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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]
Copyright of European Journal of Physics is the property of IOP Publishing and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
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DbLabel: Engineering Source
An: 152800752
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  Data: Investigation of infinitely rapidly oscillating distributions.
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  Data: <searchLink fieldCode="AR" term="%22Kim%2C+U-Rae%22">Kim, U-Rae</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Cho%2C+Sungwoong%22">Cho, Sungwoong</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Han%2C+Wooyong%22">Han, Wooyong</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Lee%2C+Jungil%22">Lee, Jungil</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> jungil@korea.ac.kr</i>
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  Data: <searchLink fieldCode="JN" term="%22European+Journal+of+Physics%22">European Journal of Physics</searchLink>. Nov2021, Vol. 42 Issue 6, p1-23. 23p.
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  Data: <searchLink fieldCode="DE" term="%22Dirac+function%22">Dirac function</searchLink><br /><searchLink fieldCode="DE" term="%22Coulomb+potential%22">Coulomb potential</searchLink><br /><searchLink fieldCode="DE" term="%22Fourier+transforms%22">Fourier transforms</searchLink><br /><searchLink fieldCode="DE" term="%22Integral+representations%22">Integral representations</searchLink><br /><searchLink fieldCode="DE" term="%22Momentum+space%22">Momentum space</searchLink>
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: 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]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of European Journal of Physics is the property of IOP Publishing and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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RecordInfo BibRecord:
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    Identifiers:
      – Type: doi
        Value: 10.1088/1361-6404/ac25d1
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      – Code: eng
        Text: English
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        PageCount: 23
        StartPage: 1
    Subjects:
      – SubjectFull: Dirac function
        Type: general
      – SubjectFull: Coulomb potential
        Type: general
      – SubjectFull: Fourier transforms
        Type: general
      – SubjectFull: Integral representations
        Type: general
      – SubjectFull: Momentum space
        Type: general
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      – TitleFull: Investigation of infinitely rapidly oscillating distributions.
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            NameFull: Kim, U-Rae
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            NameFull: Cho, Sungwoong
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            NameFull: Han, Wooyong
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            NameFull: Lee, Jungil
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          Dates:
            – D: 01
              M: 11
              Text: Nov2021
              Type: published
              Y: 2021
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              Value: 42
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              Value: 6
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            – TitleFull: European Journal of Physics
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