  The effective action is computed for the \lphi--theory at finite temperature
for small perturbations about a constant background field, using a generalized
tadpole method. We find the complete effective action, including the real and
imaginary parts, to all orders in derivatives and to order ${\cal
O}(\lambda^2)$. We demonstrate that the high $T$ approximation, where only the
zero Matsubara frequency is included, is incorrect for the imaginary part even
though it is UV-finite. The solutions of the dispersion relations show that
initial perturbations do not necessarily thermalize fast enough to be absent at
the onset of phase transition.
