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7/27/2019 Hn 3414011407 http://slidepdf.com/reader/full/hn-3414011407 1/7 Yashwant Singh and Harmendra Kumar Mandia / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 4, Jul-Aug 2013, pp.1401-1407 1401 | P age Some Double Finite Integrals Involving the Hypergeometric Functions and H -Function With Applications Yashwant Singh and Harmendra Kumar Mandia 1 Department of Mathematics, Government College, Kaladera, Jaipur, Rajasthan, India 2 Department of Mathematics, Seth Moti Lal (P.G.) College, Jhunjhunu, Rajasthan, India Abstract The aim of this paper is to evaluate four finite double integrals involving the product of two hypergeometric functions and H -function. At the end we give an application of our findings by connecting them with the Riemann-Liouville type of fractional integral operator. The results obtained by us are basic in nature and are likely to find useful applications in several fields notably electric network, probability theory and Statistical mechanics. Key words: Double finite integral, Riemann-Liouville fractional integral operator, H -function. 2000 Mathematics subject classification: 33C99 I. Introduction The H -function occurring in the paper will be defined and represented by Buschman and Srivastava [2] as follows: 1, 1, 1, 1, , , ( ; ; ) ,( ; ) , , ( , ) ,( , ; ) j j j N j j N P j j M j j j M Q M N M N a A a P Q P Q b b B H z H z 1 ( ) 2 i i z d i (1.1) where 1 1 1 1 ( ) (1 ) ( ) (1 ) ( ) j j M N A j j j j j j Q P B j j j j j M j N b a b a (1.2) Which contains fractional powers of the gamma functions. Here, and throughout the paper ( 1, ..., ) j a j p and ( 1,..., ) j b j Q are complex parameters, 0( 1,..., ), 0( 1,..., ) j j j P j Q (not all zero simultaneously) and exponents ( 1,..., ) j A j N and ( 1,..., ) j B j N Q can take on non integer values. The following sufficient condition for the absolute convergence of the defining integral for the H -function given by equation (1.1) have been given by (Buschman and Srivastava[2]). 1 1 1 1 0 Q M N P j j j j j j j j j M j N A B (1.3) and 1 arg( ) 2 z (1.4) The behavior of the H -function for small values of z follows easily from a result recently given by (Rathie [4],p.306,eq.(6.9)). We have , , 1 0 , min Re , 0 M N j P Q j N j b H z z z (1.5) If we take 1 1,..., , 1 1,..., j j A j N B j M Q in (1.1), the function , , M N P Q H reduces to the Fox’s H-function [3]. We shall use the following notation: * 1, 1, , ; , , j j j j j N N P A a A a and * 1, 1, , , , ; j j j j j M M Q B b b B

Transcript of Hn 3414011407

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Yashwant Singh and Harmendra Kumar Mandia / International Journal of EngineeringResearch and Applications (IJERA) ISSN: 2248-9622 www.ijera.com

Vol. 3, Issue 4, Jul-Aug 2013, pp.1401-1407

1401 | P a g e

Some Double Finite Integrals Involving the HypergeometricFunctions and H -Function With Applications

Yashwant Singh and Harmendra Kumar Mandia 1Department of Mathematics, Government College, Kaladera, Jaipur, Rajasthan, India2Department of Mathematics, Seth Moti Lal (P.G.) College, Jhunjhunu, Rajasthan, India

AbstractThe aim of this paper is to evaluate four finite double integrals involving the product of two

hypergeometric functions and H -function. At the end we give an application of our findings byconnecting them with the Riemann-Liouville type of fractional integral operator. The results obtained byus are basic in nature and are likely to find useful applications in several fields notably electric network,probability theory and Statistical mechanics.

