  An analysis of the different approaches used within the van Hove BCS model
for the high temperature superconductors has been done. How far the employment
of an asymptotic expression for the density of states underestimates the
thermodynamic parameters, as for example, the critical temperature, the gap at
zero temperature, and the known universal ratios of the BCS model, is
discussed. Analytical expressions obtained for some thermodynamic functions are
compared with the numerical results for which the exact 2D density of states
has been considered. Approximate analytical expressions for the temperature
dependence of the reduced gap, the superconducting electronic specific heat,
and the critical magnetic field have been obtained and compared with the
corresponding numerical results. The validity of the different approximations
and of the equations obtained for the thermodynamic functions in the framework
of the van Hove model are also analyzed.
