Explicit sequences that approach the Dirac delta function and its derivatives are often helpful in presenting generalized functions. We present a method by which a finite difference formula may be easily converted into a sequence that approaches a derivative of the Dirac delta function in one dimension. In three dimensions, we employ a sequence for the Dirac delta function based on a uniformly charged sphere of infinitesimal radius and infinite charge density and show that the charge density of an electric dipole is (in the sense of a generalized function) equal to We use this result to derive Gauss’ law in a dielectric medium directly from the charge densities, without using the potentials.
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PAPERS| May 01 2003
Derivatives of the Dirac delta function by explicit construction of sequences
Timothy B. Boykin; Derivatives of the Dirac delta function by explicit construction of sequences. Am. J. Phys. 1 May 2003; 71 (5): 462–468. https://doi.org/10.1119/1.1557302
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