Integral Involving H-function of Several Complex Varialbes and Exponential Function

Naresh Bhati; Rajeev Kumar Gupta

doi:10.17577/IJERTV13IS120050

Volume 13, Issue 12 (December 2024)

Integral Involving H-function of Several Complex Varialbes and Exponential Function

DOI : 10.17577/IJERTV13IS120050

Download Full-Text PDF Cite this Publication

Open Access
Article Download / Views: 18
Authors : Naresh Bhati, Rajeev Kumar Gupta
Paper ID : IJERTV13IS120050
Volume & Issue : Volume 13, Issue 12 (December 2024)
Published (First Online): 18-12-2024
ISSN (Online) : 2278-0181
Publisher Name : IJERT
License: This work is licensed under a Creative Commons Attribution 4.0 International License

PDF Version

View

Text Only Version

Integral Involving H-function of Several Complex Varialbes and Exponential Function

Integral Involving H-function and Exponential function

Naresh Bhati

Department of Mathematics and Statistics Jai Narain Vyas University,

Jodhpur, India

Rajeev Kumar Gupta

Department of Mathematics and Statistics Jai Narain Vyas University,

Jodhpur, India, Country

a ; A1 , A2 ,…Ar

; c1 , C1 ;

Abstract In this paper we have integrated a H-function of several

z1

j j j j

1, P

j j

1, p1

H z , z ,…, z H M , N ; m1 , n1 ;…; mr , nr

complex variables with combination of hypergeometric function and

an exponential function as product. To perform the integrations, we

1 2 r P,Q; p1 , q1 ;…; pr , qr

zr b , B1 , B2 ,…, Br

;d 1 , D1 ;

have used the definite integral listed in “Table of Integrals, Series, and Products” by Gradshteyn and Ryzhik. The integrations obtain in this

…;cr , Cr

j j j j

1,Q

j j

1, q1

(3)

paper will be useful to solve problems of mathematical, statistical and

j j

1

1, pr

r s

k

r … s1,…, sr k sk zk dsk

physical sciences where different kind of functions occurs as product. The result obtained from integration was put in compact form.

…;d r , Dr 2 i

j j 1, q 1 r

r

k 0

Keywords Multivariable H-function, Exponential function,

denote the H-function of r complex variables z1, z2 ,…, zr . Where

Hypergeometric function.

M b r Bks N 1 a r

Ak s

j j k

i 112 and s , s ,…, s ji k 1 j 1 k 1

1 2 r Q

r P r

j j k

1 b Bks a Aks

INTRODUCTION
The German mathematician Carl Friedrich Gauss (1777-1855)

(4)

j j k j j k

mk

j M 1

nk

k 1

j N 1

k 1

introduced a series called gauss hypergeometric series and represent

d k Dks 1 ck Cks

by s

j 1

j 1

(5)

j j k j j k

k k qk pk

2 F1 , ; ; x

xu

u u

(1)

1 d k Dks ck Cks

u 0

u

u!

j mk 1

where k 1, 2,…, r

j nk 1

where is the Pochhammer symbol defined by

a ,[ j 1, 2,…, P]; ck,[ j 1, 2,…, p ;k {1, 2,…, r}]
j j i

j j i

b ,[ j 1, 2,…, Q]; d k ,[ j 1, 2,…, q ;i {1, 2,…, r}]
are complex numbers,

( )u

1

1… u 1 u N

u 1

(2)

and their corresponding related coefficients

Ak ,[ j 1, 2,…, P];Ck[ j 1, 2,…, p ;k {1, 2,…, r}]

j j k

are positive real

The hypergeometric function is defined for |z| < 1, provided that is neither zero nor a negative integer. The gaussian hypergeometric function has fundamental importance in theory of special functions.

The most of functions used in physical sciences, applied

Bk ,[ j 1, 2,…, Q]; Dk[ j 1, 2,…, q ;k {1, 2,…, r}]
j j k

numbers.

And k , represent the contours start at the point k i and goes to the point k i with k R , k 1, 2,…, r . The integral in (3) is converges absolutely, under the conditions Srivastava et al. [5] if

mathematics, engineering etc., are expressible as its special cases.

