Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 304 / 3 / a

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- Solution a

Solution

The inverse Fourier transform is linear:

Φ (x) = \int_{p^{2} < Λ^{2}} \frac{d ^{d} p}{( 2 π ) ^{d}} e^{i p x} Φ (p) .

Splitting the integration domain into

p^{2} < b^{2} Λ^{2}

and

b^{2} Λ^{2} < p^{2} < Λ^{2}

, then inserting the definitions of

ϕ

and

χ

, gives

Φ (x) = \int_{p^{2} < b^{2} Λ^{2}} \frac{d ^{d} p}{( 2 π ) ^{d}} e^{i p x} ϕ (p) + \int_{b^{2} Λ^{2} < p^{2} < Λ^{2}} \frac{d ^{d} p}{( 2 π ) ^{d}} e^{i p x} χ (p) = ϕ (x) + χ (x) .

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