Key words: Double finite integral, Riemann-Liouville fractional integral operator, H -function.2000 Mathematics subject classification: 33C99

I. IntroductionThe H -function occurring in the paper will be defined and represented by Buschman and Srivastava

[2] as follows:

1, 1,

1, 1,

, , ( ; ; ) ,( ; ), ,

( , ) ,( , ; ) j j j N j j N P

j j M j j j M Q

M N M N a A a P Q P Q

b b B H z H z

1( )

2

i

i

z d i

(1.1)

where

1 1

1 1

( ) (1 )

( )(1 ) ( )

j

j

M N A

j j j j j jQ P B

j j j j j M j N

b a

b a

(1.2)

Which contains fractional powers of the gamma functions. Here, and throughout the paper ( 1, ..., ) ja j p and

( 1,..., ) jb j Q are complex parameters, 0( 1,..., ), 0( 1,..., ) j j j P j Q (not all zero

simultaneously) and exponents ( 1,..., ) j A j N and ( 1,..., ) j B j N Q can take on non integer values.

The following sufficient condition for the absolute convergence of the defining integral for the H -functiongiven by equation (1.1) have been given by (Buschman and Srivastava[2]).

1 1 1 1

0Q M N P

j j j j j j j j j M j N

A B

(1.3)

and1

arg( ) 2 z (1.4)

The behavior of the H -function for small values of z follows easily from a result recently given by (Rathie

[4],p.306,eq.(6.9)).We have

,,

10 , min Re , 0

M N j P Q

j N j

b H z z z

(1.5)

If we take 1 1,..., , 1 1,..., j j A j N B j M Q in (1.1), the function,

, M N P Q H reduces to the Fox’s

H-function [3].

We shall use the following notation: *

1, 1,, ; , , j j j j j N N P

A a A a and *

1, 1,, , , ; j j j j j M M Q

B b b B

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Vol. 3, Issue 4, Jul-Aug 2013, pp.1401-1407

1402 | P a g e

II. Preliminary ResultsThe following known results [6] will be required in the proof:

1

2 112 1

0

( 2) (1 )(1 ) 1 (1 ) , ; ;

2 1 (1 )

x a x x ax b x F dx

ax b x

= 2

2( )

2 2 22( )(1 ) (1 ) ( ) ( )a b

.

1 1(2 ) (2 )

2 2 2 21 1

1 12 2 2 2 2 2

(2.1)

Where Re( ) 0,Re(2 ) 0, a and b are constants, such that the expression [1 (1 )]ax b x is not zero and 0 1 x .

1

2 112 1

0

( ) (1 )(1 ) 1 (1 ) , ; ;

2 1 (1 ) x a

x x ax b x F dxax b x

= 2 1

1 ( 1)2 2 22

(1 ) (1 ) ( ) ( )a b

.

1 1(2 2) (2 )

2 2 2 21 1

2 2 2 2 2 2

(2.2)

Where Re( ) 0,Re(2 ) 0, a and b are constants, such that the expression [1 (1 )]ax b x is not zero and 0 1 x .

2(2 1) 1 1

2 10

' ' 2(sin ) (cos ) ', '; ; cos

2i w w w ie F e d

=

( 1)2

2 ' ' 2

' ' ' '( ) 1

2 22 ' ' ( ') ( ')

i w

w

e w w

.

' 1 ' ' ' 1(2 ' ') (2 ' ')2 2 2 2

' ' 1 ' ' 11 1

2 2 2 2

w w

w w w w

(2.3)

Where Re( ) 0,Re(2 ' ') 0w w .

2(2 1) 1 1

2 10

' '(sin ) (cos ) ', '; ; cos

2i w w w ie F e d

=

( 1)2

2 ' '

' ' ' '( 1) 1

2 22 ( ') ( ')

i w

w

e w w

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Vol. 3, Issue 4, Jul-Aug 2013, pp.1401-1407

1403 | P a g e

.

' 1 ' ' ' 1(2 ' ' 2) (2 ' ' 2)

2 2 2 2' ' 1 ' ' 1

2 2 2 2

w w

w w w w

(2.4)

Where Re( ) 0,Re(2 ' ') 0w w .

