Determination of analytical expansion from numerical field data

We introduce a method of calculation of the analytical expansion of the field near the axis that is based on an application of Greenʹs theorem. The approach is demonstrated on an example of a round electrostatic unipotential lens with field computed by the finite-element method and results are compared to methods of Hermite polynomials and wavelet transformation which are used in electron optics. The work is motivated by application to calculations of aberration coefficients where the high order axial field derivatives must be known.