We derive asymptotic series for the expansion coefficients of a function in terms
of the Pagani functions, which occur in the boundary layer solutions of the
Klein-Kramers equation. The results enable us to determine the density profile
in the stationary solution of this equation near an absorbing wall from the
numerically determined velocity distribution at the wall, with an accuracy of
about 2%. We also obtain information about the analytic behavior of the
density profile: this profile increases near the wall with the square root of the
distance to the wall. Finally, the asymptotic analysis leads to an understanding
of the slow convergence of variational approximations to the solution of the
absorbing-wall problem and of the exponents that occur when one studies the
variational approximations to various quantities of interest as functions of the
number of terms in the variational ansatz. This is used to obtain a better
variational estimate for the density at the wall.