SPECIAL FUNCTIONS AND GENERALIZED FUNCTI'

Thesis submitted for the degree of Doctor of Philosophy

at the University of Leicester

by

Fatma Al-Sirehy Department of Mathematics and Computer Science

University of Leicester 2000

UMI Number: U5B3919

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Contents

A C K N O W L E D G E M E N T S ii

A B S T R A C T iv

1 I N T R O D U C T I O N 11.1 D I S T R I B U T I O N S A N D T H E N E U T R I X C A L C U L U S 3

1 1.1 D I S T R I B U T I O N S .................................................. 31.1.2 N E U T R I X C A L C U L U S ................................... 11

2 T H E C O S IN E I N T E G R A L IT

3 T H E S I N E I N T E G R A L A N D T H E N E U T R I X C O N V O L U ­T I O N P R O D U C T 42

4 T H E N O N -C O M M U T A T I V E N E U T R I X P R O D U C T O F D IS ­T R I B U T I O N S 52

5 T H E C O M P O S I T I O N O F D I S T R I B U T I O N S 87

A C K N O W L E D G E M E N T S

I wish to express my sincere gratitude to my supervisor Dr. Brian Fisher

at the Departm ent of M athem atics, The University of Leicester for suggesting

the them e of this thesis and for his guidance and follow up through out the

work.

I am also very grateful to my local supervisor Professor Salem Sahab at the

Departm ent of M athem atics, K. A. A. U. for his encouragement and helpful

discussions.

For financial support and for making this D octorate program possible for

me and many ladies I am indebted to King Abdulaziz University.

Special gratitude for the joint supervision program office for their patience

and effort to make such a program a success.

I would like to thank the Dean of Science, the Vice Deans and the Depart­

ment of M athem atics at K. A. A. U.

I am also very grateful to rny parents and brothers. My special gratitude

is reserved to my dear husband and children.

DEDICATION

To my husband and my children

iii

A B S T R A C T

In 1950, Laurent Schwartz marked a convenient starting point for the theory of generalized functions as a subject in its own right. He developed and unified much of the earlier work by Hadamard, Bochner, Sobolev and others. Since then an enormous literature dealing with both theory and applications has grown up, and the subject has undergone extensive further development. The original Schwartz treatm ent defined a distribution as a linear continuous functional on a space of test functions.

This thesis can be considered a part of the development going in that direc­tion. It is partly an extension of earlier contributions by Fisher, Kuribayashi, Itano and others.

After introducing the background and basic definitions in Chapter One, we developed some basic results concerning the cosine integral Ci(Ax) and its associated functions Ci+(Ax) and Ci_(A x) as well as the neutrix convolution products of the cosine integral.

Chapter Three is devoted to similar results concerning the sine integral Si(Ax).

In Chapter Four, we generalize some earlier results by Fisher and Kurib­ayashi concerning the product of the two distributions and x+ A_r\ More­over, other results are obtained concerning the neutrix product of \x\x~r lnp |x| and s g n x\x \x~r In'7 \x\. Other theorems are proved about the neutrix product of som e other distributions such as x A \ n x + and xZx~r ■

Chapter Five contains new results about the com position of disributions. It involves the application of the neutrix lim it to establish such relationships between different distributions.

iv

Chapter 1

INTRODUCTION

The first to use generalized functions in the explicit and presently accepted

form was S. L. Sobolev in 1936 in studying the uniqueness of solutions of the

Cauchy problem for linear hyperbolic equations.

In 1950-1951, Laurent Schwartz’s published a monograph entitled ’’The­

ories des D istributions” . In this book Schwartz system atized the theory of

generalized functions, basing it on the theory of linear topological spaces, re­

lated all the earlier approaches, and obtained many im portant and far reaching

results. Unusually soon after the appearance o f ’’Theories des D istributions” ,

in fact literally w ithin two or three years, generalized functions attained an

extrem ely wide popularity. Since then an enormous literature dealing with

both theory and applications has grown up, and the subject has undergone

extensive further developm ent. Some m athem aticians believe that distribution

theory was one of the greatest revolutions in m athem atical analysis in the 20th

century. They think of it as the com pletion of differential calculus, just as the

other great revolution, measure theory (or Lebesgue integration theory), can

be thought of as the com pletion of integral calculus. It is sufficient just to

point out the great increase in the number of m athem atical works containing

the delta function.

1

The Dirac delta function 5(x) has a long history. Its first appearance seems

to have been in Fourier’s Theorie Analyt ique de la Chaleur , (1822). Kirchoff

[29] later defined S(x) by

6(x) = lim 7T-1 / 2//ex p ( — u2x 2).n—>00 K '

Cleary 8{x) = 0 for x ^ 0 and (5(0) = 00. He also defined

f S ( t ) d t = lim 7x~l/2ji f exp (—/j2t 2)d t ,J —00 / z —> 0 0 J —00

which im plies that

In this sense, it is apparent that the delta function is not a function in the

normal sense.

H eaviside’s function H (x ) is defined by a locally summ able function where

I I «

Considering this function, Heaviside concluded that the derivative of H is

equal in some sense to S.

Later Dirac treated the delta function as the function which is everywhere

equal to zero exept at the origin where it is infinite, in such a sense that it

satisfies

/ o o

S(x) dx = 1.- 0 0

Dirac thought of 5 as a unit point charge at the origin. Moreover, he thought

of S' the derivative of 6 as a dipole of unit electric moment at the origin since

/ c o r 00

x 8 ' ( x ) d x = l i m ^ o o ? x[exp(—/j2x 2)], d x = —l.-00 J — 0 0

Similarly, higher derivatives of 6 can be used to represent more com plicated

m ultiple-layers and have been used in the physical and engineering science for

som e tim e.

2

1.1 DISTRIBUTIONS AND THE NEUTRIX CALCULUS

In this section, we give an overview of the concepts of distributions and

neutrix calculus as used in evaluating some forms of improper integrals.

1.1.1 DISTRIBUTIONS

D E F I N I T I O N 1.1 Let f be a real (or complex) valued function defined on

the real line. Then supp f, the support of f, is the closure of the set on which

f ( x ) £ 0.

D E F I N I T I O N 1 .2 Let p be an infinitely differentiable real valued function

with compact support. Then ip is said to be a test function. The set of all test

functions with the usual definition of sum and product by a scalar is a vector

space and is denoted by V.

N ote that if p G V and g is any infinitely differentiable function, then

g p e V .

D E F I N I T I O N 1 .3 Let { p n} be a sequence of test functions in T>. Then the

sequence { p n} is said to converge to zero if there exists a bounded interval

[a,b] with supp p n C [a,b] for all n and limn_>00 p ^ ( x ) = 0 for all x and

r = 0 , 1 , 2 , . . . . If f is a linear functional on T> into the real (or complex)

numbers, we denote its value at p € V by (f , p ).

D E F I N I T I O N 1 .4 Let f be a linear functional on V . Then f is said to be

continuous z /lim n_).00( / , p n) = 0 whenever { p n} is a sequence in T> converging

to zero. A continuous linear functional on T> is said to be a distribution (gen­

eralized funct ion) . The set of all distributions is a vector space and is denoted

by V .

3

The famous m athem atician Laurent Schwartz, during the period 1945 -

1950, did an extensive work on distributions. Schwartz gave this name after

his research on generalizing the idea of electric density so that it could be

applicable to these generalized functions of electricity.

Every locally sum m able function defines a functional. In fact if / is a

locally summ able function then we can define a linear functional, which we

will also denote by / by putting

if supp tp C [a, b]. Further, the functional / is continuous and so a distribution

because if {(^n} converges to zero, then for each e > 0 there exists an N such

that \ipn \ < e for every n > N and so

if supp ipn C [a, b] for all n.

D istributions that are defined by equation (1.1) from locally summ able

functions are called regular d istributions. All distributions that are not regular

are called singular.

An exam ple of a distribution that is not regular is the D irac-delta function

6 defined on V by

This distribution can not be defined by a locally summ able function. By

changing the origin in equation (1.2), we get

( i . i )

( 1 .2 )

(.6(x - a),<p(x)) = <p(a),

where a is any real number.

4

D E F I N I T I O N 1 .5 The distribution f is said to vanish in the open neigh­

bourhood U of a point x 0 if ( / ( x ) , <p{x)) = 0 for every which has its support

in U. The support of f is the smallest closed set of points outside of which f

vanishes. For example S(x — xq) has the point x = Xq as its support.

In order to define the derivative of the distribution, we first of all consider a

continuous function / which is differentiable everywhere and whose derivative

is continuous. Its derivative / ' will define a bounded linear functional and

/ oof ' (x ) ip (x )d x

-oor n oo r ° °

= f (x ) i f (x ) - f (x)(p (x) dxJ OO i - o o

= - < / ( 2 ) V ( z ) >

for all ip € V . This suggests that we define the derivative / ' of a distribution

/ by the equation

{ f \ x ) , < p ( x ) ) = - ( f ( x ) , < p ' ( x ) ) .

It is easily seen that / ' is a distribution, since if the sequence {</?n} converges

to zero, the sequence {p>'n} also converges to zero and so

rJim ( f ' { x ),<Pn(x )) = - J i m { f ( x ) , <p'n (x ) ) = 0.

In general

and f ^ is a distribution for n = 1 , 2 , . . . .

E X A M P L E 1.1 Let x + be the locally summ able function defined by

I i , x > 0,0, x < 0.

5

Then its derivative is H eaviside’s function H , since

ro c r oo— (x + ,(p'{x)) = — x ( p ' ( x ) d x = / <p(x) dx = (H(x) , (fix)).

Jo Jo

The derivative H 1 of H is given by

r oo(H'(x), <p(x)) = ~ ( H ( x ) , <p'(x)) = - <p’(x) dx = ip(0)

Jo

and so H' — 5. In general the r-th derivative 6 ^ of S is given by

(<5<r>(x), (p(x)) = ( - l ) r (S{x), ip{T](x)) = ( - l ) V (r,(0).

E X A M P L E 1 .2 Let x x+ (A —1) be the locally sum m able function defined

byA _ / X X , X > 0 ,

+ \ 0 , X < 0 .

If A > 0, its derivative is the locally summ able function but if — 1 <

A < 0, is not a locally summ able function. If —1 < A < 0 we will still

denote the derivative of x+ by Ax^-1 but it must be defined by

roc( (^ + ) ' , >p(x)) = ~ { x x+ ,ifi'(x)) = - x xd[^(x) - v?(0)]

Joroc

= A / x x~l [ip(x) — v?(0)] dx.Jo

Thus if —2 < A < —1, we have defined x+ by

roc{xx , ip{x)) = / x x[(f(x) - < (0)] dx.

Jo

In general, it can be proved by induction that if — r — 1 < A < —r, then

x x ip(x) —'0

ro c r r 1 nri -|

(xX, ( p ( x ) ) = / x x [ip{x) ~ — (z)(0)Jo l=0 i.

dx.

6

E X A M P L E 1 .3 The locally summ able function ln x + is defined by

ln x , x > 0,In x + =

0, x < 0.

We define the distribution x +l to be the derivative of \ n x + . Thus

(x+ \ i p ( x ) ) = ( O n ^ + y ^ M ) = -< ln x+,ip'(x))

= - fJoIn xd In xdip{x)<p(x) - (p{0)

r I f -i rooJ x ~ l (p(x) — ip(0) dx + j x ~ l ip (x)dxroo 1

/ x ~ l <p(x) — ip(0)H(l — x)Jo 1

dx.

More generally, we define the distribution x^r inductively by

x + r = (—r + l ) _1( x ; r+1)'

for r — 2, 3 , . . . , see [12] and not as defined in G el’fand and Shilov [25]. D e­

noting Gelfand and Shilov’s definition of x + ~r by F (x+ , —r), it follows that

x f ( i +, - r ) + i T « r -(r - 1)!

for r = 2, 3 , . . . , where

, , , , Y . i r = 1 ,2 ,4>{r) = •!

r — 0.

It can be proved by induction that

n QQ T 2 I ™ T 1

{ x ? M x ) ) = / ^ L w - y - / ) ( o ) - ?— / - 1) ( 0 ) f f ( i - x )j o z=0 i . {I

dx

1

(r - 1)!

for r = 2, 3 , . . . , see Fisher [12].

</>(r — l)<^r ^(0)

7

Now let / be a locally sum m able function. Then

/ oo roof { - x ) < p ( x ) d x = / f ( x ) i p ( - x ) d x

-oo J — oo

= ( f (x ) ,< p { - x ) ) .

T his suggests the following definition:

D E F I N I T I O N 1 .6 Let f be an arbitrary distribution. Then f ( —x) is the

distribution defined by

( f ( - x ) ^ ( x ) ) = ( f {x ) ,< p ( - x ) ) .

E X A M P L E 1 .4 The distribution x)_ is defined by x x — (—x ) + x so that

{x* , ip (x)) = (x+ ,< p( -x ) ) .

Thusr oo

(x* , tp(x)) = / x XLp(—x ) d x Jo

for A > —1, and

(xx_ , i p ( x ) ) = [ x x \ ip { -x ) - p - ^ (0 (°)JO L , _ n v.

for —r — 1 < A < —r,

dxi = 0

roo r{ x r \ i f { x ) ) = / x ~ l \ (p ( -x ) - Lf(0)H(l - x)

Jo Ld x .

r i - r M - E ^ w ( o ) - ( x)i=0 1'

r — 1

(r - 1)!(r- dx +

(r - 1)!

for r = 2, 3 ,. . ..

The distribution |x |A is defined by

\x\x = x A + x x_ ,

the distribution sg n x |j: |A is defined by

sg n x |x |A = x A — x A

and the distribution x r is defined by

x r = x r+ + ( — l ) r x[

for r = 0, ± 1 , ± 2 , . . . . In particular

roo( x - 2r,< p (x ) )= x -

Jo<p(x) + i p ( - x ) - 2 ^ cp(2t)(0)

*=0dx,

x - 2 r + 1<X 2r+',V9(x)) = f Jo

for r = 1 , 2 , . . . .

It follows that if In |x| = In x + + In , then

(In |x |)' = x ~ l , (x~ r )' = —rx

r - 1 2 i - l

dx

- r — 1

for r = 1 , 2 , . . . , see [25].

A very im portant concept in the theory of distributions is that of conver­

gence.

D E F I N I T I O N 1 .7 Let { /„ } be a sequence of distributions in V . Then { /„ }

zs said to converge to the limit f in V if and only if

lim ( f n(x), ip{x)) = ( f {x ) , (p (x ) )

for all (p G V.

The following theorem is easily proved.

T H E O R E M 1.1 If the sequence { f n} of distributions converges to f , then

the sequence { f ^ } converges to f W for k = 1, 2 , . . . .

D E F I N I T I O N 1 .8 A sequence of test functions {<Pn} is said to be a null

sequence if

(i) the support of each ipn is contained in some fixed domain D independent

of n,

(ii) the sequence { < p ^ } converges uniformly to zero in D , as n tends to

infinity, fo r all m.

D E F I N I T I O N 1 .9 A sequence of functions { f n} is said to be regular if

(i) f n is infinitely differentiable, for all n,

(ii) the sequence { ( /n ,^ ) } converges, to a limit (f,^p) say, for each test

function ip,

(Hi) ( f , p ) is continuous in the sense that limn^ 00( / , p n) = 0 for each null

sequence of test functions {(fn), see Temple [33].

Two regular sequences {gn} and { /in} are said to be equivalent if

lim (gn - hn, cp) = 0n —> oo

for each test function ip.

10

1.1.2 NEUTRIX CALCULUS

In 1932 , see Tem ple [34] Hadamard was faced with the divergent integral

A(x) , , xL ^ dx' ( 1 -3 )

where p is a positive integer. He therefore separated the integeral

A(x)I ^ m dx’ ^ > °)

into two parts, nam ely

,l A (x ) — B (x )no-I jrP~~ 1 /2

and

dx

r l B (x )( e ^ ~ J , ’’ e x t

where

B { x ) = E*=0 L-

F(e) tends to a finite lim it F ( 0) as e tends to 0, whereas 1(e) diverges as e

tends to 0. He therefore defined F (0 ) as the finite part of the integral (1.3)

and wrote

However,

^ ,4<*>(0) ^ .4(*>(0)e-"+,+ 57(0 = ~ E -TT + Efri i \ ( p - i - i ) ,=0 | )

= A + /,(« )

and so the divergent integral

A B (x )Io xp+1/2

dx

11

has a finite part K and a divergent part /i(e ) . Thus, we can write

f.p. T 1/2

and so we should in fact have

t . P . / Jo

1 A(x)dx = F ( 0) + K .

o x p+l/2

T his was Tem ple’s interpretation in [34]. Hadam ard’s original interpreta­

tion however is entirely correct; see for exam ple The Prehistory of the Theory

of Distributions by Jesper Lutzen (Springer, 1982). We will come back to this

shortly.

In his study of asym ptotics, van der Corput [5] came across similar prob­

lems. He noticed that certain terms he had calculated just cancelled out and

so were superfluous. He called such terms negligible functions. From this he

developed the neutrix calculus. In the following chapters, we will use the neu­

trix calculus to evaluate neutrix products, neutrix convolution products and

com positions of distributions.

D E F I N I T I O N 1 .1 0 Let TV' be a set and let N be a commutative, additive

group of functions mapping TV' into a commutative, addit ive group T V ". If N

has the property that the only constant function in TV is the zero function, then

TV is said to be a neutrix and the functions in TV are said to be negligible.

E X A M P L E 1 .5 Let TV' be the closed interval [0,1] = { x : 0 < x < 1} and

let TV be the set of all functions defined on TV' of the form

where a and b are arbitrary real numbers. Then TV is a neutrix, since if

a sin x -I- bx2

a sin x + bx2 = c

for all x in TV', then a = b = c = 0.

12

Now suppose that TV' is a subspace of a topological space X having a lim it

point y which is not contained in N ' . Let TV" be the real (or com plex ) numbers

and let TV be a com m utative, additive group of functions m apping TV' into N"

w ith the property that if N contains a function f ( x ) which converges to a

finite lim it c as x tends to y, then c — 0. TV is a neutrix, because if / is in

N and f { x ) = c for all x in TV', then f ( x ) converges to the finite lim it c as i

tends to y and so c = 0.

D E F I N I T I O N 1 .11 Let f ( x ) be a real (or complex) valued function defined

on TV' and suppose it is possible to f ind a constant c such that f ( x ) — c is

negligible in TV. Then c is called the neutrix limit or N-l imit of f ( x ) as x tends

to y and we write

N — lim f ( x ) = c.x - ^ y

N ote that if a neutrix lim it c exists, then it is unique, since if f ( x ) — c and

f ( x ) — c' are in TV, then the constant function c — c' is also in TV and so c = c'.

A lso note that if TV is a neutrix containing the set of all functions which

converge to zero in the normal sense as x tends to y , then

lim f i x ) = c => N — l i m f i x ) = c.x - > y x —

In the final two exam ples, the neutrix TV we are using will have domain

TV' the positive reals and range TV" the real numbers with negligible functions

finite linear sums of the functions

cAlnr _ I6, lnr e (A < 0, r = l , 2 , . . . ) (1.4)

and all functions which converge to zero in the normal sense as e tends to 0.

13

E X A M P L E 1 .6 The G am m a function T(A) defined for A > 0 by

rooT ( X ) = t x~xe~l dt

Jo

and for — r < A < - r + 1, r € N by

r w - j f V - ' l . - -

N ote that

i=0

r - 1 ( -A*roo roo 1_f _ /V 1/ C - ' e - ' d t = / c - > [ e - t - y L i L i dt +

Je Je r. i .11z=0 L '

^ ( - i ) V ^

and it follows that

roo roor(A ) = N - l im / t dt = f.p. / t x~1e~t dt,

e—>0 ./e 70

where f.p. / 0°° dt denotes Hadam ard’s finite part of / 0°° t x~1e~i dt.

More generally, it can be shown that

roo roor (r)(A) = N - l im / t x~' lnr te~l dt = f.p. / t x~ l W te~l dt

( >0 J £ J 0

for A ^ 0, —1, — 2 , . . . and r = 0,1, 2 , . . . .

This neutrix lim it was also used to define r ^ (A ) for A = 0, — 1, —2 , . . . and

r = 0, 1, 2 , . . . , see Fisher and Kuribayashi [20].

