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 density-density, correlation function. We show that for $|k| < {\rm min} (2\pi \rho, 2 \pi \rho \beta)$, $S(k;\beta)$ is simply related to an analytic function $f(k;\beta)$ 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 in $\beta/2$ of degree $j$ divided by $(\beta/2)^j$. 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 $S(k;\beta)$ 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 $k^8$.