Conformal mapping; bilinear map to -3,-1,1

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 bilinear transformation

Let w = (az + b)/(cz + d), ad − bc ≠ 0.

• z = ∞ → w = 1: a/c = 1, so a = c
• z = 0 → w = −3: b/d = −3, so b = −3d
• z = 1 → w = −1: a + b = −(c + d)

Put a = c and b = −3d: c − 3d = −c − d, so c = d. Hence a = c = d, and b = −3a.

Take a = 1: w = (z − 3)/(z + 1).

Check: z = 0 gives −3 ✓. z = 1 gives −2/2 = −1 ✓. z = ∞ gives 1 ✓. Here ad − bc = 4 ≠ 0, so the mapping is valid and conformal everywhere except at z = −1.