Evaluation of the collision integral and solution of the basic equations of electromagnetic cascade theory

A new analytic solution is presented for the one-dimensional problem of electromagnetic cascade theory, taking into account ionization losses and the Landau- Pomeranchuk effect. The ionization losses are not assumed to be proportional to the variation in the number of particles in the shower, but are taken into account by direct approximation of the collision integral, so that the specific properties of the medium are more accurately taken into account. The proposed approximation and the solution presented in this paper can be used to obtain the distribution functions with their prescribed accuracy. These solutions are valid in a broad energy range and yield improved versions of known distribution functions.