A Parseval Goldstein type theorem on the widder potential transform and its applications

A

PARSEVAL-GOLDSTEIN

TYPE THEOREM

ON

THE

WlDDER

POTENTIAL TRANSFORM

AND ITS APPLICATIONS

O.

YREKLi

Department

ofMathematics

and

Computer

Science Ithaca

College

Ithaca,

NewYork 14850

I.SADEK Mathematical Sciences University of North Carolina

Wilmington,

N

C28403

(Received March 19, 1990 and in revised form November 20, 1990)

ABSTRACT. In

thispaperaParseval-Goldsteintypetheoreminvolvingthe Widder poten-tialtransform andaLaplace typeintegraltransformisgiven. The theoremisthen shown to yieldarelationshipbetweenthe/C-transformand theLaplacetypeintegral transform. The theorem yieldssomesimplealgorithmsforevaluatinginfiniteintegrals. Usingthe

the-oremandits

results,

anumber ofnewinfiniteintegralsof elementary and specialfunctions arepresented.

Some

illustrativeexamplesarealsogiven.

KEY WORDS AND PHRASES.

The Widderpotential

transform,

theLaplace

traform,

the

2-transform,

the

E-transform

the

modified

Bessel

function

of

the third kind or

Mac-donald’s

function.

1980 AMS

SO’BJECT

CI,

ASSIFICATION CODES:

Primary

44A10, 44A15;

Secondary

38A40,

44A35.

1.

INTRODUCTION

Widder

[1,2]

presentedasystematic account of the potential transform

( )

/()

’[f();

]

+

d

Widder pointed out that the potential transformisrelatedtothePoissonintegral

represen-tationofafunctionwhich is harmonic inahalfplaneand gave severalinversionformulae for the transform and appliedhisresults to harmonic functions. Srivastava andSingh

[3]

gavethefollowingParseval-Goldsteintype formula:

/0

’[J’(,);

1

(:)

j0

/()

’[(u);

1

a

(..)

518 O.

Y’U’REKL

AND I. SADEK

Widderpotential transform:

.[f(u),zl.T’.[g(u),zldx

f(z)’P[g(u),xldz

(1.3)

where istheLaplacetransform and

.T’

istheFourier sinetransform. Thereaxenumerous analogousresultsintheliteratureonintegraltransforms.

(See,

_{for instance, Goldstein}

[5],

Srivastava

[61,

Srivastava and Panda

[71

andYiirekli

[81.

The objective ofthispaperisto establishaParseval-Goldsteintyperelation between thepotential transform and aLaplace type integraltransform called

2-transform

where the

2-transform

isdefinedas

2[f(x);Yl

xe

-’

f(x)

dx.

(1.4)

Ifwe make a

change

of variableinthe integralon the fight-hand sideof

(1.4),

one obtains

:[f();

1

f(,/l

.

(1.

Comparing

(1.15)

withthedefinitionofLaplacetransformweobtainthefollowing

relation-ship between theLaplacetransform and the

-transform

Z:[I();

1

Wealsoobtain identitiesrelatingtheK;-transform

c[l(); ]

/-

g()

I(,)

(.7)

tohe

-transform,

where

K,,(x)

istheBessel function of the thirdkind

(it

isalsoknown

astheMeedonald

function),

mdtheLapla:etransform to the

-transform.

Usingthese

resulgs weshow howonecanextend tables ofLaplaceand I’Iankel transforms.

(See

Erdlyi

etal.

[9, 101,

Oberhettinger

[11],

OberhettingerandBadii

[12].)

or

definitionsof special functions that are usedin the paper, the reader isreferredto Oldham and Spmier

[111],

and Erdlyi et al.

[14].

Wenote that ifwewrite

.M[f(x);

9]

F()

where .,M represents any integral

trans-form,

we meanthe.M-transformof

f(x)

existsandit is

2.

A PARSEVAL-OLDSTEIN THEOREM AND

ITS

COROLLARIES

LEMMA

2.1.

We

have

1

provided that theintegralsinvolvedconvergeabsolutely.

