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INTEGRAL TRANSFORM

1. Introduction:

The classical methods of solution of initial and boundary value problems in physics and

engineering sciences have been root’s in Fourier’s pioneering work. An alternative approach

through integral transform methods emerged primarily through Heaviside’s efforts on

operational techniques. In addition to being of great theoretical interest to mathematicians,

integral transform methods have been found to provide easy and effective ways of solving a

variety of problem arising in engineering and physical science. A problem involving

derivatives can be reduced to a simpler problem involving only multiplication by polynomials in the transform variable by taking an integral transform, solving the problem in

the transform domain and then finding an inverse transform.

It is observed that the mathematical description of physical problems often leads to

differential equations. Integral transforms arise in a natural way through the principle of

linear superposition in constructing integral representations of solutions of linear differential

equations.

Definition:

By an integral transform , we mean a relation of the form

(1.1)

such that a given function f t is transformed into another function F s by means of an

integral. The new function F s is said to be the transform of f t , and , K s t is called the

kernel of the transformation. Both , K s t and f t must satisfy certain conditions to

ensure existence of the integral and a unique transform function F s

when both of the limits of integration in the defining integral are finite we called it finite

transform.

, K s t f t dt F s



If an integral equation can be so determined that

, ; f t s t F s t ds

(1.2)

Then (1.2) is termed as inversion formula of (1.1)

There are a variety of kernels that may be used to define particular integral transforms for a

wide class of functions f t .

2.LAPLACE TRANSFORMS

One of the simplest and most important integral transform is the well known Laplace

Transform. It is date back to the French mathematician Laplace who made use of the

transform integral in his work on probability theory in 1780’s. Then it occurred in Fourier’s

famous 1811 paper on heat conduction. Nonetheless, it was Oliver Heaviside, who

popularized the use of Laplace transform as a computational tool in elementary differential

equations and electrical engineering.

It is particularly useful in solving initial value problem connected with

linear differential equations (ordinary and partial).The advantage of Laplace transformation

in solving initial value problems lies in the fact that initial conditions are taken care of at the

outset and the specific particular solution required is obtained without first obtaining the

general solution of linear differential equation.

The Laplace transform of a function is defined by as follows:

0

;px L f x p e f x dx F p

(2.1)

where x > 0 , Re(p) > 0 provided that the integral exits.

The inverse Laplace transform is given by:

0 0 1;

2 2

c i

sx

c i

f x f xe L f x s ds

i

(2.2)



Existence Conditions for Laplace Transform:

; L f x p exits for all p γ, if f x is sectionally continuous in every finite interval 0 ≤

x ≤ x0 and is of exponential order γ

The inverse Laplace transform of any function is unique.

Some Important Properties for Laplace transform:

(a) Linearity property:

1 1 2 2 1 1 2 2.... ; ....n n n n L c f x c f x c f x p c F p c F p c F p (2.3)

1 1 1 1

1 1 2 2 1 1 2 2... ...n n n n L c F p c F p c F p c L F p c L F p c L F p

(2.4)

(b) Change of Scale Property:

(2.5)

Similarly

1 1;

x L F ap x f

a a

, a> 0 (2.6)

(c) First Shifting or Translation Property:

;ax L e f x p F p a (2.7)

where a is real or complex number

1 ; ax L F p a x e f x

(2.8)

(d) Second Shifting Property:

; ap L g x p e F p

(2.9)

where

f(x-a) , x>a

1

; ,a 0 p

L f ax p F a a



g(x) =

0 ,x<a

1 ;ap L e F p x f x a H x a

(2.10)

where H(x-a) is the well known Heaviside unit function

Table of Laplace Transform

S No. f(x) L{f(x)}

11

1

s

2t

2

1

s

3n

t 1

!n

n

s

4at e

1

s a5

cos t 2 2

s

s 6

sin t 2 2

s

s 7

cosh t 2 2

s

s

8 sinh t 2 2a s

9 sin

at e bt

2 2

b

s a b

10 cosat

e bt

2 2

s a

s a b

11 sint at

2

2 2

2as

s a

12

cost at

2 2

22 2

s a

s a

13

c H t

1, 0

, 0ca

c s

ec

s



14 0

J t 2

1

1 s 15

0 J t

2 2

22 2

s a

s a

16

t J at

1 2

2 2

2 1 2a

p a

17logx

' 1 log p

p

Laplace Transform of Derivatives:

