 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ib/paper-1/2a/b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

For

Re z > 0

∣ z + 1 ∣^{2} - ∣ z - 1 ∣^{2} = 4 Re z > 0,

w = (z - 1) / (z + 1)

lies in the unit disc. Its inverse is

z = (1 + w) / (1 - w)

, whose real part is

(1 - ∣ w ∣^{2}) /∣1 - w ∣^{2} > 0

. Both derivatives are nonzero, proving this Möbius transformation is conformal.

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