  We present a detailed finite-temperature Hartree-Fock-Bogoliubov (HFB)
treatment of the two-dimensional trapped Bose gas. We highlight the numerical
methods required to obtain solutions to the HFB equations within the Popov
approximation, the derivation of which we outline. This method has previously
been applied successfully to the three-dimensional case and we focus on the
unique features of the system which are due to its reduced dimensionality.
These can be found in the spectrum of low-lying excitations and in the
coherence properties. We calculate the Bragg response and the coherence length
within the condensate in analogy with experiments performed in the
quasi-one-dimensional regime [Richard et al., Phys. Rev. Lett. 91, 010405
(2003)] and compare to results calculated for the one-dimensional case. We then
make predictions for the experimental observation of the quasicondensate phase
via Bragg spectroscopy in the quasi-two-dimensional regime.
