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

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- Solution c

Solution

The Fourier coefficient of the velocity-integrated density is its mixed transform at zero velocity frequency. Thus

ρ_{t} (k) = f (t, k, 0) = f_{0} (k, k t) .

The constant

ρ_{\infty}

contributes only to the zero Fourier mode, where it equals

f_{0} (0, 0)

. Therefore

r_{t} (k) = f_{0} (k, k t) - 1_{{k = 0}} f_{0} (0, 0) = {f_{0} (k, k t), 0, k \neq = 0, k = 0.

It is important to remove the zero mode, which is conserved rather than mixed away.

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