Cmjv05i04p0390

JOURNAL OF COMPUTER AND MATHEMATICAL SCIENCES An International Open Free Access, Peer Reviewed Research Journal www.compmath-journal.org

ISSN 0976-5727 (Print) ISSN 2319-8133 (Online) Abbr:J.Comp.&Math.Sci. 2014, Vol.5(4): Pg.390-401

On Some Definite and Indefinite Integrals Involving I-Function of Two Variables S. S. Srivastava and Anshu Singh Department of Mathematics, Govt., P. G. College, Shahdol, M. P., INDIA. (Received on: August 13, 2014) ABSTRACT The object of this paper is to evaluate some definite and indefinite integrals involving the I-function of two variables. As we know that Integrals are useful in connection with the study of certain boundary value problems. It is also helpful for obtaining the expansion formula. It also used in the study of statistical distribution, probability and integral equation. Looking importance and usefulness of integral in various fields we have established some new integrals of various types, which will be helpful in the study of boundary value problems, expansion formula, statistical distribution, probability and integral equation.

Keywords: Mellin-Barnes type integral, *multiplication formula for the Gamma-function, boundary value problems, expansion formula, statistical distribution,integral equation.

The I-function of one variable introduced by Saxena2, will be represented as follows:

2.1 INTRODUCTION 2.1.1. Definition of I Function:

n [(aj, αj)1, n], [(aji, αji)n + 1, pi] ] = (1/2πi) ∫ θ(s) xs ds I pm, [x| [(b i, q i: r j, βj)1, m], [(bji, βji)m + 1, qi] L m Π Γ(bj – βjs)

θ (s) =

j=1

r Σ i=1

qi Π

n Π Γ(1 – aj + αjs) j=1

Γ(1 – bji + βjis)

j=m+1

(1.3.1)

pi Π

, Γ(aji – αjis)

j=n+1

Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411)

391

S. S. Srivastava, et al., J. Comp. & Math. Sci. Vol.5 (4), 390-401 (2014)

integral is convergent, when (B >0, A ≥ 0), where n

pi

m

qi

Β = Σ αj – Σ αji + Σ βj – Σ βji , j=1

j=n+1

pi

A=

j=1

(1.3.2)

j=m+1

qi

Σ αji – Σ βji , j=1

j=1

|arg x| < ½ Bπ, ∀ i ∈ (1, 2, …, r). pi (i = 1, 2, …, r); qi(i = 1, 2, …, r); m, n are integers satisfying 0 ≤ n ≤ pi, 0 ≤ m ≤ qi, (i = 1, 2, …, r); r is finite αj, βj, αji, βji are real and positive and aj, bj, aji, bji are complex numbers such that αj (bh + v) ≠ Bh (aj – 1 – k), for v, k = 0, 1, 2, ….. h = 1, 2, …, m; j = 1, 2, …, r; L is a contour runs from σ – i∞ to I [ xy ] = I 0, n

: m1, n1 :m2, n2 [ x| pi, qi: r : pi´, qi´: r´ :pi´´, qi´: r´´ y

σ + i∞ (σ is real), in the complex s-plane such that the poles of s = (aj – 1 – v) | αj j = 1, 2, …, n; v = 0, 1, 2, …. lie to the left hand side s = (bj + v) | βj j = 1, 2, …, m; v = 0, 1, 2, …. and right of L. The I–function of two variables introduced by Sharma & Mishra9, will be defined and represented as follows:

[(aj; αj, Aj)1, n], [(aji; αji, Aji) n + 1, pi] [(bji; βji, Bji)1, qi]

: [(cj; γj)1, n1], [(cji´; γji´) n1 + 1, pi´ ]; [(ej; Ej)1, n2], [(eji´´; Eji´´) n2 + 1, pi´´] : [(dj; δj)1, m1], [(dji´; δji´) m1 + 1, qi´ ]; [(fj; Fj)1, m2], [(fji´´; Fji´´) m2 + 1, qi´´] ∫

