  We study the transverse momentum dependence of the Landau-Pomeranchuk-Migdal
effect in QED, starting from the high energy expansion of the solution of the
Dirac equation in the presence of an external field. The angular integrated
energy loss formula differs from an earlier expression of Zakharov by taking
finite kinematical boundaries into account. In an expansion in powers of the
opacity of the medium, we derive explicit expressions for the radiation cross
section associated with N=1, 2 and 3 scatterings. We verify the Bethe-Heitler
and the factorization limit, and we calculate corrections to the factorization
limit proportional to the square of the target size. A closed form expression
valid to arbitrary orders in the opacity is derived in the dipole
approximation. The resulting radiation spectrum is non-analytic in the coupling
constant which is traced back to the transverse momentum broadening of a hard
parton undergoing multiple small angle Moliere scattering. In extending the
results to QCD, we test a previously used dipole prescription by comparing to
direct pQCD results for N=1 and 2. For N=1, the QCD dipole prescription
reproduces exactly the Bertsch-Gunion radiation spectrum. For N=2, we find a
sizeable correction which reduces to a multiplicative factor 17/8 at large
separation.
