International Journal Of Computational Engineering Research (ijceronline.com) Vol. 3 Issue. 3
Function Transform And Its Inversion With “A Study of H Properties” 1,
Dr. A.K. Ronghe, 2, Mrs. Kulwant Kaur Ahluwalia 1
2,(
,(S.S.L. Jain, P.G. College, Vidisha, MP (India) Pin-464001) Mata Gujri Mahila Mahavidyalaya, Jabalpur, MP (India) Pin-482002)
function transform and developed its inversion formula, Abstract: In this research paper we have defined H function transform comes out as Fox’s H-function and it is further detected that particular cases of H transform defined by Gupta and Mittal [6.7], G-function transform defined by Bhise [1] and other etc. 1. Introduction And Preliminaries: The Mellin transform of the
1
0
function is defined as follows: H
a j , Aj ; j , a j , Aj 1, n ,n x 1 H m ax p ,q bj , B j , bj , B j , j 1, m
dx . d m 1, q
a
n 1, p
[1.1]
Where,
b m
j 1 q
j m 1
b Provided min Re j 1 J m j
j
Bj
1 a n
j 1
j
1 b B a p
j
j
j
j n1
Aj j
Aj
1 aj Re 1min J n aj
Re
j
,
[1.2]
[1.3]
function associated in the definition of And other convergence conditions will be those of the H H function, see [(3), (8), (9)] etc. Integral involving product of Hyper geometric function and H function given as follows,
x x 1
0
1 2
, F1 ; 1 x
a j , Aj ; j , a j , Aj 1, n m ,n H p ,q ax bj , B j , bj , B j ; j 1, m
dx m 1, q n 1, p
;1,1 , ;1,1 m 1, n 2 a H p 3, q 3 ;1 , b j , j , bj , B j , j 1, m a j , Aj ; j , a j , Aj , ,1 1, n n 1, p ,1,1 , ,1,1
m 1, q
Provided Re 0, Re 0, Re 0 , and other convergence conditions stated in definition of
[1.4]
H function, see [(8) and (9)].
H and 2 F1 , occurring in the left hand side of (1.4) in term of function, given by Inayat Hussain [8] and [9], Mellin Barnes contour integral with the help of definition of H series form of 2 F1 and using the property of Gamma-function, we arrive at the R.H.S. of (1.4) after a little Proof: To establish (1.4), we first express.
simplification.
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Function Transform And Its… H
A Study Of
Function Transform: 2. The H
An integral transform of function f x whose kernel is
H function defined by Inayat Hussain is called
H function transform, which is defined as follows: a j , Aj ; j , a j , Aj 1, n m , n x H p ,q x bj , B j , bj , B j , j 0 1, m
f x dx ,
m 1,q n 1, p
Provided Re 0, Re 1 0 ,
[2.1]
H function transform as follows: f x or f x ;
We may represent
3.
[2.2]
Special Cases: If 0, j j 1, in (2.1) we get Fox’s H-function transform defined by Gupta and Mittal (6) in
(i)
1970 is as follows:
H
m ,n p ,q
0
aj ,j x bj , j
1, p
1, q
f x dx
[3.1]
If 0, j j 1, , a j c j d j , b j d j , m m 1, n 0, p m, q m 1 in (2.1)
(ii)
we get G-function transform defined by Bhise [1] in 1959 as follows:
G
m 1,0 m , m 1
0
cj d j 1,m x dj ,1 1,m
Note: A number of other transform involving various special functions which are the special cases of
H function given by Saxena [11] can also be obtained from H function transform, but we do not record them here, due to lack of space for that see [10].
H function transform: Multiplying on both sides of equation (2.1) by and integrating with respect to “ ” between the limits 0, , we have,
4.