E X A M P L E 1 .7 The B eta function is usually defined by

£ (A ,/ i ) - [ l t x~ l ( i - t y ~ l dt Jo

for A, (j, > 0, but more generally, if

Br,s(K V) —

14

it can be proved that

B r s(A, /i) = N —lim [ tA_1 lnr t( 1 — t ) ^ 1 lns(l — t) dte->0 Je

for A, /i / 0, — 1, —2 , . . . and r, s = 0,1, 2 , . . . .

Again

B-,-)S(A, ji) = f.p. [ t x 1 lnr t ( l — t)^1 1\ns(l — t)dt . Jo

T his neutrix lim it was also used to define B rjS(A, fi) for A, fi = 0, — 1, —2 , . . .

and r , s = 0 , 1 , 2 , . . . , see Fisher and Kuribayashi [19].

We finally note that if the set of negligible functions is changed, then the

neutrix lim its will change. For exam ple, if we look at Hadam ard’s integral

(1.3) again we can write

[ ' f a - [ 1 ~ f a , y ' j4(*)(Q) ( -p+i+l/2 _A XP+'/2 A xP+'/2 aX + 1 ) ^

Using the set of negligible functions (1.4) it follows that

N - l im / ' = F{0) + K .t - * 0 J t x P + 1 / 2

However, if we use the set of negligible functions consisting of

(eA — 1) lnr_1 e, lnr e (A < 0, r = 1, 2 , . . . ) (1.5)

we get

N —lim / — yr- dx = F ( 0). c —>0 J e x P + l / 2

The former seems to be the correct answer using the standard set of negligible

functions (1.4), otherwise incorrect answers would be given in Exam ples 1.6

and 1.7 if we used the set of negligible functions (1.5).

In addition to this introduction, there are four chapters in this thesis.

15

Chapter Two involves the neutrix convolution products which are intro­

duced for the cosine integral Ci(Ax). Moreover, some new results are estab­

lished in this regard.

In Chapter Three sim ilar results concerning the sine integral Si(Ax) are

obtained.

In Chapter Four, some earlier results by Fisher and Kuribayashi concerning

the neutrix products are generalized.

In Chapter Five, new results are established regarding the com position of

distributions such as ( #+) and ( |x |_A)2r//A.

16

Chapter 2

THE COSINE INTEGRAL

The cosine integral C i(x) is defined for x > 0 by

r ooC i(x) = — / w_ 1 coswdw. (2.1)

J x

This integral is divergent for x < 0. Equation (2.1) however can be rewritten

in the form

rooCi(x) = — I u ~ l [cosu — H(1 — u)]du + H ( l — x ) \n \x \ . (2.2)

J x

The integral in this equation is convergent for all x and ln |x | is a locally

sum m able function on the real line. We will therefore use equation (2.2) to

define Ci(j;) on the real line as a locally summ able function.

More generally, see [8], if A ^ 0, we define Ci(Ax) by

rooCi(Xx) = — u -1 [cos ii — H(1 — u)] du + H(1 — Xx) In \Xx\. (2.3)

J Xx

In particular, if A > 0, it follows that

rooCi(Ax-) = — u ~ ] [cos(Au) — H ( 1 — Aw)] du + H ( 1 — Xx) In |Air| (2.4)

J X

and if A < 0, it follows that

Ci(Ax) = f w- 1 [cos(Aw) — H (1 — Aw)] du + H(1 — Xx) In \Xx\ (2.5)J — oo

/ oow_1[cos(Aw) — H ( 1 + Aw)] du + H ( l — Xx) In |Ax|. (2.6)

- x

17

If A ^ 0, we define Ci+(A, x) to be the locally sum m able function given by

Ci+(A, x) = i f (x) Ci(Ax). (2.7)

It follows that if A > 0, then'OO

u _1[cos(Aw) — H ( 1 — Aw)] du 4- f f ( 1 — Ax) In |Ax|, x > 0,Ci+(A, x) =

0, x < 0.(2 .8 )

A lternatively, as a generalization of equation (2.1), equation (2.8) can be ex­

pressed in the simpler form✓ rOO

Ci+ (A, x) = I Jx “ C0< Xu) du> * > 0 ’ (2.9){ r oo

~L u ~ l c c

o,x < 0.

If A < 0, it follows that■oo

u -1 [cos(Au) — i f (1 + Aw)] du + In |Ax|, x > 0,Ci+(A, x) =

[ 0, x < 0.rO

= —c — u - 1 [cos(Au) — 1] du + In |Ax|, x > 0,J — X

r x■c+ / w_1[cos(Aw) — 1] du + In |Ax|, x > 0, (2.10)

Jo

where

rooc = / u -1 [cos(Aw) — i f (1 + Aw)] du, A < 0,

Joroo

= / w-1 [cos(Aw) — i f (1 — Aw)] dw, A > 0,Jo

roo= / w_1[cosw — i f ( l — w)] du.

Jo

We next define the locally sum m able function Ci_(A, x) for A ^ 0 by

Ci_(A, x) = i f ( - x ) Ci(Ax) = C i ( A x ) - C i + ( A , x ) (2.11)

so that if A > 0,

Qj m x \ _ / ~ c ~ J w_1[cos(Aw) - 1] du + In |Ax|, x < 0, (2 12)~~ \ X 0, x > 0.

18

If A < 0, it follows that

q ^ _ / J u ~ 1[cos(Xu) — H ( 1 — An)] du + H ( 1 — Xx) In |Ax|, x < 0,

1 0, x > 0,

/ X

u~ l cos(Aw) du, x < 0. (2.13)-oo

For future reference, we note that if we replace x by — x in equation (2.3),

we see that C i(A (—x)) = C i((—X)x) — C i(-X x ) and so if we replace x by — x

in equation (2.7) we get

Ci+(A, ( - x ) ) = H ( - x ) C i(A (—x)) = H { - x ) C i((-A )x ) .

It follows that

Ci+(A, ( - x ) ) = C i_(( —A), x) (2.14)

for all A. Sim ilarly

C i-(A , ( - x ) ) = Ci + ( ( - A) , x ) (2.15)

for all A.

We will now find the derivative [Ci(Ax)]' of Ci(Ax) as a distribution for

A / 0. We let V be the space of infinitely differentiable functions w ith com pact

support and let V be the space of distributions defined on V . P utting

roof ( x ) = / u -1 [cos u — H ( l — u)] du

J Ax

and letting ip be an arbitrary test function in V , we have

( lf{x)]' ,<p(x)) = ~ ( f ( x ) , p ' ( x ) )

/ oo roo(f'(x) / u ~ x [cos u — H(1 — u)] du dx

-oo J Xx

/ oo riL/Au ~ 1 [cos u — H ( 1 — u)] / d x d u

-OO J — oo

/ oou ~ 1 [cos u — H ( 1 — u)](p(u/X) du

-ooroo

= — / u ~ l cos(Au)[(^(u) — ip(—u)] du +Jo

roo+ / u ~ l [H( 1 — Au)(p{u) — H ( 1 + Au)(p(—u)] du

Jo= — (cos(A x)x-1 — H ( 1 — A x)x-1 , (^(x)), (2.16)

19

where cos(A x)x_1 is the product of the infinitely differentiable function cos(Ax)

and the distribution x ~ l , see G el’fand and Shilov [25]. It now follows from

equation (2.3) that

°o t _ 1 \ 2i[Ci(Ax)]' = cos(Xx)x~1 = x ~ l + —— —j— x 2*- 1 , (2-17)

i= 1 v U -

since x lx ~ l = x l~~l for z = 1 , 2 , . . . . More generally, we have

oo / -| \ i \ 2i[Ci(Ax)](2’’_1) = (2r - 2)!x~2r+1 + £ I ^ - 1yjS2<~2r+1, (2-18)

[Ci(Ax)](2r) = - ( 2 r - l ) ! x - 2r + f ; 5y f ^ y j x 2i- 2r, (2.19)

for r = 1 , 2 , . . . .

We next find the derivative of Ci+(A, x) for A > 0. Using equation (2.9)

we have

([Ci+(A,x)]',<p(x)) = -(C i+ (A ,x ),y ? '(x ))r oo r oo

= / p'(x) / u ~ l cos(Au) du dxroc ru

= / n _ 1 cos(An) / (p '(x )dxdu Jo Jo

roc= u ~ ] cos (An) [(/?('(/) — y>(0)] du

Jor oo

= / u ~ 1 [cos(Xu)(p(u) — 77 ( 1 — it)y?(0)] du +Jo

roc— <£>(0) / u _1[cos(Au) — / / ( l — An)] du +

roc+</?(0) J u~ l [H( 1 — u) — 77(1 — An)] <7m

= ( c o s ( A x ) j : ~ 1 - (c - In |A | )5 (x ) , cp(x)).

It follows that

[Ci+(A,x)]' = cos(A x)x^ 1 - (c - In |A |)5(x) (2.20)oo / i Ai \ 2i

= * ; , + £ U ^ * ? ~ 1 - ( c - 1n | A | W s ) , (2.21)i = 1 \ Z l r

20

since x lx + l — x l+ 1 for i = 1 , 2 , . . . , see G el’fand and Shilov [25]. More gener­

ally, we have

[Ci+(A, x)]" = - x ~ 2 + g 2i(2i — 2) \x2^~2 “ (c “ ln l-M)5'^ ) (2 -22)

and

[Ci+ (A, x )](2r~ 1) = (2r - 2)!x+2r+1 + g +

+ E M ^ < 5 (2r- 2- 2)(x) - (c - In |A |)^ 2r- 2>(x),i=i 2*

(2.23)

[Ci+ (A,x)]<*> = - ( 2 r - l ) ! x ; 2- + f 2ty ] ‘A22‘) , x 2i- 2- +

+ E ^~19'>‘A2‘ <5(2r~2l~ 1>(x) - (c — In |A|)<5*2r_1*(x) (2.24)1=1 2z

for r = 2 , 3 , . . . .

For the case A < 0, we put

_ I J u ~ l [cos(\u) — 1] du, x > 0,

\ 0, x < 0.

Then for arbitrary test function p in T>, we have

([ /(* )] '. ¥>(*)) = - ( / ( * ) > ¥>'(*))r oo /-x

= — y>'{x) / u -1 [cos(Au) — 1] d udx7 0 JOroo roo

= — j u _1[cos(Au) — 1] J p ' ( x ) d x d uroo

= / u _1[cos(Au) — l](/?(u) du Jo

roo= / u _1[cos(Au)v?(u) — H ( l — u)(/9(0)] du +

Joroo

— / u ~ l [p(u) — H(1 — u)p(0)] du Jo

= (cos(A x)x^1 — x ^ 1, p(x ) ) (2.25)

and it follows that equations (2.20) to (2.24) also hold for A < 0.

21

It now follows from equations (2.11), (2.17) and (2.20) that for all values

of A ^ 0

for r = 2, 3 , . . . .

C o n v o lu t io n o f d is tr ib u t io n s

The classical definition of the convolution product of two functions / and g

is as follows:

D E F I N I T I O N 2 .1 Let f and g be functions. Then the convolution product

f * g is defined by

fo r all points x for which the integral exists.

It follows easily from the definition that if f * g exists then g * / exists and

[C i_(A ,x)]' = — cos(Xx)x_ l + (c — In A)5(x) (2.26)

(2.27)

x 2f 2 + (c — In A)£'(:r) (2.28)

and more generally

[Ci_(A, x ) p r“ 1)

+ E — T ~ A (2r- 2,- 2)(x) + (c - In X)Si2r~2\ x ) ,V I

[Ci_ (A, *)]<*> = -

i — 1

f * 9 = 9 * f (2.29)

22

and if ( / * g)' and f * g' (or / ' * g) exists, then

( / * # ) ' = / * # ' (or f ' * g ) - (2-30)

D efinition 2.1 can be extended to define the convolution product / * g of

two distributions / and g in V with the following definition, see G el’fand and

Shilov [25].

D E F I N I T I O N 2 .2 Let f and g be distributions in V . Then the convolution

product f * g is defined by the equation

( ( / * <?)(*), <p) = ( f ( y ), <p(x + y ) ) )

fo r arbitrary ip in V , provided f and g satisfy either of the conditions

(i) ei ther f or g has bounded support,

(ii) the supports of f and g are bounded on the same side.

In the following, the locally sum m able functions sin±(Ax) and cos±(Ax) are

defined by

sin + (Ax) = H (x ) sin(Ax), sin_(Ax) = H ( —x) sin(Ax),

c o s + ( \x ) = H (x )cos (X x) , cos_(Ax) = / / ( —x) cos(Ax).

It follows as above that

sin+(A (—a;)) = s in_( ( —A)x), sin_(A (—a;)) = sin+ (( —A)x),

cos+ (A (-x ) ) = cos_ ( ( —A)x), c o s_ (A (-x )) = cos+ ((-A)ar).

We need the following theorem s and corollaries which were proved by Fisher

and Al-Sirehy, see [8].

23

TH EO REM 2.1 If X + 0, than

Ci+(A, x) * x r+ = ----- -j— XI f r + A)x*+ H-----x r+1 Ci+(A, x) (2.31)r + 1 \ z / r + 1

/o r r = 0 ,1 , 2 , . . . , where

r xa*(x, A) = / (—u)1 cos(Xu) du

Jo

fo r i = 0 ,1 , 2 , . ... /n particular,

Ci+ (A, x) * / / ( x ) = —A-1 sin+ (Ax) + x Ci+(A, x), (2.32)

Ci+(A, x) * x + = — ^[A“2/ / ( x ) — A~2 cos+(Ax) + A-1 xsin+(Ax)] +

+ | x 2 Ci+ (A, x). (2.33)

C O R O L L A R Y 2.1 If X ^ O , ^ e n

[cosfA xJx/1] * x+ = (c —In |A|)x+ — X i •')ar -* - i(^ } \ ) x \ + x r Ci+(Ax) (2.34)i= 0 \ v

fo r r = 1 , 2 , . . . . /n particular,

[cosfAxJx/1] * H( x) = (c — In |A |) / / (x ) + Ci+(A, x), (2.35)

[cos(A .t)x /1] * x + = (c — In |A|)x+ — A-1 sin+ (Ax) + x Ci+(A, x). (2.36)

T H E O R E M 2 .2 I f X / 0, then

Ci_(A, x) * x l = E f r + ( - l ) ’ar- , (x , X)x’_ + Ci_(A, x)r + 1 \ 1 / r + 1

(2.37)

for r — 0 ,1 , 2 , . . . . In particular,

Ci_(A, x) * H ( —x) = A-1 sin_(Ax) - x Ci_(A, x), (2.38)

Ci_(A, x) * x_ = - |[A _2/ / ( - x ) - A-2 cos_(Ax) + A_1x sin_(Ax)] +

+ x 2 Ci_(A, x). (2.39)

24

C O R O L L A R Y 2 .2 If A ^ 0, then the convolution product [cos(Ax)x_s] * x r_

exists fo r r = 0 ,1 , 2 , . . . and s = 1 , 2 , . . . . In particular,

[cos(A x)xI1] * H ( —x) = (c — In |A|) H ( —x) + Ci_(A, x ) : (2.40)

[cos(A x)xI1] * — (c — In |A|)x_ + A-1 sin_(Ax) — x Ci_(A, x) (2.41)

and in general

[cosfA x)^!1] * x T_ = (c — In |A|)af + ( —l ) r_zar_ i_ i(x , \ ) x l_ +i=0 \ 1/

+ ( —l ) rx r Ci_(A, x) (2.42)

f o r r = 1, 2 , . . . .

The above definition of the convolution product is rather restrictive and so

a neutrix convolution product was introduced in [10] and was later modified

in [17]. In order to define the neutrix convolution product we first of all let r

be a function in V satisfying the following properties:

( i ) t (x ) = t ( - x ),

(ii) 0 < t (x ) < 1,

(iii) t (x ) = 1 for \x\ <

(iv) t (x ) = 0 for \x\ > 1.

The function rn is now defined by

rn(x) = <1, |.t | < n ,

r ( n nx — n n+l ), x > n,r { n nx 4- nn+ ), x < ■n

for all real n > 0.

The following definition of the neutrix convolution product was given in

[10].

25

D E F I N I T I O N 2 .3 Let f and g be distributions in T>' and let f n = f r n for

n > 0. Then the neutrix convolution product f ® g is defined as the neutrix

l imit of the sequence { f n * g ] , provided that the limit h exists in the sense that

N —lim ( /n * g , p ) = ( h , p )n —►oo

fo r all <p> in V , where N is the neutrix, see van der Corput [5], having domain

N ' the posi t ive reals and range N" the real numbers, with negligible functions

f inite linear sums of the functions

n x lnr_1 n, lnr n , (A > 0, r = 1, 2 , . . . )

and all functions which converge to zero in the usual sense as n tends to

infinity.

We now increase the set of negligible functions given here to include finite

linear sums of the functions

n /icos(An), n Msin(An), n MCi[A(a + n)] ( p ^ O ) .

Note that in this definition the convolution product f n * g is defined in

G el’fand and Shilov’s sense, the distribution f n having bounded support.

It is easily seen that any results proved with the original definition hold

with the new definition. The following two theorems were proved in [10], the

first showing that the neutrix convolution product is a generalization of the

convolution product.

T H E O R E M 2 .3 Let f and g be distributions in V satisfying ei ther condition

(a) or condition (b) of G e l ’fand and Shilov’s definition. Then the neutrix

convolution product f © g exists and

f ® 9 = f * 9-

26

T H E O R E M 2 .4 Let f and g be distributions in V and suppose that the

neutrix convolution product f ® g exists. Then the neutrix convolution product

f ® g' exists and

( /© < ? ) ' = / © < / • (2.43)

Note however that equation (2.29) does not necessarily hold for the neutrix

convolution product and that ( / © g)' is not necessarily equal to f ® g , but

we do have the following lemma, see Fisher and Al-Sirehy [8].

L E M M A 2.1 . Let f and g be distributions in V and suppose that the neutrix

convolution product f ® g exists. / / N — lim ((/r^ )* g , ip) exists and equals (h, (f)71—> OO

for all (p in V , then f ® g exists and

( /© < ? ) ' = f ' ® 9 + h. (2.44)

P R O O F . Using equation (2.30) we have

( { f n * g) ' , v) = { ( f ' r n ) *g , g>) + ( ( / < ) * g, <p)

and it follows that

N - l i m ( ( / V n) *g, tp) = - N - l i m ( f n *g, ip' ) - N - l i m ( ( / r r'J *g, ip) ,n —7 0 0 71—7 0 0 71—7 0 0

proving the existence of f © g and equation (2.44).

The next theorem was proved in [8].

T H E O R E M 2 .5 If A / 0, then

c i +(A,a- ) © v = — r r E ( r + 1U i*‘ (2.45)r + 1 f o V * J

fo r r — 0 , 1 , 2 , . . . , where

L 2t = 0, L2t+1 = (2 i + 1 ) ! ( - 1 ) ’A -2- 2

27

for i = 0, 1, 2 , . . . . In particular,

Ci+(A, x) © 1 = 0,

Ci + ( X , x ) @ x = — \ \ ~ 2.

(2.46)

(2.47)

P R O O F . We put [Ci+(A, x)]n = Ci+(A, x ) r n(x). Then the convolution product

[Ci+ (A, x)]n * x r exists by Definition 2.1 and

rn rn + n ~ n[Ci+(A, x )]n * x r = / Ci+(A, t ) (x — t ) r dt + / Tn(t) Ci+ (A, t ) ( x — t ) r dt.

Jo Jn(2.48)

If A > 0, we have

rn rn roo/ Ci+(A, t ) (x — t ) r dt = — (x — t ) r u~ l cos(Au) du dt

Jo Jo Jtrn ru

= — u ~l cos(Au) / (x — t ) r d t d u +Jo Jo

roo rn— / u~l cos(Au) / (x — t ) r d t d u

Jn Jo

[ (—u)r~l cos(Xu) du + lJ 01 E i + 1 i ^r + ! \ z

+ ( _ n )r- l+l Ci(Xn) (2.49)

We now put

rnI t = / (—u)1 cos (Xu) du

Jo

= A-1 ( —n )1 sin(An) + A- I z [ ( —u) l~l s i n(Xu)duJo

= X~l ( - n ) 1 sin(An) - iX~2( - n ) l~ l cos(An) — i(i — 1)A-2 I*_2

for i > 2. In particular

I0 = X ! sin(An), / L = — A ^ s in fA n ) - A 2 cos(An) + A

and so

Lj = N — lim /j = — z(z — 1)A 2L*_2

28

for i > 2 and

L q = 0, L\ = A 2.

