  We develop theory for a flattened dipolar Bose-Einstein condensate (BEC)
produced by harmonic confinement along one direction. The role of both
short-ranged contact interactions and long-ranged dipole-dipole interactions
(DDIs) is considered, and the dipoles are allowed to be polarised along an
arbitrary direction. We discuss the symmetry properties of the condensate and
the part of the excitation spectrum determining stability, and introduce two
effective interaction parameters that allow us to provide a general description
of the condensate properties, rotons, and stability. We diagonalize the full
theory to obtain benchmark results for the condensate and quasiparticle
excitations, and characterize the exact mean field stability of the system. We
provide a unified formulation for a number of approximate schemes to describe
the condensate and quasiparticles, including the standard quasi-two-dimensional
(quasi-2D) approximation, two kinds of variational ansatz, and a Thomas-Fermi
(TF) approximation. Some of these schemes have been widely used in the
literature despite not being substantiated against the exact theory. We provide
this validation and establish the regimes where the various theories perform
well.
