 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/iii/paper-12/4/a/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

For

∣ w ∣ \leq 1/2

, the power series of the principal complex logarithm is absolutely convergent and gives

∣ Log (1 - w) ∣ \leq \sum_{j = 1}^{\infty} \frac{∣ w ∣ ^{j}}{j} \leq \sum_{j = 1}^{\infty} ∣ w ∣^{j} = \frac{∣ w ∣}{1 - ∣ w ∣} \leq 2∣ w ∣.

Thus $C_{1} = 2$ works. The value

1 - w

stays in the right half-plane, so there is no logarithm branch ambiguity. This is the bounds for the principal logarithm near one estimate used to control product tails.

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