The H-function of several complex variables, defined H. M. Srivastava and R. Panda in series of their research papers [5]. The

arg z .

k 2 k

j
1. k pk k Q
  j
  
  k
  
  k 1, 2,…, r
  
  qk k
  
  (6)
  
  H-function defined and represented in terms of a multiple Mellin-
  
  Where A
  
  k j
  
  j 1
  
  C
  
  j 1
  
  B
  
  j 1
  
  D 0
  
  j
  
  j 1
  
  Bernes type contour integral as
  
  Ak
  
  Ak Bk
2. nk
Bk ck

pk

Ck

N P

k j

j 1

j N 1

j j

M

j 1

j

j M 1

mk

j j 1

qk

j

j nk 1

(7)

Dk

Dk 0

, (k 1, 2,…, r)

j 1

j j

j mk 1

where M, N, P,Q, m , n , p , q are positive integer and restricted by the

kcu yu k a byu

k k k k

y a by u u

…

s ,…, s

0 N P ,

Q M 0 , and

q m 0 ,

p n 0,

k {1, 2,…, r}

and

k 0 u 0

k !u!2 ir

1 r

k k k

0 u 1 r

inequalities (6) suitably constrained values of the complex variables

s … s cS1 zS1 y1 S1 (a by)1 S1 …cSr zSr yr Sr (a by)r Sr dS …dS dy

z1, z2 ,…, zr . The points

zk 0, k 1, 2,.., r

1 1

and many exceptional

r r 1 1 r r 1 r

parameter values, being tacitly excluded. From Srivastava and Panda [5], we have

By interchanging the order of integration and summation, we get

u u

H z , z ,…, z o z e1 …. z er lim

z 0 , where

cu k

1 2 r 1 r

1 j mr j

r … s1,…, sr 1 s1 …r sr

k 0 u 0 k !u!2 i

r

d k

u 1 r

e lim Re j

k 1, 2,…, r

(8)

r

u k S

k k

i i

i

u i Si

S1 S1

Sr S1

1 j mr

D

y

1 a by

i 1

dy c1 z1 …cr z1 dS1..dSr

j

0

r r

Lemma:

cu k

u k i Si u i Si 1

a i

u u

…

1 i 1

r

From the table of integration, series and products I.S. Gradshteyn,

k 0 u 0 k !u!2 i r

u k i Si 1

i

u 1 r b 1

M.I. Ryzhik [3, (2007): Eq. 3.194 (7), p.316] We need the following

u k r S 1 r r

integration formula

i i

u i Si u k i Si 1

i 1

u

i 1

r S

i 1

xm a bx dx

n amn1 m 1n m 1

m 1

(9)

S S S S

i 1

i i

0 b
[n m 1 0, a 0;b 0]
n

s1,…, sr 1 s1 …r sr c1 1 z1 1 …cr r z1 1 dS1..dSr

a 1 cu k au u k

r r

i Si i Si

ai

u u

…

1 i 1

r
MAIN RESULTS:
b 1

k 0 u 0

bk u k !u!2 i r

i Si

bi

In this section we have obtain som integrals involving the product

u

u k r

1 r 1

r r

i Si 1 u u k i Si i Si 1

of the basic hypergeometric function with H-function of several

i 1 i 1 i 1

complex variable and exponential function.

u r S

i 1

i i

s ,…, s s … s cS1 zS1 …cSr zSr dS ..dS

Theorem 1:

If we take

1 0 ;

a 0; b 0 ;

0, 0 ;

0; 0; and

1 r 1 1

r r 1 1

r r 1 r

i o,i 0 , (i 1, 2,…, r) then the following integration hold

by virtue, interpreting of equation (1) and (3), we obtain the required result.

y a by e y

0

F , ; ; cy a by

2 1

a 1

a

ca

M 1, N 1;m , n ;..;m , n

c1z1 y 1 (a by) 1

a

e b F , ; ;

