On the Composition of the Distributions X<sup>-1</Sup> Ln|x| and X+<sup>r</Sup>

Abstract

Let F be a distribution and let f be a locally summable function. The distribution F(f) is defined as the neutrix limit of the sequence {F-n(f)}, where F-n(x) = F(x) * delta(n)(x) and {delta(n)(x)} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function (x). The distribution (x(+)(r))(-1) ln \x(+)(r)\ is evaluated for r = 1, 2.....