Research Article On the Composition of Distributions x-sln|x| and |x|

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 composition of the distributions x−sIn|x| and |x| is evaluated for s= 1,2, . . . ,μ > 0 and μs = 1,2, . . . .

In the following, we let Ᏸ be the space of infinitely differentiable functions with compact support, let Ᏸ(a,b) be the space of infinitely differentiable functions with support contained in the interval (a,b), and let Ᏸ be the space of distributions defined on Ᏸ.
We define the locally summable functions x λ + , x λ − , x λ + lnx + , x λ − lnx − , |x| λ , and |x| λ ln|x| for λ > −1 (see [1]) by The distributions x λ + and x λ − are then defined inductively for λ < −1 and λ = −2, −3, ... by 2 International Journal of Mathematics and Mathematical Sciences It follows that if r is a positive integer and −r − 1 < λ < −r, then for arbitrary ϕ in Ᏸ.In particular, if ϕ has its support contained in the interval [−1,1], then , We define the distribution x −1 ln|x| by and we define the distribution x −r−1 ln|x| inductively by for r = 1,2,....It follows by induction that where In the following, we let N be the neutrix, see [2], having domain N the positive integers and range N the real numbers, with negligible functions which are finite linear B. Jolevska-Tuneska and E. Özc ¸aḡ 3 sums of the functions n λ ln r−1 n, ln r n, λ > 0, r = 1,2,... (10) as well as all functions which converge to zero in the usual sense as n tends to infinity.Now let ρ(x) be an infinitely differentiable function having the following properties:

is a regular sequence of infinitely differentiable functions converging to the Dirac delta-function δ(x).
Next, for an arbitrary distribution f in Ᏸ , we define for n = 1,2,....It follows that { f n (x)} is a regular sequence of infinitely differentiable functions converging to the distribution f (x).
The following definition was given in [3].
Definition 1.Let F be a distribution and let f be a locally summable function.Say that the distribution F( f (x)) exists and is equal to for all test functions ϕ with compact support contained in (a,b).
Theorem 2. The distribution (x r ) −s exists and We need the following lemma which can be easily proved by induction.
Proof.We will suppose that r < μs < r + 1 for some positive integer r.We put and note that on using the substitutions u = nx μ and v = nt.