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Investigation of infinitely rapidly oscillating distributions.

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Title: Investigation of infinitely rapidly oscillating distributions.
Authors: Kim, U-Rae1 (AUTHOR), Cho, Sungwoong1 (AUTHOR), Han, Wooyong1 (AUTHOR), Lee, Jungil1 (AUTHOR) jungil@korea.ac.kr
Source: European Journal of Physics. Nov2021, Vol. 42 Issue 6, p1-23. 23p.
Subjects: Dirac function, Coulomb potential, Fourier transforms, Integral representations, Momentum space
Abstract: We rigorously investigate the rapidly oscillating contributions in the sinc-function representation of the Dirac delta function and the Fourier transform of the Coulomb potential. Beginning with a derivation of the standard integral representation of the Heaviside step function, we examine the representation of the Dirac delta function that contains a rapidly oscillating sinc function. By contour integration, we prove that the representation satisfies the properties of the Dirac delta function, although it is a function divergent at nonzero points. This is a good pedagogical example demonstrating the difference between a function and a distribution. In most textbooks, the rapidly oscillating contribution in the Fourier transform of the Coulomb potential into the momentum space has been ignored by regulating the oscillatory divergence with the screened potential of Wentzel. By performing the inverse Fourier transform of the contribution rigorously, we demonstrate that the contribution is a well-defined distribution that is indeed zero, even if it is an ill-defined function. Proofs are extended to exhibit that the Riemannâ€"Lebesgue lemma can hold for a sinc function, which is not absolutely integrable. [ABSTRACT FROM AUTHOR]
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Database: Engineering Source
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Abstract:We rigorously investigate the rapidly oscillating contributions in the sinc-function representation of the Dirac delta function and the Fourier transform of the Coulomb potential. Beginning with a derivation of the standard integral representation of the Heaviside step function, we examine the representation of the Dirac delta function that contains a rapidly oscillating sinc function. By contour integration, we prove that the representation satisfies the properties of the Dirac delta function, although it is a function divergent at nonzero points. This is a good pedagogical example demonstrating the difference between a function and a distribution. In most textbooks, the rapidly oscillating contribution in the Fourier transform of the Coulomb potential into the momentum space has been ignored by regulating the oscillatory divergence with the screened potential of Wentzel. By performing the inverse Fourier transform of the contribution rigorously, we demonstrate that the contribution is a well-defined distribution that is indeed zero, even if it is an ill-defined function. Proofs are extended to exhibit that the Riemannâ€"Lebesgue lemma can hold for a sinc function, which is not absolutely integrable. [ABSTRACT FROM AUTHOR]
ISSN:01430807
DOI:10.1088/1361-6404/ac25d1