i = 1 ✓. z = −1 gives ∞ ✓.