Analytic properties of the structure function for the one-dimensional one-component log-gas

Abstract

The structure function S(k; β) for the one-dimensional one-component log gas is the Fourier transform of the charge charge, or equivalently the density density, correlation function. We show that for |k| < min(2πp, 2πpβ), S(k; β) is simply related to an analytic function f(k; β) and this function satisfies the functional equation f(k; β) = f( - 2k/β; 4/β). It is conjectured that the coefficient of kJ in the power series expansion of f(k; β) about k = 0 is of the form of a polynomial in β/2 of degree J divided by (β/2)J. The bulk of the paper is concerned with calculating these polynomials explicitly up to and including those of degree 9. It is remarked that the small k expansion of S(k; β) f..