For the one-dimensional (1D) Hubbard model, we extended the bosonizati
on technique, away from half filling, in such a way that a general for
mula is obtained for the zero temperature equal-time single-particle G
reen function with validity over the whole doping range. With our meth
od, we can calculate, for the first time, the one-body Green function
and thus, the spectral function in both the Tomonaga-Luttinger and Lut
her-Emery universality classes. The method also gives a tool to determ
ine the coefficients of the correlation functions which cannot be dete
rmined with previously used methods.