Investigation of infinitely rapidly oscillating distributions
| Autor: | Jungil Lee, Wooyong Han, Sungwoong Cho, U. Rae Kim |
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| Rok vydání: | 2021 |
| Předmět: | |
| Zdroj: | European Journal of Physics. 42:065807 |
| ISSN: | 1361-6404 0143-0807 |
| DOI: | 10.1088/1361-6404/ac25d1 |
| Popis: | We study the rapidly oscillating contributions in the sinc-function representation of the Dirac delta function and the Fourier transform of the Coulomb potential. Starting from the 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 employing the contour-integral approach, we prove that the representation satisfies the properties of the Dirac delta function, although the representation is a function divergent at nonzero points. This is a good pedagogical example to show the difference between a function and a distribution. The rapidly oscillating contribution in the Fourier transform of the Coulomb potential into the momentum space has been ignored in most textbooks because only the regulated screened potential of Wentzel was employed to compute the amplitude free of such a rapidly oscillating contribution. By performing the inverse Fourier transform of the rapidly oscillating contribution rigorously, we demonstrate that the contribution is a well-defined distribution that is zero, even if it is an ill-defined function. We extend our proofs carried out for those oscillating contributions to show that the Riemann-Lebesgue lemma can hold for a sinc function, which is not absolutely integrable. |
| Databáze: | OpenAIRE |
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