  The field-driven reorientation transition of an anisotropic ferromagnetic
monolayer is studied within the context of a finite-temperature Green's
function theory. The equilibrium state and the field dependence of the magnon
energy gap $E_0$ are calculated for static magnetic field $H$ applied in plane
along an easy or a hard axis. In the latter case, the in-plane reorientation of
the magnetization is shown to be continuous at T=0, in agreement with free spin
wave theory, and discontinuous at finite temperature $T>0$, in contrast with
the prediction of mean field theory. The discontinuity in the orientation angle
creates a jump in the magnon energy gap, and it is the reason why, for $T>0$,
the energy does not go to zero at the reorientation field. Above the Curie
temperature $T_C$, the magnon energy gap $E_0(H)$ vanishes for H=0 both in the
easy and in the hard case. As $H$ is increased, the gap is found to increase
almost linearly with $H$, but with different slopes depending on the field
orientation. In particular, the slope is smaller when $H$ is along the hard
axis. Such a magnetic anisotropy of the spin-wave energies is shown to persist
well above $T_C$ ($T \approx 1.2 T_C$).