1 2 ' 2 1; 0 0 ..... 0 0n n n n n n L f x p p F p p f p f pf f (2.11)

1 1; ; 1n

nn n

n

d L F p x L F p x x f x

dp

n=1, 2, 3,…. (2.12)

Laplace Transform of Integrals:

0

;

x F p L f u du p

p

(2.13)

1

; p

f x L F u du x

x

(2.14)

Multiplication and Division by Powers of x:

; 1 1n

n nn n

n

d L x f x p F p F p

dp (2.15)

1;

nn n

n

d L p F p x f x f x

dx

(2.16)



Initial Value Theorem:

Let f(x) be continuous for all x ≥ 0 and be of exponential order as x ≥ ∞. Also suppose that

f(x) is of class A, then

0

lim lim ; x p

f x pL f x p

(2.17)

Final Value Theorem:

Let f(x) be continuous for all x ≥ 0 and be of exponential order as x ≥ ∞. Also suppose that

f(x) is of class A, then

0

lim lim ; x p

f x pL f x p

(2.18)

The final value theorem is useful because it gives the long term behavior without having to

perform partial fractions decompositions. If a function’s poles are in the right hand plane the

behavior of this formula is undefined.

The Convolution of Two Functions:

The convolution (f ∗g)of two functions f and g define by the equation

0 0

x x

f g x f u g x u du f x u g u du (2.19)



using the convolution we can write

1

0

. ;

x

L F p G p x f u g x u du f g (2.20)

The convolution f*g is also known as Faltung or Resultant of f and g.

Some Extensions of Laplace Transform

Bilateral Laplace Transform:

The bilateral Laplace transform of a function f: R→R is defined by

sx

f L s e f x dx

(2.21)

The domain of Lf

(s) is the set of all complex numbers s = r+it, r,t Є R, for which the

integral exists. If r=0 and the function f is integrable, then the bilateral Laplace transform

exists for all t Є R.In this Case, the bilateral Laplace transform is called the Fourier transform

of f .If the domain of f is [0,∞), the bilateral Laplace transform becomes the unilateral

Laplace transform or simply Laplace transform.

Sumudu Transform:

Watugala introduced the Sumudu transform defined by



0

1 t u F u e f t dt

u

(2.22)

for any function f(t), or by the conversion rule

0

!n

n

n

F u n a u

(2.23)

This transform also follows all the properties of Laplace transform like linearity, change of

scale, shifting property etc. Sumudu transform can be applied to ordinary differential

equation, control engineering, partial differential equation and convolution type integral

equation.

Laplace Transform of Some Other Function

Laplace Transform of Sine and Cosine Integrals:

2log 1

; 2

p

L Ci x p p

(2.24)

11 1; tan L Si x p

p p

(2.25)

where Ci(x) and Si(x) are cosine integrals and sine integrals defined by the equations

cos

x

t Ci x dt

t

(2.26)

0

sin x

t Si x dt

t (2.27)

Laplace transform of Error function:



2

4

2;

p pe erf

L erf x p p

(2.28)

where erf(x) is the error function, which arises in heat conduction problems, is defined by

2

0

2x

t erf x e dt

(2.29)

Laplace transform of Heaviside’s unit step function:

;ap

e L H x a p

p

(2.30)

Where H(x-a) is unit step function defined by

(2.31)

Laplace Transform of Generalized Hypergeometric Function:

1 1

111 10

,... ; 1 , ,... ;1

,... ; ,... ;

p p st c

p q p qcq q

a a c a ac z e t F zt dt F

b b b b s s

(2.32)

Laplace transform of G-function:

1 11 , , 1

, 1,

1 10

,... 1 , ,...

,... ,...

x p p x m n m n

p q p q

q q

a a a a

x e G x dx Gb b b b

(2.33)

0,

1,

x a H x a

x a



Laplace transform of H-function:

1, 1,, 1 , 1

, 1,

01, 1,

, , , ,

, ,

x j j j j

p ptx m n m n p q p q

j j j jq q

a a

x e H zx dx t H zt b b

(2.34)

Inverse Laplace Transform of Generalized Hypergeometric Function:

1 11

1 1

1 1

,... ; ,... ;11

1, ,... ; 1, ,... ;

p p t

p q p q

q q

a a a a L F z F z e

s b b b b s

(2.35)

Laplace transform of Fractional Integrals:

