  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 Goldstone modes and
determine M in the thermodynamic limit both by extrapolation and by chiral
perturbation theory. This enables us to calculate the corresponding critical
amplitude. At T_c the critical scaling behaviour of the magnetization as a
function of H is used to determine another critical amplitude. In both cases we
find negative corrections-to-scaling. Our low-temperature results are well
described by the perturbative form of the model's magnetic equation of state,
with coefficients determined nonperturbatively from our data. The O(2) scaling
function for the magnetization is found to have a smaller slope than the one
for the O(4) model.