=

∫

L1 L2

φ1(ξ, η) θ2(ξ) θ3(η)xξ yη dξ dη,

(1.3.3)

where

φ1 (ξ, η) =

n Π Γ ( 1 − aj + αjξ + Ajη) j=1 r

Σ [ i=1

θ2 (ξ) =

pi

qi

Π Γ(aji - αjiξ − Αjiη) Π Γ (1 − bji + βjiξ + Bjiη) j = n+ 1

j=1

m1 Π Γ (dj − δjξ) j=1 q r´

,

i´

Σ [

Π

i´ = 1

j=m +1 1

n1

Π Γ (1 − cj + γjξ)

j=1

,

pi´

Γ (1 − dji´ + δji´ξ) Π

Γ (cji´ − γji´ξ) ]

j=n +1 1

Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411)

S. S. Srivastava, et al., J. Comp. & Math. Sci. Vol.5 (4), 390-401 (2014)

θ3 (η) =

m2 n2 Γ (fj − Fjη) Π Γ (1 − ej + Ejη) Π j=1 j=1 r ´´

qi´´

Σ [

Π

i´´ = 1

j=m +1 2

Γ (1 − fji´´ + Fji´´η) Π

V=− Σ j=n+1

and

j=1

j =1

The contour L2 is in the η−plane and runs from – ω∞ to + ω∞, with loops, if necessary, to ensure that the poles of Γ (fj − Fjη) (j=1,....., n2) lie to the right, and the poles of Γ (1 − ej + Ejη) (j = 1, ..., m2), Γ ( 1 − aj + αjξ + Ajη) (j = 1, ..., n) to the left of the contour. Also

j=1

j =1

n1 pi´ δji´ + Σ γj − Σ γji´ > 0, j =1

j = m1 + 1

q qi m2 i´´ Aji − Σ Bji − Σ Fj − Σ

pi

with loops, if necessary, to ensure that the poles of Γ(dj−δjξ) (j = 1, ..........., m1) lie to the right, and the poles of Γ (1 − cj + γjξ) (j = 1, ..., n1), Γ ( 1 − aj + αjξ + Ajη) (j = 1, ..., n) to the left of the contour.

qi´ qi pi´ + Σ γji´ − Σ βji − Σ δji´ < 0, j=1 j =1 j=1 qi qi´´ pi´´ + Σ Eji´´ − Σ Bji − Σ Fji´´ < 0, j =1 j=1 j=1

qi qi´ m1 Σ αji − Σ βji + Σ δj − Σ

j=n+1

Γ (eji´´ − Eji´´η) ]

j=n +1 2

pi

U=

,

pi´´

x and y are not equal to zero, and an empty product is interpreted as unity pi, pi´, pi´´, qi, qi´, qi´´, n, n1, n2, nj and mk are non negative integers such that pi ≥ n ≥ 0, pi´ ≥ n1 ≥ 0, pi´´ ≥ n2 ≥ 0, qi > 0, qi´ >0, qi´´ > 0, (i = 1, …, r; i´ = 1, …, r´; i´´ = 1, …, r´´; k = 1, 2) also all the A’s, α’s, B’s, β’s, γ’s, δ’s, E’s and F’s are assumed to be positive quantities for standardization purpose; the definition of Ifunction of two variables given above will however, have a meaning even if some of these quantities are zero. The contour L1 is in the ξ−plane and runs from – ω∞ to + ω∞,

pi R = Σ αji j=1 pi S = Σ Aji j=1

392

j = m2 + 1

Fji´´

n2 pi´´ + Σ Ej − Σ j =1

(1.3.4)

j = n1 + 1

j = n2 + 1

Eji´´ > 0,

| arg x | < ½ Uπ, | arg y | < ½ Vπ.

Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411)

(1.3.5)

393

S. S. Srivastava, et al., J. Comp. & Math. Sci. Vol.5 (4), 390-401 (2014)

2.1.2 REVIEW OF LITRATURE Ronghe1, Saxena2, Sharma3, 5 6,7 Goyal , Mohan , Srivastava , Shalesh8 and several other authors haveevaluated some definite, indefinite and double integrals involving theI-function and other generalized hypergeometric functions 4

provided that Ď > â&#x2C6;&#x2019;1.

