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Defining Compositions of \(x_+^\mu ,\,|x|^\mu ,\,x^{-s}\), and \(x^{-s}\ln |x|\) as Neutrix Limit of Regular Sequences

Abstract

In this paper the compositions \((x_+^\mu )_-^{{-}s},\, (x_+^\mu )_+^{{-}s},\, (|x|^\mu )_-^{{-}s}\) and \((|x|^\mu )_+^{{-}s}\) of distributions \(x_+^\mu ,\,|x|^\mu \) and \(x^{{-}s}\) are considered. They are defined via neutrix calculus for \(\mu >0, \, s=1,\,2,\ldots \) and \(\mu s\in {\mathbb {Z}}^+.\) In addition, the composition of \(x^{{-}s}\ln |x|\) and \(x_+^r\) is also defined for \(r,\,s\in {\mathbb {Z}}^+.\)

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Acknowledgments

B. Jolevska-Tuneska was supported by Tubitak (Scientific and Technological Research Council of Turkey).

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Correspondence to Emin Öz c̣ ağ.

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Öz c̣ ağ, E., Lazarova, L. & Jolevska-Tuneska, B. Defining Compositions of \(x_+^\mu ,\,|x|^\mu ,\,x^{-s}\), and \(x^{-s}\ln |x|\) as Neutrix Limit of Regular Sequences. Commun. Math. Stat. 4, 63–80 (2016). https://doi.org/10.1007/s40304-015-0076-8

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Keywords

  • Composition of distributions
  • Dirac delta function
  • Pseudo-function
  • Neutrix calculus
  • Hadamard finite part
  • Regular sequence
  • Delta sequence

Mathematics Subject Classification

  • 46F10
  • 46F30
  • 41A30