 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ib/paper-2/12b/b/i/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Liouville theorem states that every bounded entire function is constant.

Because

C

is simply connected, the harmonic function

u

v

, so

F = u + i v

is entire. Positivity gives

∣ e^{- F} ∣ = e^{- u} < 1.

Liouville's theorem makes

e^{- F}

constant. Differentiating this nonzero constant gives

F^{'} = 0

, so

F

, and hence

u

, is constant.

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