  We solve the Schwinger-Dyson equations for QED in 2+1 or 3+1 dimensions in
the presence of a strong homogeneous external magnetic field. The magnetic
field is assumed strong enough, so that the lowest Landau level approximation
holds, but the usual assumption of a momentum-independent self-energy is not
made. In 2+1 dimensions, the scaling with logarithm changes to a square root
dependence on the magnetic field, but the most spectacular result takes place
in 3+1 dimensions, where the constant mass approximation turns out to be
unreliable and the (momentum-dependent) dynamical mass is larger by several
orders of magnitude compared to what has been found till now using the constant
mass approximation.
