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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 [8]).

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 [1]).

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 [10], and the more general form [([[delta].sup.(s)(x)).sup.r] was considered by Kou and Fisher in [11]. Recently the rth powers of the Dirac distribution and the Heaviside function for negative integers have been defined in [13] and [14], 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 [9].

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 1.1. 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 [1], 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

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

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

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

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

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

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

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

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

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

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

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

[12] 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.

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

[14] 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
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Author:Fisher, Brian; Ozcag, Emin
Publication:Advances in Dynamical Systems and Applications
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Geographic Code:7TURK
Date:Jun 1, 2011
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