 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-327/3/c/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

For any smooth amplitude

v (x, λ)

D_{x}^{α} (e^{iλ \cdot x} v) = e^{iλ \cdot x} (λ + D_{x})^{α} v

because

D_{x} = - i \partial_{x}

. Applying each term of the differential operator under the oscillatory integral gives

Q (x, D) A_{1} (x) = \frac{1}{( 2 π ) ^{n}} \int e^{iλ \cdot x} Q (x, λ + D_{x}) a_{1} (x, λ) d λ .

The identity is justified distributionally by regularizing the frequency integral and integrating by parts.

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