  The infrared behavior of QED changes drastically in the presence of a strong
magnetic field: the electron self-energy and the vertex function are infrared
{\em finite}, in contrast with field-free QED, while new infrared divergences
appear that are absent in free space. One famous example of the latter is the
infrared catastrophe of magnetic Compton scattering, where the cross section
for scattering of photons from electrons which undergo a transition to the
Landau ground state {\em diverges} as the frequency of the incoming photon goes
to zero. We examine this divergence in more detail and prove that the
singularity of the cross section is {\em removed} as soon as proper account is
taken of all quantum electrodynamical processes that become indistinguishable
from Compton scattering in the limit of vanishing frequency of the incident
photon.

Astrophysics;High Energy Physics - Phenomenology