Let the Laplace transform of

, ,

0, x x I f

and, ,

, x x J f

exist. Then

, ,

0,

0

; , , , ; , x L x I f s S s x f x dx

(2.36)

for Re(s) > s0

≥0 and for non integer values of λ-β, λ-η, β-η, and

, ,,

0

; , , , ; , x L x J f s T s x f x dx

(2.37)

for Re(s) >s0

≥0 and μ> max {0, β-η}-1, where



2 2

2 2

2 2

, , , ; ,

1, 1; 1, 1;

1 , 1; 1, 1;

1 , 1; 1, 1;

S s x

x F sx

s F sx

s x F sx

(2.38)

and

2 2

, , , ; ,

1 11, 1; 1, 1;

1 1

T s x

x F sx

(2.39)

Here

, ,

0, x x I f

and, ,

, x x J f

are fractional integral operators defined in appendix.

3.MULTIDIMENSIONAL LAPLACE TRANSFORM

The multidimensional Laplace transform will be defined in the following manner:

1 1

1 1

...1 1

0 0

, ..., ; , ...,

... ,..., ... s s

s s

x x s s

L f x x

f x x e dx dx

(3.1)

Where Re(λ i

) >0, (i=1,…,s) and the function f(x1

,…,xs

) is so chosen that the above multiple

integral is absolutely convergent. We will represent the equation (3.1 ) symbolically as



1 1 1,..., ,..., ; ,..., s s s L f x x (3.2)

Parseval Goldstien Theorem:

The following analogue of the well known Parseval Goldstein theorem for the

multidimensional Laplace transform:

If

1 1 1 1 1,..., ,..., ; ,..., s s s L f x x (3.3)

and

2 1 2 1 1,..., ,..., ; ,..., s s s L f x x

(3.4)

then

1 1 2 1 1 2 1 1 1 1

0 0 0 0

... ,..., ,..., ... ... ,..., ,..., ... s s s s s s f x x x x dx dx f x x x x dx dx

(3.5)

provided that the various integrals involved in the equations (3.3 ), (3.4) and (3.5) are

absolutely convergent.

A Well-Known Property:

The following s-dimensional analogue of the well-known property of the Laplace transforms:

If



1

2

ipx F f x F p e f x dx

(-∞<p<∞) (4.1)

where f(x) is a function defined on (-∞,∞) and be piecewise continuously differentiable and

absolutely integrable in above interval. Here 2

ipxe

is known as the kernel of the Fourier

transformation

By changing the value of kernel of Fourier transform we can write

(i) Fourier Sine Transform:

0

2; sin

s s F f x p F p f x px dx

,p > 0 (4.2)

(ii) Fourier Cosine Transform:

0

2; cosc c

F f x p F p f x px dx

(4.3)

where function f(x) is defined for x >0 and be piecewise continuously differentiable and

absolutely integrable in (0, ∞)

Inverse Fourier Transform:

(i) The inverse Fourier transform of F(p) , is denoted by f(x) and given as



1

2

ipx f x e F p dp

(4.4)

(ii)

The inverse Fourier sine transform of Fs

(p) is given by

0

2( ) s s f x F p in px dp

(4.5)

(iii)

The inverse Fourier cosine transform of Fc

(p) is given by

0

2( ) cos

c f x F p px dp

(4.6)

Our definition of the Fourier transform and inverse transform is not unique.

Elementary properties for Fourier Transform:

(a)Linearity Property

1 1 2 2 1 1 2 2.... ; ....n n n n F c f x c f x c f x p c F p c F p c F p

(4.7)

(b) Change of scale property

(4.8)

(c) Shifting Property

; ipa F f x a p e F p (4.9)

1

;p

F f ax p F a a



Modulation Theorem:

If f(x) has the Fourier transform F(p),then

2

F p a F p a F p

(4.10)

The Convolution Theorem for Fourier Transform:

Statement:- Let

(i) f(x) and g(x) and their first order derivatives are continous on (-∞,∞)

(ii) f(x) and g(x) are absolutely integrable on (-∞,∞)

(iii) F(p) and G(p) are Fourier transforms of f(x) and g(x) respectively.

Then the Fourier transform of the convolution of f(x)and g(x) exists and is the product of

the Fourier transforms of f(x) and g(x)

( ) F f g F f x F g x (4.11)

Parseval’s identity for Fourier transform:

If F(p) and G(p) are the complex Fourier transforms of f(x) and g(x) respectively ,then

(i) F p G p dp f x G x dx

(4.12)

(ii)

2 2

F p dp f x dx

(4.13)

where * signifies the complex conjugate. By using Parseval’s identity we can evaluate

many integrals.

