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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-3/30d/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 30D / a / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-29

The phase function

cos t

attains its global maximum

1

on

[0, 3 π]

at the stationary endpoint

t = 0

and at the interior point

t = 2 π

. At either point,

cos (t_{0} + u) = 1 - \frac{u ^{2}}{2} + O (u^{4}) .

By Laplace's method at a nondegenerate stationary endpoint, the endpoint contributes

e^{x} \frac{π}{2 x} .

By Laplace's method, the interior maximum contributes

e^{x + 2 π} \frac{2 π}{x} .

The remaining parts of the interval are exponentially smaller. Summing the two contributions according to Laplace's method with multiple global maxima gives

I (x) \sim e^{x} \frac{π}{2 x} (1 + 2 e^{2 π}) .

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