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

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 (2πρ, 2πρβ), 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) for the two-dimensional one-component plasma shares some properties in common with those of the one-dimensional one-component log-gas, but these break down at order k8.