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Multiplication of the distributions (x±i0)z

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EN
Abstrakty
EN
In previous work of the author, a convolution and multiplication product for the set of Associated Homogeneous Distributions (AHDs) with support in ℝ was defined and fully investigated. Here this definition is used to calculate the multiplication product of homogeneous distributions of the form (x±i0)z, for all z∈C. Multiplication products of AHDs generally contain an arbitrary constant if the resulting degree of homogeneity is a negative integer, i.e., if it is a critical product. However, critical products of the forms (x+i0)a.(x+i0)b and (x−i0)a.(x−i0)b, with a+b∈Z−, are exceptionally unique. This fact combined with Sokhotskii–Plemelj expressions then leads to linear dependencies of the arbitrary constants occurring in products like δ(k).δ(l), η(k).δ(l), δ(k).η(l) and η(k).η(l) for all k,l∈N (η≜1πx−1). This in turn gives a unique distribution for products like δ(k).η(l)+η(k).δ(l) and δ(k).δ(l)−η(k).η(l). The latter two products are of interest in quantum field theory and appear for instance in products of the partial derivatives of the zero-mass two-point Wightman distribution.
Wydawca
Rocznik
Strony
15--27
Opis fizyczny
Bibliogr. 28 poz.
Twórcy
  • Belgian Institute for Space Aeronomy, Ringlaan 3, B-1180 Brussels, Belgium
Bibliografia
  • [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 9th ed., Dover, New York, 1970.
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  • [11] G. R. Franssens, The convolution of associated homogeneous distributions on R. II, Appl. Anal. 88 (2009), 333-356.
  • [12] G. R. Franssens, Convolution product formula for associated homogeneous distributions on R, Math. Methods Appl. Sci. 34 (2011), 703-727.
  • [13] G. R. Franssens, Multiplication product formula for associated homogeneous distributions on R, Math. Methods Appl. Sci. 34 (2011), 1460-1471.
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Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-9350615c-f0fb-4fca-98c6-05236f7b49c7
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