Abstract
The formalism of the elementary solution of the Vlasov kinetic equation described in this paper enables us to investigate rather easily macroscopic properties of thermal charged-particle beams in the paraxial region. The distribution function can be found readily for plane electron or ion sources perpendicular to the axis of the beam with arbitrary velocity distribution. Plane sources with inhomogeneous emissivities are included into this conception if the phase trajectories describe precisely enough the considered situation. In principle, it is possible to construct the distribution function for any geometric configuration with axial symmetry (in the paraxial region) with the aid of the elementary solution and the convolution.
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