 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ib/paper-1/12f/b/solution

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

Write the Laurent expansion on the punctured disc as

f (z) = \sum_{n = - \infty}^{\infty} a_{n} z^{n} .

For every integer

k \geq 1

∣ a_{- k} ∣ = \frac{r ^{k}}{2 π} \int_{0}^{2 π} f (r e^{i θ}) e^{ik θ} d θ \leq \frac{r ^{k}}{2 π} (\int_{0}^{2 π} ∣ f (r e^{i θ}) ∣^{2} d θ)^{1/2} (\int_{0}^{2 π} 1 d θ)^{1/2} \leq \frac{r ^{k}}{2 π} .

Letting

r ↓ 0

shows that

a_{- k} = 0

f has a removable singularity at 0 .

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