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

The structure function S(k; beta) for the one-dimensional one-component log -gas is the Fourier transform of the charge-charge, or equivalently the den sity-density, correlation function. We show that for \k\ < min(2 pi rho, 2 pi rho beta), S(k; beta) is simply related to an analytic function f(k; bet a) and this function satisfies the functional equation f(k; beta) = f( -2k/ beta; 4/beta). It is conjectured that the coefficient of k(j) in the power series expansion of f(k; beta) about k = 0 is of the form of a polynomial i n beta /2 of degree j divided by (beta /2)(j). The bulk of the paper is con cerned with calculating these polynomials explicitly up to and including th ose of degree 9. It is remarked that the small k expansion of S(k; beta) fo r the two-dimensional one-component plasma shares some properties in common with those of the one-dimensional one-component log-gas, but these break d own at order k(8)