Fourier Transform of Derivatives:



If (i) f, f ’,……, f

(n-1)

are continuous on (-∞,∞)

(ii) f

n

is piecewise continuously differentiable and absolutely integrable on (-∞,∞)

Then the Fourier transform of the function

n

n

d f

dx is (-ip)n times, the Fourier transform of

function f(x)

; ;nn F f x p ip F f x p (4.14)

Fourier transform method is useful in solving some partial differential equations subject

to boundary conditions.

Relationship between Fourier Transform and Laplace Transform:

Let us define a function f(x) as follows:-

, 0

0, 0

xe x x f x

x

;ipx

F f x p e f x dx

0

0

ipx ipxe f x dx e f x dx

0

0

0ipx ipx xe dx e e x dx

0

ip xe x dx

0

sxe x dx



where

;ip s L x s

so we can write

; ; F f x p L x s (4.15)

5. MELLIN TRANSFORM

This transform is not very useful as Fourier and Laplace transformations in a direct

manner. It is quite effective, in the derivation of certain properties of integrals, in

summing series and in statistics. Generally we can consider the mellin transform as a sort

of indirect tool in applications.

Definition:- The Mellin transform of any function f(x) is defined as

1

0

; p F p M f x p x f x dx

(5.1)

Here x p-1 is known as kernel of the Mellin transform and 0< x <∞

Elementary properties of Mellin Transform:

(a) Scaling of the original variable by a positive number

; pM f ax p a F p ,a > 0 (5.2)

(b)

Raising of the original variable to a real power

1;

a pM f x p a F

a

, a is real (5.3)

(c) Multiplication of the original function by some power of a



;aM x f x p F p a (5.4)

(d) Multiplication of the original function by logx

log ;k

k

k

d M x f x p F p

dp

, k is a positive integer (5.5)

(e) Derivation of the original function:-

; 1k

k

k k

d M f x p p k F p k

dx

(5.6)

where the symbol (p-k)k is defined for positive integer k by

1 ..... 1k

p k p k p k p

Result (c) and (e) can be used in various ways to find the effect of linear

combinations of differential operatiors such that

m

k d x

dx

, k,m are integers

Some of them are

; 1

k k k d

M x f x p p F pdx

(5.7)

; 1

k k k

k k

d M x f x p p k F p

dx

(5.8)

The Mellin Transform of Integrals:

Case I : If ;M f x p F p then we can write

(a) 0

1; 1

x

M f u du p F p p

(5.9)



(b)

0 0

1; 2

1

y x

M dy f u du p F p p p

(5.10)

(c)

; 1

n

n

pM I f x p F p n

p n

(5.11)

where Inf(x) is n

th

repeated integral of f(x) i.e.

1

0

x

n n I f x I f u du

Case II:

(a) 1; 1 x

M f u du p F p p

(5.12)

(b)

1

; 21

x y

M dy f u du p F p p p

(5.13)

(c)

;n

pM I f x p F p n

p n

(5.14)

where

1n n

x

I f x I f u du

Inverse Mellin Transform:

If the integral 1

0

k x f x dx

is bounded for some K > 0 and if

1

0

;p F p M f x p x f x dx

(5.15)

Then the Mellin inverse transform f(x) of F(p)is given by

1 1;

2

c i

p

c i

f x M F p x x F p dpi

,where c> k (5.16)



Convolution for Mellin Transform:

If F(p) and G(p) are the Mellin transform of the functions f(x) and g(x) respectively.

Then

1

;2

c i

c i

M f x g x p F z G p z dz i

(5.17)

Inverse Mellin Transform of Two Functions:

If F(p) and G(p) are the Mellin transforms of the functions f(x) and g(x) respectively,

then

1

0

;x du

M F p G p x f g uu u

(5.18)

Mellin Transform of Some Other Function

Mellin Transform of a G-Function:

1 1 11 ,

,

10

1 1

1,...