PROOf: Usingthe definition of the

-transform

weobtain

:2[:2[f(x); z];

y]

ze xe-r’’’

f(x)

dx dz

(2.2)

Nowtheresult follows from

(1.1)

and

(2.3).

THEOPM

2.1. Wehave

x

2[f(Y);

z]

2[g(z);

z]

dz _Y

f(!/) P[g(z);

V] dv.

(2.4)

providedha[ [heinegsinvolvedconverge absolutely.

PROOF:

Usinghe definition of he 2-rgormweobn

Chgingheorder ofintegration, whichispeissiblebyhe hypothesis, d henusing [hedefinitionof[he

2-rsform

wefind[ha

,[f(N);

]

2[g(z);

]

d N

f(v)

[[g(z);

]; ] d.

(2.6)

Nowthertionfofiows from

mma

2.1.

MARK

2.1.

We

have

=

,[g(u);

=l

z[/(z);

=l a=

g(N)

[f(z); ]

,

(2.7)

sincetherelation

(2.4)

issymmetdcwithrespectto

f

d_9. Usingtherelations

(2.4)

d

(2.7)

weobtn the Pe-Golteintypeformula

(1.2).

Thus,

Threm 2.1 generizes

relation

(1.2).

COROLLARY

2.1. Wehve

provided that he

intes

involvedconverge eblutely.

PROOf: heidengigy

(2.8)

follows iediately

er

lein

h()

[(); ]

in

gherelation

(2.4).

COROLLARY

2.2.

We

have

(2.9)

provided that theintegralsinvolvedconverge absolutely.

PROOF: We

set

f(!/)

e

-=’’’

inTheorem 2.1. Then

LT,2[f(tJ);

x]

!/e-(=’+=’)’’

1

.(

+

x

)"

Now

theassertion

(2.9)

follows from

(2.4)

and

(2.10).

THEOIZEM

2.2. If

Rev

>

-I

1

,ICy[!/,+’

f(!/);

z]

2"z,+1/2

2 x2"-2 2

/(!/);

-’

provided that theintegralsinvolvedareabsolutely convergent.

(2.0)

520 O. Y0tEKLi AND I. SADEK

PROOF: Weset

g(u)

u"J,,(zu)

inTheorem2.1,where

J,,

isthe Besselfunctionof the firstkindof orderv. Usingrelation

(1.6)

and thenmakinguseof theLaplacetransform table

(see

[9,

formula

(30),

p.

185])

wehave

1

[,

(zu1/2

1 z

z__

(2.12)

-

Cxp--Now inorder toevaluate thepotentialtransform of the function

g(u)

we useLernrna2.1 andobtain

The;-transformontheright-handsideof

(2.12)

maybe evaluatedby usingtherelation

(1.6)

and then theLaplaceransform table

(see

[9,

formula

(20),

p.

146]).

Thus

P[(,I;

1

(-

(.

Substitutingthe results

(2.12)

and

(2.14)

into

(2.4)

of Theorem2.1 gives

/0

()/0

Z

--2--I

y+

K(zy) f(y)

dy exp

--x

:=[y(y);

x]

dx.

(2.15)

Nowtheassertionfollows bymakingthechangeof variable x

t/2

andthenbyusing the definitions ofthe K:-transform and the

:2-transform.

Itiswell known that

(r)1/2

e-’,

(2.16)

:

()

:_

()

(see

[13,

p.

306]).

Using

(2.11)

and

(2.16)

weobtainthe identitiesinthefollowing corollary:

COROLLARY

2.3.

We

have

providedghag_he

inte

_involvedareablutely

convergent.

.

EXAMPLES

We shN1 illustrate he above

resets

by

sever

exples.

In

thefollowing exple

EXAMPLE

.1. Weshow that

[.-’;z]

=.-,z-r

5

+

r

5

+

+

()

provided that

Re

p

>

IRe

v[-

].

Wet

f(y)

y--]

inThrem 3.2. Mingud identity

(1.6)

weobtn

Substituting

(3.2)

into

(2.11)

wefind

provided that

Re(p-

u) >

.

Nowforma

(3.1)

follows

ter

evMuatingthe

-trform

ontheright sideof equation

(3.3).