Thus

Z/2i = 0, Z/2z+i — (2z + 1)!( —l ) lA 2l 2

for i = 0, 1, 2 , . . . . It now follows from equation (2.49) that

1 ' (r + TN - l i m / Ci+(A, t)(x - t )r dt = |L r- i £ l . (2.50)

n -> oo 7 o r +

Further, with n > X 1 and K = sup{| Ci+ (x)| : x > 1} we have

r n + n ~ n/ rn(t) Ci+ (A, t)(x — t)r dt < X ( n + n~n + \x\)rr i

J n

for each fixed x and so

rn+n */ Tn(t) Ci+ (A, t ) ( x — t ) r dt = 0. (2.51)

J n

'U + n nlim

Equations (2.45), (2.46) and (2.47) now follow immediately from equations

(2.48), (2.49), (2.50) and (2.51) for the case A > 0.

If A < 0, equation (2.48) still holds but this time we have

/ Ci+ (A, t ) ( x — t ) r dt = (In |A| — c) / (x — t ) r dt +Jo Jo

M l rt+ / (x — t ) r / u - 1 [cos(Au) — 1] du dt +

Jo Jorn

+ / (x — u) r In u du Jo

= J, + J2 + j 3. (2-52)

Now(x — n )r+l — x r+lrn

/ (x - t y Jo

dt'o r + 1

and it follows that

N — lim J[ = 0. (2.53)n —> oo

29

N ext

Jo = [ u ^cosfAu) — 1] f (x - t ) r d t d u J 0 J u

— — — [(oo — n ) r+l - (x — u )r+1]n_1[cos(Au) — 1] du

E ( r ( - « ) r' !+1* ‘ Jo u _ 1 [cos(Am) - 1] d u +

1 . r f r + l \ rny ( ) x 1 (—u)r~l [cos(Xu) — 1 ] du

n \ 2 / JOr + 1 i=0

Y . ( ' W ( —n) T ,+1[Ci+(A, n) + c - In |An|] +

r + 1 , t s v i

and it follows that

N - l i m J2 = —r y ( r+ \Lr_ix\ (2.54)n->°o r + l y % J

Finally we have

= ------- -j— j- J In u d[(x — n )r+1 — x r+l

Inn +1 +1 1 ^ ( r + l \ x l ( - n ) r Z+1 - [ (x - n) - x r+l ] H-----------y — ----- --------r + 1 r + I i / r — i + I

and it follows that

N — l im J 3 = 0. (2.55)n —> oo

It is easily seen that equation (2.51) still holds and equations (2.45), (2.46)

and (2.47) now follow from equations (2.48), (2.51), (2.52), (2.53), (2.54) and

(2.55) for the case A < 0.

C O R O L L A R Y 2 .3 If A / 0, then

/ l A r + l r / , i \

Ci+(A, x) ® x r_ = ~'r ~ i f % ){L r- iXl - ar- i (x, X)x\ ] +

; x r+1Ci+(A,x) (2.56)r + 1

30

for r = 0, 1, 2 , . . . . In particular,

Ci+(A, x) © H ( —x) = A-1 sin+ (Ax) — x Ci+(A, x), (2.57)

Ci + ( A , x ) © x _ = \ [ \ ~ 2H ( —x ) + A-2 cos+ (Ax) — \ ~ l x sin+ (Ax)] +

+ \ x 2 Ci+(A, x). (2.58)

P R O O F . Since the neutrix convolution product is distributive with respect

to addition, we have

Ci+ (A, x) ® x r = Ci+(A, x) * x r+ + (—l ) r Ci+(A, x) © x r_

and equation (2.56) follows from equations (2.31) and (2.45). Equation (2.57)

follows from equations (2.32) and (2.46) and equation (2.58) follows from equa­

tions (2.33) and (2.47).

T H E O R E M 2 .6 If X + 0, then

[cos(Aj;)x^1] © x r = (c — In |A|)xr — f L r_ i _ i x l (2.59)i=0 V2/

for r — 0, 1, 2 , . . . . In particular

[cos(A:r)x^1] © 1 = c — In |A|, (2.60)

[cos(Ax)x^1] © x = (c — ln |A |)x. (2.61)

P R O O F . We have

[Ci+(A, x ) t „ ( x ) \ * x r = [ Ci(A t ) ( x - t ) r d r n ( t )J n

rn + n~ n= — Ci(An) (x — n) r — cos(A t ) t ~ l (x - t ) r r n ( t ) d t

J nrn + n~ n

+ r Ci(At ) ( x — t ) r ~ l Tn ( t ) d t . (2.62)

31

Now

and so

rn+n 71/ cos (Xt ) t ~l (x - t ) r Tn ( t ) d t

J n< n n 1 (|rr;| + 2n) 1

rn+n 71lirn / cos(A£)t_1(rr - t ) rrn{t) dt = 0.

J Ti(2.63)

Further

rn+n n rn+n 71/ Ci(At ) ( x - t ) r~ rn(t) dt < ( | x | + 2n )r~1 |Ci(At)|

Jn J n

n+n 71dt.

If A > 0, we put K = sup{| Ci(x)| : x > 1}. Then with n > A we have

rn+n 71/ \ C \ ( \ t ) \ d t < K n ~ n.

J n

If A < 0, we put

du.K \ — [ u McosfAw) — 1|Jo

Then with n > A- 1 , we have

r n + n ~ n/ | Ci(At)| dt < [\c\ + | ln(2An)| + K \ + 2 ln(2n)]n_n.

J n

It follows that in either case

limT l—tO O

rn+n 71/ Ci(Xt ) (x — t ) r ~ l Tn ( t ) dt = 0.

J n(2.64)

It now follows from equations (2.62), (2.63) and (2.64) that

N — lim[Ci+ (A, x ) t ' u ( x ) \ * x r = 0. (2.65)

Equation (2.59) now follows from Lemma 2.1, equation (2.65) and the equation

r — 1[cos(Aa:)x+ 1 — (c — In |A|)5(x)] © x r = r Ci+ (A, x) © x r 1 = — I \ L r_i^ix

i= 0

Equations (2.60) and (2.61) follow immediately.

32

COROLLARY 2.4 If A ^ 0, then

r - 1 / 7.

[ c o s C A x ) ^ 1] © x l = ( - l ) r 1 5 3 . [ i r - i - 1X! - a r - i _ i ( x , A ) x ' + ] +i = 0

+ ( c - In |A|)xr_ + ( - l ) r- ‘x r Ci+(A, x) (2.66)

fo r r = 0 , 1, 2 , . . . . In particular

[cosfAxJx"1] © H { - x ) = (c - In | A | ) # ( - x ) - Ci+(A, x), (2.67)

[cosfAx)^^1] © x_ = (c — In |A|)x_ — A-1 sin+(Ax) + x Ci+(A, x). (2.68)

P R O O F . We have

(cos(Ax)x^1] © x r__ = ( —l ) r[cos(Ax)x^1] © x r — ( —l ) r[cos(Ax)x+1] * x r+

and equation (2.66) follows from equations (2.34) and (2.59). Equations (2.67)

and (2.68) follow immediately.

T H E O R E M 2 .7 / / A / 0, then

Ci_ (A, x) © x r = ---- i - £ ( r + A ( - 1 y - ' L r - i j (2.69)r + 1 z=0 V 1 )

fo r r — 0, 1, 2 , . . . . In particular,

Ci_ (A, x) ® 1 = 0, (2.70)

Ci_ (A, x) ® x = \ \ ~ 2. (2.71)

P R O O F . Replacing A by —A in equation (2.45) we get

Ci+ ( ( —A), x ) © V = - - 2 - £ ( r + b' + 1 1=0 \ 1 )

and equation (2.69) follows by replacing x by — x in this equation. Equations

(2.70) and (2.71) follow immediately.

33

COROLLARY 2.5 I f \ ± 0, then

Ci(Ax) © x r = 0 (2.72)

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

P R O O F . We have

Ci(Ax) © x r = Ci+(A, x) © x r + Ci_(A, x) © x r

and equation (2.72) follows from equations (2.45) and (2.69).

C O R O L L A R Y 2 .6 If X ^ O , then

Ci_ (A, x) © x r+ = ----- L r ^ ( r + 1 ) [ ( - l ) ’--<Lr_jx < + ( - l ) ia r -<(a:,A)®L]r + 1 i = 0 V 1 )

+ — !— x r+1C i_(A ,x) (2.73)r _|_ i > ' '

fo r r — 0 , 1 , 2 , . . . . In particular,

Ci_(A, x) © H( x ) = — A-1 sin_(Ax) + x Ci_(A, x), (2-74)

C i _ ( A , x ) © x + = |[A -2 / / ( x ) + A-2 cos_(Ax) — A_1x s in _ (A x ) ]+

+ | x 2 Ci_(A, x). (2.75)

P R O O F . Equation (2.73) follows on replacing A by —A and then x by —x in

equation (2.56). Equations (2.74) and (2.75) follow immediately.

C O R O L L A R Y 2 .7 If X ^ 0, then

C i ( A x ) © x r+ = — ^ £ ( r + b K - i r - W + ar. , ( x , A)*-] + r + i i=0 \ i )

+ — — x r+1 Ci(Ax) (2.76)r + 1

fo r r = 0,1, 2 , . . . . In particular,

Ci(Ax) © H( x ) = - A -1 sin(Ax) + x Ci(Ax), (2.77)

Ci(Ax) © x + = ^[A~2 cos(Ax) - A_1x sin(Ax)] + \ x 2 Ci(Ax). (2.78)

34

P R O O F . Equation (2.76) follows from equations (2.31) and (2.73), equation

(2.77) follows from equations (2.32) and (2.33) and equation (2.78) follows

from equations (2.33) and (2.35).

C O R O L L A R Y 2 .8 If A / 0, then

( _ l ) r + l r / , -1 \

Ci(Ax) ® x r_ = — — — . ) [Lr- iXl - ar_i(x, A)xl] +

+ Y ^ y - x r+1Ci(Ax) (2.79)

fo r r = 0,1, 2 , . . . . In particular,

Ci(Ax) ® H { —x) — A-1 sin(Ax) — x Ci(Ax), (2.80)

Ci(Ax) ® = \ [X~2 cos(Xx) — \ ~ lx sin(Ax)] + \ x 2 Ci(Xx). (2.81)

P R O O F . Equation (2.79) follows from equation (2.76) on replacing A by —A

and x by —x. Equations (2.80) and (2.81) follow similarly from equations

(2.77) and (2.78).

C O R O L L A R Y 2 .9 If A / 0, then

[cosfAx)^’! 1] © x r = (c — In |A|)xr — f ( —1 )r~l L r_ i _ i x l (2.82)z=o Vv

for r — 0,1, 2 , . . . . In particular

[cosJAx)^!1] © 1 = (c — In |A|), (2.83)

[cos(A x)xI1] © x = (c — ln |A |)x. (2.84)

P R O O F . Equation (2.82) follows on replacing A by —A and x by — x in equa­

tion (2.59). Equations (2.83) and (2.84) follow immediately.

35

C O R O L L A R Y 2 .1 0 / / A / 0, then

[cos(Ax)x l ] © x r = 0 (2.85)

fo r r = 0 , 1 , 2 , . . . .

P R O O F . We have

[cos(Ax):r_1] ® x r = [cosfAx)^^1] ® x r — [cos(Ax)xI1] © x r = 0

on using equations (2.59) and (2.82).

C O R O L L A R Y 2.11 I f A ^ 0 ; then

(cos(A x)xI1] © x \ — — [( — l ) r_lZ/r_j_iXz — (—l ) zar_ i_ i(x , A)xz_] +

[cos(Ax)a;_1] © H( x ) = (c — In |A|) H( x) — Ci_(A, x), (2.87)

[cos(Ax)x_1] © x + = (c — In |A|)x+ + A 1 sin_(Ax) — x Ci_(A, x). (2.88)

P R O O F . Equation (2.86) follows on replacing A by —A and then x by — x in

equation (2.66). Equations (2.87) and (2.88) follow immediately.

C O R O L L A R Y 2 .1 2 If X / 0, then

+ (c — In |A|)x+ — x r Ci_(A, x) (2.86)

fo r r = 0 , 1 , 2 , . . . . In particular

[cos(Ax)x !] © x r+ = [(— l ) r lL r_l- \ x l — ar_ i_ i(x , X)xl] +

+ x r Ci(Ax) (2.89)

fo r r — 0,1, 2 , . . . . In particular

[cos(Ax)x l ] © H ( x ) = Ci(Ax), (2.90)

[cos(Ax)x 1] © x + = - A 1 sin(Ax) + x Ci(Ax). (2-91)

36

P R O O F . Equation (2.89) follows from equations (2.34) and (2.86). Equation

(2.90) follows from equations (2.35) and (2.87). Equation (2.91) follows from

equations (2.36) and (2.88).

C O R O L L A R Y 2 .1 3 If A + 0, then

[cos(Ax)x_1] © x r_ = ( - l ) r Y [ L r - i -1%1 - CLr- %- i(re, A)rcl_] +z = 0 \ V

— (c — In |A|)rc+ + (—l ) rx r Ci_(A, re)

fo r r — 0, 1, 2 , . . . . In particular

[cos(Ax)x_1] © H ( —x) = — (c — In |A |) i / (—re) — Ci_(A, re),

[cos(Arc)rc-1 ] © = —(c — In |A|)re_ + A-1 sin+ (Arc) + x Ci+(A, x).

P R O O F . Equation (2.92) follows on replacing A by —A and then x by

equation (2.89). Equations (2.93) and (2.94) follow similarly.

T H E O R E M 2 .8 / / A / 0, then

i r / r + l \ x r © Ci+(A, x) = ----- — £ . \L r - ix '

r + 1 f ^ o \ 1 Jfor r — 0 , 1, 2 , . . . . In particular

1 ® Ci+(A, x) = 0,

rc©Ci+(A, re) = — \ A- 2 .

P R O O F . Put (xr )n = x rrn (x). Then the convolution product (xr)n *Ci.

exists by Definition 2.1 and if n > |rc|, we have

(xr)n * C i + (A,x) = f C i+ [A, ( x - t ) ] f d t +J —n

+ [ rn(t) Ci+[A, (re - t )}tr dt.J —n —n~n

37

(2.92)

(2.93)

(2.94)

—x in

(2.95)

(2.96)

(2.97)

. (A,x)

(2.98)

If A > 0, we have

rx rx rooCi+[A, (x - t )] t r dt = - / t r j u~l cos(Au) du dt

•x+nu 1 cos(Aw) / t r d t d u +

0 J x —uroo rx

u 1 cos(Au) / t r d t d ux+n J —n

( T + l A rx+nY ( y u)r 1 c o s (Xu) d u - \ -

XrJr^f Ci[A(x + n)]. (2.9£

If we now put

Ii = / ( —u )1 cos(Aw) du,rx+n

it follows as in the proof of Theorem 2.5 that

I q — A-1 sin[A(x + n)],

I\ = — A_1nsin[A(:r + n)] — A-2 cos[A(x + n)\ + A- 2 ,

It = A-1 (—n )1 sin[A(x + n)\ — zA~2( —n )I_1 cos[A(x + n)] — i( i — l)A _2/j_ 2

for i > 2 and so

for z = 0 , 1 , 2 , . . . . It follows from equations (2.99) and (2.100) that

Further with K — sup{| Ci(Ax)| : Xx > 1} and n > A 1 — x we have

N —lim It = Lt (2 .100)

N L r- i x \ (2.101)

nTn(t) Ci+[A, (x — t)]tr dt < K ( n + n u)rn n

—n —n

and it follows that

(2 .102)

38

Equation (2.95) now follows for A > 0 from (2.98), (2.101) and (2.102).

Now suppose that A < 0. Equation (2.98) still holds but this time we have

f Ci+[A, (x — t)]tr dt = (In |A| — c) f tr dt +J - n J —n

/ X P X — t

t r j u ~ l [cos(\u) — 1] du dt +-n JO

/ X

t r ln(x — t) dt-n

= J \ + J 2 + J3, (2.103)

where

= Eur+l _ ( _ n )r+l]t (2.104)r + 1

rx+n rx —uJ2 = u ~ 1[cos(Xu) — 1] / tr d t d u

Jo j - n1 pX ~\~7l

= -- / u~l [(x - u )r+1 - ( - n ) r+1][cos(Au) - 1] dur + 1 Jo

"1 fX~\~TL

= ----------- / u ~l [(x - u )r+l — x r+1][cos(Au) - 1] dur + 1 Jo

V +1 - (—n) r+1] J « _1[cos(Au) — 1] dur + 1

1r + 1

r / r 1 \ . rx+n J x j Zcos(^w) ^ w +

1 / r + l \ x z( —rc — n ) r 1+1

r + 1 ^ S V * / r - i + l +

l . u r + i _ ( _ ri)r+i] / m-1[cos(Aw) - l] du. (2.105) r + 1 do

— r in ( x - t ) d { t r+i - x r+i )+ 1 J — n

ln(x + n) , _ (—n )r"*"ij + - J — ( X (x - - x ^ 1) dtr + 1 L r + l d - V

1P(gtlO[ar + l _ ( _ n ) r+ l] + 1 f +V ‘[(x - v r l ~ V+ 1 } d vr + 1 r + 1 J o

ln(g + n ) [a;r+1 _ (_ n)r+I] +

1 ^ ( r + l \ x ' ( - x - n)— ( 2. 106)

r + l , t o V * > r - i + 1

39

It follows from equations (2.103), (2.104), (2.105) and (2.106) that

f X Ci + [ \ , ( x - t ) } t r dt = ? r+‘ ~ (~ n)r+1 Ci+ [A, {x + n)\ +J-n r + 1

1 ( r + l \ • rx+n— 7 XI I / / (—u)r~l c o s ( \ u ) d u . (2.107)+ 1 i=0 \ 1 /r +

It can be proved as above that

rx~\~TiN —lim / (—u)r~l cos(Xu) du = Li (2.108)

n—>oo J o

for i = 0, 1, 2 , . . . and it follows from equations (2.107) and (2.108) that

N — lim f Ci+[A, (x — t )] t r dt = -------— I I L r^ x l . (2.109)n —to c J - n r + 1 “ J \ i J

Also as above, we can prove that

lim [ Tn{t) Ci+[A, (x — t)]tr dt = 0. (2.110)n —> o o J _ n _ n — n

Equation (2.95) now follows for A < 0 from equations (2.98), (2.99) and

(2.100). Equations (2.94) and (2.95) follow immediately.

The proofs of the following corollaries follow easily.

C O R O L L A R Y 2 .1 4 If A / 0, then

f _ i y + 1 r / r 4 - l \x r_ © Ci+(A, x) = - T ~ X ) [Lr- iXl ~ ar- i {x, X)x\ ] +

r + 1 V 1 Jx r+l C i+(A ,x)

r + 1

fo r r = 0,1, 2 , . . . . In particular,

H ( —x ) ® C i + (X,x) = A-1 sin+ (Ax) — x C i+ (A, x),

X - ® C i + (X,x) = I[X~2H ( - x ) + A-2 cos+ (Ax) - A_1x sin+ (Ax)] +

+ | x 2 Ci+(A, x).

40

for r = 0, 1, 2 , . . . . In particular

C O R O L L A R Y 2 .1 5 If X ± 0, then

x r © [cos(Ax)x^1] — (c — In |A|):z;r — ^ ^ [ \ L r_ i_ ix l

1 © [cos(Ax)x+1] = (c — In |A|),

x © [cos(Ax)x^1] = (c — ln |A |)x.

C O R O L L A R Y 2 .1 6 If X + 0, then

x T_ © [ ^ ( A x ) ^ " 1] = ( ~ l ) r_1 [Lr- i - i x l - ar- i - i ( x , X ) x \ ] +

+ (c — In |A|):rf. + (—l ) r~l x r Ci+ (A, x)

for r — 0, 1, 2 , . . . . In particular

H ( —x) © [cos(Ax)x^1] = (c — In |A |) /f (—x) — Ci+(A, x),

x_ © [cos(Aa;)a;+1] = (c — In |A|)x_ - A-1 sin+ (Ax) + x Ci+(A, x).

Theorem 2.9 and its corollary follow as above.

T H E O R E M 2 .9 If A / 0, then

for r = 0 , 1 , 2 , . . . . In particular

1 © Ci_(A, x) = 0,

x © Ci _(A, x) = \ A"2.

C O R O L L A R Y 2 .1 7 I f X ^ O , then

x r © Ci(Ax) = 0

fo r r = 0,1, 2 , . . . .

