  We use a functional integral formalism developed earlier for the pure
Luttinger liquid (LL) to find an exact representation for the electron Green
function of the LL in the presence of a single backscattering impurity. This
allows us to reproduce results (well known from the bosonization techniques)
for the suppression of the electron local density of states (LDoS) at the
position of the impurity and for the Friedel oscillations at finite
temperature. In addition, we have extracted from the exact representation an
analytic dependence of LDoS on the distance from the impurity and shown how it
crosses over to that for the pure LL.
