The composition of the distributions x- -ms ln x - and x+ r-p/m

Abstract

Let F be a distribution and 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{ δn(x) and {δn(x)} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function δ(x). The distribution x-sIn X- is denoted by Fs(x) and then Fms(x+rp/m) is evaluated for r,s= 1,2,..., and m = 2,3,..., where 1 ≤ p < m and p and m are coprime. © 2005 Taylor & Francis Ltd.