On the composition of the distributions xλ+ and x μ+

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+ )λ + and (|x|μ)λ + are evaluated for λ<0, μ>0 and λ,λμ =−1,−2, . . . . © 2005 Elsevier Inc. All rights reserved