Balanced Colombeau products of the distributions $x_{\pm}^{-p}$ and $x^{-p}$

Abstract

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 Colombeau algebra of generalized functions, which is most relevant algebraic construction for tackling nonlinear problems of Schwartz distributions.