Solution of the Pierce problem near a corner of a rectilinear flow with a polygonal cross-section

An ill-posed problem of synthesis of the Pierce electrodes for a cylindrical beam with a polygonal cross-section is considered. It is assumed that a beam of charged particles is extracted from a space-charge-limited planar diode and the Pierce electrodes outside of the beam ensure its zero angular divergence. A mathematical statement of the problem presumes a computation of the electrostatic potential outside of the beam that should match the Child-Langmuir 1D potential inside of the beam. An exact solution is first obtained for the potential outside of the beam near its right angle. The solution involves double analytic continuation and a numerical integration of the hypergeometric function and can be used as a benchmark for testing numerical codes. It is shown that equipotential surfaces have fractures that can be pushed away from the corner of the beam by means of smoothing the beam corners. This solution is then generalized to an angle of arbitrary magnitude.