Regularization using different surfaces and the second-order derivatives of 1/r

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We study the dependence on the surface Σ: g(x) = 1, where g(x) is continuous in ℝn\{0} and homogeneous of degree 1, of the distributional regularization ℛΣ(K(x))∈D′(ℝn), of a homogeneous function K defined in ℝn\{0}. We also consider the dependence on Σ of the regularization PfΣ(K(x)) given by the Hadamard finite part of the limit when this limit does not exist. We study, in particular, the results on surface dependence for the kernels the ordinary second-order derivatives in ℝ3, and recover the formulae for the second-order generalized derivatives obtained by Hnizdo [V. Hnizdo, Generalized second-order partial derivatives of 1/r, Eur. J. Phys. 32 (2011), pp. 287-297]. © 2013 Copyright Taylor and Francis Group, LLC.

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