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Taming Hypersingular Integrals using Dimensional Continuation

We use the method of dimensional continuation to isolate singularities in integrals containing products of Green's functions or their derivatives.

Published onJan 27, 2020
Taming Hypersingular Integrals using Dimensional Continuation
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Abstract

We use the method of dimensional continuation to isolate singularities in integrals containing products of Green's functions or their derivatives. Rules for the extraction of the finite part of so-called hypersingular integrals are developed, which should be useful in methods based on boundary integral techniques in science and engineering. In applications to potential theory, electromagnetic scattering, and crack dynamics in continuum mechanics, boundary integrals now can be readily evaluated using computational techniques without recourse to complex analysis or contour distortions since the hypersingularities occurring in intermediate steps of the computations can be isolated and ignored while taking the finite parts of the integrals into account in a consistent manner. We have also identified new forms of the Dirac δ function in D dimensions, which are useful and convenient in the calculations. A summary of the integrable singular integrals is given in tabular form. We extend the considerations to a wider class of Green's functions and present a theorem, with additional results arising from it, that shows that hypersingular integrals associated with three-dimensional potential problems can be reduced to one-dimensional finite integrals rather than two-dimensional integrals, again leading to direct evaluations in such cases. These calculations are compared with existing results to show the efficacy of the approach.

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Li, Z., & Ram-Mohan, L. R. (2012b). Taming hypersingular integrals using dimensional continuation. Physical Review E85(1). https://doi.org/10.1103/physreve.85.016706

*denotes a WPI undergraduate student author

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