41

Chapter 3

THE SINE INTEGRAL AND THE NEUTRIX CONVOLUTION PRODUCT

The sine integral Si(Ax), see Sneddon [31], can be defined on the real line for

A ^ 0 byrXx rx

Si(Ax) = / u~ 1s ' m u d u = / u~ l sin(Aw) du. (3-1)Jo Jo

The function Si+ (A,x) is defined by

Si+ (A, x) = H( x ) Si(Ax) = { / T - ' s i n ( A u ) du, x > 0 , ( 3 2)[ 0, x < 0

and Si _(A, x) is defined by

Si_ (A, x) = H ( - x ) Si(Ax) = { - [ u ~' sin(A“ } du’ X < 0> (3.3)( 0, x > 0.

For future reference, we note that if we replace x by — x in equation (3.1)

we see that

Si(A(—x)) = S\ ( ( - X) x ) = S i(—Xx)

and so if we replace x by —x in equation (3.2) we get

Si+(A, ( - * ) ) = H ( - x ) S i { X{ - x ) ) = H ( - x ) S i ( ( -A )x ) .

42

It follows that

Si+(A, ( - x ) ) = S i _ ( ( - A ) , z )

for all A ^ 0. Similarly,

S i-(A , ( - x ) ) = Si+ ( ( - X ) , x )

for all A / 0. The derivative of Si(Ax) is given by

[Si(Ax)]' = sin(Ax)x_1

and further,

[Si+ (A,x)]; = sin(Ax)x^1,

[Si_(A,x)]' = — s m( Xx) xZ1-

In the following results, which were proved in [4],

r xbi(x, A) = / (—u)1 sin(Au) du

Jo

for i = 0, 1, 2 , ___

Si+(A, x) * x r+ = -----------------( + 1 A)xYr + 1 i = 0 \ 1 J

x r+l Si+(A, x),r + 1

[sin(Ax)x~1] * x \ = - ^2 ( '■ A)xl+ + x r Si+(A, x),

Si _(A, x) * x T_ = f - T 2 ( r' + 1 ) ( - l ) ,6r_ i (x,A)x*_ +7 + I z=0 \ 1 J

• ' ’ S L (A ,x ) ,r + 1

[s in (A x)x:1] * x l = ( - i r ‘ £ ( ' ) ( - + V — i( * ,A ) x t +

+ ( —x )r Si_ (A, x).

r - 1 / r s

43

(3.4)

(3.5)

(3.6)

(3.7)

(3.8)

(3.9)

(3.10)

(3.11)

(3.12)

To prove the next theorem we now increase the set of negligible functions

given above to include finite linear sums of the functions

cos(An), sin(An), Si[A(cx + n)] (fi ^ 0).

T H E O R E M 3.1 If X ^ 0, then the neutrix convolution product x r ® Si+(A,x)

exists and

V © S i + (A)x) = - ^ - ^ f r + 1 ) Lr_Ix 1 (3.13)T + i i=0 V 1 )

fo r r = 0 , 1, 2 , , where

L 2t = ( - l ) t(2j)!A'2' - 1, L2t+i = 0

/o r i = 0 , 1 , 2 , . . . . /n particular

1 © Si+(A, x) = - A " 1, (3.14)

x ® S i + (X,x) = —X~l x. (3.15)

P R O O F . We put (xr)n = x rTn(x). Then the convolution product (xr)n*Si(Ax)

exists by Definition 2.2 and if \x\ < n,

(xr )n * S i+(A,x) = [ Si+(A , (x - t ) ) t r dt + [ r n(t) Si+(A, (x - t ) ) t r dt.J—n J—n—n~n

(3.16)

We have

/ X / *x — i

Si+(A, (x - t ) ) t r dt = t r i T 1 sin(Au) d u d t-n J—n J 0

rx+Tl rx — u= / u _ 1 sin(Au) / t r d t d u

JO J-n“J /«j_J-^

= ------- / u -1 sin(Au)[(x — u)r+l — (—n )r+1] dur + \ Jo

1 r x + n / V T l \ i i -

— J u _ 1 s i n ( A u ) ^ ( . \ x l (—u)r ~l du +

(—n )r+1 rx+nr + 1 Jo

u 1 sin(Au) du

x r+lSi+ (A, (x + n ) ) ---------------------- + ( ~ UY zsin(Au ) d u +

r + 1 fr'oK i J Jo

Si+(A, (x + n)). (3.17)

r + 1

( —n ) r+1

r + 1

44

Puttingr x + n

I i = (—u)1 sin(Au) du,J o

it follows easily that

h — —A-1 c o s (A (x + n ))+ A _1, I x = A_1(:r-l-n) cos(A(x-fn)) — A-2 s in(A (x+n))

and

Ii = ( — l ) z_1A_1(a; + n )% cos(A(x + n)) + (—l ) lA_2z(x + n ) l~l sin(A(x + n))

- l ) I i - 2

for i > 2. Thus

N — lim /j = Li (3.18)n —»oo

for i = 0, 1, 2 , . . . and it follows from equations (3.17) and (3.18) that

N - l i m r Si+(A , ( x - t ) ) f d t = ^ - Y ( r + ) Lr- ix i . (3.19)n - > o o J - n r + 1 “ J \ i J

Further, with K = sup{| Si+ (A, x) | : Xx > 0}, we have

[ Tn(t) Si+ (A, (x — t ) ) t r dt < K ( n + n~n)rn~Tl J — n —n ~ n

and it follows that

lim [ Tn(t) Si+ (A, (x — t ) ) t r dt = 0. (3.20)Tl—t OO J _ n _ n — n

Equation (3.13) now follows from equations (3.16), (3.19) and (3.20). Equa­

tions (3.14) and (3.15) follow immediately.

C O R O L L A R Y 3 .1 If A / 0, then

/ _ i \ r + l r / , i \

x C © S i + (A,x) = ■■■■;— £ U L r - i X 1 - br- i (x , A)xV] +r + 1 ,=0 V I

/ _ 1 Ar +1V ’ x r Si+ (A, x) (3.21)

r + 1

45

for r = 0, 1, 2 , . . . . In particular,

H ( —x ) ® Si+ (A,x) = — x) — A-1 cos+ (Ax) — x Si+(A, x), (3.22)

X- © Si_|_(A, x) = |[A _1x cos+ (Ax) + A-2 sin+(Ax)] — A_1x_ +

+ | x 2 Si+(A, x). (3.23)

P R O O F . We have

(—l ) rx r_ © Si+(A, x) = x r © Si+(A, x) — x+ * Si+(A, x)

and equation (3.21) follows from equations (3.9) and (3.13). Equations (3.22)

and (3.23) follow immediately.

C O R O L L A R Y 3 .2 If A / 0, then

x r © [sin(Ax)x~1] = - ^ 2 ( r ^\Lr- i - i x l (3.24)*=o Vv

fo r r — 0, 1, 2 , . . . . In particular,

1 ® (sin(Ax)x^1] = 0, (3.25)

x © (sin(Ax)x^1] = —A- 1 . (3.26)

P R O O F . Using equations (3.7), (2.43) and (3.13), we have

[xr © Si+(A, x)]' = x r © [sir^AxJx^1]

= - Er +1 ,t s V 1 J l=0 wr + l f i

giving equation (3.24). Equations (3.25) and (3.26) follow immediately.

C O R O L L A R Y 3 .3 If X + 0, then

x r_ ® [sin(Ax)x“ 1] = ( - l ) r_1 f-1 [ Lr - i - i x l - 6r_i_i(x, A)x*+] +z = 0 V V

+ (—l ) r_1x r Si_j_(A, x) (3.27)

46

fo r r — 0, 1, 2 , . . . . In particular,

H ( —x) © [sin(Ax)x^1] = - Si+ (A, x), (3.28)

x_ © [sin(Ax)x“ 1] = A~1H ( - x ) + A-1 cos+(Arr) + x Si+(A, x). (3.29)

P R O O F . Using equations (2.43), (3.7) and (3.21) we have

[xr_ © Si+(A, x)xf_1 = x r_ © [sin(Ax)x^1]

^ ( r + ^ [ iLr-iX1* 1 - ( - 1 ) 7’"* s in ( A x ) ^ +

—ibr- i (x, A)xY x] +' +

f _ i y + i+ ( —l ) r+1x r Si+(A, x) H------ -j—j— sin(Ax)x^.

Noting that

E ( r + 1) ( - i ) ' + ( - i r +1 = ( i - i )r+1 = o,t = 0 V 1 J

and equation (3.27) follows. Equations (3.28) and (3.29) follow immediately.

T H E O R E M 3 .2 / / A / 0, then

x r © Si_(A, x) = - j - £ ( T + X) ( - 1 y - X r - i x ' (3.30)r + 1 i= 0 V I )

fo r r — 0, 1, 2 , . . . . In particular

1 ® Si_(A, x) = A"1, (3.31)

x ® S i - ( \ , x ) = X~lx. (3.32)

P R O O F . Replacing A by —A in equation (3.13) gives us

* r © Si+(( —A), x) = - L . t | W

and equation (3.30) follows on replacing x by — x in this equation. Equations

(3.31) and (3.32) follow immediately.

47

C O R O L L A R Y 3 .4 / / A + 0, then

x r © Si(Xx) = 0 (3.33)

fo r r = 0 , 1 , 2 , . . . .

P R O O F . We have

x r © Si(Ax) = x r @ Si_|_(A, x) 4- x r © Si_(A, x)

and equation (3.33) follows from equations (3.13) and (3.30).

C O R O L L A R Y 3 .5 If A ^ 0, then

< © S i _ ( A , x ) = © ^ ^ ( r + 1) [ ( - i ) ’'-*£,r_Ix i - ( - l ) i6r_l (a;,A)x*_] +r 1 2=0 V 1 J

r + 1+ - j -y S i_ (A ,x ) (3.34)

/o r r = 0, 1, 2 , . . . . In particular

H( x ) © Si_(A, x) = A-1 / / ( x ) + A-1 cos_(Ax) + x Si_(A, x), (3.35)

x + © Si_(A, x) = |[A -1 j;cos_(Ax) + A-2 sin_(Ax)] + \ ~ lx + +

+ £ x 2 Si_(A, x). (3.36)

P R O O F . We have

x \ © Si_ (A, x) = x r © Si_ (A, x) — ( — l ) rx r_ * Si_(A, x)

and equation (3.34) follows from equations (3.11) and (3.30). Equations (3.35)

and (3.36) follow immediately.

The following two corollaries follow similarly.

48

C O R O L L A R Y 3 .6 If A ± 0 then

x r+ © Si(Ax) = Z ( r +. X) { { - l y - ' L r - t - 6r- , ( x , A)]x! +

x r+lr + 1

for' r = 0,1, 2 , . . . . In particular

Si(Az) (3.37)

H ( x ) ® S i ( X x ) = A 1 cos(Ax) + x Si(Ax), (3.38)

x + © S i ( A x ) = ^[X~lx cos(Xx) + A~2 sin(Ax) + x 2 Si(Ax)]. (3.39)

C O R O L L A R Y 3 .7 / / A / 0, then

X r_ © Si(Ax) = E fr + !) [ L r - i - br - ( * . A)]x‘ +r + 1 1=0 \ 2 /

/ _ l^r+ix r+1 Si(Ax) (3.40)

r + 1

fo r r — 0 , 1, 2 , . . . . In particular

H ( —x ) ® S i ( X x ) = — A-1 cos(Ax) — x Si(Ax), (3-41)

© Si(Ax) = |[A _12‘cos(Ax) + A-2 sin(Ax) + x 2 Si(Ax)]. (3.42)

C O R O L L A R Y 3 .8 If X ^ 0, then

x T © [s in (A x)#!1] = ]T ( r \ ( — \ ) r~lL r_i_iXl (3.43)i= 0 \ v

fo r r — 0 , 1 , 2 , . . . . In particular

1 © [sir^Aj;)^!1] = 0, (3.44)

x © [sinfAa;)^!1] = —A- 1 . (3.45)

49

P R O O F . Using equations (3.8), (2.43) and (3.30), we have

[xr © Si_(A, x)]' = — x r © [sin(Ax)x_1]

- 1 +r + 1 v *r - 1 / \

= e , ( - r wi= 0 \ v

and equation (3.43) follows. Equations (3.44) and (3.45) follow immediately.

C O R O L L A R Y 3 .9 If X ^ 0, then

x r © [sin(Ax)x_1] = 0 (3.46)

for r = 0, 1, 2 , -----

P R O O F . Equation (3.46) follows immediately on using equations (3.43) and

(3.24).

C O R O L L A R Y 3 .1 0 If X + 0, then

© [sinO*)*:1] = £ ^ ) [ ( - l ) r- L r_t_1x1 + ( - l )V , - i (x ,A )x l_] +

—x r Si_(A, x) (3.47)

for r — 0,1, 2 , . . . . In particular

H( x ) © [sin(Ax)x,I 1] = — Si_(A, x), (3.48)

x + © [sin(Ax)xI1] = - \ ~ l [H(x) + cos_(Ax)] - xS i _ ( A, x ) . (3.49)

P R O O F . Equation (3.47) follows similarly on using equations (2.43) and

(3.34). Equations (3.48) and (3.49) then follow immediately.

50

COROLLARY 3.11 I f A + 0, then

x \ © [sin(Ax)x_1] = - J 2 f r) [br- i - i ( x , A) + ( - l ) r_!Lr_ i_1]a:• + x r Si(Ax)i—0 \ V

(3.50)

fo r r = 0, 1, 2 , . . . . In particular

H( x ) © [sin(Ax)x_1] = Si(Ax), (3.51)

x + © [sin(A:r):r_1] = A-1 cos(Ax) + xSi (Xx) . (3.52)

P R O O F . Equation (3.50) follows similarly from equations (2.43) and (3.37).

Equations (3.51) and (3.52) follow immediately.

The proof of the final corollary follows similarly from equations (2.43) and

(3.40).

C O R O L L A R Y 3 .1 2 If A / 0, then

x r_ © [sin(Aj:)x_1] = ( - l ) r_1 lL r - i - \ ~ K - i - i (x, A)]x' +i=o Vv

- ( - l ) V S i ( A : r ) (3.53)

fo r r = 0,1, 2 , . . . . In particular

H ( —x) © [sin(Ax)x_1] = — Si(Ax),

x_ © [sin(Ax)j;_1] = A-1 cos(Ax) + x Si(Ax).

51

Chapter 4

THE NON-COMMUTATIVE NEUTRIX PRODUCT OF DISTRIBUTIONS

The product of an arbitrary distribution by an ordinary infinitely differentiable

function is defined as follows.

D E F I N I T I O N 4 .1 Let f be a distribution and let g be an infinitely differ­

entiable function. Then the product f g = g f is defined by

( f g , T ) = ( g f , v ) = (f,9<p)

for all (p in T>.

It then follows easily by induction that

i = 0 W

for r — 1 , 2 , . . . .

This suggests the following extension of Definition 4.1, see for example [11].

52

D E F I N I T I O N 4 .2 Let f be the r- th derivat ive of a locally summable func­

tion F in L p(a,b) and let g ^ be a locally summable funct ion in L q(a,b) with

1 / p + 1 /q = 1. Then the product f g = g f on the interval (a, b) is defined by

z = 0 W

It should be noted that the product is not in general associative, see

Schwartz [32].

For our next definition we let p(x) be a fixed infinitely differentiable func­

tion in T> having the following properties:

(i) p(x) = 0 for |x| > 1,

(ii) p(x) > 0,

(iii) p(x) = p ( - x ) ,

(iv) J p(x) dx — 1.

The function 5n is then defined by 5n(x) = np(nx) for n = 1 , 2 , . . . . It

follows that {5n} is a regular sequence of infinitely differentiable functions

converging to the Dirac delta-function

Now let / be an arbitrary distribution in V and define the function f n by

fn(x) = f * Sn(x) = (f ( t ) , 5 n(x - t))

for n = 1 , 2 , . . . . It follows that { / n} is a regular sequence of infinitely differ­

entiable functions converging to the distribution / .

Fisher [11] generalized Definition 4.1 as follows:

D E F I N I T I O N 4 .3 Let f and g be arbitrary distributions in V and let

fn{x) = ( / * <Sn)(z), 9n = (g * $n)(x).

53

We say that the product f • g of f and g exist and is equal to the distribution

h on the interval (a, b) if { f ngn} is a regular sequence converging to h on the

open interval (a, b).

The product which is defined above is clearly commutative if it exists. Later

Fisher [13] gave the following non-commutative definition of the product:

D E F I N I T I O N 4 .4 Let f , g be distributions in V and gn(x) = (g * 6n)(x).

We say that the product f • g of f and g exists and is equal to the distribution

h on the interval (a, b) if

) h ^ { } ( x ) g n{x),<p(x)) = {h(x) , tp(x))

for all in V with support contained in the interval (a, b).

It was proved in [13] that if f g exists by Definition 4.2 then it exists by

Definition 4.4 and f g = f - g -

Definition 4.4 was generalized by Fisher [14] with the following.

D E F I N I T I O N 4 .5 Let f and g be distributions i n V and gn(x) = (g*(5n)(x).

We say that the neutrix product f o g of f and g exists and is equal to the

distribut ion h on the interval (a, b) if

N - l i m ( f ( x ) g n(x),(p(x)) = (h(x) , (p(x))T l-^ O O

for all funct ions <p in T> with support contained in the interval (a,b), where N

is the neutrix, see van der Corput [5], having domain N' = {1, 2 , . . ., n , . . .}

and range the real numbers, with negligible functions finite linear sums of the

funct ions

n x \nr~l n, In r n : A > 0, r = l , 2 , . . .

and all funct ions which converge to zero in the normal sense as n tends to

infinity.

54

This definition of the neutrix product is in general non-commutative. It is

obvious that if the product / • g exists then the neutrix product f o g exists

and / • g = f o g. The next theorem was proved in [13].

T H E O R E M 4 .1 Let f and g be distributions and suppose that the neutrix

products f o g and f o g ' exist on the interval (a , b ). Then the neutrix product

f o g exists and

( f ° g)' = f ° 9 + / ° g'

on the interval (a, b).

The following theorem was proved in [23].

T H E O R E M 4 .2 Let f and g be distributions in V and suppose that the neu­

trix product f ogb) (or f ( l) 0 gJ exists on the interval (a, b) for z = 0 , 1, 2 , . . . , r.

Then the neutrix product f o g (or f o g exists on the interval (a, b) and

orr /

r ] Vv_ ;v r f(d n „l(r-*)t

on the interval (a, 6).

The next theorem was proved in [18].

T H E O R E M 4 .3 The neutrix product X+ 1//2 o x f 1//2 exists and

x rf l/1 o x f ~ 1/2 = x + l + arS(x) (4.1)

for r — 0, ± 1 , ± 2 , . . . , where

a0 = 2[ln 2 — c(p)],

ar = a _ r = 2 In 2 - c(p) -1

. . 2z - 1i=i

55

for r = 1 , 2 , . . . and•I

c (p) = / In t p ( t )d t .Jo

The next theorem is a generalization of Theorem 4.3 and was proved in [6].

T H E O R E M 4 .4 The neutrix product o x “A_1 exists and

o xf_x~l — x + l + ai(A)£(x) (4-2)

for A ^ 0, ± 1 , ± 2 , . . . , where

a i(A) = A + 1) + 4 0, — A) — 2ci(p)

= “ [7 + | ^ ( - A ) + |^ ( A + 1) + 2c(p)], (4.3)

where

■ mand 7 denotes Euler ’s constant.

We now generalize Theorem 4.4

T H E O R E M 4 .5 The neutrix product x+ o x +x~r exists and

4 O x 7 “r = x ; r + ar(A)<5(r- 1)(z) (4.4)

for r — 1 , 2 , . . . and A ^ O , ±1 , ± 2 , . . where

( - l ) r [7 + 2c(p) + 4'0(A + 1) + - r + 1) - 4>{r - 1)]ar(A) = I )!----------------- +

T(A + 1) rj Z ( \ + r - l \ ( - l ) J2 f(A + r) V j J r - j - 1

56

P R O O F . We first of all suppose that — 1 < A < 0. We require to evaluate

N —lim ( x i ( x +A r )n ,^ (x ) ) ,

where

(£+A_r)n = £+A -r*£n(^)

( / ' ? {X ~ i ) ‘ A_1^ r_1)(«) dt, x > 1 /n ,1 A + r J —i n

— <

r(A + r) J —\ / n

r a + ' V ^ / (x ~ <) ~A~ 1'5nr~ 1)W rff> - i / n < * < ! / « ,i (A + rj J—i / n0, x < —1/n.

