Linear fractional map z = 0,1,-1 to i,2,4

Linear fractional mapping

A linear fractional (bilinear, Möbius) mapping is w = (az + b)/(cz + d), where a, b, c, d are complex constants and ad − bc ≠ 0. It maps circles and lines to circles or lines, and is fixed by three points and their images.

Solution

Take z₁ = 0, z₂ = 1, z₃ = −1 and w₁ = i, w₂ = 2, w₃ = 4.

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(1 + 1)/[(z + 1)(1 − 0)] = 2z/(z + 1).

Left side: (w − i)(2 − 4)/[(w − 4)(2 − i)] = −2(w − i)/[(w − 4)(2 − i)].

(w − i)/(w − 4) = −(2 − i) z/(z + 1)  =: S
w − i = S(w − 4)   →   w = (i − 4S)/(1 − S)

Substitute S = −(2 − i)z/(z + 1) and multiply numerator and denominator by (z + 1):

numerator   = i(z + 1) + 4(2 − i)z = (8 − 3i) z + i
denominator = (z + 1) + (2 − i) z  = (3 − i) z + 1

Answer: w = [(8 − 3i) z + i] / [(3 − i) z + 1]

Check

• z = 0: w = i/1 = i ✓
• z = 1: (8 − 3i + i)/(3 − i + 1) = (8 − 2i)/(4 − i) = 2 ✓
• z = −1: (−8 + 3i + i)/(−3 + i + 1) = (−8 + 4i)/(−2 + i) = 4 ✓