ON THE COMPOSITION OF THE DISTRIBUTIONS xλ + lnm x+ AND xμ +

Abstract

Let F be a distribution and let f be a locally summable function. The neutrix composition F(f), of F and f, is defined as the neutrix limit of the sequence {Fn(f)}, whereFn(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 neutrix composition of the distributions xλ + lnm x+ and xμ + is evaluated for −1 <λ<0, μ >0, λμ = −1,−2,... and m =0 ,1,2,....