We note that since x+ • x + x~r = on any closed interval not containing

the origin, we need only consider ip € V with supp <p C [—1,1]. W ith such a

y?, we have by Taylor’s Theorem

v i x ) = g +7 1z=0

where 0 < £ < 1 and so

( x + ( x+x r )n, ip(x)) = / x x (x+x r)„lp ( x ) d x

= [ ' X X+ '(x~+ X~ r ) n d x +Jo

f x x[xr (x~x~r )n}(p{r](£x)dx . (4.5) *! Jo

i-ol rl

+■rl

We have

( - i r ‘r(A + r) cJ 1 r)»l' dx

r(A +1)

= x x+l [ (x — t )~x~l 5[l~l\ t ) d t d x +Jo J —l / n

+ [ ' x x+i f lln ( x - t ) - x^'5{; - l)( t ) d t d xJ 1 /n J — l / n

57

= J l \ t ) J x x+l(x — t) x l d x d t +

H- [ [ x x+l(x — t )~x~l dx dtJ — l / n J 0

= n r~l~ l [ p(r~l \ v ) [ ux+1(u — v) ~x~l d u d v +Jo Jv

— ( —1 )rn r~l~l [ [ ux+l(u + v ) ~x~1 du dv, (4.6)Jo Jo

where the substitutions nt = v and nx = u have been made in the first integral

and nt = —v and nx — u in the second integral.

We have

[ ux+l(u — u)~A_1 du — ( — l ) r [ ux+l(u + v ) ~x~l du =Jv Jo

= f ux+l[(u — v ) ~ x~l — ( — l ) r(u + ^)_A_1] du +J v

— ( —l ) r [ ux+l(u + u)_A_1 du Jo

and it follows for the cases i = 0 , l , . . . , r — 2 that

N — lim nn—> oo

r —i —l J ux+l(u — v) x 1 du — ( — l ) r J ux+l(u + v) x 1 duj

= N — lim n r_*_1 [ ux+l[(u — v ) ~x~l - ( — l ) r(u + ^)_A_1] du n —> OO J y

°° / \ l \ rn , ,= N — lim n r~l~1 Y ] ( ) [ ( - i y - ( - l ) r]vj ul~i~l du

n —>oo r- V 7 / J v

r — i — 1 \ r — 12F(A + r) r — 1

V(r — i — l ) ( r — 1 ) !T( A + 1 )

It follows that

N — lim [ x x (x+x~r )nx l dx = --------- ;---- (4.7)n —»oo J -1 ^ 'r r — I — 1

for i = 0 , 1 , . . . , r — 2, since it is easily proved by induction that

[ v rp r\ v ) dv = \ { — l ) r r \ .Jo

58

W hen i = r — 1, we have on making the substitution u — v j y

f u x+r~l {u — v ) ~ x~l du = v r~l f y~r ( l — y ) ~x~l dyJv Jv/n

= y T 1 f , 2/_r[(1 - y)~x~l - Y i ( A- *) ( - y ) J\ dv +Jv/n i=o V 3 )

+ « r" 1£ ( - i ) ,' f A . X) P. l r r dy3 = 0 J I J v / n

— V 1 [ y r [(1 - y ) A 1 - Y , ( A - 1 \ - y y ] d y +J v / n j = 0 \ J J>h

+ vt~ x 2 ( - i ) j ( ~ x ~ b +j = 0

/ - a - r

j j - r + 1

r — 1(—v ) r (lnu — Inn).

It follows that

N — lim / ux+r 1(u — v) x 1 du = v r 1B ( —r + 1, — A) +n —► oo J v

- X - F

r — 1(—v ) r Mnu

see Fisher and Kuribayashi [19], and so

N —lim f p(r ^(u) [ ux+r l (u — v) x l d u d v = n - » o o Jo Jy

- A - 1r

^ ( r - 1 )![|?>(r - 1) + c(p)\

s ( —1) (r — l ) \ B ( —r + 1, —A) +

(—1)T (A + r) r(A + i) 12

since it is easily proved by induction that

</>(r - 1) + c ( p ) \ , (4.8)

[ v r In vp^r\ v ) dv = i ( —1 )rr\(p(r) + (—l ) rr!c(p). Jo

59

Further, making the substitution u = v ( y 1 — 1), we have

l n u » r- ' ( u + v ) - ' - ' d u = v r~l [ y ~ r {\ - y ) x+T- ' d yJO Jv/fa+v)

'l - r f / , i / A + r — 1

r —

= V ' [ t f- ' [(i -i / )A+f- l - E A + r ) +Jv / (n+v) j = 0 \ J J

+ « r" 1 2 ( - l y ( X + r ~ l ) f 1 y j ~r d y \ J J J v / { n + v )

1 / ' , 2 / 'r [(1 - V ) * * - 1 - £ { X + r “ l ) ( - y ) J] d y +Jv/ (n+v) j=Q \ J J

+ < /- ! £ ( - 1 ) ' ^ + T ~ ' ) 1 ~ (n ! V + 1)r~ '~ 1 +h ’ \ j J j - r + 1

~ ( X ^ r j ^ ( - u ) r“ 1[lnu - ln(n + u)].

It follows that

N —lim [ ux+r 1(u + v) x 1 du = v r 1B ( —r + 1, A + r) +n —>00 Jo

and so

Nn —►00

lim [ p r ^(u) [ ux+r 1(u + v) x 1 d u d v = 00 Jo Jo

= l ) ! B ( - r + l ,A + r) +

- ( A r - 1 ! ) (r ~ ^ r ~ ^ +

= \ ( — l ) r~ l (r - 1 ) ! £ ( —r + 1, A + r) +

r (A + r ) f| 0 ( r - l ) + c(p)], (4.9)F(A + 1)

60

since it was proved in [19] that

£ ( 0 , / i ) = - 7 - (4.10)

for /i / 0, ± 1 , ± 2 , . . . . It follows that

B ( —r + 1, A + r) = * [] W - 1) - 7 - V>(A + 1)],

B ( - r + l , - A ) = (f J [ (_ ) }+ 1} W ~ 1) ~ 7 ~ ^ (~A - r + 1)]

+ [</>(r - 1) - 7 — i/'(—A - r + 1)](r — 1)!T(A + 1)

and so

B ( —r + 1 , —A) — ( —l ) r£?(—r + 1, A + r) =

= (r J i ^ r ( X + 1) ^ r M X + “ r + 1)]- (4 -n )

It now follows from equations (4.6), (4.8), (4.9) and (4.11) that

N - l i m ^ ^ | f x x (x+x r )nx r l dx = n-> oo 1 A + 1 J - 1v_ l i m E A ± E f 1n—too r(A - | -1) J — ]

— \ ( r - 1)![J3(—r + 1, - A) - ( - 1 )vB ( - r + 1, A + r)] +

- | ( r - i ) ! E f A + r _ A +n I r o I3 = 0 \

T(A + r)r(A + i)

( — l ) r_1 (r — l)!r(A + r)

[<f>(r- l ) + 2c(p)]

br(X), (4.12)

where

r(A +1 )

l \ \ _ ( - l ) r [7 + M p ) + ^ ( A + 1) + I Ip{ -X - r + 1)] f MA) - yjj +

T(A + 1) g / ' A + r - l ^ ( - I p '2T(A + r) “ J \ j J r - j - 1

61

It was proved in [12] that

r oo{ x ~ r , i p ( x ) ) = X

JoV (X) - g ^ P - x ' - - x ) d x +

t = 0 V. (r - 1)!

(f)(r - 1)■ip( r -

l ) (0)(■r - 1)!

for all ip in V . In particular, if the support of <p is contained in the interval

[—1,1], we have

r_1 v?(z)(0){ x + r , < f ( x ) ) = j x r [ < p ( x ) ~ Y ,i= 0

■xl\ z=0 I x r dx +

0)

r - 1 , J i= f X r \<p(x) - g

J ° 1 i —0

VW(0)■x

(r - 1)!

f z 1)

</’(>' ~ 1) (r - 1)!

(r- ^(O). (4.13)

Since the sequence of continuous functions { x r (x+A r)n} converges distri-

butionally to x ~x on the closed interval [0,1], it follows on using equations

(4.7), (4.12) and (4.13) that

N - l im ( j : A (x~x~r)n, p (x ) ) = N - l i i n ] T ^ f x++l (x~x~r)n dx +u —>oo n —> oo r “ T ?! 7 — 1

z = ( J

+ lim — f x A[xr(x+A r )n](p{r)( ^x)dxi - * o o 71! </ _ 1 J1 rl r — 2- <p{r)( ( , x ) d x - Y ,• JO ■ — n

¥>«(<))r'. Jo ;rj i!(r - i - 1)

+ ( - l ) r- 16r(A)V(r- 1)(0)

+

f - r r / \ V (°) l ^ J s r/ % ip(x) - 2 2 — n— x J dx ~ z 2J0 i = 0 *’ i=0

^W(O)

i!(r - I - 1) +

(r — 1)! r v ’ (r — 1)!

+ ( - l ) ’- 16r(A)v?(r- 1)(0)

(xTr,^(x-)> + ( - i r 1ar( A ) ^ - 1)(0),

62

giving equation (4.4) on the interval [—1,1] and hence on the real line when

- 1 < A < 0.

Now suppose that (4.4) holds when —k < A < —k + 1 and r = 1, 2 , . . . , for

some positive integer k. This is true for k = 1. Differentiating equation (4.4)

with —k < A < —k + 1, we get

Ax^-1 o x + x~r — (A + r ) x \ o x~tX~r~l = — r x + r~1 + ar(A)tf^(a;).

It follows from our assumptions that

Xx+_1 o x~x~r = Xx~r~1 + [ (A + r )a r+i(X) + a r ( A ) ] 5 ( r ) ( x ) .

We have

(A + 7')flr-(-i (A) + nr( A) =

= ^ r. [A ~ r ^ T' + 5 “ r ) + \ i>(* + ! ) + 2c(p)] +

r (A + 1) ^ l / A + r\ ( - i ) j (_ l ) r (A + r)0(r)2r(A + r) \ j ) r - j r!

+ 7 7 ^ 1 ) ! [7 + 5 - r + 1) + i ^ (A + 1) + 2c(p) \ +

, r(A + l) Tj i ( \ + r - q (-1)-1 (—\ ) r<j>(r — 1)2F(A + r) \ j J r - j - 1 (r - 1)!

Noting that

(A + r ) ^ ( - A - r) = 1 + (A + r)ip( — \ - r + 1),

A?/>(A + 1) = 1 + A,0(A),

^ /A + r - l \ ( - i y = /A + r \ ( - l ) ' f V,2 /A + r - l \ ( - 1 ) '

V / r ~ jVo w ' / r - i V j ) r - j ~ i ’

it follows that

(A 4- r )a r+i(A) T &r (A) = Aflr_).i(A — 1)

and we see that equation (4.4) holds when —k — 1 < A < —k.

63

Equation (4.4) therefore holds by induction for negative A / — 1, —2, . . .

and r = 1, 2 , . . . . A similar argument shows that equation (4.4) holds for

positive A ^ 1, 2 . . . . This completes the proof of the theorem.

C O R O L L A R Y 4 .1 For A 7 0, ± 1 , ± 2 , . . . we have

x x_ o xZX~r = xZr - { - l ) rar (X)S^r- 1)(x). (4.14)

P R O O F . Equation (4.14) follows immediately on replacing x by —x in equa­

tion (4.4).

In the next corollary, the distribution (x + z0)A is defined by

(x + z0)A = x A + ezA7T A

for A / 0, ± 1 , ± 2, . . . and

(x + i0)~r = x~T + i ~ 1)r,” A (r~ 1)Or) (4 -15)(r - 1)!

for r = 1 , 2 , . . . , see [25].

C O R O L L A R Y 4 .2 For A / 0, ± 1 , ± 2 , . . . and r — 1 , 2 , . . . we have

(x + z0)A o (x + z0)_A_r = (x + z0)"r. (4.16)

P R O O F . The neutrix product is distributive with respect to addition and so

(x + z0)A o (x + z0)~A~r = £ A O x - x~r + ( - l ) rx A O xZX~r +

+ ( - l ) re~lX7Tx x+ o xZX~r + elX*xx_ o x ~x~r. (4.17)

Further, it was proved in [12] that

XX+ o X~_X~T = ( - 1 )r- ' x i O x ~ X~r = _ ^ | ^ M 5(r-D (:c) (4-18)

64

for A ^ 0, ± 1 , ± 2 , . . . . It follows from equations (4.4), (4.14), (4.15), (4.17)

and (4.18) that

(x + i0)x O (x + i 0 ) - x~r = x ~ r + = (x + i0)~r,

proving equation (4.16).

We finally note that the following results can be proved similarly.

|x |A o (sgn x\x\~x~2r+l) = x ~2r+l, (4-19)

(sgnx|;r|A) o |x |~ A_2r+1 = x~2r+1, (4.20)

|x |A o |x |_A~2r = x ~2r, (4.21)

( s g n x |x |A) o (sgn x\x\~x~2r) = x~2r (4.22)

for A / 0, ± 1 , ± 2 , . . . and r = 1 , 2 , . . . .

To prove the next theorem, we need to prove the following three equations.

J3(0, A) = £(0 , A + 1) + A-1 , (4.23)

-£?i,o(0, A) = 5 i to(0, A 1) + A 1 -0(0, A + 1), (4.24)

£o,i(0,A) = £ 01( 0 , A + 1 ) - A - 2 (4.25)

for A / 0 , ± l , ± 2 , . . . .

To prove (4.23), we note that

t ~ l ( l - t ) x~ l = r ‘ ( 1 - t ) x + (1 - *)A_1

and it follows that

r l —l / n£ ( 0 , A) = N —lim / r ' ( l - f)A_1 </tn - > o c J i / n

rl — 1 / n rl — l / n= N —lim / r ' ( l - t ) x dt + N - l i m / (1 - t ) x~ l dt

rl—>00 J \ / n n —>oc J l / n

= B(Q, A + 1) + A *,

65

proving equation (4.23).

Similarly, we have

■1 —1 / n£ i )0(0, A) = N — lim [ t M n ^ l — t ) x 1 dt

n - > oo J i / nr l — 1 / n rl — 1 / n

= N —lim / t ~ l l n t ( l — t ) x dt + N — lim / l n t ( l — t )A_1n —>00 7 i / n n - > o o 7 l / n

= Bi o(0, A + 1) — N — lim A-1 \ n t ( l — t ) x’ n —►oo L

+ N —lim A-1 f 1 1,H t ~ l ( \ - t ) x dtn —> oo 7 i / n

1 —1 / n 1 / n

1 / n

= 5 i o(0 5 A + 1) + A 1R(0, A + 1),

proving equation (4.24). Finally, we have

•1 —1 / n£?oi(0,A) = N — lim [ t 1( l — t ) x 1 ln ( l — t) dt

’ n —>oo J i f n

= N — lim [ t - 1 ( l — t ) x ln ( l — t) dt +n —>oo 7 i /n

+ N —l i m / (1 — t )A_1 ln ( l — t) dtn —>oo 7 i/n

— B 0 i(0, A + 1) — N — lim A 1 (1 — t ) x ln ( l — t)’ n —> oo

1 —1 / n 1 / n

/■I — 1 / nN —l i mA-1 / (1 — t ) x~l dt

J 1 / nn —>oo 7 l / n- 2— B 0>1 (0, A + 1) — A

proving equation (4.25).

We now prove the following theorem.

T H E O R E M 4 .6 For A / 0, ± 1 , ± 2 , . . we have

( x | ln x + ) o x ~ A_1 = x + l lnx+ + a( \ ) 5 ( x ) , (4.26)

= x +x 1 o ( x i l nx+) , (4.27)

where

a (A) = | [ # 0,1 (0> A 4- 1) — ^ i,o (0, A + 1) — R i;o(0, — A)] +

+ ci(p)[B(®i ~ ~ + ^(0) A + 1)] — C2{p)

66

and

Ci{p) = [ \ nup( u ) du , c2(p) = [ In2up(u) du. Jo Jo

P R O O F . The proof of this theorem follows the same general strategy as that

of Theorem 4.5. We first of all suppose that — 1 < A < 0 and put

(x +A x)n = x +x ! *(5n(x)

f (x — t )~x~1Sn(t) dt, x > 1 /n ,J — 1/nf (x — t )~x~l Sn(t) dt, — 1 / n < x < 1 /n ,

J — l / n

0, x < —1/n.

Then

•1/n/ i />i/ n rxx . In x + {x~x~1)n dx = / x x \ n x {x — t )~x~l 5n(t) dt dx +

-1 Jo J — l / n

+ [ x x \ nx [ [x — t )~x~l 5n[t) d t d x J l / n J — l / n

= J 8n(t) J x x In x(x — t )~x~l dx dt +

+ / (t) [ x x In x(x — t )~x~1 dx dtJ —l / n Jo

r 1 rn= / p(v ) / ux \nu{u — v ) ~x~l d u d v +

Jo Jv

— Inn / p(v) / ux(u — v ) ~x~l du dv +Jo Jv

+ f p(v) [ ux In u(u + x ) _A_1 du dv +Jo Jo

— Inn [ p(v) [ ux(u + x ) _A_1 du dv, (4.28)Jo Jo

where the substitutions nt — v and nx = u have been made in the first integral

and nt = —v and nx = u in the second integral.

Making the substitution u = v / y , we have

rn r l/ ux In u{u — v) ~x~1 du = In v / y ~ l (l — y)~ ~l dy +

Jv J v / n

- [ y ~ l In y ( l - 2/)~A-1 dyJ v / n

’ v / n

I' v / n

67

= In v f y 1 [(1 — y) A 1 — 1] dy — In v( \n v — Inn) +Jv / n

- f y ~ l ln y [ ( l - y ) ~ x~l - 1] dy + | ( l n u - ln n ) ;Jv/ n

and it follows that

N77—7 OO

rn r 1— lim / u \ nu( u — v ) ~x~l du = \ nv I y ~ l [(l — y ) -A_1 — 1] dy +i—»oo . / t; 70

- In2 v - j y ~ l In y[( 1 - y )_A-1 - 1] dy + \ In2 vJ 0

= B ( 0, —A) In v — £i,o(0, —A) — \ In2 v. (4.29)

Next, we have

r 7i ° ° / — \ — l \ rnJ u (u — v)~ ~l du = Y2 f u ~ l l( u

7 = 0 OO

= Y J-——(t;_z — n~l) + In n — In v

and it follows that

N — lim Inn [ ux(u — v) ~x~l du = 0. (4.30)77—7 0 0 Jv

Further, making the substitution u = v ( y ~ l — 1), we have

dur n

/ ux In u(u + v)~ -1 J o

= \ n v [ y ~ l { l - y ) x d y - f y ~ l In y ( l - y ) x dy +J v / ( n + v ) J v / ( n + v )

+ f y ~ l { 1 - y ) X ln (l - y ) d yJ v / { n - \ - v )

r 1= In i; / 2/- 1 [(l ~ y ) X ~ 1] dy — lnv[ln v — ln(n + u)] +

J v / ( n + v )

— f y ~ l In y[( 1 — y ) x — 1] dy + |[ ln v — ln(n + v)}2 +J v / ( n + v )

+ f y ~ l ( i ~ y ) x ln ( i - y ) d yJ v / ( n + v )

68

Nn —Kx>

and it follows that

rn Nhm J ux In u(u + v)~x~l du = In v J y - 1 [(l — y ) x — 1] dy — In2 v +

- V~l In y[( 1 - y )x - 1] dy + \ In2 v +

+ / 2/_1(l — 2/)A ln (l — y) dy J o

= B (0 , A + 1) In w — F?io(0, A + 1) +

+-£?o,i(0> A + 1) — | In2 v. (4.31)

Finally, with n > v, we have

PTX OO / \ 1 \ p i )

J UX(vl + v)~X~l du = 53 f j v ~X t l J VLX+ldu-\-

~ y i'

~ / - A - l W , ,+ 2_j^ J — (v — n ) + Inn — Inn

and it follows that

N —lim Inn [ ux (u + v ) ~x~l du = 0 . (4.32)n -> 00 L Jo

It now follows from equations (4.28), (4.29), (4.30), (4.31) and (4.32) that

N —lim / x + \ n x + (xZx~1)n dx =T l—tO O 7 — 1

— 2 ^ 0 ,i(0, A + 1) — 1 ^ 0 (0 , A + 1) — 5 ^0 (0 , —A)] +

+ ci(p)[F?(0, — A) -F B(0, A + 1)] — C2 (/?)

