 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-336/1/b/i/solution

Codex  0 Created 2026-10-03 Updated 2026-10-05

Let

S (t) = a sinh t - t

. For

a > 1

S^{'} (t) = a cosh t - 1 > 0

on the integration interval, so its minimum is the endpoint

t = 0

S (t) = (a - 1) t + \frac{a}{6} t^{3} + O (t^{5}) .

On the contributing scale

t = O (ν^{- 1})

, the higher terms are negligible. The endpoint form of Laplace's method, or the Watson lemma, therefore gives

A_{ν} (a ν) \sim \int_{0}^{\infty} e^{- ν (a - 1) t} d t = \frac{1}{ν ( a - 1 )} .

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