Investigation of infinitely rapidly oscillating distributions

U. Rae Kim, Sungwoong Cho, Wooyong Han, Jungil Lee

Research output: Contribution to journalArticlepeer-review

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 Riemann-Lebesgue lemma can hold for a sinc function, which is not absolutely integrable.

Original languageEnglish
Article number065807
JournalEuropean Journal of Physics
Volume42
Issue number6
DOIs
Publication statusPublished - 2021 Nov

Keywords

  • Dirac delta function
  • Fourier transform
  • Heaviside step function
  • rapidly oscillating distribution
  • Rutherford scattering

ASJC Scopus subject areas

  • Physics and Astronomy(all)

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