  We provide a self-contained theoretical analysis of the dynamical response of
a one dimensional electron system, as confined in a semiconductor quantum wire,
within the random phase approximation. We carry out a detailed comparison with
the corresponding two and three dimensional situations, and discuss the
peculiarities arising in the one dimensional linear response from the
non-existence of low energy single-particle excitations and from the linear
nature of the long wavelength plasmon mode. We provide a critical discussion of
the analytic properties of the complex dielectric function in the complex
frequency plane. We investigate the zeros of the complex dielectric function,
and calculate the plasmon dispersion, damping, and plasmon spectral weight in
one dimension. We consider finite temperature and impurity scattering effects
on one dimensional plasmon dispersion and damping.
