# A result on the neutrix composition of the delta function.

1 Introduction

In distribution theory, no meaning can be generally given to expressions of the form F(f (x)), where F and f are distributions. However, the compositions [delta](f) and [delta]'(f), where f = 0 is a surface in three-dimensional space, appear in wave propagation problems, see [2,3]. Furthermore, in physics, one finds the need to evaluate [[delta].sup.2] when calculating the transition rates of certain particle interaction (see ).

The technique of neglecting appropriately defined infinite quantities was devised by Hadamard, and the resulting finite value extracted from a divergent integral is referred to as the Hadamard finite part. In fact, his method can be regarded as a particular application of the neutrix calculus developed by van der Corput (see ).

Using the concepts of a neutrix and neutrix limit, the first author gave a general principle for the discarding of unwanted infinite quantities from asymptotic expansions, and this has been exploited particularly in connection with multiplication, convolution and composition of distributions, see [4,5]. Using Fisher's definition, Koh and Li gave meaning to [[delta].sup.r] and [([delta]').sup.r] for r = 2, 3,..., see , and the more general form [([[delta].sup.(s)(x)).sup.r] was considered by Kou and Fisher in . Recently the rth powers of the Dirac distribution and the Heaviside function for negative integers have been defined in  and , respectively.

In the following, we let D be the space of infinitely differentiable functions [phi] with compact support and let D[a, b] be the space of infinitely differentiable functions with support contained in the interval [a, b]. We let D' be the space of distributions defined on D and let D'[a, b] be the space of distributions defined on D[a, b]. Now let [rho](x) be a function in D having the following properties:

(i) [rho](x) = 0 for [absolute value of x] [greater than or equal to] 1,

(ii) [rho](x) [less than or equal to] 0,

(iii) [rho](x) = p(-x),

(iv) [[intergral].sup.1.sub.-1] [rho](x) dx = 1 .

Putting [[delta].sub.n](x) = n[rho](nx) for n = 1, 2,..., it follows that {[[delta].sub.n](x)} is a regular sequence of infinitely differentiable functions converging to the Dirac delta-function [delta](x). Further, if F is a distribution in D' and [F.sub.n](x) = <(F(x - t), [[delta].sub.n](x))>, then {[F.sub.n](x)} is a regular sequence of infinitely differentiable functions converging to F(x).

Now let f(x) be an infinitely differentiable function having a single simple root at the point x = [x.sub.0]. Gel'fand and Shilov defined the distribution [[delta].sup.(r)] (f(x)) by the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for r = 0,1, 2,..., see .

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  and was originally called the composition of distributions.

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(/(x)) exists and is equal to h on the open interval (a, b) if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all [phi] in D[a, b], where [F.sub.n](x) = F(x) * [[delta].sub.n](x) for n = 1, 2,... and N is the neutrix, see , having domain N' the positive integers and range N" the real numbers, with negligible functions which are finite linear sums of the functions

[n.sup.[lambda]] [ln.sup.r-1] n, [ln.sup.r] n : [lambda] > 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 h on the open interval (a, b) if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all [phi] in D[a, b].

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).

The following theorems were proved in [6, 7, 12] respectively.

Theorem 1.2. The neutrix composition [[delta].sup.(s)](sgn x [[absolute value of x].sup.[lambda]]) exists and [[delta].sup.(s)](sgn x [[absolute value of x].sup.[lambda]]) = 0

for s = 0, 1, 2, ... and (s + 1)[lambda] = 1, 3, ... and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for s = 0,1, 2,... and (s + 1)[lambda] = 2, 4,....

Theorem 1.3. The neutrix composition [[delta].sup.(rs-m)] [[x.sup.1/r.sub.+]/(1 + [x.sup.1/r.sub.+])] exists and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for r, s = 1, 2,... and m = 1, 2,..., rs, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

if rs - m + 1 < rk + r. In particular, we have

[delta][[x.sup.1/r.sub.+]/(1 + [x.sup.1/r.sub.+])] = 0

for r = 2, 3, ... and

[delta][[x.sub.+]/(1 + [x.sub.+])] = 1/2 [delta](x).

Theorem 1.4. The neutrix composition [[delta].sup.(s)] [[ln.sup.r] (1 + [x.sup.1/r.sub.+])] exists and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for s = 0, 1 , 2, ... and r = 1, 2, ....

2 Main Results

We now need the following lemma, which can be easily proved by induction. Lemma 2.1. We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for s = 0,1, 2,....

We now prove the following theorem.

Theorem 2.2. The neutrix composition [[delta].sup.(rsm-pm-1)] [[x.sup.1/rm.sub.+]/[(1 + [x.sup.1/r.sub.+]).sup.1/m]] exists and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

for r, s, m = 1, 2,... and p = 0,1,..., rs - 1.

