On the Neutrix Composition of the Delta and Inverse Hyperbolic Sine Functions

Definition 1.1. 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 h on the open interval (a, b), with −∞ < a < b < ∞, if

for all φ in 𝒟[a, b], where F_n(x) = F(x)*δ_n(x) for n = 1,2, … and N is the neutrix, see [10], having domain N^′ the positive and range N^′′ the real numbers, with negligible functions which are finite linear sums of the functions 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 h on the open interval (a, b) if

for all φ in 𝒟[a, b].

Theorem 1.2. The neutrix composition δ^(s)(sgn x|x|^λ) exists and

for s = 0,1, 2, … and (s + 1)λ = 1,3, …, and for s = 0,1, 2, …, and (s + 1)λ = 2,4, ….

Theorem 1.3. The neutrix compositions δ^(2s−1)(sgn x|x|^1/s) and δ^(s−1)(|x|^1/s) exist and

for s = 1,2, ….

Theorem 1.4. The neutrix composition $$ exists and

for s = 0,1, 2, … and r = 1,2, …, where

Theorem 1.5. The neutrix composition δ^(s)[ln ^r(1 + |x|)] exists and

for s = 0,1, 2, …, and r = 1,2, ….

In particular, the composition δ[ln (1 + |x|)] exists and

Theorem 1.6. The neutrix composition δ^(s)[ln (1 + |x^1/r|)] exists and

for s = 0,1, 2, … and r = 2,3, …, where m is the smallest non-negative integer greater than (s − r + 1)r⁻¹.

In particular, the composition δ^(s)[ln (1 + |x^1/r|)] exists and

for s = 0,1, 2, …, r − 2 and r = 2,3, … and for r = 2,3, ….

Theorem 2.1. The neutrix composition $$ exists and

for s = 0,1, 2, … and r = 1,2, …, where In particular, the neutrix composition δ(sinh ⁻¹x₊) exists and

Proof. To prove (2.1), we first of all evaluate

We have It is obvious that for k = 0,1, 2, ….

Making the substitution $$, we have for large enough n

where It follows that and by applying the neutrix limit we obtain for k = 0,1, 2, ….

When k = sr + r, we have

Thus, if ψ is an arbitrary continuous function, then We also have and it follows that If now φ is an arbitrary function in 𝒟[−1,1], then by Taylor′s Theorem, we have where 0 < ξ < 1, and so on using (2.3) to (2.14). This proves (2.1) on the interval (−1,1).

It is clear that $$ for x > 0 and so (2.1) holds for x > −1.

Now, suppose that φ is an arbitrary function in 𝒟[a, b], where a < b < 0. Then,

and so

It follows that $$ on the interval (a, b). Since a and b are arbitrary, we see that (2.1) holds on the real line. This completes the proof of the theorem.

Corollary 2.2. The neutrix composition $$ exists and

for s = 0,1, 2, … and r = 1,2, ….

In particular, the composition δ(sinh ⁻¹|x|) exists and

Proof. To prove (2.19), we note that

and (2.19) now follows as above.

Equation (2.20) follows on noting that in the particular case s = 0, the usual limit holds in (2.10). This completes the proof of the corollary.

Theorem 2.3. The neutrix composition δ^(2s−1)[sinh ⁻¹(sgn x · x²)] exists and

for s = 1,2, …, where

Proof. To prove (2.22), we now have to evaluate

We have Making the substitution t = n(sinh ⁻¹x²), we have for large enough n where It follows that and so by using the neutrix limit, we have for k = 0,1, 2, ….

When k = 2s, we have

Thus, if ψ is an arbitrary continuous function, then If now φ is an arbitrary function in 𝒟[−1,1], then by Taylor′s Theorem, we have where 0 < ξ < 1, and so on using (2.25) to (2.31), proving (2.22) on the interval (−1,1). However, it is clear that $$ for |x| > 0 and so (2.22) holds on the real line, completing the proof of the theorem.

Corollary 2.4. The composition δ^′[sinh ⁻¹sgn x · x²)]   exists and

Proof. To prove (2.34) note that in the particular case s = 1, the usual limits hold and then (2.34) is a particular case of (2.22). This completes the proof of the corollary.