InteM

trfforms evMuatedin Exples 3.2,3.4 d 3.8, din the appendix,to

the best of the author’sknowledge, M1new.

EXAMPLE

3.2. Weshow that

g

+

i.;

.()--

r(

+

)

+

s__,l

(a.)

where

Re

>

-2 d

S,

is the Lommel function.

Wet

I()

sin inhrem .2. Minguseof

(1.6)

d hen tables ofLaplace

ransfor

(see [12,

foula7.1,

p.4l)

weobtn

1[

1]

+sin;

=

sin;

+

(a’)

where

1/4a.

Substitutingthe function

f

into

(2.11),

dusing

(.g)

d

(1.6),

wefind

L

=[=+’+

=;

C

Now

(3.)

fonowf,om

tm

of

Lp=

t,fo,=

( [12,

fo=3.1,

p.22]).

Usingthetechqueofxple 3.2, weprovidetion

resets

inthe appendix.

In

the following expleweobtn awell known result

(Erd]yi

[9,

forma

(30)

p.

153])

aspeciceofExple3.2.

EXAMPLE

3.3. Weshow that

[sin(ag’);,]:{(-

C(,))cost+

(-

S(,))sin,},

(3.7)

where t=

z/(ga),

=d

C(t)

=d

S(t)

ethe esnelintegs.

We

t

-1/2

in

(3.g).

Using

(2.16)

=d the definition of the -tr=sfo we

obtn

[sina;z]=S

a-’,

(z)

g

(3.s)

Itfollows fromafoaontheLoel function

( [12,

p. glfi])that

s_,,(t)

[3(t)+

_(t)-

(t)

_(t)],

(3.)

where

J(t)

is the Bessel fction of order d

J(t)

is the Anger-Weber functionof order

.

However,

wehave

522 O.

Y’U’REKL

AND I. SADEK

see

[13,

p.

306]

and

.11/2

(t)

t

{[C(t

S(t)]

cost

+

[C(t)

+

S(t)]

sint}

(3.11)

J_1/2(t)

V/--

{[C(t)+

S(t)]

cost-

[C(t)-

S(t)]

sint}

(3.12)

seeOberhettingerandBadii

[12,

p.

415].

Now,

substituting

(3.10), (3.11)

and

(3.12)

into

(3.9)

and thenusing

(3.8)

weobtain

formula

(3.7).

EXAMPLE

3.4. Weshow that

(z)

..

(x+4ax

:)

-1/2;z

=2

(ra)-nr

+

exp

a

K

a

provided that -1

<

Rev

<

1,

Rea

>

0.

We set

f(y)

y-exp(-ay

)

in Theorem 2.2. Makinguseof

(1.6)

and thenusing

tables ofLaplacetransforms

(see [12,

formula5.3,

p.37]),

weobtain

2-"F

( )

x-"

(4ax2

+ 1)

"-1/2

(3.14)

provided that

Re

v

<

1. Usingtables of Hankel transforms

(see

[10,

formula

(24),

p.

132])

weobtain

1 ,r rz 1/2

provided that

Rea

>

0 and -1

<

lieu

<

1. Nowformula

(3.13)

follows fromsubstituting

(Z.l)

=d

(Z.)

i=to

(z0)

=d th

i

(1.).

EXAMPLE

3.5.

We

showtha

4

(az-).

where

Eft(x)

istheeorfction.

We set

f(y)

y-z

sin ay in Corolly 2.3. Minguof

(1.6)

d then tables of

inteM

trffos

(s [12,

formda7.76,p.

66])

weobtn

1

sin;

=

sinl;

2

Itfollows fromebles ofLaplace

ros

(s

[12,

fortune

7.,

p.

4])

ACKNOWLEDGEMENT

We would like to thank Thomas K. Boehme for his interest

andvaluable suggestions.

We

arealso indebtedtoJohnMaceliforacarefulreadingof the preprint. Referee’sremarksarealso

acknowledged.

REFERENCES

1. D.

V.

Widder,

A

transformrelated to the Poissson integral forahalf-plane, Duke Math.

J.,

33,

(1966),

355-362.

2. D. V. Widder,

An

IntroductiontoTransformTheory, Academic

Press,

New

York,

1971.