Proof. We will first of all prove equation (2.1) on the interval [-1,1]. To do this, we need to evaluate

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It is seen immediately that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

for r, s, m = 1, 2,... and k = 0,1, 2,.... Substituting [nx.sup.1/rm]/[(1 + [x.sup.1/r]).sup.1/m] = t, i.e.,

x = [t.sup.rm]/[n.sub.rm][(1 - [t.sup.m]/[n.sup.m]).sup.r],

we have

dx = [rmt.sup.rm-1] dt/[n.sup.rm][(1 - [t.sup.m]/[n.sup.m]).sup.r+1].

Then for n > 1, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

on using Lemma 2.1, for k = 0,1, 2,..., s - 1 and r, s, m = 1, 2,... and p = 0, 1, 2, ..., rs - 1. Next, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and so if [psi](x) is an arbitrary continuous function, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.4)

for r, s, m = 1, 2,.... Further,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.5)

for r, s, m = 1, 2, .... Now let [phi] be an arbitrary function in D[-1,1]. By Taylor's theorem, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where 0 < [xi] < 1. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

on using equations (2.2), (2.3), (2.4) and (2.5). This proves that the neutrix composition [[delta].sup.(rsm-pm-1)][[x.sup.1/r.sub.+]/(1 + [x.sup.1/r.sub.+])] exists and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

on the interval [-1,1] for r, s, m = 1, 2, 3,... and p = 0,1, 2,..., rs - 1. []

Replacing x by -x in Theorem 1.4, we get the following corollary.

Corollary 2.3. The neutrix composition [[delta].sup.(rsm-pm-1)] [[x.sup.-1/rm.sub.-]/[(1 + [x.sup.1/r.sub.-]).sup.1/m]] exists and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for r, s, m = 1, 2, ... and p= 0, 1, ... , rs - 1.

Corollary 2.4. The neutrix composition [[delta].sup.(rsm-pm-1)] [[[absolute value of x].sup.1/rm]/ [(1 + [absolute value of x].sup.1/r]).sup.1/m]] exists and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.6)

for r, s, m = 1, 2,... and p = 0,1,..., rs - 1.

Proof. Noting that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we see that equation (2.6) follows. []

Acknowledgment

The first author is supported by Tubitak. The second author acknowledges support under Research Project Number 10 G 702001, awarded by the Hacettepe University Scientific Research Unit.

References

 J. G. van der Corput, Introduction to the neutrix calculus, J. Analyse Math., 7 (1959), 291-398.

 F. Farassat, Introduction to generalized functions with applications in aerodynamics and aeroacoustics, NASA Technical Paper 3428, (1996), 1-45.

 F. Farassat, The integration of [delta]'(f) in a multidimensional space, J. Sound Vibration, 230(2)(2000), 460-462.

 B. Fisher, On defining the change of variable in distributions, Rostock. Math. Kolloq., 28(1985), 75-86.

 B. Fisher, On defining the distribution [[delta].sup.(r)](f(x)), Rostock. Math. Kolloq., 23 (1993), 73-80.

 B. Fisher, The delta function and the composition of distributions, Dem. Math. 35(1)(2002), 117-123.

 B. Fisher and T. Kraiweeradechachai, A neutrix composition involving the delta function, submitted.

 S. Gasiorowicz, Elemetary particle physics, J. Wiley and Sons Inc. New York, 1966.

 I. M. Gel'fand and G. E. Shilov, Generalized Functions, Vol. I, Academic Press, 1964.

 E. L. Koh and C. K. Li, On Distributions [[delta].sup.k] and [([delta]').sup.k], Math. Nachr., 157 (1992), 243-248.

 H. Kou and B. Fisher, On Composition of Distributions, Publ. Math. Debrecen, 40(3-4) (1992), 279-290.

 T. Kraiweeradechachai, S. Orankitjaroen, B. Fisher and E. Ozcag, Further results on the neutrix composition of the delta function, East-West J. Math., 11(2)(2009), 151-164.

 E. Ozcag, Defining the kth Powers of the Dirac Delta Distribution for Negative Integers, Appl. Math. Letters, 14 (2001), 419-423.

 E. Ozcag, I. Ege and H. Gurcay, On Powers of the Heaviside Function for negative integers, J. Math. Anal. Appl., 326 (2007), 101-107.

Brian Fisher

Department of Mathematics

University of Leicester

Leicester LE1 7RH, England

fbr@le.ac.uk

Emin Ozcag

Department of Mathematics

Hacettepe University

Beytepe, Ankara, Turkey

ozcag1@hacettepe.edu.tr

Received October 31, 2010; Accepted November 11, 2010

Communicated by Dragan Djurcic
Author: Printer friendly Cite/link Email Feedback Fisher, Brian; Ozcag, Emin Advances in Dynamical Systems and Applications Report 7TURK Jun 1, 2011 1719 Exponents of convergence and games. Some integral inequalities with maximum of the unknown functions. Convergence (Mathematics) Distribution (Probability theory) Functional equations Functions Functions (Mathematics) Probability distributions Sequences (Mathematics)