  Isolated patches of spatially oscillating pattern have been found to emerge
near a pattern-forming instability in a wide variety of experiments and
mathematical models. However, there is currently no mathematical theory to
explain this emergence or characterise the structure of these patches. We
provide a method for formally deriving radial amplitude equations to planar
patterns via non-autonomous multiple-scale analysis and convolutional sums of
products of Bessel functions. Our novel approach introduces nonautonomous
differential operators, which allow for the systematic manipulation of Bessel
functions, as well as previously unseen identities involving infinite sums of
Bessel functions. Solutions of the amplitude equations describe fully localised
patterns with non-trivial angular dependence, where localisation occurs in a
purely radial direction. Amplitude equations are derived for multiple examples
of patterns with dihedral symmetry, including fully localised hexagons and
quasipatterns with twelve-fold rotational symmetry. In particular, we show how
to apply the asymptotic method to the Swift--Hohenberg equation and general
reaction-diffusion systems.
