Two results on the composition of distributions

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 {Fn(f)}, where Fn(x) = F (x) ∗ δn(x) and {δn(x)} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function δ(x). The distributions (|x|µ)−s and (sgnx|x|µ)−s are evaluated for s = 1, 2,..., µ> 0 and µs 6 = 1, 2,...