  In classical semi-infinite Coulomb fluids, two-point correlation functions
exhibit a slow inverse-power law decay along a uniformly charged wall. In this
work, we concentrate on the corresponding amplitude function which depends on
the distances of the two points from the wall. Recently [L. \v{S}amaj, J. Stat.
Phys. {\bf 161}, 227 (2015)], applying a technique of anticommuting variables
to a 2D system of charged rectilinear wall with "counter-ions only", we derived
a relation between the amplitude function and the density profile which holds
for any temperature. In this paper, using the M\"obius conformal transformation
of particle coordinates in a disc, a new relation between the amplitude
function and the density profile is found for that model. This enables us to
prove, at any temperature, the factorization property of the amplitude function
in point distances from the wall and to express it in terms of the density
profile. Presupposing the factorization property of the amplitude function and
using specific sum rules for semi-infinite geometries, a relation between the
amplitude function of the charge-charge structure function and the charge
profile is derived for many-component Coulomb fluids in any dimension.