III. Main IntegralsWe shall evaluate the following four finite double integrals involving hypergeometric functions and

H -function.First Integral1 2

1 2 12 1

0 0

( 2) (1 )(1 ) [1 (1 )] , ; ;

2 1 (1 ) x a

x x ax b x F ax b x

. (2 1)2 1

( ' ' 2)(sin ) (cos ) ', '; ; cos2

i w w w ie F e

. 1 1 1 1 1, 2 2 *, *(1 ) [1 (1 )] (sin ) (cos )

m n w i w w A p q B H zx x ax b x e d dx

=

2 2

1 2

22

1 12( ) ( )( )(1 ) (1 ) 2 2 2 2

H H a b

( 1)/ 2

3 42 ' ' 2

' ' 2' 1 ' ' ' 12

2 ( ' ') ( ') ( ') 2 2 2 2

i w

w

e H H

(3.1)

Where 1 2 3, , H H H and 4 H are given as follows:

1 1 1 1

1 11 1 1

2 2 2 ;2 ;1 , 1 ; ;1 , 1 ; ;1 , *, 3 21 3, 3 1

*, 1 2 ;2 ;1 , ; ;1 , ; ;12 2

2(1 ) (1 )

Am n p q

B

z H H

a b

(3.2)

1 1 1 1

1 11 1 1

2 2 2 ;2 ;1 , 1 ; ;1 , 1 ; ;1 , *, 3 22 3, 3 1

*, 1 2 ;2 ;1 , ; ;1 , ; ;12 2

2(1 ) (1 )

Am n p q

B

z H H

a b

(3.3)

1 1 1 1

11 1 1

' '/2 ' ' 2 ;2 ;1 , 1 ; ;1 , 1 ; ;1 , *, 3 23 3, 3 ' ' 12

*, 1 2 ' ';2 ;1 , ; ;1 , ; ;12 2

2

i w w w w w w w Am n p q w B w w w w w w

ze H H

(3.4)

1 1 1 1

11 1 1

' '/2 ' ' 2 ;2 ;1 , 1 ; ;1 , 1 ; ;1 , *, 3 24 3, 3 ' 1 '2

*, 1 2 ' ';2 ;1 , ; ;1 , ; ;12 22

i w w w w w w w Am n p q w

B w w w w w w

ze H H

(3.5)

(i) Re( ) 0,Re( ) 0,| arg | , 0,2

w z

(ii) 1Re(2 2 ) 0, 0, j j j j

j j

b b B B

(iii) 1Re 2 ' ' 2 0, 0. j j j j

j j

b bw w B B

Second Integral

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Vol. 3, Issue 4, Jul-Aug 2013, pp.1401-1407

1404 | P a g e

1 21 2 1

2 10 0

( 2) (1 )(1 ) [1 (1 )] , ; ;

2 1 (1 ) x a

x x ax b x F ax b x

(2 1) 2 1

2 1

( ' ')(sin ) (cos ) ', '; ; cos2

i w w w i

e F e

1 1 1 1 1, 2 2 *, *(1 ) [1 (1 )] (sin ) (cos )

m n w i w w A p q B H zx x ax b x e d dx

=

2 1

5 6

22

2( ) ( ) (1 ) (1 ) 2 2 2 2

H H a b

( 1)/ 2

7 82 ' 1

' ' 2' 1 ' ' ' 22

2 ( ' ') ( ') ( ') 2 2 2 2

i w

w

e H H

(3.6)

Where 5 6 7, , H H H and 8 H are given as follows:

1 1 1 1

1 11 1 1

2 2 2 ;2 ;1 , 2 ; ;1 , 2 ; ;1 , *, 3 25 3, 3 3

*, 3 2 ;2 ;1 , 1 ; ;1 , ; ;12 2

2(1 ) (1 )

Am n p q

B

z H H

a b

(3.7)

1 1 1 1

1 11 1 1

2 2 2 ;2 ;1 , 2 ; ;1 , 2 ; ;1 , *, 3 26 3, 3 3

*, 3 2 ;2 ;1 , ; ;1 , 1 ; ;12 2

2(1 ) (1 )

Am n p q

B

z H H

a b

(3.8)

1 1 1 1

11 1 1

' '/2 2 ' ' 2 ;2 ;1 , 2 ; ;1 , 2 ; ;1 , *, 3 27 3, 3 ' ' 32

*, 3 2 ' ';2 ;1 , 1 ; ;1 , ; ;12 22

i w w w w w w w Am n p q w B w w w w w w

ze H H

(3.9)

