  The computation of two and three point functions in the Coulomb gas free
field approach to string theory in the SL(2,R)/U(1) black hole background is
reviewed. An interesting relation arises when comparing the results obtained
using two different screening operators. The formalism is then modified to
study string theory propagating in AdS$_3$ which is considered as the direct
product of the SL(2)/U(1) coset times a timelike free boson. This
representation allows to naturally include the spectral flow symmetry and
winding number in vertex operators and correlation functions. Two and three
point tachyon amplitudes are computed in this new scenario and the results
coincide with previous reports in the literature. Novel expressions are found
for processes violating winding number conservation.