,...1

m n s

j j p j j s m n

p q q pq

j j

j m j n

b s a sa a

x G x dxb b

b s a s

(5.19)

Mellin Transform of H-function:

1, 1 11 ,

,

01,

1 1

1,

, 1

m n

j j j j x j j p j j s m n s

p q q p

j j q j j j j

j m j n

b s a sa x H ax dx a

b b s a s

(5.20)

Mellin Transform of p

ψq

-function:



1,

, 1; 1 p

p q p qM x s H s Mf s (5.21)

6.HANKEL TRANSFORM

Hankel Transform arises naturally in solving boundary value problems formulated in

cylindrical coordinates. It also occurs in applications such as determining the

oscillations of a heavy chain suspended from one end.

Definition:- Hankel transform H υ{f(x);p} or F υ(p) or F(p) is defined by

0

, H f x p F p xJ px f x dx

(6.1)

where the function f(x) is defined for all positive values of the variable x and J υ

(px)

is the Bessel function of first kind of order υ. Here xJ υ

(px) is known as the kernel of

the Hankel transform.

Elementary properties of Hankel Transform:

Hankel transform also follows the linearity property and change of scale property.

Relation Between Hankel and Laplace Transform:

By definition we have

0

,ax ax H e f x p xJ px e f x dx

0

axe xJ px f x dx

; L xJ px f x a (6.2)



Inversion Formula for the Hankel Transformation:

If F υ

(p) is the Hankel transform of the function f(x) then

0

f x pJ px F p dp

(6.3)

is known as the inversion formula for the Hankel transform of F υ

(p) and is written as

1 ; f x H F p x

Hankel Transform of Derivatives of Functions:

Let F υ

(p) and F υ

’

(p) be the Hankel transform of order υ of f(x) and f’(x)

respectively.Then

1 1' ; ' 1 1

2

p H f x p F p v F p v F p

v

(6.4)

Parseval’s Theorem for Hankel Transform:

If F υ

(p) and G υ

(p) are the Hankel transforms of the functions f(x) and g(x)

respectively, then

0 0 xf x g x dx pF p G p dp

(6.5)

Hankel Transform of Some Other Transform

Hankel Transform of a G-Function:



11 ,

,

10

2 121 2 ,2 1

2 2,2 2

,...

,...

1 , 2, ,12 22 2 2 22 2

2,

x pm n

p q

q

p q pm n

p q

q

a a x J x G x dx

b b

aG

b

(6.6)

Hankel Transform of H-Function:

1,,

,

1,

,

,,

j j pm n

p q

j j q

a

H H t xb

1,, 1

2,

1,

1 1 1 1, , , , ,2 2 4 2 2 4 2 2

,

j j pm n

p q

j j q

a H

x xb

(6.7)

7.H-Function Transform

The kernel of this transform is the H-function. Since most of the important function are

special cases of the H-function,various integral transform involving these functions as

kernels are special cases of this transform

1 1,

,

0 1 1

, ,..., ,

, ,..., ,

p pm n

p q

p p

a a s s H st f t dt

b b

(7.1)

where this function is the well known Fox H-function defined in the appendix.

Special cases of the H-function Transform

(i) Generalized Hankel Transform

If we put m = 1, n = p = 0, q = 2, b1 = υ, β1 = 1, b2 = -λ+μν and β2 = μ in ( 7.1) it

reduces to the Generalized Hankel transform defined by Ramkumar as



(iii) Meijer’s G-function Transform

If we replace m by m+1 ,n by 0, p by m and q by m+1 in (7.1) and further put α’s and

β’s equal to 1 and therein and take a1

= η1

+α1

,…, am

=ηm

+αm

; b1

= η1

,…,

bm

= ηm

, bm+1

= e, it reduces to the following transform introduced by Bhise

1 11,0

, 1

10

,...,

, ..., ,

m mm

m m

m

s s G sx f x dxe

(7.6)

(7.6) itself is a general transform and reduces to well known Integral transforms viz.

Laplace transform, Meijer transform, Varma transform etc. as pointed out by Bhise.

Inversion Formula:

If y

k-1

f(y) Є L(0,∞), f(y) is of bounded variation in the neighborhood of the point

y = t, and

1 1,

,

0 1 1

, ,..., ,

, ,..., ,

p pm n

p q

p p

a a s s H st f t dt

b b

Then

0 0

2

f t f t

1 1

1 1

11

21

lim

q p

j j j j j jc i

j m j n k

m n

c i j j j j j j

j j

b k a k

t F k dk i

b k a k

(7.7)

here



1

0

k F k s s ds

8. H Function Transform

The H function transform will be defined and represented in the following form:

1, 1,, : , ; ,

,, ; , ;

