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Multiplication of the distributions (x±i0)z

Franssens G. R

Journal of Applied Analysis

2014

Vol. 20, nr 1

15--27

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.