Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 1 / 12F / b

Solution

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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