Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 327 / 1 / b / i

Let

v (λ) = - i \int_{1}^{\infty} \frac{e ^{- iλ x}}{x ^{2}} d x .

φ

λ

give

\int_{R} φ^{'} (λ) v (λ) d λ = - i \int_{1}^{\infty} \frac{1}{x ^{2}} (\int_{R} φ^{'} (λ) e^{- iλ x} d λ) d x = - \int_{1}^{\infty} \frac{1}{x} (\int_{R} φ (λ) e^{- iλ x} d λ) d x = ⟨ u, φ ⟩ .

Equivalently,

v^{'} = - u

⟨ u, φ ⟩ = \int_{R} φ^{'} (λ) v (λ) d λ .

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