On the composition of the distributions $x^{-1}ln^m|x|$ and $x^r$

Abstract

Let F be a distribution and f a locally summable function. The dis- tribution F(f) is defined as the neutrix limit of the sequence ${F_n(f)}$, where $F_n(x) = F(x)ast delta_n(x)$ and ${delta_n(x)}$ is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function $delta(x)$. The composition of the distributions $x^{-1} ln^m |x|$ and $x^r$ is evaluated for r,m = 1, 2, 3 . . ..