Inversion formula for
k
0
k d
0
k
0
x
[4.1] x dx d , Now changing the order of integration and using the definition of Mellin transform of H-function given in (1.1) we get, a j , Aj ; j , a j , Aj 1, n n 1, p m ,n H p ,q x f bj , B j , bj B j , j 1, m m 1, q
0
k d k 1 x k 1 f x dx , 0
[4.2]
Where,
b m
k 1
J 1 q
J m 1
Let
0
j
j 1 k
1 b
j
1 a n
J 1
j 1 k
0
k 1 x k 1 f
j 1 k
a J
k d F x ,
Then, F x
j
p
J n1
j
j
j 1 k
[4.3]
[4.4]
x dx ,
[4.5]
OR
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A Study Of
F x
x dx
x k 1 f
Function Transform And Its… H [4.6]
, k 1 Now using the definition of inverse Mellin transform we get, 0
x 0
f
f
x 0
2
1 2
xk F
c
x
k 1
c
[4.7]
dx
Where, F x is given by (4.5) and k 1 is given by (4.2). Provided and other convergence conditions 1 a b 1 c
J min Re J
1 J m
Re 1min J m
J
J
H function.
are stated in the definition of
4. Special Cases:
If 0, j j 1 in (4.7) we get inversion formula for Fox’s H-function transform, which
(i)
developed by Gupta and Mittal [7] as follows:
x 0
f
f
x 0
2
c xk F x 1 dx , 2 c 1 k 1
[4.8]
Where,
b m
k 1 1
J 1
q
J m 1
j
j 1 k
1 b
j
1 a n
J 1
j 1 k
j
j 1 k
a 1
p
J n 1
j
1
j 1 k
[4.9]
(ii) j j unity, a j c j d j , b j d j , m m 1, n 0 p m, q m 1, in (4.7), we get an inversion formula for G-function transform which developed by Bhise [1] as follows:
f x 0 f x 0 2
k c x F x 1 dk , 2 c 2 k 1
Where,
d m
k 1 2
J 1
j
1 k
n
J 1
c m
J 1
j
dj
j
1 k
1 k
1
1
function transform: 5. Properties of H Ist property: f x : and H f x : then If, H 1 2 2 C f x C f x : C C H 1 2 2 2 1 1 2 2
[5.1]
Where C1 and C 2 are arbitrary constant. IInd Property:
If H
f x :
x :
and H
f x : , then
6. Property:
If H
f x :
Re 0, th
1 s
and H
1
then 1
5
1
1
1
[5.2]
, where
[5.3]
IV
Property:
If H
f x : and H f 12 :
1
[5.4]
Proof of all properties given in [10, P, 110]
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A Study Of
Function Transform And Its… H
7. Acknowledgement: The author are thankful to prof. R.D Agrawal (SATI Vidisha) for useful disscusions.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
Bhise, V.M., (1959); Inversion Formula For Generalized Laplace, Transform, Vikram, Univ. 3, (P- 5763) Bhise. V.M. (1967);Certain Properties Of Meijer Laplace Transform, Comp. Math. 18, (P- 1.6).Bushman, R.G. And The H-Function Associated With A Certain Class Of Shrivastava, H.M. (1990); Feynmann Integrasl. J. Phys.; A Math, Gen (23), (P- 4707-4710). Fox, C. (1961); The G Function And H-Function As Symmetrical Fourier Kernels, Trans. Amer. Math. Soc. 98. (P- 395-429). Gupta, K.C. (1964); On The Inverse Meijer Transform Of G-Function. Collect Math 16. (P- 45-54). Gupta K.C. And The H-Function Transform, J. Austral Math. Soc. Mittal P.K. (1970); 11 (P- 142148). Gupta K.C. And; The H-Function Transform II. J Austral. Math. Soc. Mittal P.K. (1971) 12. (P- 444450). Inayat Hussain A.A. (1987); New Properties Of Hypergeometric Series Dvivable From Feynmann Integrals : I Transformation And Reduction Formulae J. Phys. A Math. Gen 20 (P- 4109-4117). Inayat-Hussain A.A. (1987); New Properties Of Hypergeometric Series Derivable From Feynmann Integral II Generalization Of The K.K. (2013);
H Function J. Phys. A. Math. Gen (20), (P- 4119-4129).Ahluwalia
[11]
Comparative Study Of The I-Function, The H Function, The Generalized Legendre’s Associated Function And Their Applications, Ph.D. Thesis, Univ. Of B.U. Bhopal, India. Saxena, R.K. J. Ram And Unified Fractional Integrals Formulae For The
[12]
And Kalla, S.L. (2002);
[10]
Generalized
H Function Rev. Acad. Canor Cienc. 14 (P- 97-101)
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