The Kirkwood–Salsburg equations for a bounded stable Kac potential. I. General theory and asymptotic solutions

We derive for Kac potentials of the form γsψ (λx) an expansion, for all the distribution functions, in powers of γs (s=dimension) and prove for z<?cr that the expansion is at least asymptotic. The coefficients in the expansion are shown to be solutions of linear operator equations similar to the Kirkwood–Salsburg equation. We also explicitly obtain a rather simple expression for the coefficients of γs and show that they are given by solving the Ornstein–Zernicke integral equation with the choice of −βψ (y) for the direct correlation function.