Solutions of the Percus shielding equation

We consider the Percus shielding approximation (PSA) for the density profile of a hard sphere fluid in the presence of a hard spherical inclusion or plane wall. Two calculations are reported. For an inclusion which is equal to a fluid particle, an equation of state is obtained from the numerical solution of the integral equation by requiring self-consistency. The results are inferior to the predictions of the Percus–Yevick equation. For the case of a plane wall, where the PSA is good, the exact solution of the PSA is found in terms of elementary functions. The recurrence relations for polynomial coefficients in the solution are given in some detail.