1 1 1 1

11 1 1

' '/2 2 ' ' 2 ;2 ;1 , 2 ; ;1 , 2 ; ;1 , *, 3 28 3, 3 ' 3 '2

*, 3 2 ' ';2 ;1 , ; ;1 , 1 ; ;12 22

i w w w w w w w Am n p q w B w w w w w w

ze H H

(3.10)

The integral (3.6) is valid if the following set of (sufficient) conditions are satisfied:

(i) Re( ) 1, Re( ) 1,| arg | , 0,2

w z

(ii) 1Re(2 2 ) 2, Re 0, j j j j

j j

b b B B

(iii) 1Re 2 ' ' 2 2, 0. j j j j

j jb bw w B B

Third Integral1 2

1 2 12 1

0 0

( 2) (1 )(1 ) [1 (1 )] , ; ;

2 1 (1 ) x a

x x ax b x F ax b x

(2 1) 1 12 1

( ' ')(sin ) (cos ) ', '; ; cos

2i w w w ie F e

1 1 1 1 1 1, 2 2 *, *(1 ) [1 (1 )] (sin ) (cos )

m n w i w w A p q B H zx x ax b x e d dx

=2 2

1 22 1 1( ) ( )( )(1 ) (1 ) 2 2 2 2

H H a b

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Vol. 3, Issue 4, Jul-Aug 2013, pp.1401-1407

1405 | P a g e

( 1) /2

7 82 ' ' 1

' '' 1 ' ' ' 12

2 ( ') ( ') 2 2 2 2

i w

w

e H H

(3.11)

Where 1 2 7, , H H H and 8 H are mentioned in (3.2), (3.3), (3.9) and (3.10) respectively and set of conditionsare as follows:

(i) Re( ) 0,Re( ) 1,| arg | , 0,2

w z

(ii) 1Re(2 2 ) 0, 0, j j j j

j j

b b B B

(iii) 1Re 2 ' ' 2 0, 0. j j j j

j j

b bw w B B

Fourth Integral

1 21 2 2 1

2 10 0

( 2) (1 )(1 ) [1 (1 )] , ; ;

2 1 (1 ) x a

x x ax b x F ax b x

(2 1) 12 1

( ' ' 2)(sin ) (cos ) ', '; ; cos

2i w w w ie F e

1 1 1 1 1 1, 2 2 *, *(1 ) [1 (1 )] (sin ) (cos )

m n w i w w A p q B H zx x ax b x e d dx

=

2 1

5 6

22

1 12 2( ) ( )(1 ) (1 ) 2 2 2 2

H H a b

( 1)/ 2

3 42 ' ' 2

' ' 2' 1 ' ' ' 12

2 ' ' ( ') ( ') 2 2 2 2

i w

w

e H H

(3.12)

Where 5 6 3, , H H H and 4 H are mentioned in (3.4), (3.5), (3.7) and (3.8) respectively and set of conditions areas follows:

(i) Re( ) 0,Re( ) 1,| arg | , 0,2

w z

(ii) 1Re(2 2 ) 0, 0, j j j j

j j

b b B B

(iii) 1Re 2 ' ' 2 2, 0. j j j j

j j

b bw w B B

Proof: To establish (3.1), express the H -function on the left hand side using (1.1)in single Mellin-Barnescontour integrals and interchanging the order of integration which is justifiable due to absolute convergence of the integrals, we have:

=

1 1

11 1 1

0

1 1

11

(1 )2

1

j

j

n m A

j j j j j j s sq p B

L j j j j

j m j n

a s b s x x

wb s a s

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1406 | P a g e

12 2 12 1

2 (1 )[1 (1 )] , ; ;

2 1 (1 ) s x a

ax b x F dxax b x

1 1 1

22 2 1

2 10

' ' 2(sin ) (cos ) ', '; ; cos2

i w w s w w s w w s i

e F e d

Evaluate the inner integral with the help of (2.1) and (2.3) and then applying (1.1), we get R.H.S of (3.1)in terms

of product of H -function.The proof of second, third and fourth integrals are similar to the first with the only difference that here we makeuse of known integrals, (2.1), (2.4) for second result (2.1) and (2.4) for third result and (2.2) and (2.3) for fourthresult respectively instead of (2.1) and (2.3).

