Linear fractional map -2, 0, 2 to ∞, 1/2, 3/4

Find w = (az + b)/(cz + d) with −2 → ∞, 0 → 1/2, 2 → 3/4.

Conditions

• z = −2 maps to ∞: the denominator vanishes, −2c + d = 0, so d = 2c.
• z = 0 maps to 1/2: b/d = 1/2, so d = 2b. Hence 2c = 2b and c = b.
• z = 2 maps to 3/4: (2a + b)/(2c + d) = 3/4.

Take b = c = 1. Then d = 2. The third condition gives (2a + 1)/(2 + 2) = 3/4, so 2a + 1 = 3 and a = 1.

Answer: w = (z + 1)/(z + 2)

Check

• z = −2: denominator 0, so w = ∞ ✓
• z = 0: 1/2 ✓
• z = 2: 3/4 ✓

Also ad − bc = 2 − 1 = 1 ≠ 0, so the transformation is non-degenerate.