 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-4/31c/c/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

Write

λ = 1 + δ

, where

δ ↓ 0

, and put

x = λ (1 - u)

. The integration range is

0 \leq u \leq 1 - 1/ λ = δ + O (δ^{2})

, and

((1 - u)^{- 2} - 1)^{1/2} = (2 u)^{1/2} + O (u^{3/2}) .

Consequently

\int_{1}^{λ} (\frac{λ ^{2}}{x ^{2}} - 1)^{1/2} d x = \frac{2 2}{3} δ^{3/2} + O (δ^{5/2}) .

Substitution into the quantization condition gives

λ = 1 + α_{n} ϵ^{2/3} + O (ϵ^{4/3}), α_{n} = [\frac{3 π}{2 2} (n + \frac{3}{4})]^{2/3} .

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