A dual integral formulation for the Laplace problem with a smooth boundary is derived by using the contour approach surrounding the singularity. It is found that using the contour approach the jump terms come half and half from the free terms in the L and M kernel integrations for the two-dimensional case, which is different from the limiting process by approaching an interior point to a boundary point where the jump terms come totally from the L kernel only. The definition of the Hadamard principal value for hypersingular integral at the collocation point of a smooth boundary is extended to a generalized sense for both the tangent and normal derivatives of double-layer potentials in comparison with the conventional definition. For the threedimensional case, the jump terms come one-third and two-thirds from the free terms of L and M kernels, respectively.
Chen, Jeng-Tzong; Kuo, Shyh-Rong; Chen, Wei-Chih; and Liu, Li-Wei
"On the Free Terms of the Dual BEN for the Two and Three-Dimensional Laplace Problems,"
Journal of Marine Science and Technology: Vol. 8:
1, Article 2.
Available at: https://jmstt.ntou.edu.tw/journal/vol8/iss1/2