  We analyze the problem of microwave absorption by the Heisenberg-Ising magnet
in terms of shifted moments of the imaginary part of the dynamical
susceptibility. When both, the Zeeman field and the wave vector of the incident
microwave, are parallel to the anisotropy axis, the first four moments
determine the shift of the resonance frequency and the line width in a
situation where the frequency is varied for fixed Zeeman field. For the
one-dimensional model we can calculate the moments exactly. This provides exact
data for the resonance shift and the line width at arbitrary temperatures and
magnetic fields. In current ESR experiments the Zeeman field is varied for
fixed frequency. We show how in this situation the moments give perturbative
results for the resonance shift and for the integrated intensity at small
anisotropy as well as an explicit formula connecting the line width with the
anisotropy parameter in the high-temperature limit.