- a(A). (4.33)

Now let (p be an arbitrary function in V with support contained in the

interval [—1,1]. By the Mean Value Theorem

ip(x) = (p(0) + xcp’f a ) ,

69

where 0 < £ < 1 and so

r 1(x+ \n x + (x~x~l )n,ip(x)) = / x x \ n x ( x l x~l )ni f ( x ) dx

Jo

= (/?(0) J x A Inx ( x+x~l )n dx + / x x \nx [ x ( x+x~1)n](p'(£x) dx.

Since the sequence of continuous functions { x (x + A_1)n} converges distribu-

tionally to x ~ x on the closed interval [0,1], it follows on using equation (4.33)

that

N — l im (x i In x + (xTA-1)n, tp(x )) — N —limy?(0) [ x x ln x (x T A_1)n dx +n —> oo n —KX) Jo

+ lim [ x x \ nx \ x ( xZX~1)n]ip'(£x ) dxn —>oo Jo 1 '

= a(X)ip(0) + [ Inx(p'(t;x)dx Jo

= a(X)ip(0) H- [ x ~ l \nx[(p(x) — ip(Q)}dx Jo

= a(X)(p(0) + (x^1 ln x + , (p(x)),

giving equation (4.26) on the interval [—1,1] and hence on the real line when

— 1 < A < 0 and r = 1 , 2 , . . . , .

Now suppose that equation (4.26) holds when —k < X < —k + l . I f —h — 1 <

X < —k, then the product (xA+1 ln x + )x^A_1 exists by Definition 4.2 and

(xA+1 ln x + )x^A_1 = In x + .

By Theorem 4.1 we can differentiate this equation to get

[(A + l ) x A l n z + + x A] o a;~A_1 - (A + 1)(£++1 l nx+) o £ ~ A = x + l

and it follows from our assumption and equation (4.2) that

(A + 1)(xa ln x + )o x +A 1 = ( A + l ) : ^ 1 In:r+ + [ (A + l)a (A + l ) —ai(A)]<5(:r). (4.34)

70

Now

(A + l)a(A “I- 1) — ai(A) —

= | (A + l ) [ 5 O)1(0, A + 2) — 5 1|0(0, A + 2) — 5 i ;o(0, —A — 1)] +

+ (A + l ) c i (p) [5(0, —A — 1) + 5 ( 0, A + 2)] +

— (A + l )c2 (/?) + 2c\(p) — | [ 5 ( 0 , —A) + 5 ( 0 , A + 1)]

= |(A H- l ) [ 5 0)i(0, A + 1) — 5 i )0(0, A + 1) — 5 i )0(0, —A)] +

+(A + l )c i (p )[5 (0 , —A) + 5 ( 0 , A + 1)] — (A + l ) c 2(p)

= (A + l)a(A),

on using equations (4.3), (4.23), (4.24) and (4.25). Equation (4.26) follows by

induction for negative A ^ — 1, —2 , . . . .

A similar proof shows that equation (4.26) holds for positive A ^ 1, 2 , ___

To prove equation (4.27), we differentiate equation (4.2) partially with

respect to A. This gives

x+ o (x^A_1 In x+) = (xA ln x + ) o x^A_1 — a/1(A)5(x)

= x + l ln x + -f ^[50 ,1(0, —A) — 5 i ;0(0, A + 1) — 5 ^0(0, —A)]#(a;)

+ c i(p ) [5 (0 , -A ) + 5 (0 , A + l)]£(x) _ c2{p)$(x), (4.35)

on using equations (4.3) and (4.26). Replacing A by —A — 1 in equation (4.35)

gives equation (4.27). This completes the proof of the theorem.

C O R O L L A R Y 4 .3 For A / 0, ± 1 , ± 2 , . . . we have

(.xx__ l nx_) o x I A_1 = x I A_1 o x A l nx_ = xZ1 l nx_ + a(A)5(x). (4.36)

71

P R O O F . Equation (4.36) follows immediately on replacing x by — x in equa­

tions (4.26) and (4.27).

To prove the next theorem we need the following lemmas which are easily

proved.

L E M M A 4 .1 . For r = 0,1, 2 , . . . we have

L E M M A 4 .3 Let p be in V with support contained in [—1,1]. Then

L E M M A 4 .2 For k = 0,1, ±2 , ± 3 , . . . and p = 1 , 2 , . . . we have

(k + 1 )l+1(p — i)!

(x 2r+l \np \x\, p (x ) ) = f x —2r+ l r 2 r ~ 2 (Z?h)(Cn ilnp |x| p(x) — ^2 — — x l dx +

i = 0

y>(2i+1)(0)p!

(2i + l)!(2r — 2i — 3)p+1 ’

2 v — v?(20(0)p!

t o (2i)!(2r - 2 i - l f +1

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

The following neutrix products were proved in [9].

for A ^ 0, dzl, z t 2 , . . . and r — 1 , 2 , ___

We now give the following generalization of these equations.

T H E O R E M 4 .7 For A / 0, ± 1 , ± 2 , . . . , r = 1 , 2 , . . . and p, q = 0, 1, 2 , . . . we

have

( |x |A lnp |x|) o (sgn x |i; |_A_2r+1 In9 |x|) = x -2r+1lnp+9 |x|, (4.41)

(sgn x\x \x lnp |x|) o ( |x |-A_2r'f l In9 \x\) = x~2r+1lnp+q |x|, (4.42)

(|a;|A lnp |x|) o (|a;|~A~2r In9 \x\) = x~2r\np+q |x|, (4.43)

(sgn x|a:|A lnp \x\) o (sgn x|a:|_A_2r In9 |x|) = x _2rlnp+9 |x|. (4.44)

P R O O F . We first of all prove equations (4.41) to (4.44) for the case q = 0

and suppose that —1 < A — m < 0 for some non-negative integer m, and put

(z + A-r)n = X~X~T * Sn(x)

+ Ar ,m~ / Vn " <)“A+m'5ir+m)(«) dt, X > 1/n,1 (A + r) J —i / n

( - i : f ( " - < ) - > * " « “ > « ) * M < l / » ! 4 -4511 (A + r) J - i / n

0, x < —1/n.

Then,

( - i r T ( A i r ) F . |Al _. _A_r.

r ( A - m ) |X| ln |X|(X+ ^ dX =

= f 1/n x x+i f X (x - t )~x+mlapxS^+m\ t ) dt dx +Jo J — l / n

•1 /n+ [ ' x x+> ( ' " ( x - t ) - x+mlnpx S l r+m'>{t)dtdx +J 1 /n J — 1/n

+ ( _ ! y f ° \x \x+i T ( x - t y x+mlnp\ x \ 6 ^ +m)( t ) d t d xJ —l / n J — l / n

= J S ^ +m\ t ) J x x+,( x — t ) ~ X+mlnpx d x dt ++ f S{nr+m)(t) j ' x x+,( x - t y x + m \np x d x d t +

■1/n

73

+ [ e +m)« f x x+1(x — t) A+mlnp x d x d t +J — \ / n JO

+ ( - l / 8%+m){t) J" \x\x+i{x - t )~x+m lnp \x\ d x d t— I/7

= I\ + I2 H- 3 + ( —1)^4- (4.46)

Making the substitutions nt = v and nx = u in the first term of (4.46), we

obtain

Ii = n r~l~l [ p(r+m\ v ) [ uA+i(ln u — In n)p(u — v) ~X+m du dv Jo Jv

= E E (fe) ( _ A / m ) ( - 1 )p~k~j n r- i- 1 \np~h n x

x [ Uj p(r+m)(u) r um+i- n n h u d u d v .Jo Jv

It follows from Lemmas 4.1 and 4.2 that

N — lira 11 = ( ~ 1)r+m+"P! + / ' h r+mp(r+m,W ^n->00 (1 — r + l )p+1 \ r + m J Jo

(r + m)\p\ / - A + m \ <4 _2 (r — z — l ) p+1 \ r + m J

for z = 0, 1, 2 , . . . , r — 2.

Similarly, replacing t by —t in I2l and making the above substitutions, we

have

I2 — { — \ ) r+mn r~l~ l j p <r+rn\ v ) J tzA+*(ln u — In n)p(u 4- z;)~A+m du dv

and it follows that

N - l h n / 2 = - 0 / r + m)^ +T f - A + m ) (4.48)n —>00 2(r — z — l )p+1 \ r + m J

for i — 0,1, 2 , . . . , r — 2.

Further, with the above substitutions, we obtain

• 1 /n _f_ /-Z

'0/3 = ( —1 y +m J 5l[+m>>(t) J x x+l(x + t) x+m \np x d x d t

= ^ p (r+m) ^ ^ + * ^ + ^ ) - A + m l n P ( w / n ) f a f a

74

and it follows im m ediately that

N — lim J3 = 0 (4.49)n—»oo

for i = 0, 1, 2 , . . . , r — 2.

Similarly

N — lim / 4 = 0 (4.50)n—>00

for i — 0, 1, 2 , . . . , r — 2.

It now follows from equations (4.46) to (4.50) that

N — lim [ |x |Alnp|x |(2:TA_r) x l dx = — ------- —— -— -, (4.51)n-> 00 J —l / n ' ' 'V + >n (r — i — 1)P+1 V '

since/ - A + m \ _ ( - l ) r+mr(A + r)\ r + m ) (r 4- m)!T(A — m)

We now note that

N — lim [ |x |Alnp|x | ( s g n x |x r A_2r+1) x l dx =n—>00 J — \

= N — \ i m [ |a:|Alnp|x|[(xTA_2r+1) — (xZX~2r+l)n\xl dxn —>00 7 - 1 71

= N — lim 2 [ |7:|Alnp|x|(a;TA_2r‘f l ) x l dxn-> 00 J —l / n ' ' + , n

- | 4 ' 5 2 )

on using equation (4.51), for i = 1 , 3 , 5 , . . . , 2r — 3, since the integrand is even

and

N — lim [ |x |Alnp|:r|(sgn:z:|:r|-A-2r+1) x 1 dx = 0 (4.53)n—>00 7 — 1

for i = 0, 2 , 4 , . . . , 2r — 2, since the integrand is odd.

Now let ip be an arbitrary function in V with support contained in the

interval [—1,1]. By Taylor’s Theorem, we have

where 0 < f < 1. Then

( |x |Alnp |^|(sgn x |x l_A_2r+1)n, ip(x)) =

= J |j:|A lnp |x |( sg n x |x |_A_2r+1)n(^(x) d x

2r~ 2 £o(*)('n'\ r l = —r]— / \x\x \np \ x \ ( s g n x \ x \ ~ x~2r+1)nx l d x -\-

i=o 1• J - 1

+ ( 2 r ~ T j l / \x \X nP \x \(s^n x \x \~X~2r+1)nx ‘2r~ 1^ 2r~ 1\ ^ x ) ^x -

Since the sequence { |x |Ax 2r~ 1(sg n x |x |_A_2r+1)n} converges to 1 on the interval

[—1,1], we have on using equations (4.52) and (4.53)

N —lim(|a:|A lnp |x|(sgn x\x\~x~2r+1)n, P i 00)) =n—> oo

2r~2 (01 r1= N — lim ^2 — — J x l \x\x lnp |x |(sgnx|a:|_A_2r+1)n dx +

+ n1 (2, ~ 1)! / . , M* lnP M(sgn x\x\~x- 2r+1)nx 2r- l ip{2r- 1)^ x ) dx

= I____ [ 1 inp b l ^ - D ( ^ ) dx _ 2 y P '¥ 2i+1)(0)_____(2r — 1)! 7 - i { i X ) d X — (2r — 2i — 3)p+1(2i + 1)!

f 1 - 2 r + l i p I iT / \= / x lnp |x| ip(x) — V ----- ------ dx +J—i 1 ~L i\ J

o v ? p!^(2l+1)(0)2 ^ O - o \P +lS ( 2 r - 2 z - 3 ) p+1(2z + l)!

= (x “2r+1 lnp |x|, < (a;)).

It follows that

(\x\x lnp |x|) o (sgn j;|x|-A_2r+1) = x~2r+l lnp |a;| (4.54)

for A > —1, A / 0 , 1 , 2 , . . . , r = 1 , 2 , . . . and p = 0,1, 2 , . . . .

We now consider the product (sg n x |x |A lnp |a;|) o |x |_A“2r+1. It follows from

equation (4.46) that

( _1I ..Li :* i t r ) f sgnx \ x \ x\np\x\(x~x~r)nx l dx = I x + I2 + h ~ ( ~ l ) lh— Tfl) J — l / n

76

and so

N —lim / s g n x |x |Alnp|x |(x+A r) x l dxn —>00 J — l / n 71

p \

■i/n n (r — i — 1)p+1 '

Then

N —lim / sgn x|a;|Alnp|x|(|j: |_A_2r+1) x l dxn —>00 J — l

= 2 N —l i m / sgn x\x\X\np\x\(xZx~2r+1) x l dxn 00 J — l / n n

2 p\(2r - i - 2)p+1

for i = 1 , 3 , 5 , . . . , 2r — 3 and

N —lim / sgn x\x\x\np \x\(\x\~x~2r+l)„xl dx = 0n —>00 J — i

for i = 0, 2, 4 , . . . , 2r — 2. Thus

(sgn x\x\x lnp |x|) o |x |_A_2r+1 = x~2r+1 lnp |x| (4.55)

for A > —1, A / 0 , 1 , 2 , r = 1 , 2 , . . . and p = 0,1, 2 , ___

Next, we consider the product (\x\x lnp \x\) o |x |~A~2r. It follows from above

that

(—l ) r+mr(A + r ) ri . |AlT(A — m) 7-1/

and so

/ |x |Alnp|j:|(x+A r)nx l dx = h + I2 + h + ( - l ) * ^ J — l / n

N —lim / |x |Alnp|x |(x , A r) x l dx ^n-> 00 J — l / n n■ i/n 71 (r — i — 1)p+1 '

Then

N —lim / |x |Alnp|a:|(|j;| A 2r) x l dx =n—>00 J — I

= 2 N —l i m / |x|Alnp|x|(xTA_2r) x l dxn-> 00 y _ i / n 71-1/n

2 p \

(2r - i — 1)p+1

77

for i = 0, 2 , 4 , . . . , 2r — 2, since the integrand is even, and

N —lim [ \x\X\np\x\(\x\~x~2r)nx l dx = 0n—► oo J — i

for i = 1, 3, 5 , . . . , 2r — 1, since the integrand is odd. It now follows as above

that

N —lim (|x |A lnp | x | ( | x r A_2r)„) = (x~2r lnp |x|, p ( x ))n—>oo

for arbitrary p in V and so

( |x |A lnp |x|) o \x\~x~2r = x~2r lnp |x| (4.56)

for A > —1, A 7 0,1, 2 , . . . , r — 1, 2 . . . and p = 0 , 1, 2 , . . . .

Finally, it follows similarly that

(sgn x\x\x lnp |x|) o (sgn x\x\~x~2r) = x~2r lnp |x| (4-57)

for A > —1, A / 0,1, 2 , . . . , r = 1, 2, . . . and p — 0 , 1, 2, . . . .

Now suppose that (4.54) to (4.57) hold for — m < A < — m + 1 and p =

0, 1, 2, . . . . This is true when m = 1. Also suppose that

(\x\x ln \x\) o (sgn x|:r|_A_2r+1) = x~2r+l ln |x|, (4.58)

(sgn x|a;|A \nk |x|) o |x |_A_2r+1 = x~2r+1 lnfc |x|, (4.59)

( |x |A \nk |x|) o |x |_A~2r = x~2r \nk \x\, (4.60)

(sgn x\x\x lnfc |x|) o (sgn x\x\~x~2r) = x~ 2r \nk \x\ (4-61)

for — m — 1 < A < —m and some k. This is also true for all m when k = 0.

Then with —m — 1 < A < —ra, we have

(|:c|A+1 \nk+l |x|) o (sgnx\x\~x~2r) = x~2r+1 ln*+1 \x\

for k = 0,1, 2 , . . . .

78

Differentiating this equation we get

[(A + 1) sgn x\x\x ln^-1-1 |x| + (k + 1) s g n x |x |A \nk |x|] o (sgn x\x\~x~2r) +

— (A + 2 r )( |x |A+1 lnfc+1 |x|) o |x |~ A_2r_1

= (—2r + l ) x _2r \nk+1 \x\ + (k + l ) x _2r \nk \x\.

Using our assumptions, Theorem 4.1 and equations (4.56) and (4.61), we see

that the neutrix product (sgn x\x\x lnfc+1 |x|) o (sgn x |x |_A_2r) exists and

(sgn:r|j;|A \nk+1 |x|) o (sgn x\x\~x~2r) = x~ 2r ln/c+1 |x|

for k = 0, 1, 2 , . . . . It follows by induction that equation (4.57) holds for — m —

1 < A < — m and p = 0,1, 2 , . . . . It then follows by induction that (4.57) holds

for A < —1, A / —2, —3 , . . . and p = 0,1, 2 , . . . . Equations (4.54), (4.55) and

(4.56) follow similarly for A < —1, A ^ —2, —3 , . . . , and p — 0,1, 2 , . . . .

Finally, suppose that (4.41) holds for A / 0, ± 1 , ± 2 , . . . , p = 0,1, 2 , . . . and

some q. This is true when q — 0. Then differentiating equation (4.41) partially

with respect to A, we get

(|.t |a lnp+1 \x\) o (sgn x |x |_A_2r+1 In'7 |x|) +

— (\x\X lnp \x\) o (sgn x\x\~X~2r+l ln9+1 |x|) = 0

and so

(\x\X lnp |.t|) o (sgn x\x\~X~2r+1 ln9+1 |x|) =

= (|^r|A lnp+1 \x\) o (sgn x\x\~X~2r+l In9 |x|).

Equation (4.41) now follows by induction for A / 0, ± 1 , ± 2 , . . . and p , q =

0 , 1 , 2 , . . . .

Equations (4.42), (4.43) and (4.44) follow similarly for A ^ 0, ± 1 , ± 2 , . . .

and p , q = 0,1, 2 , . . . . This completes the proof of the theorem.

79

T H E O R E M 4 .8 For A / 0, ± 1 , ± 2 , . . . and r — 1 , 2 , . . . we have

( x i In x + ) O * : * - • = - ^ ^ M [ 2c(p) + i p ( \ + r) - r ' ( l ) ] ^ - 1)(^) (4.62)

where

e(p) = [ In tp(t) dt. Jo

P R O O F . We will first of all suppose that — 1 < A < 0. Then £ + ln :r + and

xZx~l are locally summable functions and

- A - r r (A + 1) / - A - i \ f r - lX " ' = r ( x _ X 1) rT(A + r)

Thus

( * : A- r)„ = * : A- r * sn(x) = £ /n(t - dt

for r = 1, 2 , . . . and so

f j l T i y C X> X+(X- X~r )nX'dx =r 1 / n

— J x x+l \n x j (t — x)~x~l 8^~x\ t ) dt dx

— J J x x+l \ nx ( t — x ) ~x~l dx dt

= f l/U t lS l [ - l )(t) f 1 v x+i \n( t v ) ( l - v ) - x- ] d v d t Jo Jo

= B(A + z + l , —A) J t l In t 8^~x\ t ) dt +

+jBi!o(A + i + 1, — A) J t*8 ^( t ) dt, (4.63)

where the substitution x = t v has been made, B denotes the Beta function

and in generalQP+Q

B v,q{ A,/l) = dpxgq^ B ( \ , [ i ) .

80

Making the substitution n t = y we have

f 1,n^ ~ 1\ t ) d t = [ ' y ' p ^ ( y ) d y , (4.64)JO JO

[ t l In tS^~l\ t ) dt = — nr~l~l \ n n [ y %p^T~l\ y ) dy +Jo Jo

+ n r~l~1 [ y x \nyp( r~1\ y ) dy (4.65) Jo

for i = 0,1, 2 , . . . .

In particular, when z = r — 1, it is easily proved by induction that

[ y r~1P(r~l){ y ) d y = ± ( - l ) r_1(r - 1)!, (4.66)J 0

[ y r~1\ n y p {r~1)( y ) d y = ( - l ) r_1(r - l ) ! [§ 0 (r - 1) + c(p)\ (4.67)J o

for r = 1 , 2 , . . . .

