  We obtain a closed-form analytical expression for the zero temperature
Fourier transform of the $2k_F$ component of the density-density correlation
function in a Luttinger liquid with different spin and charge velocities. For
frequencies near the spin and charge singularities approximate analytical forms
are given and compared with the exact result. We find power law like
singularities leading to either divergence or cusps, depending on the values of
the Luttinger parameters and compute the corresponding exponents. Exact
integral expressions and numerical results are given for the finite temperature
case as well. We show in particular how the temperature rounds the
singularities in the correlation function.
