A new method of analysis of critical-point singularities from power series expansions: Application to eigenvalue problems and virial series

Abstract

A new method is developed for studying real singularities of functions from their power series expansions. It is based on an appropriate calculation of the least-squares deviation of the Taylor coefficients from their asymptotic behavior. Under certain conditions the procedure is shown to lead to the widely used ratio method and it is applied to perturbation-theory problems and virial series. The convergence radii of the Rayleigh–Schrödinger perturbation series for the ground-state energy levels of the plane and linear polar rigid rotators in uniform electric fields are calculated. The occurrence of the singularities in the compressibility factor for hard disks and spheres predicted by the Percus–Yevick and scaled-particle theory is verified. The critical parameters characterizing the Kirkwood ‘‘phase transition’’ near close-packing density and the asymptotic form of the equation of state in the neighborhood of the singularities are verified. Present results elucidate a critical-exponent paradox.