Further results on the neutrix composition of distributions involving the delta function and the function cosh − 1 + ( x 1 / r + 1 )

Abstract: The neutrix composition F(f (x)) of a distribution F(x) and a locally summable function f (x) is said to exist and be equal to the distribution h(x) if the neutrix limit of the sequence {Fn(f (x))} is equal to h(x), 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 function cosh−1 + (x + 1) is defined by cosh−1 + (x + 1) = H(x) cosh−1(|x| + 1), where H(x) denotes Heaviside’s function. It is then proved that the neutrix composition δ(s)[cosh−1 + (x1/r +1)] exists and


Abstract:
The neutrix composition F(f (x)) of a distribution F(x) and a locally summable function f (x) is said to exist and be equal to the distribution h(x) if the neutrix limit of the sequence {Fn(f (x))} is equal to h(x), 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 function cosh −

Introduction
In the following, we let D be the space of infinitely differentiable functions φ with compact support and let (ii) the sequence {< fn , φ >} converges to a limit < f , φ > for every φ ∈ D; (iii) < f , φ > is continuous in φ in the sense that limn→∞ < f , φn >= 0 for every sequence φn → 0 in D.
There are several methods for consructing a sequence of regular functions which converges to δ(x). For example, let ρ(x) be a function in D having the following properties: Putting δn(x) = nρ(nx) for n = 1, 2, . . . , it follows that {δn(x)} is a regular sequence of infinitely differentiable functions converging to the Dirac delta-function δ(x). Further, if F is a distribution in D ′ and Fn(x) = ⟨F(x − t), δn(x)⟩, then {Fn(x)} is a regular sequence of infinitely differentiable functions converging to F(x).
Antosik [2] defined the composition g(f (x)) as the limit of the sequence {gn(fn)} providing the limit exists. By this definition he defined the compositions √ δ = 0, √ δ 2 + 1 = 1 + δ, sin δ = 0, cos δ = 1 etc. Using the definition of Antosik, it is not possible to define the compositions for many pairs of distributions. Fisher gave a general principle, by using the neutrix calculus developed by Van der Corput [3], for the discarding of unwanted infinite quantities from asymptotic expansions and this has been exploited in context of distributions, [4]. The technique of neglecting appropriately defined infinite quantities was devised by Hadamard and the resulting finite value extracted from divergent integral is referred to as the Hadamard finite part, see [5]. In fact his method can be regarded as a particular application of the neutrix calculus. Now let f (x) be an infinitely differentiable function having a single simple root at the point x = x 0 . Gel'fand and Shilov defined the distribution δ (r) (f (x)) by the equation for r = 0, 1, 2, . . . , see [6].
In order to give a more general definition for the composition of distributions, the following definition for the neutrix composition of distributions was given in [4] and was originally called the composition of distributions.

Definition 1.
Let F be a distribution in D ′ and let f be a locally summable function. We say that the neutrix composition F(f (x)) exists and is equal to for n = 1, 2, . . . and N is the neutrix, see [3], having domain N ′ the positive integers and range N ′′ the real numbers, with negligible functions which are finite linear sums of the functions n λ ln r−1 n, ln r n : λ > 0, r = 1, 2, . . . and all functions which converge to zero in the usual sense as n tends to infinity.
In particular, we say that the composition F(f (x)) exists and is equal to Note that taking the neutrix limit of a function f (n), is equivalent to taking the usual limit of Hadamard's finite part of f (n).
It was proved in [7] that if the composition F(f (x)) exists by Gel'fand and Shilov's definition, then it exists by Definition 1 and the two are equivalent.
We now prove the following improvement of Theorem 3: