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Results on singular products of the distributions $x_{\pm }^{-p}$ and $x^{-p}$ for natural $p$ are derived, when the products are balanced so that their sum exists in the distribution space. These results follow the pattern of a known distributional product published by Jan Mikusiński in 1966. The results are obtained in the Colombeau algebra of generalized functions, which is the most relevant algebraic construction for tackling nonlinear problems of Schwartz distributions.

Models of singularities given by discontinuous functions or distributions by means of generalized functions of Colombeau have proved useful in many problems posed by physical phenomena. In this paper, we introduce in a systematic way generalized functions that model singularities given by distributions with singular point support. Furthermore, we evaluate various products of such generalized models when the results admit associated distributions. The obtained results follow the idea of a well-known...

The differential $ℂ$-algebra $𝒢\left({ℝ}^m\right)$ of generalized functions of J.-F. Colombeau contains the space ${𝒟}^{\text{'}}\left({ℝ}^m\right)$ of Schwartz distributions as a $ℂ$-vector subspace and has a notion of ‘association’ that is a faithful generalization of the weak equality in ${𝒟}^{\text{'}}\left({ℝ}^m\right)$. This is particularly useful for evaluation of certain products of distributions, as they are embedded in $𝒢\left({ℝ}^m\right)$, in terms of distributions again. In this paper we propose some results of that kind for the products of the widely used distributions $x_{\pm }^a$ and ${\delta }^{\left(p\right)}\left(x\right)$, with $x$ in ${ℝ}^m$,...

MSC 2010: 46F30, 46F10 Modelling of singularities given by discontinuous functions or distributions by means of generalized functions has proved useful in many problems posed by physical phenomena. We introduce in a systematic way generalized functions of Colombeau that model such singularities. Moreover, we evaluate some products of singularity-modelling generalized functions whenever the result admits an associated distribution.

The non-commutative neutrix product of the distributions $ln{x}_{+}$ and $x_{+}^{-s}$ is proved to exist for $s=1,2,...$ and is evaluated for $s=1,2$. The existence of the non-commutative neutrix product of the distributions $x_{+}^{-r}$ and $x_{+}^{-s}$ is then deduced for $r,s=1,2,...$ and evaluated for $r=s=1$.