Assessing the ability of the first-order (left) and our novel (right) tight-winding approximations to provide a gravitational potential for the target density (31) of a logarithmic spiral with pitch angle α and exponentially declining amplitude. From the density that is actually generated by the approximate potential, I plot the radial profiles of the relative amplitude error (top) and the errors in phase (middle) and pitch angle (bottom), which for the first-order methods are also the respective offsets between the potential and density because to first order, ψ_Ψ = ψ_target.