The inverse <Fourier transform> is linear:
$$
\Phi(x)=\int_{p^2<\Lambda^2}\frac{d^dp}{(2\pi)^d}
e^{ipx}\widetilde\Phi(p).
$$
Splitting the integration domain into $p^2<b^2\Lambda^2$ and $b^2\Lambda^2<p^2<\Lambda^2$, then inserting the definitions of $\widetilde\phi$ and $\widetilde\chi$, gives
$$
\Phi(x)=\int_{p^2<b^2\Lambda^2}\frac{d^dp}{(2\pi)^d}e^{ipx}\widetilde\phi(p)
+\int_{b^2\Lambda^2<p^2<\Lambda^2}\frac{d^dp}{(2\pi)^d}e^{ipx}\widetilde\chi(p)
=\boxed{\phi(x)+\chi(x)}.
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-304/3/a/solution