Further, putting

K = - J r ^ ) s'tp{ lpir ' 1 ) { x ) l } > 0 '

we obtain

| ( x I A_r)n| = ■- f n x+l (u — n x )_A_1nr_1^ r~ 1Hu) dur(A + r) Jnx

r 1 ~\~Tix< — AK n x+r / (u — nx)~x~l du

J nx

= K n x+r

and so when i = r, we obtain

/ oo r l / nx i l n x + ( x l ~r )nx r dx < \x Inx(xZX~r )nx r \ dx

-oo Jo< K n ~ l \nn. (4.68)

Now let p be an arbitrary function in V. Then by Taylor’s Theorem, we

have

<p(x ) = +

81

where 0 < f < 1 and so

r_1 ^ ) ( o )( x + \ n x + , ( x _ x r)n(f(x)) = ^ 2 ^ —.— [ x + \ n x + ( x _ x r )nx l dx +

i = 0 11 J - o o1 r°°

H— - / x A In x + (xZX~r)nXT(P {£x)dx . (4.69)T ! J — oo

Since

/ o °x \ Inx +( x Zx~r)nx r dx < sup{\ip^r\ x ) \ } K n ~ 1 Inn,

- o o X

it follows from equations (4.63) to (4.69) that

N —lim ^ /x | i x (x A ln x + , (x_x r)n(p(x)} = r(A + r ) ,„ A . , _ _ A_r ,n —> oo T ( A + 1)

= ( - l ) r_1B(A + r, -A)[i 4>(r - 1) + c(p)](/j(r_1)(0) +

+ K - i r ' f M A + r, -A)</9(r_1)(0). (4.70)

U sing the well-known result (see e.g. [1, Equation 6.3.1])

H r )(r - 1)!

it follows that

W r - l ) + r ' ( l ) , (4.71)

r(A + 1 |B,,.<A + r . - A ) - r(A + , ) r ( - A , ' r '(A + ’') r 'WT(A + r) ’ ’ (r — 1)! Lr(A + r) (r — 1)!

7r cosec(7rA)(r — 1)!

since (see e.g. [1, Equation 6.1.17])

.ty(A + r ) - 0 ( r - l ) - r ' ( l ) ] > (4.72)

r ( —A)T(A 4- 1) = — 7r cosec(7rA).

Further,

r(A + l) , r(A + l ) r ( -A ) 7rcosec(7rA)r(A + r) } (r — 1)! (r - 1)! ’ j

82

and equation (4.62) now follows from equations (4.70), (4.72) and (4.73) for

the case —1 < A < 0.

Now let us suppose that equation (4.62) holds when — k < A < — k + 1 and

r = 1 , 2 , . . . , where A; is a positive integer. This is true when k = 1. Thus if

— k — 1 < A < — k, it follows from our assumption that

a ^ 1 \ n x + ox-_x~l ~T = ^ |^ ^ y p [ 2 c ( p ) + ^(A + 1 + r) - r ' ( l ) ] 5 (r_1)(a:),

for r = 1 , 2 , . . . . It follows from Theorem 4.1 that

[(A -I- l):r+ ln x + + £+] o xZx~r~l + (A + r + l):r++1 ln x + o xZx~r~2 =

= T V Mp) + ^ x + r + V ~ r'C1)]^ 0 ^ )

= (A + 1 ) 4 ln x+ o xZx~r~l - ^ cosec(7rA) ^(r) +2r!

(A + r + l)7r cosec(7rA). . . ( / , _ w / iMrW/ \+ ------------- ^ ------- — [2c(p) + V>(A + r + 2) - r'(l)]<5(r>(2;).

Thus

(A 4- 1)£+ In x + o xZX~r~l =

= - (A + 1)7r2^ SeC(7r-A )[2c(p) + -0(A + r + 2) - r'(l)](5M (^) +

+ T ° SeC( f )-[r- 1 + ^(A + r + 1) - ^(A + r + 2)]«5<'>(z)2 (r — 1)!

= - (A + 1)7r2^°SeC(7rA)[2c(p) + V>(A + r + 1) - r'(l)]<5(r)(z),

since

ip( A + r + 2 ) - ( A + r + I ) -1 = ^(A + r + 1)

and so

r ~ x + iP(\ + r + 1) - ip(X + r + 2) = *+ ^ ■

Equation (4.62) now follows by induction for A < 0, A ^ - 1 , - 2 , . . . and

r = 2 , 3 , . . . .

83

To cover the case r = 1, we note the product (xA+1 ln x + ).a:_A 1 exists by

Definition 4.2 and

(xA+1 ln x + ).x _ A 1 = 0 (4.74)

for all A.

Let us suppose that equation (4.62) holds when —k < \ < —h+ 1 and r = 1,

where k is a positive integer. This is true when k = 1. Thus if —A: — 1 < A < —h,

it follows from our assumption that

(xA+1 ln x +) o xZx~2 = \ 7rcosec(7rA)[2c(p) + i p ( X + 2) — r'(l)]£(a;).

It follows from equation (4.74) and Theorem 4.1 that

[(A + l)a;A ln x+ + :rA] o x I A_1 + (A + l ) ( x A+1 ln x + ) o xZX~2 = 0

= (A + l) ( z + ln x + ) o x I A_1 — 17r cosec(7rA)(5(x) +

+ i(A + 1) 7T cosec(7rA)[2c(p) + ij){\ + 2) — r'(l)]5(a;)

= (A + 1)(£+ ln x + ) o xZx~l +

+ \ (A + l)7r cosec(7rA)[2c(p) + ip( A + 1) — r'(l)]<5(:c).

Equation (4.62) now follows by induction for A < 0, A / — 1, —2 , . . . and r = 1.

Now let us suppose that equation (4.62) holds when k — 1 < A < k and

r = 1, 2 , . . . , where A; is a positive integer. This is true when k = 0. Then for

an arbitrary function p in V we have

where x(x) = X(p(%) is also in V. It follows from our assumption with k — 1 <

A < k that

N —lim ((xA ln z + ), {x_x r =n —►oo

( —l ) r7r cosec(7rA)

—A—r—1

[2c(p) + ^(A + r + 1) - r ' ( l ) ] x (r)(0)

(—l ) r7r cosec(7rA)[2c(p) + ip(\ + r + 1) - r '(l)]¥>(r- I)(0)

2 (r - 1)!

84

and so

N ^ -H m ((^ +1 ln x + ), (xZX~r~l )ny (x ) ) =

( ~ l ) r7rcosec(7rA) r . . , /x . .. / x= ---------2 r _ [2c(p) + V>(A + r + 1) - T (1)]^ ( >(0).

Equation (4.62) now follows by induction for A > 0, A ^ 1 , 2 , . . . and r =

1 , 2 , . . . , completing the proof of the theorem.

C O R O L L A R Y 4.4 For A / 0, ±1, ± 2 , . . . and r = 1 , 2 , . . . we have

( x i l n * - ) o ^ A- r = (~ 1)2r^ C° Sie)C!(OTA) [2c(/,)+ ^(A +r)-r'(l)]< $(r - 1>(x). (4.75)

P R O O F . Equation (4.75) follows on replacing x by — x in equation (4.62).

T H E O R E M 4.9 For A / 0, ±1, ± 2 , . . . and r = 1 ,2 , . . . we have

o ( x z x~r l n z _ ) = - ^ g W ) + t p ( - \ - r + 1) - r'(l)]<5(r“1)(a:).

(4.76)

P R O O F . Differentiating the equation

A —X—r - 7T C O S e c ( 7r A ) r / r _ n /

2(r — 1)! ( l ) ’

which was proved in [14] partially with respect to A we get

/ A i \ - A - r a / —X—r 1 \ 71-2 cot(7rA) cosec(7rA) ./r_ lW N(x+ \ n x + ) o x _ —x + o ( x _ l nx_) = ------- —------- — ---------5( '(x)

and on using equation (4.62) it follows that

x^.o(a;IA_r l nx_ ) = -----—----- [n cot(7rA) + 2c(p )+ ^ (A + r) —r^ l)]^ ^ " 1 ^).2{r — 1).

(4.77)

Taking logs and differentiating the identity

r ( —A)T(A + 1) = (—l ) r_1r ( —A — r + 1)T(A + r) = —7r cosec(7rA)

gives

— ( —X — r + 1) + ip ( \ + r) = —7rcot(7rA) (4.78)

and equation (4.76) follows from equations (4.77) and (4.78).

85

C O R O L L A R Y 4.5 For A / 0, ±1, ± 2 , . . . and r = 1 , 2 , . . . we have

x x o (x~x~r \ n x + ) =

= (~ 1)2(T-C-T )C!(7rA~[2c(p) + " r + 1} “ r '(1)]<5(r_1)(:r)-(4 -79)

P R O O F . Equation (4.79) follows on replacing x by — x in equation (4.76).

We finally note that if we replace A by —A — r in equation (4.79), we get

i:A_ro(i+lni+) = -^|^3 yyp[2c(p) + ij}{\ + 1) - r'(l)]<S(r-1)(a:)

and we see that the product of the distributions x+ln:r+ and xZx~r is com­

m utative only when r = 1.

86

Chapter 5

THE COMPOSITION OF DISTRIBUTIONS

In the following the function Sn(x) is defined as in Chapter 4 and F is a

distribution in V , Fn{x) = (F * 6n)(x). We now define the com position of a

distribution F and a locally summable function / .

D E F IN IT IO N 5.1 Let F be in V and let f be a locally summable function.

We say that the distribution F ( f ( x ) ) exists and is equal to the distribution

h{x) in V on the interval (a, b) if

N - l im [ Fn(f (x)) ip(x) dx = (h(x), tp(x))n —> oo J a

for all (p in T> with support contained in the interval (a,b), where N is the

neutrix defined in Chapter 4-

The following two theorems were proved in [15] and [16] respectively:

T H E O R E M 5.1 The distributions ( x t )^ and {x+)^_ exists and

(x » )x_ = (x^)x_ = 0

for p > 0 and \ p ^ — 1, —2 , . . . and

( r f ) i = ( - i ) v ( ^ ) A„ =

for p > 0 ; A / —1, —2 , . . . and Xp = —1, —2 , ----

87

THEOREM 5.2 The distribution (x2)+s 1//2 exists and

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

2 \ - s - l / 2 _ i ^ j —2 s —1(25)

[In 2 — c(p)](5^2s^(x)

Before proving the next theorem we note the following lemm a which can

be proved easily.

L E M M A 5 .1 Let ip be a function in V with support contained in the interval

[—1,1]. Then

(;T+r, tp(x)) = / 1 r k l ) - ^Jo 1 i=0

r_1 ,/i(0

11 d x ~ T i? V(i)( 0)

r j i!(r - i - 1)+

r 1V r- 1(o) (5.1)(r - 1)

f o r r = 1 , 2 , . . . .

We now prove the following theorem.

T H E O R E M 5 .3 Le£ F (x ) denote the function x+ 1//2. T/ien the distribution

F ( x 2f ) exists and

F(x?) = + ( - i r 1[ln2 -c(p)+^(r - l)]5(r, 1)(:c) (g 2)

/or r = 1 , 2 , . . . .

P R O O F . We have

Fn{x) = <I - i / n( x - t ) 1/2Sn(t )d t ,J - l , n ( x - t ) - 1/2Sn(t)dt,

o,

x > 1 /n ,■1/n < x < 1/n ,

x < —1/n.

88

It follows that

J F n ( x 2£ ) x l d x = J F n ( x 2r) x l dx + J F n ( 0 )x l d x

px r= / ( x 2r — t ) ~ 1/ 2x l5n (t) d t d x +

JO J — l / n

+ [ [ ( x 2r — t ) ~ 1/ 2x l5n (t) dt d x +J n ~ l / 2r J — l / n

+ f [ (—t ) ~ l / 2x l5n ( t ) d t d x J — l J — l / n

— [ $n(t) [ ( x 2r — t ) ~ l !2x l d x dt +Jo J t l / 2r

+ [ $n{t) [ {x2r — t ) ~ l !2x l d x dt +J - l / n Jo

+ 7 7 f { - t ) ~ 1/2Sn ( t ) d t (5.3)l + 1 J — l / n

— [ $n(t) [ [(x2r — t) l/2 + (x2r + t) l l 2\x l d x d t +J 0 7 fi/2r

r l / n r t xl 2r+ / $n(t) / (x2r + t )~ ll2x % dx dt +

f /n U % ( t ) d tJoi + 1 Jo

— Ii + I2 + I3

P utting nt = u and n l/2rx = v, we have

Ii = n

/ ± \ /•! /*n*'“•

= 2n(r- i - 1)/2r V 2 / u2kp(u) V' irh- r+i dv du“ \ 2k Jo v ’ JuVir

r 1 r n x!2r

( r - z - l ) / 2 r J j [(v2r — u)~1!2 + (v2r + U)~ll 2]vl dv du

00 / _ JA „n i / 2r

/c=000 / 1 \ „ ( —4r/c—r + i + l ) / 2 r - 4rA;—r - N + l ) / 2 r

= 2n<— 1‘>/* g ( J j ^ _-4rfc_ r + < + 1----------- «“ />(«) «*«•

Since the non-zero powers n are negligible, it follows that

N —lim /x = -------- -— - [ p(u) du = --------- :---------------------------- - (5.4)n -> 00 r — z — 1 70 r — z — 1

for i = 0 , 1 , . . . , r — 2.

89

N ext we have

I2 = n r~l~1 2r / p(u) / (v2r + u)~ ll2v l dv du Jo Jo

and it follows that

N —lim / 2 = 0 (5.5)n —>00

for i = 0, 1, . . . , r — 2.

Finally we have

h = t l M l [ \ - m p(u )d uZ "h 1 JO

and it follows that

N — lim /3 = 0 (5.6)n—¥ 00

for i = 0 , 1, . . . , r — 2.

It now follows from equations (5.4), (5.5) and (5.6) that

[ Fn( x + ) x l dx = -------- — - (5.7)J- 1 r — z — 1

for i = 0 , 1, . . . , r — 2.

We now consider the case i — r — 1. We have

[ ' ( x » - dx = ln[l + ( l - * ) 1/2] - l n ^Jtl/ 2r r

/ V - r ^ v - ' d * =Jo r

[ (—t ) ~ 1/ 25n( t ) d t = n 1//2 [ (—u)~1/2p(u)du.J—l/n J — 1

It follows from equation (5.3) that

1 /•!/r1 1 r l / n ,/ Fn(x2I ) x r~l dx = - ln[l + (1 - t ) l l2]8n(t) dt +

J - 1 r J - l / n1/1

2r 7- 1/

1 /* 1 / 71 ( _____________________ /* 0

' \tl\t\Sn(t) d t --------------------/ (—u )_1/2 dur • / —1

90

and so

N

since

\ — \ i m [ Fn(x2J)xr 1 dx = -----------— , (5.8)n -> oo 7 - 1 T

/ 1 / n /*1 r lIn |£|5„(t) dt = / In M p(u) du — Inn / p(u) du.

- 1 / n 7 - 1 7 - 1— 1 / n

W hen z = r, we have

, — 1/2 r „ —1/2 r r x 2 r[ \xrFn( x + ) \ d x = [ [ x r (x2r — t) l l 25n{ t ) d t d x

Jo Jo J — l / n

= n~ ll2r [ [ v r (v2r — u)~1/2p ( u ) d u d v Jo 7-1

= 0(rz_1/2r)

and it follows that if ip is an arbitrary continuous function then

pn~l / 2rlim / x rFn(x ir’) ,0 (7;) dx = 0. (5.9)n - * o o 7 0 "r

Further,

/ xrFn(a;2r) (7:) d x = J x rip(x) J (—t) l 2Sn(t) d t d x

= rz1//2 J x rip(x) J (—u)~ l!2p ( u ) d u d x

and it follows that

N — lim [ x rFn(x2P)ip(x) dx = 0. (5.10)n —> oo 7 — 1

If now 77 is chosen so that n -1//2r < rj < 1, then we obtain

f V \xrFn(x2Z ) \ d x = [ [ x r (x2 r - t ) 1/26n(t) d t d xJ n ~ l / 2r J n ~ ll 2r J —l / n-1/7

00 / — 1 \ r1 r7? l-u V = 2 1 / _ . dx du

frZ \ i J 7 - i Jn- l/2r nlx 2riOO ( _L\ /”1 / 7T7 — 77 Z 7"

■ § ( i' ) / - / “> H l-2„ ~ J“

91

It follows that-T)

and so if ^ is a continuous function then

^ F^ X+ ) ^ X) \ dX = ° ( Tl)- (5 -U )

Now let (p be an arbitrary function in V with support contained in the

interval [—1,1]. By Taylor’s Theorem we have

<p(x) = E :t— r~Lx' + 7 V ,*! r\

where 0 < f < 1. Then

(Fn(x+),p(x)) = J ^Fn(x+)ip(x) dx

= ^ P _ J 0 ) f Fn(x2 ) x l dx + j_ f x rFn(x+)(p{r){£x )dx +i = o i . J - 1 r . J - i

1 rn~^l'i r 1 rr]H— - x rFn(x+)ip(r)({;x)dx + — xrFn(x+)ip(r\ £ x ) dx +

r\ Jo r\ Jn~ 1/2r

+ ~ f x rFn(x2r)(p(r\ £ x ) d x .T\ Jrt

Taking the neutrix lim it as n tends to infinity, using equations (5.7), (5.8),

(5.9), (5.10) and (5.11) and noting that the sequence { x rFn( x + ) } converges

uniformly to 1 on the interval [77, 1], it follows that

N-ton<FB(x?),V(x)> = - E 1} + ln2- ^ ’(O) +

+ ■7 / ^ r\ € x ) dx + 0 (77)

+ - [ [ (p{r)( & ) d x , r\ Jo

92

since 77 can be made arbitrarily small. Thus, on using Lemma 5.1, we have

r 1 r r ~ l nN —lim (Fn{x 1 ) , (pix)) = x r \(p(x) — — -— x l] dx +

n - > 00 "r J 0 L “ z! J

<pW(0) l n 2 - c ( p ) (r_ 1}

= { x 7 M x ) ) + ( - i r - H l n 2 - c ( p ) + ^ - l ) ] (g(r_1)(:c)!^ ))|

proving equation (5.2) on the interval [—1,1]. However, since F(x*£) = £+r on

any closed interval not containing the origin, we have proved equation (5.2)

on the real line.

C O R O L L A R Y 5.1 The distribution F ( x 2T) exists and

F(3» ) = a - + l n 2 - c ( p ) + ^ ( r - _l ) (;(r_ 1)(x) (5 12)

fo r r = 1 , 2 , . . . .

P R O O F . Replacing x by —x in equation (5.2) we have

F { ( - x ) X ) = ( - x ) ; r + ( ~ 1)r 1 lln 2 ~ c{p) + r<t>{r -

and so

F(x?) = a:r + ln2-c(p) + r^(r-l)tf(r_1)(g)[

proving equation (5.12).

C O R O L L A R Y 5 .2 Let G denote the function x _ 1 2. Then the distribution

G ( —x + ) exists and

G ( - a ? ) = g r + ( - D r- 1[ ln2 - c(p) + r 0 ( r - l ) ]a (r - 1) ( j ) (g 13)

for r = 1,2, —

93

PROOF. Since G ( —x) = F(x) , we have

G(-x:?) = F(s?) = + ( - i r 1P ° 2 - y + ^ ( r - l ) ] (

from equation (5.2), proving equation (5.13).

C O R O L L A R Y 5 .3 The distribution G ( —x 2f ) exists and

G (_ a * ) = ^ + l ^ - c ^ + r ^ - l ) ^ - , )r\

fo r r = 1, 2 , . . . .

P R O O F . Replacing x by —x in equation (5.13) we obtain

G(-(-*)?) = (-*); ' + ( - 1)r~‘P n 2 - c ( p ) + r 4 r - l ) ] 6(r

and so

G (_ x 2r) = x -r + l ^ - C ^ + r f l r - l ) ^r!

proving equation (5.14).

T H E O R E M 5 .4 Let Fr (x) denote the function \x\2r x where

Then the distribution Fr( |x |_A) exists and

Fr (\x\~x) = x~2r

for r = 1 , 2 , . . . .

P R O O F . We have

F r,n(^) <

f - { n/n (x - t ) 2r/x8„( t)d t ,

f - i/„(x - <)2r/A<5»M dt + S l / n {t - x f r l x5n (t) dt , I - / nn (t - x ) 2r/x5n (t) dt, -

r~1}(x),

(5.14)

l)( - 4

0 < A < 1.

(5.15)

x > 1 / n ,

x\ < 1 / n ,

v < —1 / n .

94

Since Frjn(|x | A) is an even function, it follows that

f 1 F (Ix\~x)x l dx = { ^7-1 r ( 2 Jo Fr n(\x\~x)x l dx, i even.

