Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 45 / 4 / b / Solution

Write $K=k_{\max }r$. Using the transform in part (a), the band-limited linear density correlation isTwo integrations by parts give the antiderivative $-k^2\cos (kr)/r+2k\sin (kr)/r^2+2\cos (kr)/r^3$. Subtract its value at zero to obtain, for $r>0$,The origin is a removable limit. The bracket starts with $K^4/4$, soThis is also the density variance obtained directly by setting $\sin (kr)/(kr)=1$ in the integral. The sharp spectral edge produces oscillatory correlations, with leading large-$r$ term $-A{k}_{\max }^2\cos (k_{\max }r)/(2{\pi }^2{r}^2)$. A negative correlation at some separation is compatible with a nonnegative power spectrum.