 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/ii/paper-3/30a/b/solution

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

Use disjoint small neighborhoods of the finitely many nondegenerate stationary points

t_{j}

, apply the stationary phase formula in each, and integration by parts elsewhere. The leading contributions add:

I (λ) = j \sum f (t_{j}) \frac{2 π}{λ ∣ g ^{''} ( t _{j} ) ∣} e^{iλ g (t_{j}) + iπ sgn (g^{''} (t_{j})) /4} + O (λ^{- 1}) .

Interference between these oscillatory contributions can cancel the leading coefficient at particular values of

λ

 New to topics? Read the docs here!