 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2010/ii/paper-4/31c/b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

The unique minimum is at zero with phase

t^{2}

cos 0 = 1

. The interior Laplace method gives

\int_{- 2}^{1} cos t e^{- λ t^{2}} d t \sim \frac{π}{λ} .

For an additional check, extending to the real line gives the exact Gaussian Fourier integral

π / λ e^{- 1/ (4 λ)}

. The omitted tails are exponentially small, so the displayed leading term follows directly.

 New to topics? Read the docs here!