H

1 1 r r

a 1

ca

b 1 2 1 b

P 2,Q 1; p1 , q1 ;…, pr , qr

H

M , N ;m1 , n1 ;..;mr , nr

P,Q; p1 , q1 ;…, pr , qr

dx

1

e b F , ; ;

k o u 0

c z y

(a by) b

2 1

b

1 r

1 2 r

j

j j j

r r

r r

a1 1 b 1 c z

u k ; ,.., ;

(10)

a1 1 b 1 c z

1 1

u k ; ,.., ; a ; A , A

,…A

;

2, P 1

H

1 1 1 r

M 1, N 1; m1 , n1 ;..;mr , nr

P 2,Q 1; p1 , q1 ;…, pr , qr

u u k 1;

;

a r r b

r cr zr

u u k 1;1 1,..,r r ;

k o u 0

r r r

a ; A1 , A2 ,…Ar

; u; ,.., ; c1, C1

;…; cr, Cr

1

1 1

r

r r

a b cr zr 1 1,..,r r

u; ,.., ;c1 , C1

;…;cr, Cr

1 2 r

1 r j j

1, p

j j 1, p

j j j j

2, P 1

1 r j j

1, p

j j 1, p

b , B , B

,…, B

;d

, D

;…;d

, D

1

1 2 r

1

1

1

r r

r j j j j 1,Q 1 j j 1, q j j

1, qr

bj , Bj , Bj

,…, Bj

1,Q 1

; d j , Dj

1, q1

;…;d j , Dj

1, qr

Hence theorem 1 is proved.

The above integral will be convergence for condition (6), (7) and (8).

Theorem 2:

If we take

1 0;

a 0;b 0;

0, 0;

0; 0 and

Proof:

i o,i 0;

(i 1, 2,…, r)

then the following integration hold

y

y a by e F , ; ; cy a by

y

2 1

0

y a by e

0

2 F1 , ; ; cy a by

H c z y1 (a by)1 ,…, c z yr (a by)r dy

1

1

1 1

r r

c1z1 y (a by)

1

a

M , N ;m , n ;..;m , n a

ca

k H

1 1 r r

dx

e b F , ; ;

Now we replace

e y

by y and express the hypergeometric

P,Q; p1 , q1 ;…, pr , qr

c z yr (a by)r

b 1 2 1 b

k 0 k !

r r

a 1 1 b1 c z

2 k u u ;

(11)

function (1) and H-function of several complex variables (3), then

1 1 ;

we get

P 1,Q 2; p1 , q1 ;…, pr , qr 1 1 r r

k o u 0

H M 1, N 1; m1 , n1 ;..;mr , nr

a r r br c z

,…,

1 k u; ,.., ;

k yk

cu yu a byu

1 2 r

1 1

r r 1 r

r r

y a by

u u

…

s ,…, s

aj ; Aj , Aj

,…Aj

;cj , Cj

;…;c j , Cj

k !

k 0

u 0

u!2 ir

1 r

2, P 1

1, p1

1, pr

0 u 1 r

b , B1 , B2 ,.., Br

;1 u; ,.., ; d 1 , D1 ;..;d r , Dr

s … s cS zS y S (a by) S …cS zS y S (a by) S dS ..dS dy

j j j j

1 r j j j j

1 1 r r

1 1 1 1 1 1

1 1

r 1 1 1 1 1

r 1 1 r

2,Q 1

1, q1

1, qr

The above integral will be convergence for condition (6), (7) and (8).

c z y1 (a y)1

y a y

e y

F , ; ; cy H M , N ; m1 , n1 ; m2 , n2 1 1

dx

Theorem 3:

If we take

1 0;

a 0;b 0;

0, 0;

0

0; 0 and

2 1

P,Q; p1 , q1 ; p2 , q2

c z y2 (a y)2

2 2

1 1

a 1e a

F , ; ; ca H M 1, N 1; m , n ; m , n a c1 z1

i o,i 0;

(i 1, 2,…, r) then the following integration hold

2 1

k o u 0

1 1 2 2

2 2

P 2,Q 1; p1 , q1 ; p2 , q2

a2 2 c z

y a by e y

F , ; ; cy a by

2 k u ;;a ; A1 , A2

;c1 , C1

;c2 , C2

0 ,

j j j

2, P 1

j j 1, p

j j 1, p

2 1

ca

c z y

1 (a by)1

1 1 2 2

1 2

(15)

