Band-limited linear density correlation ($\xi (r)=\frac{A}{2{\pi }^2{r}^4}[-K^2\cos K+2K\sin K+2(\cos K-1)]$)

For an isotropic matter power spectrum $P(k)=Ak$ on $0\le k\le k_{\max }$ and zero elsewhere, put $K=k_{\max }r$. Angular integration of the Fourier transform gives $\xi (r)=A(2{\pi }^{2}r{)}^{-1}{\int }_0^{k_{\max }}{k}^2\sin (kr)dk$, and two integrations by parts yield the displayed answer. Its apparent origin singularity is removable, with $\xi (0)=A{k}_{\max }^4/(8{\pi }^2)$. The sharp cutoff creates oscillatory, including negative, correlations without violating positivity of the spectrum.