Correction to the classical RPA dielectric function in all three dimensions for a charged particle medium

The classical dielectric function has been studied extensively in the Random Phase Approximation (RPA), which yields the Debye-Huckel result. The present work extends the accuracy of the calculations beyond the RPA for a charged particle medium of either one component (with an oppositely charged 'jellium' background) or two components of opposite charge. The calculations will be carried out in one, two, and three dimensions, with particular emphasis on the two dimensional results. Numerical values of the dielectric function will be obtained for the second order results (the RPA values are referred to as the first order results). Surprisingly, these values differ only slightly between the dimensions for a given value of a particular parameter within the region of applicability. A diagrammatic method is also developed to provide a systematic procedure for approaching higher order results.