Extension of Gravity Field Functionals Through Higher-Order Radial Integration

Higher-order radial integration extends the conventional differential hierarchy of gravity field functionals toward negative integer orders. In this theoretical study, we distinguish uniquely referenced radial integrals from non-unique radial primitives and derive spatial, spherical harmonic, and surface kernel representations. Regular power-law primitives are separated from exceptional logarithmic and polynomial-logarithmic terms. For an exterior harmonic field, classical Euler operator identities explain harmonicity after the radial scaling of derivatives and regular particular primitives, while finite reference normalization terms generally prevent the corresponding complete scaled integral from being harmonic. Integration from infinity exists precisely in the regular degree sector. Consequently, the second particular radial primitive is the highest order that remains regular for every degree of a conventional disturbing potential expansion beginning at degree two. Explicit integrated Hotine and Stokes spectral coefficients are complemented with the Poisson kernels. Synthetic calculations verify the identities and kernel coefficients, reproduce cancellation near the reference radius, and test a hypergeometric alternative over a stated parameter grid. The construction defines analytically constrained spectral transformations, while a possible improvement of geophysical interpretations relative to conventional filtering is not investigated in the presented study.