Boundary conditions and generalized functions in a transition radiation problem

The aim of this work is to show how all the components of the electromagnetic field involved in the transition radiation problem can be obtained using distribution functions. The handling of the products and derivatives of distributions appearing in the differential equations governing transition radiation, allows to obtain the necessary boundary conditions, additional to those implied by Maxwell's equations, in order to exactly determine the longitudinal components of the electromagnetic field. It is shown that this method is not only useful but it is really convenient to achieve a full analysis of the problem.