IV. Special Cases(i) Putting a b in (3.6), we get

1 21 2 1

2 10 0

( 2)(1 ) [1 ] , ; ;

2 x x b F x

(2 1) 2 12 1

( ' ' 2)(sin ) (cos ) ', '; ; cos

2i w w w ie F e

1 1 1 1 1 1, 2 2 *, *(1 ) [1 ] (sin ) (cos )

m n w i w w A p q B H zx x b e d dx

=

2 1

5 62

21 12

( ) ( ) (1 ) 2 2 2 2 H H

b

( 1) /2

7 82 ' ' 1

' '

' 1 ' ' ' 222 ( ') ( ') 2 2 2 2

i w

w

e

H H

(4.1)

Where 5 6 7, , H H H and 8 H are integrals mentioned in the second integral are satisfied and the set of conditions mentioned with second integrals are also satisfied.

(ii) If we put 1 j j A B in (3.1), the result will be changed into the Fox’s H -function instead of the H -

function.

V. ApplicationWe shall define the Riemann-Liouville fractional derivative of function ( ) f x of order (or

alternatively, th order fractional integral) ([7], p.181;,[8],p.49) by

11 ( ) ( ) ,Re( ) 0( )

( ) , 1 Re( )( ) ,

x

aq

qa xq

x t f t dt

a x d D f x q q

dx

D f x

(5.1)

Where q is a positive integer and the integral exists. For simplicity the special case of the fractional derivative

operator a x D , when 0a will be written as x D . Thus we have

0 x x D D (5.2)

The preliminary result (3.1), after interchanging the order of integration which is justified due to absoluteconvergence of the integrals can be rewritten as the following fractional integral formula:

2(2 1)

2 10

' ' 2(sin ) (cos ) ', '; ; cos

2

i w w w ie F e

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1407 | P a g e

.

1 1 1 1 1 1

1 2 12 1

1

, 2 2 *, *

2 (1 )[1 (1 )] , ; ;

2 1 (1 )

(1 ) [1 (1 )] (sin ) (cos ) x

m n w i w w A p q B

x a x ax b x F

ax b x D d

H zx x ax b x e

Using (5.1) and evaluating the integral with the help of (2.3), we get

=

2 2

1 2

22

1 12( 1) ( ) ( )( )(1 ) (1 ) 2 2 2 2

H H a b

( 1)/ 2

3 42 ' ' 2

' ' 2' 1 ' ' ' 12

2 ( ' ') ( ') ( ') 2 2 2 2

i w

w

e H H

(5.3)

Where Re( 1) 0 and all the conditions of validity mentioned with (3.1) are satisfied. The fractional

integral formula given by (5.3) is also quite general in nature and can easily yield Riemann-Liouville fractionalintegrals of a large number of simpler functions and polynomials merely by specializing the parameters H occurring in it which may find applications in electromagnetic theory and probability.

References[1] Braakshma, B.L.J.; Asymptotic expansions and analytic continuations for class of Barnes integrals,

Camp. Math. 15, (1963), 233-241.

[2] Buschman, R.G. and Srivastava , H.M.; The H function associated with a certain class of Feynmanintegrals, J.Phys.A:Math. Gen. 23, (1990), 4707-4710.

[3] Fox, C.; The G -function and H function as symmetric Fourier kernels, Trans. Amer. Math. Soc. 98,(1961), 396-429.

[4] Rathie,A.K. ,A new generalization of generalized hypergeometric functions Le Mathematiche Fasc.II,52,(1997),297-310.

[5] Ronghe, A.K.; Double integrals involving H -function of one variable, Vij. Pari. Anu. Patri. 28(1),(1985), 33-38.

[6] Sharma, G. and Rathie, A.K.; Integrals of new series of hypergeometrical series, Vij. Pari. Anu. Patri.34(1-2), (1991), 26-29.

[7] Srivastava,H.M.,Gupta, K.C. and Goyal,S.P.; The H-Functions of One and Two Varibale withApplications,South Asian Publishers, New Delhi and Madras, (1982).