On the neutrix composition of the distributions x^−s ln^m|x| and x^r

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) * δ _n (x) and {δ _n (x)} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function δ(x). The composition of the distributions x ^−s ln ^m |x| and x ^r is proved to exist and be equal to r ^m x ^−rs ln ^m |x| for r, s, m = 2, 3….

Keywords:

• distribution
• delta-function
• composition of distributions
• neutrix
• neutrix limit

AMS Subject Classification:

• 46F10