  We calculate by a new way the two point function on the boundary of AdS
spacetime in 1+2 dimensions for the massless conformal real scalar field. The
result agrees with the answer provided by the Boundary-limit Holography and
Witten recipe. This is done in Poincar\'{e} coordinates. The basic ingredients
of this new method are conformal techniques, quantum fields defined on a half
of Minkowski spacetime and a limit inspired by the Boundary-limit Holography.
We also show that a state in AdS, the global vacuum, in three dimensions
induces a state on the conformal boundary of AdS spacetime, which in turn
induces a state on the BTZ black hole. On the other hand the same state in AdS
induces a state on the BTZ black hole which in turn induces a state on its
conformal boundary. The two ways of getting the state on the conformal boundary
of the BTZ black hole coincide for the massless conformal real scalar field. We
point out that the normalizable modes in the AdS/CFT correspondence for the BTZ
black hole give a similar contribution as the non-normalizable modes used in
the Witten prescription. We also give some clues on why the Witten and the
Boundary-limit Holography prescription coincide.
