  Previous functional integral methods for translationally invariant systems
have been extended to the case of a confining trap potential. Essentially all
finite-temperature properties of the repulsive Bose gas in a paraboloidal trap
can be determined this way. New analytical results reported here are for the
finite-temperature two-point correlation functions below the critical
temperatures in d=3,2 and 1 dimensions. Only for d=3 are correlations both long
range and coherent - at long range consistent with the existence of a coherent
state of the condensate in the trap. The two-dimensional condensate is
marginally stable in the sense that correlations decay by a power law.