Also

l * / A r X ~ X

^0 J —l / n[ Fr n (\x\ x)x l dx — [ [ x l (x x — t )2r/ x5n(t) dt dx +

JO ’ Jo J - l / n/ •n1/ A r l / n or / \

+ / / x l (t — x~x) 5n( t ) d t d x +Jo Jx~x

+ [ [ x l (x~x — t ) 2r XSn(t) d t d xJn * / A J — l / nn x'x J — l / n

= / 1 + / 2 + / 3 . (5-16)

Putting nt = u and n ~ l/ xx = v, we have

Ii = n^+1_2r^ A J j v l {v~x — u)2r Xp(u) d u d v

and it follows that

N —lim /i = 0 (5.17)n —>00

for i = 0, 2 , . . . , 2r — 2.

N ext we have

■ i ri

ro Al­

and it follows that

N — lim F = 0 (5.18)

J2 = n l+l 2r^ x [ [ v l {u — v x)2r Xp(u) d u d v Jo Jv~x

n —>00

for i = 0, 2 , . . . , 2r — 2.

Finally we have

1/A /-1A = n (*+1 2r) /x J J y t^v X — u } 2r/ X d u d v

= p (u ) p ^2 r / Aj (_ „ )* „ -* -+ * * + * d „ d«i= V ( 2r/ AN\ ---------------- 1---------------- [ „ - * _ n ( i + l - 2 r ) /A l j 1 < - u f p < u ) du

k - 2 r + \ k + i + U U - i v ' ^ v 'k=0

95

and it follows that

N —lim h = - -------b - r (5.19)n—too 2r — I — 1

for i = 0, 2 , . . . , 2r — 2. It now follows from equations (5.16), (5.17), (5.18) and

(5.19) that

J t Fr<nX\x\~x)x2' dx = — 2 L _ (5.20)

for i = 0 , 1, . . . , r — 1.

W hen i — 2r we have

r n l / x rn1//A rx~x or / \/ x 2rFr n(\x\~x) dx = / x 2r(x~ — t) 5n( t ) d t d x +

Jo ’ Vo J —l / nI! / * 1 / 7

,’1 rn *=

/•n ' , o_ /\+ / / x 2r(t — x ~ ) Sn( t ) d t d x

Jo Jx~x

J J v 2r(v~x — u)2r Xp ( u ) d u d v

+ n 1/x [ [ v 2r(u — v~ x)2r Xp(u) d u d v Jo Jv~*

= 0 ( n x/A)

and it follows that if ip is an arbitrary continuous function then

,1/A

r0

Similarly

rn1/*N —lim / x 2rFr)n( |x |_A)'0(a;) dx = 0. (5.21)

n-> oo Vo

N — lim [ x 2rFr)n(\x\ x)ip(x) dx = 0. (5.22)n —> oo J —n i / \ ’

If now rj is chosen so that n 1//A < p < 1, we have

I"1 \x2rFr n (\x\~x) dx = [ [ x 2r(x~x - t f r/X5n(t) d t d xJrV/^ * V n 1/^ J — l / n

= E ( 2r/ AV ‘ f ( - u y p { u ) j p u x x iXd x d u

96

It follows that

N — \ i m [ x 2rFr n (\x\~x) dx = r\ n — > 0 0 J n i /A ’

and so if ip is a continuous function then

N — lim [ x 2rFr n(\x\~x)'ip(x) dx = Oirj). (5.23)n —> 0 0 Jn i / a ’

Similarlyr — nl/X

N —lim / x 2rFr n (\x\~ )ip(x) dx = 0{rf). (5.24)n —> 0 0 J--q ’

Now let if be an arbitrary function in V with support contained in the

interval [—1,1]. By Taylor’s Theorem, we have

2r—l

where 0 < £ < 1. Then

(Fr,n(|x |" A), f ( x ) ) = j Fryn(\x\~x)if(x) dx

2r - l , „ ( * )

X 2 r

= 5 Z f Fr,n{\x\ X)xl dx +«! ./-1

——7 / 3;2’'Fr,n( |x |_A)<;9(2r,(^x)da: +(2r)!

(27)! . /J m a;2r-Fnn(l:,:r A)v,<2r)(?2:) +

— !— • f x 2rFrin(\x\~x)ip(2r'l (£x )d x + (2r)! Jn

(27)! / „ i/ a +1 p — yi 1 ^

— — / x 2rFr)n( |x |_A)(^(2r)(^ x)dx +(2r)! V—77

(27)! L i x2ri?r'"(lx N ) <'9<2r)(£;E)

97

Taking the neutrix lim it as n tends to infinity, using equations (5.20), (5.21),

(5.22), (5.23) and (5.24) and noting that the sequence ( x 2rFr)n(|a;|-A)} con­

verges uniformly to 1 on the intervals [77, 1] and [—1, —77], it follows that

r~1 2( ^ ( 0),ts (2i)!(2r — 2i — 1)

' (2 0 ! j v ^ ^ ^ (2 r)! / - i T (£ )

Since 77 can be made arbitrary small, it follows on using Lemma 4.3 that

N - l im ( F r>n(|x | A,(p(x)) = - J 2 /o~~\ 1 /o-T — +

N —lim (Fr)n(|x| x),(p(x)) = - (c\ t(r\----^ — TT +r_1 2 ^ ( 0 )^ (2z)!(2r — 2i — 1)

>! 2r —1 <pW(0)

0

{x~2r,<p(x)}.

+ [ x - 2r[ip(x) - £ — p - A dx>7 — 1 »_n I-

T H E O R E M 5 .5 Let F\ (x ) denote the function \x\ X where s — 1 < A < s.

Then the distribution F \ ( |x |2r/ A) exists and

Fx{\x\2r/X) = x~2r (5.25)

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

P R O O F . P utting

F x , n { x ) = Fx{x) * Sn(x),

we have

T ( - A + s ) r ( - A + i )

F \ , n ( x ) =

= <

f l { n/n ( x - t ) A+s x>(t) dt, x > l / n ,j - i , n ( z - t ) - x+a- 1* l r 1)( t ) d t +

+ ( - l ) s_1 Jx1/n(£ - x ) ' x+s~15ff;~V(t) dt , |x| < 1/ti,( - 1 ) 5-1 - a:)_A+s_1^ _1) w dt , rr < -1 / t i .

98

Since F\^n{\x\2r/ X) is an even function, it follows that

f 1 FXn(\x\2r/>')xi dx = I . fl „ ,°2r/M i , * 0d d ’ (5.26)J-1 ’ Vl 1 ' ( 2 Jo F\^n{x 1 )x dx, i even.

We have

n j i ~ A / 2 r „ g 2 r / \

= / (z2r/A - i ) - A+s- 1i i5is- 1)(< )^ d x +7 0 V - l / n

— A/ 2 r i /

— (—i y [ [ (t — x 2r/x)~x+s~1x l5 y ~ 1\ t ) dt dx +7 0 7 x 2r/ A

+ / ' f 11" (x2r/A — t)“A+s_1xi5)),_1)(t) d t d xJ 7T. A/ 2 r- J — l / n

— I\ — ( — 1)SI2 + 3- (5.27)

Putting nt = u and n xl2rx = v, we have

/*1 pXp“T I x — n Al2r_l_1P 2r j J (v2r/x — u)~x+s~l v lp(s~l \ u ) d u d v

and it follows that

N — lim I\ = 0 (5.28)n —> o o

for i = 0, 2 , . . . , 2r — 2.

N ext we have

I2 = n x(2r 1 1 2r [ [ (u — v 2r/ x) x+s l v lp s l \ u ) d u d v Jo Jv2r/X

and it follows that

n—>ooN —lim / 2 = 0 (5.29)

for i = 0, 2 , . . . , 2r — 2.

99

Finally we have

ix/2r p\I3 = n x(2r~l~1 2r J (v2r!x — u) A+S 1 p s l \ u ) v l d u d v

— n H2r - i~i)/2r J p(s“ 1)(w) ^ {v2r!x — u) A+5 l v l d v d u

( - X + s - l \ ns f 1 k (a-i), \ j= A > I - — / (—u) p1 Hu) du\ k J —2rX + 2r s — 2r — 2r k + zA + A J - i

~ / - A + 3 - l \ n A ( 2r - z - l ) / 2r !- A ; I —- — ---- — ----- r / (—u) P Hu) du

\ k J —2rX + 2r s — 2r — 2r k 4- zA + A J - i

and so

N - l i m / s = - 75------ F( i H H t Z u (5 ’3°)n-> oo (2 r — z — l ) r ( —A + 1)

for i = 0, 2 , . . . , 2r — 2, since

J (—u)s~l p s~l\ u ) d u = ( s — 1)!.

It now follows from equations (5.26) to (5.30) that

£#ww*'V*--sr!jri (mi)for z = 0,1, 2 , . , . , r — 1.

W hen z = 2r we have

T (-A + S) /•»-V2r 2 |2r/A\ I in - T + T ) /o I* * W I* I

—A/2 r= / / x 2r(x2r/ x — )""A+s_1|^ s_1^(t)| dt dx +

J — l / nrTl /> 1J fi \ 1

+ [ f x2r( t - x 2r/x)~ \5is- 1]{ t ) \ d t d xJo J x 2r/ X

r \ r v 2r/ X= n~x/ 2r / (u2r//A — u)~~A+s_1z;2r| / / s~ 1)(zz)| du dv +

Vo y~i

+ n~ x!2r [ [ (u — v 2r/ x) A+s 1v 2r\p(*~1\ u ) \ d u d v Jo J v 2r/ X

= 0(n~x/2r)

100

and it follows that if 'ijj is an arbitrary continuous function, then

lim [ %2rF\ n(\x\2r x)il){x) dx = 0. (5.32)n —>oo Jo v '

Similarly

n ^ o f _ n-x,2, x2rF\n ( \x \2r,x)i>{x) dx = 0. (5.33)

If now r] is chosen so that n~x/ 2r < r\ < 1, then we obtain

r ( - A + S ) n | 2r p / I | 2r / A \ | d x -

r ( - A + 1) y„-A/ar >'aX

= f \ n f n, x2r(x2r/A- t r x+s- l \5(r 1]( t ) \ d t d xJ n - x/ 2r J - l / n

= J J x/2 ^2 ^jx2 rs~r~ri^ xn s~l~1\ulp s~1\ u ) \ d x du

- g C - A + ' ; - A A (, , , ,“ J \ 2 y 2rs — 2r — 2ri + A J - i

It follows that, n

Tl—>00Jija, I-** \x2rFM 2r/x)\dx = owand so if ip is a continuous function then

Jim £ _ x/2t x 2rFXt„(\x\2r/x)ip(x) dx = 0(r j ) . (5.34)

Similarly

r — n ~ x/ 2rlim / x 2rFAjn(|a;|2r/A)i/;(a;) dx = 0 (77). (5.35)77. ► oo J — rj

Now let (/? be an arbitrary function in V with support contained in the

interval [—1,1]. By Taylor’s Theorem, we have

where 0 < f < 1. Then

(FXjn( \x\2r/X), ( f(x) ) = J FX,n(\x\2r/X)(p{x) dx

= Z I ' FKn{ \ x \ ^ ) x ' dx +i=0 1 • J ~ l

1 rn~x/ 2r— — / x 2rFx,n{\x\2r/x)ipi2r){£ x )d x +(zrj! Jo

[ x 2rFXjn(\x\2r/x)<p{2r)(£x) dx +J n~x/ 2r

f 1 x 2rFKn(\x\2’-fx)V^ ( t [ x ) d x +Jf]

[ x 2rFKn(\x\2r/x)(p{2r)(£,x)dx +

p—n~x/2r/ x 2r FKn{\x\2r/x)(pi2r) (£x) dx +

J - r \

x 2TFx n(\x^rlx)ifi(2r){ ix ) dx.

(2 r )

1(2 r)

1(2r)

1(2 r)

1(2r)

Taking the neutrix lim it as n tends to infinity, using equations (5.26) and (5.31)

to (5.35), and noting that the sequence { x 2r FX:Tl(\x\2r/x} converges uniformly

to 1 on the intervals [77, 1] and [—1, —77], it follows that

r —1N —lim (F A;n( |x |2r/A, (f(x)) = - J 2

2(pW(0)+ [ ^ 2r\ ^ x ) dx

Jn0 (2i)!(2r - 2i - 1) (2r)\ Jn

+0(r] ) + 0 (t]) + —l— J ip(2r\ £ x ) d x .

Since 77 can be m ade arbitrarily small, it follows on using Lemma 5.2 that

2¥3(2i>(0)r — 1N —lim {F \in(|:r|2r/ A, <p(x)) = - £

fr! (2i)!(2r - 2r - l )+

i = 0

+L x-2 r

-2 r ^ V ^ O ) ,v ( x ) ~ T — x dx

i = 0

proving equation (5.25).

102

T H E O R E M 5 .6 Let F \ tS(x) denote the function xZx lns x - , where 0 < A < 1

and s = 0,1, 2 , . . . . Then the distributions FA>s(x + A), FX}S(xrJ x) and F \ )S( |x |r/A)

exist and

FA;S(x+/A) = — —A + l)c j(p )]£ (r-1)(x), (5.36)r ■ j =o

Fx,s ( x J X) = - X > o , 5-,(A , -A + l)c J- ( p ) ] ^ - 1>(a;), (5.37)r - j =o

f * , ( M r/A) = [1 + (~ l r ' ]A f [ B o , s - A ^ ~A + I M p ) ] ^ - 1 ) (5.38)r■ j=ofo r r = 1 , 2 , . . . and s = 0,1, 2 , . . . , where B ( a , 0) denotes the Beta function,

QP+Q

Bp,q{a i (3) — Q p Q q ^ ^ ® ' ^

and

Cj(p) = f Ini tp(t) dt.J 0

In particular

W _,X) = ( - i y - ' n \ c o SeC( . \ ) s{r_ 1){x l

F M r/X) = [l + ( - l ) r- ' ] ^ ^ ( ^ ^ - D (:c) (5 .39)

for r = 1 , 2 , . . . .

P R O O F .W e have

[F a,s (^)]n <

f , y n(t — x) Alns (t — x)6n(t) dt, \ x \ < l / n ,S - i /n t t - x )~x \ns (t - x)Sn(t) dt, x < - \ j n , (5.40)

0, x > 1 / n

and it follows that if n A//r < a , then

i ~ x/ r r0f x'{F^s(xTi x)]n dx = r x'[FKs(xr/X) } n d x + f x'[Fx,s(0)]ndx

J - a JO j - a

= h + I 2 • (5-41)

103

On using (5.40)we have

- n ~ x^T r l / nrn 1 pL/n , _ \I i = / x 1(t — x r x) Ins (t — x r! )5n( t ) d t d x

JO Jxr/X

— f $n(t) [ x l (t — x r!x) Alns (t — x r/ x) dx dt Jo Jo

— ~ f $n(t) f t x(<l r+1)/ruA(l+1)/r l ( l — u) A[ln t + ln ( l — w)]s du dt r Jo Jo

= — ^ 2 J t x l+l~r)/r \nj tSn(t) J wA +1^r_1( l — u)~X lns~J( l — u ) d u d t

~ J 2 + ! ) / r> - A + 1) f t x{l r+1)/r In-7 tSn(t) d t ,r j = o \J / Jo

on putting x = ( tu)x/ r .

N ext, putting nt = u, we have

■ l/nr l / n p 1/ tA(<-r+1)/r in-' t5n(t) dt = n ^ r - ‘_1)/r / t / ( i+1- r>/r (ln?; - ln n )V M <fo

70 70

and it follows that

for i = 0,1, 2 , . . . , r — 2 and

N —lim /! = 0 (5.42)n —>oo

\ 5 / ^N - l h n h = - £ L )5 o ,s-,(A , -A + l ) c » (5.43)

7=0

for i = r — 1.

Next, on putting nt = v, we have

’0 / - l / nn

i/nx zt“A lns dt dx

__( _ i y + + 1 r1 r1

= -—: ?2A / v _A / r!_A(ln u — In n )sp(v) dv2 + 1 70 70

and it follows that

N — lim / 2 = 0 (5.44)n —>00

for i = 0 , 1 , . . . , r — 1.

104

W hen i — r we have- A / r

'0

and consequently, for any continuous function 'ip, we have

[ \xr [Fx,s (xr+ X)]„| dx = 0 ( n x/r lns n)J 0

lim [ x rF\ s (xr/ x)'ip(x) dx = 0. (5.45)n —» o o J q ’

Finally we have

'° , . . r° . . r l /nJ [Fx,s (xrJ X)]n'ip(x) dx = J 'tp(x) J t AlnstSn( t ) d t d x

r0 r 1= n x *p(x) / v~ x( \nv — \ n n ) sp(v) dv dx

J-a Jo

and so

N - l i m [ [FA)S(x+/A)]n^ (x ) dx = 0. (5.46)n —> o o J —a

Now let <p be an an arbitrary function in V w ith support contained in

—a,a\. By Taylor’s Theorem we have

r — 1

v>(x) = Y l — i— x’ +rX .

where 0 < f < 1. Then with n x r < a, we have

([C\,s (z+/A)]n,</>( x)) = J _ j F x , s (xT/ x)]nip(x)dx

= J A F ' K s ( x T ) \ n d x +1 = 0 L ‘ u

+ r! Io lF *>s(x + X)]nxr(P{r)( t x ) d x +

+ [ lFx,s {xrl X)]n(p{x)dx.J —a

Taking the neutrix lim it as n tends to infinity and using equations (5.40) to

(5.46), it follows that

N n- H m ( [ F AiS( x + /A ) ] n , < f ( x ) ) = Xlfi ^ ^ J ] F j B 0, s - j ( X , -A + 1 ) c j ( p )

( - l ) r_1A sy I J 2 -A + l ) c j {p ) {5 {r 1)(x) , ip(x))

j =0

105

giving equation (5.36) on the interval [—a, a]. Since [—a, a] can be any closed

interval containing the origin, equation (5.36) follows on the real line.

To prove equation (5.37), we note from equation (5.40) that

( ~ \x \r X)~X dt, —n ~x/r < x < 0 ,

[^a,s(^- )]n = | f ^ n t~x lns tSn(t) dt, x > 0,( 0, x < —n~x/r

and it follows that

[ x l [Fx>s(xrJ X)]n dx = [ x l {Fx,s (\x\r/x)]n dx + [ x l [FXyS{0)]n dxJ - a J - n ~ x / T JO

= ( - 1 y r Vr x'[FKs(xrfx)}n dx + ( - 1 Y [ ° ^ [ ^ . . ( o ) ] ^JO J - a

= ( - l ) * ( /1 + / 2).

Equation (5.37) now follows as above.

Finally, to prove equation (5.38), we note from equation (5.40) that

[ f » , . ( i* r " ) i . - ( < £ ■ ' * ‘ w <I 0, \x\ > n A/T

and it follows that

f a x'{Fx,s (\x\r/X)]ndx = f x i [FKs(\x\T/x)}n d x + r ' x \ FK, ( x r/X)]ndxJ — a J —n ~ x / r JO

= [ l + ( - l ) ‘] / l .Equation (5.38) now follows as above.

C O R O L L A R Y 5 .4 Let G x,s(x) denote x^Alns x +; where 0 < A < 1. Then

the distributions G x s (—x r/ x), G X)S(—x r_/x) and G x s (—|x |r//A) exist and

G xA - 4 X) = (~ T 1A £ - A + l)c ;,(p)^(r- 1)(x), (5.47)3 =0

A 5G xA - x F = 4 £ A, -A + 1 )c3( p ) S ^ l \ x ) , (5.48)

T ■ 3 = 0

GA,s ( - k | r/A) = [1 + ( ~ | )r 1]A £ B o,s -A^ - A + l)c ,(p )g (r- 1>(^)(5.49)r! j =o

106

fo r r = 1 , 2 , . . . and s = 0,1, 2 , . . . . In particular

G a ,o( - ^ A ) = E ^ ! £ M 5 ( r - D (:c )i

GAo(_ | x r /A) = ! l ± l ± l ! ^ ) ^ - . ) ( l )I ,

f o r r = 1 , 2 , . . . .

P R O O F . We have

G x,s ( - x ) = Fx,s (x)

and so

[GA,,(-V +/A)]n = [Fa,s(V+/a)]„,

[Ga,s(-V_/a)]„ = [FA,S(V_/A)]n,

[GA,s ( - i x r / A)]„ = [FA,s( |x r /A)]„.

Thus equations (5.47), (5.48) and (5.49) follow from equations (5.36), (5.37)

and (5.38) respectively.

107

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I l l