01, 1,

, , , ,

;, , ,

j j j

j j j

j j j j j N N P M N a A M N

P Q P Q b B

j j j j jM M Q

a A a H f t s H st f t dt

b b B

(8.1)

where f(t) Є Λ, and Λ denotes the class of functions for which

1 2

, 0

,w w t

O t t f t

O t e t

Provided that the existence conditions (A-7) to (A-10) for the

H -function are satisfied

and:

1

2 2 2

1

Re Re 1 0

1Re 0 Re 0 Re Re 1 0

min

min

j

j M j

j

j N j

b

and

aw or w and w

Special Cases of The H -function Transform:

(a)The H-function transform

, : , 1,,

,, ; ,

01,

, ,

;,

j j

j j

j jM N a P M N

P Q P Q b

j j M

a H f t s H st f t dt

b

(8.2)

(b)The P Q transform

: , ; 1,

: , ;

01,

, , :;

, , :

j j j

j j j

j j ja A P

P Q P Qb B

j j j Q

a A f t s st f t dt

b B

(8.3)

where

P Q is defined in Appendix



(c) The P Q F -transform

: , 1,

: ,0

1,

, :

;, :

j j

j j

j ja A P P Q P Q

b B j j Q

a A F f t s F st f t dt

b B

(8.4)

(d) The generalized Bessel transform

, ,

0

; J f t s J st f t dt

(8.5)

where,

J

is defined in Appendix

9.Meijer and Varma Integral Transform

We consider the Meijer transform Mk,m

defined by

1 2 2

, 1 2,

0

k xt

k m k mM f x xt e W xt f t dt

(9.1)

with k,m Є R containing the Whittaker function Wl,m(z) in the kernel.This function isgiven by

2 1 2

,

1, 2 1;

2

z m

l mW z e z m l m z

(9.2)

where ψ(a,c;x) is the confluent hypergeometric function of Tricomi:

2,1

1,2

11, ;

0,11

aa c x G xt

ca a c

which has the integral represntaion

11

0

1, ; 1

c a xt aa c x e t t dt a

when k=-m,since ψ(a,a+1;z)=z-a and 1 2 2

1 2,

m x

k mW x x e

then we find that the

transform M-m.m coincides with the laplace transform



10.Generlized Whittaker Transform

We consider the integral transform

2, ,

0

k k xt W f x xt e W xt f t dt

(10.1)

with ρ,γ Є C and k Є R, containing the Whittaker function in the kernel.It is known

that the Meijer transform and the Varma transform are particular cases of this

generlized Whittaker transform,

k W

:

1 2 ,

1 2

, k m

k

k mM f x W f x

, ,

1 2

, k m

m

k mV f x W f x (10.2)

11.pF

qTransform:

This type of integral transform is given by

1

1 1

0

1

,..., ; ,..., ;

p

i

i p q p q p qq

j

j

a

F f x F a a b b xt f t dt

b

(11.1)

with ai

, b j

ЄC (Re(ai

) >0;i=1,2…,p;j=1,2,…,q), for which we take pЄ N and either

q=p, q=p+1 or q=p-1.This transform, containing the generalized hypergeometric

function as the kernel, can be generalize into the transforms

1

F1

,1

F2

and2

F1

by giving values to p and q.

12.The Generlized Stieltjes Transform

This transform is given as



, ,

1 1

1 f x

2 1

0

1; 1; 1; f t t t

F dt

x x x

(12.1)

with α, β,η Є C (Re (β+η)+1>0, Re(β)+1>0).When α = 0, by the known relation

2 1 1 0, ; ; ; 1

b F a b a z F b z z

above transform takes the form

, ,0 1 0

0

1 1; f t t t

f x F dt

x x x

1

0

1t

f t d t t x

(12.2)

and becomes the Stieltjes transform for β=0

0

1 f x f t dt

t x

13.The Wright Transform

We consider the integral transform of the form

1,

01,

,

,

i i p

p q p q

j j q

a

f x xt f t dt b

(13.1)

containing the Wright function p

ψq

as kernel of the transform.Using the following

result

1, 1,1,

, 1

1, 1,

, 1 ,

, 0,1 1 ,

i i i i p p p p q p q

j j j jq q

a a

z H z b b

above transform converts the H-transform of the form

1,1,

, 1

01,

1 ,

0,1 1 ,

i i p p

p q p q

j jq

a

f x H xt f t dt b

(13.2)

integral transform final

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