  In this paper, we describe a new class of fast solvers for separable elliptic
partial differential equations in cylindrical coordinates $(r,\theta,z)$ with
free-space radiation conditions. By combining integral equation methods in the
radial variable $r$ with Fourier methods in $\theta$ and $z$, we show that
high-order accuracy can be achieved in both the governing potential and its
derivatives. A weak singularity arises in the Fourier transform with respect to
$z$ that is handled with special purpose quadratures. We show how these solvers
can be applied to the evaluation of the Coulomb collision operator in kinetic
models of ionized gases.
