Padé Summation of the Cauchy Dispersion Equation*

Abstract

A method is presented, based on recognition of the Cauchy dispersion equation as a series of Stieltjes, for the bounded extrapolation of long-wavelength refractivity data into the far ultraviolet. The extrapolation is performed with the [n, n−1] and [n,n] Padé approximants, which improve the convergence of the Cauchy equation and provide upper and lower bounds to its exact sum. Illustrative applications are given for atomic hydrogen, the inert gases, and molecular hydrogen, oxygen, and nitrogen. Comparison with the exact dispersion curve for atomic hydrogen and with available theoretical and experimental ultraviolet-dispersion data for the inert and diatomic gases indicates that the Padé approximants converge rapidly to accurate refractivity values. In addition, recognition of the Cauchy equation as a series of Stieltjes provides nontrivial constraints on the Cauchy coefficients; these afford a test of their accuracy and allow estimates of higher-order coefficients from measured refractivity data.