Exact results for some Madelung type constants in the finite-size scaling theory

Abstract

A general formula is obtained from which the madelung type constant: $$ C(d|ν)=\int_0^\infty dx x^{d/2-ν-1}[(\sum_{l=-\infty}^\infty e^{-xl^2})^d-1-(\fracπx)^{d/2}] $$ extensively used in the finite-size scaling theory is computed analytically for some particular cases of the parameters $d$ and $ν$. By adjusting these parameters one can obtain different physical situations corresponding to different geometries and magnitudes of the interparticle interaction.