The recently developed multi-term approach for solving the space-dependent electron Boltzmann equation is adopted to study, under the extreme field conditions occurring in the cathode region of the dc glow discharge, the energy distribution and all important macroscopic quantities of the electrons. It is shown that even under the difficult conditions in the cathode region an efficient determination of significant components of the electron velocity distribution by the multi-term treatment of the kinetic equation is possible. These significant components can be calculated with remarkable accuracy and with relatively less computational expenditure by a multi-term approach with the inclusion of about ten coefficients of the Legendre polynomial expansion of the velocity distribution. Based on the results for a typical example, main features of rite electrons in the cathode region are presented and discussed.