Friedel oscillations in a Luttinger liquid with long-range interactions

We introduce a path-integral approach that allows us to compute charge-density oscillations in a Luttinger liquid with impurities. We obtain an explicit expression for the envelope of Friedel oscillations in the presence of arbitrary electron-electron potentials. As examples, in order to illustrate the procedure, we show how to use our formula for contact and Coulomb potentials.