On the Non-Commutative Neutrix Convolution of tan − 1 x and x r

. We deﬁne regular distributions tan − 1+ x and tan − 1 − x from the function tan − 1 x. We then evaluate some convolutions and neutrix convolutions of these distributions and the functions x r + , x r − and x r .


Introduction
In distribution theory, the convolution of distributions is rather restricted, see for example Gel'fand and Shilov [4]. Jones [6] extended the definition of the convolution of two distributions to cover certain pairs of distributions which could not be convolved in the sense of Gel'fand and Shilov. For instance, he proved that (1) 1 * sgn x = x.
x 2r+1 for r = 0, 1, 2, . . . . He later made some minor changes in Jones's definitions by using neutrix calculus, see [5]. He proved in [2] that The idea of using neutrix calculus has opened up the research area in another direction for the field of distributional convolutions. In this research, it is of particular interest to evaluate convolution of distributions for a certain class of distributions.
In the following the locally summable functions x r + , x r − , r = 0, 1, . . . , tan −1 + x and tan −1 − x are defined by where H denotes Heaviside's function. Note that If f and g are locally summable functions then the classical definition for the convolution f * g of f and g is as follows: Definition 1.1. Let f and g be functions. Then the convolution f * g is defined by for all points x for which the integral exists.
It follows easily from the definition that if the classical convolution f * g of f and g exists, then g * f exists and (5) f * g = g * f.
Further, if (f * g) and f * g (or f * g) exist, then We now let D be the space of infinitely differentiable functions with compact support and let D be the space of distributions defined on D.
The classical definition of the convolution can be extended to define the convolution f * g of two distributions f and g in D with the following definition, see Gel'fand and Shilov [4]. Definition 1.2. Let f and g be distributions in D . Then the convolution f * g is defined by the equation for arbitrary ϕ in D , provided that f and g satisfy either of the following conditions: (a) either f or g has bounded support, (b) the supports of f and g are bounded on the same side.
It follows that if the convolution f * g exists by Definition 1.2, then (5) and (6) are satisfied.
We need the following lemmas for proving the results on the convolution.
Proof. It is obvious when m = 0 and m = 1 that so that equations (8) and (9) hold when m = 0.
When m ≥ 2, we have In particular, we have and on assuming that equation (8) holds for some m, it follows that equation (8) holds for m + 1. Equation (8) follows by induction. Equation(9) follows similarly. This completes the proof of the lemma.
where the sum is empty when k = 0 and (16) where the sum is being empty when k = 0, 1.
Proof. Integrating by parts by parts and then making the substitution t = tan u, we have It follows that for k = 0, 1, 2. . . ..
It now follows from equations (10) and (17) that Similarly, it follows from equations (9) and (18) that This completes the proof of the lemma.

Proof. It is obvious that
proving equation (19).

The Neutrix Convolution
The definition of the convolution of distributions is rather restrictive and so the non-commutative neutrix convolution of distributions was introduced in [1]. In order to define the neutrix convolution product we first of all let τ be a function in D satisfying the following properties: The function τ n is then defined by The following definition was given in [1].
Definition 2.1. Let f and g be distributions in D and let f n = f τ n for n = 1, 2, . . . . Then the neutrix convolution f g is defined as the neutrix limit of the sequence {f n * g}, provided that the limit h exists in the sense where N is the neutrix, see van der Corput [5], having domain N = {1, 2, . . . , n, . . .} and range N , the real numbers, with negligible functions being 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, if lim n→∞ f n * g, ϕ = h, ϕ for all ϕ in D, we say that the convolution f * g exists and equals h.
Note that in this definition the convolution f n * g is as defined in Gel'fand and Shilov's sense since the distribution f n having compact support. Note also that because of the lack of symmetry in the definition of f g, the neutrix convolution is in general non-commutative.
The following theorem was proved in [1], showing that the neutrix convolution is a generalization of the convolution.
The next theorem was also proved in [1].
Theorem 2.2. Let f and g be distributions in D and suppose that f g exists, then the neutrix convolution f g exists and Note however that (f g) is not necessarily equal to f g, but we do have the following theorem, which was proved in [2].
We also need the following lemmas for proving results on the non-commutative neutrix products.
Proof. Note that when n tends to infinity, we would have
We now prove the following theorems.

Proof.
We have Equation (53) Proof. Note that if x is fixed and n tends to infinity, we would have The proof of the lemma then follows.

Proof.
We have from equation (2)