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 $r,\,s\in {\mathbb {Z}}^+.$

Keywords

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