 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/ib/paper-4/17b/b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

On the circular arc,

∣ z ∣ = R

, the reverse triangle inequality gives

∣1 + z^{n} ∣ \geq R^{n} - 1

2 π R / n

, so the ML inequality gives

\int_{γ_{R}} f (z) d z \leq \frac{2 π R}{n} \frac{R ^{m}}{R ^{n} - 1} = O (R^{m + 1 - n}) ⟶ 0.

The strict assumption

n > m + 1

is exactly what makes the exponent negative; it also ensures convergence of the positive-real-axis integral at infinity.

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