  We investigate numerically the three-dimensional O(2) model on 8^3-160^3
lattices as a function of the magnetic field H. In the low-temperature phase we
verify the H-dependence of the magnetization M induced by the Goldstone modes
and determine M in the thermodynamic limit on the coexistence line both by
extrapolation and by chiral perturbation theory. We compute two critical
amplitudes from the scaling behaviours on the coexistence line and on the
critical line. In both cases we find negative corrections to scaling. With
additional high temperature data we calculate the scaling function and show
that it has a smaller slope than that of the O(4) model. For future tests of
QCD lattice data we study as well finite-size-scaling functions.
