A paraxial solution to the exterior problem of the theory of 2D and 3D electron beam formation

For several reference 2D and 3D nonparaxial beams considered in coordinate systems associated with the geometry of a flow, a solution determining the configuration of forming electrodes is obtained in the paraxial approximation. The approximate solution is compared with the exact solution if the latter exists. The comparison is performed for extrapolation to the cases when this approach seems the only one possible.