M , N ; m , n ;..; m , n

a

1 1 1

a

1 k u; 1 ;

H 1 1

r r dx

e b F , ; ;

;bj , Bj , Bj

; ;d j , Dj

;d j

, Dj

1 2

1 1

2 2

P,Q; p1 , q1 ;…, pr , qr

c z yr (a by)r

b 1 2 1

b

1, 2

1,Q 1 1,..,r

1, q1

1, qr

r r

a1 1 b1 c z

2 k u u ;

(12)

Eq. (12) reduce to (16) as follows

1 1 ;

H

M 1, N 1; m1 , n1 ;..; mr , nr

P 2,Q 1; p1 , q1 ;…, pr , qr

1 1,…,

r r

k o u 0

ar r br c z

1 k u; ,.., ;

r r 1 r

y

M , N ;m , n ;m , n

c1z1 y

1 (a y) 1

y a y e

2 F1 , ; ; cy H

1 1 2 2 dx

a ; A1 , A2 ,…Ar

; u; ,.., ;c1 , C1

;…; cr , Cr

P,Q; p1 , q1 ; p2 , q2

c z y2 (a y)2

j j j j

2, P 1

1 r j j

1, p

j j 1, p

0 2 2

b , B1 , B2 ,…, Br

;d 1 , D1

1

1

;…;d r , Dr

r

a 1e a

F , ; ; ca H M 1, N 1;m1 , n1 ;m2 , n2

j j j j 2,Q 1 j

j 1, q j j

1, qr

2 1

k o u 0

P 2,Q 1; p1 , q1 ; p2 , q2

a1 1 c z

2 k u ; 1 1, 2 2 ;

(16)

1 1

The above integral will be convergence for condition (6), (7) and (8).

a2 2 c z

1 k u; 1, 2 ;

2 2

a ; A1 , A2

;; ,.., ;c1 , C1

;c2 , C2

j j j

2, P 1

1 r j j

1, p

j j 1, p

Theorem 4: 1 2

If we take

1 0;

a 0;b 0;

0, 0;

0;

0; and

b , B1 , B2

;d 1 , D1

;d 2 , D2

j j j

2,Q 1

j j 1, q j j 1, q

i o,i 0;(i 1, 2,…, r) then the following integration hold

1 2

2 1

y a by e y

0

F , ; ; cy a by

Eq. (13) reduce to (17) as follows

c z y1 (a by)1

c z y1 (a y)1

H M , N ; m , n ;..; m , n dx a e F , ; ; ca

y a y e

2 F1 , ; ; cy HP,Q; p1 , q1 ; p2 , q2

c z y (a y) dx

1 1

1 1 r r

1

a

b

y

M , N ;m1 , n1 ;m2 , n2 1 1

2 2

P,Q; p1 , q1 ;…, pr , qr

c z yr (a by)r

b 1 2 1

b 0

2 2

r r

a 1e a

F , ; ; ca H M , N 2; m1 , n1 ; m2 , n2

a1 1 b 1 c z

2 k u u ;

2 1

k o u 0

P 2,Q 1; p1 , q1 ; p2 , q2

1 1 ;

(13)

2 k u ; ,

; u k ; , ;

H M , N 2; m1 , n1 ;..; mr , nr

1 1,…,r r

a1 1 c z

1 1 2 2 1 2

(17)

k o u 0

P 2,Q 1; p1 , q1 ;…, pr , qr

ar r b r c z

b , B1 , B2 ,…, Br ;

1 1

a2 2 c z

r r j j j j

1,Q

2 2

b , B1 , B2

;1 k u; , ;

u k ; ;a ; A1 , A2 ,…Ar

;c1 , C1

;…; cr , Cr

1 2

j j j

1 1

1,Q 1 2

2 2

,…,

j j j j

2, P 1

j j 1, p

j j 1, p

aj ; A

, Aj

;cj , Cj

; cj , Cj

1 r

1 r

j 2, P 1

1, p

1, p

1 k u; ,.., ;d 1 , D1

;…; d r , Dr

1 1

1 2

2 2

1 r j j

1, q1

j j

1, qr

d j , Dj

1, q

;d j

, Dj

1, q

The above integral will be convergence for condition (6), (7) and (8). The integrals (13) to (15) can be proved on lines similar to those of integral (12)
PARTICULAR CASES:
If we take r 2, 0 and b 1 in eq. (10), (11), (12) and (13) then Eq. (10) reduce to (14) as follows

