Solutions of the Laplace equation in cylindrical coordinates, reduced to two-dimensional harmonic potentials

The solutions of the Dirichlet problem for the Laplace equation in cylindrical coordinates are considered. The approaches are studied that allow such problems for axisymmetric and transaxial corpuscular optical systems to calculate two-dimensional harmonic potentials to find which methods of the theory of functions of a complex variable (TFCV) are used. A simple analytical formula is derived that accurately describes the electrostatic potential of a quadrupole field with electrodes in the form of a circular cylinder. Analytical formulas are found that describe with quite high accuracy the electrostatic potential of the field of a multi electrode axial symmetric cylindrical lens or mirror. Analytical expressions are also obtained that describe with good accuracy the electrostatic potential of a three-electrode transaxial lens. The found analytical formulas for potentials are in good agreement with the results obtained by other methods.