3. H. M. Srivastava and S. P. Singh,

A

note on the Widder transform related to the

Poisson integralfor ahalf-plane, Internat.

J.

Math. Ed. Sci.Tech., 16,

(1985),

675-677.

4. H. M.Srivastavaand

O.

Yfirekli,

A

theoremonWidder’s potential transform andits applications,

J.

Math. Anal. andAppl.,to appear.

5. S. Goldstein, Operational representationsof Whittaker’s Confluent Hypergeometric Functionand Weber’s parabolicfunction, Proc. London. Math.Soc.

(2),

34,

(1932),

103-125.

6. H.

M.

Srivastava,

Some

theorems on Hardy transform Nederl. Akad. Wetensch.

Indag.

Math.,

30,

(1968),

316-320.

7. H. M.Srivastava and

R. Panda,

Certain multidimensionalintegraltransformations: and

II,

Nederl. Akad. Wetensch. Indag.

Math.,

40,

(1978),

118-131 and 132-143. 8.

O.

Yfirekli,

A

parseval type theorem applied to certain integral

transforms,

IMA J.

Appl.

Math.,

42,

(1989),

241-249.

9. A. Erdlyi etal., Tables ofIntegralTransforms

I,

McGraw Hill, New

York,

1954. 10. A. Erdlyi et

al.,

Tables ofIntegralTransforms

II,

McGraw

Hill, New

York,

1954.

11. F. Oberhettinger,Tables of BesselTransforms, Springer-Verlag,

New York,

1972.

12. F. Oberhettingerand K. Badii,Tables ofLaplace

Transforms,

Springer-Verlag, New

York,

(1973).

13. J. Spanier andK. B.

Oldham, An

AtlasofFunctions,HemispherePublishing

Corpo-ration,

New York,

1987.

14.

A.

Erdlyi et

al.,

HigherTranscendentalFunctions

I, II,

III,

Springer-Verlag, New York, 1953and 1954.

APPENDIX

A.

SOME

K:-TRANSFORM PAIRS

Thefollowingformulae

(A.1)

through

(A.5)

areconsequences ofTheorem 2.2. The techniques ofExample3.1 and 3.2areused to obtainthese results.

,,

y’+1/2

sinay2;z

2(2a)

-+1/2

F(v +2)z

+’]

S__,],]

aa

(A.1)

where

Re

>-2.

c

+1/2

o;

-()-"-1/2

r(,+

1)/

s__,1/2

(a.)

where

Re

u>-1.

1

z-1/2

where

Ci(x)

and

si(x)

are cosineandsineintegrals,respectively.

C2[V-1/2sinavZ;z]

=z-]-Ci

a

a sin

aa

-si

a

c

g

+

zl

_,

[-Ci

()cos

()-si

()sin

()]

(A.4)

/Cv

!/V+1/2sin2a!/2;

z

2"-4(2a)

-"-

r(v

+

3)z

v+

s_v_,1/2

aa

(A.5)

where

Re

v

>

-3.

B.

SOME

LAPLACE TRANSFORM PAIRS

Thefollowingformulae

(B.1)

through

(B.5)

result from Theorem 2.2and Corollary

2.3. The techniques ofExample3.4 and3.5areused to obtain these results.

.

[z_2g(4az2

z)

t_fi+S;

z]

z’’

t’

(v+12

z z

-p

exp(aa)W,,1/2(aa

)

(B.1)

where

2Re/

<

1

-[Rev[

Rea

>

0 and

Wt,,.

isthe Whittaker function.

a-

z1/2

+

z1/2

(B.2)

where

Re

p

>

IRe

vl-

,

D,

istheparabolic cylinderfunctionand

mFn

isthe hypergeo-metricfunction.

sn

(rCt=

())

]

r(

)

x1/2

(x2

_

+

x

+1/2

sin

-+v

_

s

-+

(B.3)

where -1

<

Re

p

<

2 and

Y.

isthe Besselfunctionofthe secondkindof order v.

whereRev

>

-1.

x-’ exp

P

-v,

xx

;z

r’

22"+3

a1/2

F

_v

+

S_2._.,

1/2

az1/2

(B.5)