  We provide analytic expressions for the Green's functions in
position-frequency space as well as for the tunneling density of states of
various Luttinger liquids at zero temperature: the standard spinless and
spinful Luttinger liquids, the helical Luttinger liquid at the edge of a
topological insulator, and the Luttinger liquid that appears either together
with an ordering transition of nuclear spins in a one-dimensional conductor, or
in spin-orbit split quantum wires in an external magnetic field. The latter
system is often used to mimic a helical Luttinger liquid, yet we show here that
it exhibits significantly different response functions and, to discriminate, we
call it the spiral Luttinger liquid. We give fully analytic results for the
tunneling density of state of all the Luttinger liquids as well as for most of
the Green's functions. The remaining Green's functions are expressed by simple
convolution integrals between analytic results.
