The numerical integration of expressions containing strong singularities or strong near-singularities has long been a challenging problem in the electromagnetics community. Much attention has been paid to this problem, as strong ๐๐/๐น๐น๐๐ singularities routinely appear when implementing electromagnetic simulation techniques like the Method of Moments (MoM). To date, several techniques, from singularity extraction to singularity cancellation, have been employed to deal with problems that require the evaluation of 2D strongly-singular integrals. However, no single technique has been proposed that can deal with both strong singularities and strong near-singularities in a fully-numerical manner for arbitrary 2D domains. Moreover, it has been claimed that the Helmholtz-type strongly singular integral found in the MoM is convergent in a principal value sense, but this convergence value has yet to be proven mathematically. In this work, we will conduct the convergence proof and introduce a โpolar scalingโ change of variables method that may be used to evaluate Helmholtz integrals with both strong and weak singularities/near-singularities. The technique is fully-numerical and can in principle be applied to any planar or curved polygon and any non-singular basis function. We will also provide numerical results showing useful convergence behavior for integrals involving both exact and near-singularities.
