  A single impurity in the 1D Luttinger model creates a local modification of
the charge density analogous to the Friedel oscillations. In this paper, we
present an exact solution of the case $g={1\over 2}$ (the equivalent of the
Toulouse point) at any temperature $T$ and impurity coupling, expressing the
charge density in terms of a hypergeometric function. We find in particular
that at $T=0$, the oscillatory part of the density goes as $\ln x$ at small
distance and $x^{-1/2}$ at large distance.
