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 $|k| < {\rm min} (2πρ, 2 πρβ)$, $S(k;β)$ 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 $k^j$ 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 $k^8$.

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