Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 7 / 2 / c / Solution

The Fourier coefficient of the velocity-integrated density is its mixed transform at zero velocity frequency. ThusThe constant ${\rho }_{\infty }$ contributes only to the zero Fourier mode, where it equals ${\hat{f}}_0(0,0)$. ThereforeIt is important to remove the zero mode, which is conserved rather than mixed away.