1 2
CONCLUSION:

The H-function of several complex variables, is quite basic in nature. Therefore, on specializing the parameters of this function, we may obtain various other special functions such as Meijers G-function, Foxs H-function, Wrights generalized hypergeometric function, Wrights generalized Bessel function, Whittaker function, generalized hypergeometric function, Mac-Roberts E-function, modified Bessel function, Bessel function of first kind, binomial

c z y1 (a y)1

function, exponential function, etc. as its special cases, and therefore,

y a y e y F , ; ; cy H M , N ; m1 , n1 ; m2 , n2 1 1

dx

2 1

0

P,Q; p1 , q1 ; p2 , q2

c z y2 (a y)2

2 2

a1 1 c z

various unified integrals can be obtained as special cases of our results.

a 1e a

F , ; ; ca H M 1, N 1;m1 , n1 ;m2 , n2 1 1

(14)

This template, modified in MS Word 2007 and saved as a Word 97-

2 1

r r

k o u 0

P 2,Q 1; p1 , q1 ; p2 , q2 a2 2 c z

2003 Document for the PC, provides authors with most of the

u k ; ;a ; A1 , A2

; ; , ;c1 , C1

; c2 , C2

formatting specifications needed for preparing electronic versions of

,

j j j

2, P 1

1 2 j j 1, p j j

1, p2

1

1 2 their papers. All standard paper components have been specified for

u k 1;

1 2

1 1

2 2

three reasons: (1) ease of use when formatting individual papers, (2)

,

;bj , Bj , Bj

;d j , Dj

; d j

, Dj

automatic compliance to electronic requirements that facilitate the

1 1 2 2

1,Q 1 1, q1

1, qr

concurrent or later production of electronic products, and (3)

Eq. (11) reduce to (15) as follows

conformity of style throughout a conference proceeding. Margins, column widths, line spacing, and type styles are built-in; examples of the type styles are provided throughout this document and are identified in italic type, within parentheses, following the example.

Some components, such as multi-leveled equations, graphics, and tables are not prescribed, although the various table text styles are provided. The formatter will need to create these components, incorporating the applicable criteria that follow.

REFERENCES

Ashiq Hussain Khan, Neelam Pandey, Integrals Involving H-function of Several Complex Variables, International Journal of Scientific and Research Publications, Volume 7, Issue 2, February 2017.
Bateman Project, (1953) Higher Transcendental Function, Vol. I.
Gradshteyn, I.S. and Ryzhik, I.M., (2007), Table of Integrals, Series and Products, Edited by A. Jeffrey, Daniel Zwillinger, Academic Press in an imprint of Elsevier,
Mathai, A. M. and Saxena, R. K. The H-function with Application in Statistics and Other Disciplines, Wiley Eastern Limited, New Delhi, Bangalore, Bombay, (1978).
Srivastava, H. M. and Panda, R., Some expansion theorems and generating relations for the H -function of several complex variables. I and II, Comment. Math. Univ. St. Paul. 24, fasc. 2, 119-137 (1975); ibid. 25, fasc. 2, 167-197.
Panda, R. & Srivastava, H. (1976). Some bilateral generating functions for a class of generalized hypergeometric polynomials. Journal fÃ¼r die reine und angewandte Mathematik, 976 (283-284), 265-274. https://doi.org/10.1515/crll.1976.283-284.265
Rainville, E. D., (1971), Special Functions, Chelsea Publication Company,
Bronx, New York.
Srivastava, H. M., K.C. Gupta and S. P. Goyal, (1982), The H-Functions of One and Two Variables with Applications, South Asian Publishers, New Delhi, Madras.
https://doi.org/10.1002/asna.2113060427.
Srivastava, H.M., (1972), A contour integral involving foxs H-function,

Indian J. Math.