From Kuipers12:

1 â&#x2C6;&#x2019; x 1 + x P

=

In our investigation we shall need the following results: From Erdelyi10: /

( / ) ( / ) , ( / )

(sinθ) e

dθ

Ď&#x20AC;Î&#x201C;( )e / , (2.2.2) + + 1 â&#x2C6;&#x2019; + 1 2 Î&#x201C;( )Î&#x201C;( ) 2 2 provided that Re (ν) > 0;

=

(sinθ) cosuθ dθ

Ď&#x20AC;u Ď&#x20AC;Î&#x201C;(1 + Ď )cos ( 2 ) = Ď +u Ď â&#x2C6;&#x2019;u , 2 Î&#x201C;(1 + 2 )Î&#x201C;(1 + 2 ) provided that Ď > â&#x2C6;&#x2019;1.

=

ಥడభ

: ( ) ]

â&#x2C6;&#x2018; ,

dx

ŕą&#x152; ๨ ๨ ŕą&#x2018; ๪ ๪ (

)

ŕ°Ž(ŕą&#x17E;๨జ๥๪) ŕą? ! ŕą&#x2019; ๪ ! ( ๨

ಥజఎŕą&#x17E;๨జఎ๥๪¹๣జభ ) ŕ°Ž

,

(2.2.6) where h and k are positive integers, P < Q (or P = Q + 1 and |c| < 1), U < V(or U = V + 1 and |d| < 1), no one of the β and δ are zero negative integer and Re (Ď&#x2030;) > 0.

From Erdelyi14:

( , ) 1 â&#x2C6;&#x2019; x 1 + x P (x) dx

=

( ) ( ) ( )

,

, ; 1], F [ , ;

(2.2.7)

where Re(Ď ) > â&#x2C6;&#x2019; 1, Re(Ď&#x192;) > â&#x2C6;&#x2019; 1.

(2.2.3)

1 â&#x2C6;&#x2019; x P (x) dx

=

(sinθ) sinuθ dθ

( / ) ( / )

, (2.2.8)

( / ) ( / ) (

!" # " ) ( )

where Re(2Îą + δ) > 0, Re(2Îą â&#x2C6;&#x2019; δ) > 0.

Ď&#x20AC;u Ď&#x20AC;Î&#x201C;(1 + Ď )sin ( 2 ) = Ď +u Ď â&#x2C6;&#x2019;u , 2 Î&#x201C;(1 + 2 )Î&#x201C;(1 + 2 )

: sin x e F [ ]

ŕą&#x;๣ŕ˛&#x2DC;

(2.2.1)

provided that Ď > 0, Ď&#x192; > 0.

From Nielsen11:

provided that Re (Ď&#x192;) > - 1.

From Mishra13:

F [

sin θ cos θ dθ

= (1/2)

(2.2.5)

2.2 FORMULAE USED:

(x) dx

/ [ ] [ ] [ ] [ ] ,

] [ ] [ ] [ ] [

( , )

From Rainville21: (2.2.4)

,

1 â&#x2C6;&#x2019; x 1 + x [P

(x)] dx

Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411)

394

S. S. Srivastava, et al., J. Comp. & Math. Sci. Vol.5 (4), 390-401 (2014) $!"!# ( ) ( ) , !( ) ( )

=

(2.2.9)

where Re(Îą) > â&#x2C6;&#x2019; 1, Re(β) > â&#x2C6;&#x2019; 1; and

1

â&#x2C6;&#x2019; x 1 +

,

,

x P x P x dx

= 0,

(2.2.10)

for m â&#x2030; n, Re(Îą) > â&#x2C6;&#x2019; 1, Re(β) > â&#x2C6;&#x2019; 1; From MacRobert15:

sin (2n + 1)θ sinθ dθ =

% â&#x2C6;&#x161; ( ) ( )

( ) ( )

,

(2.2.11)

where Re(3 â&#x2C6;&#x2019; 2u) > 0, n = 0, 1, 2, â&#x20AC;Ś;

cos nθ sin θ/2 dθ

# â&#x2C6;&#x161; ( ) ( )

=

( ) ( )

/

,

(2.2.12)

(cosθ) e ( ) dθ (2.2.13)

&- S x = â&#x2C6;&#x2018; # X A , , ! (n = 0, 1, 2, â&#x20AC;Ś), (2.2.16) where m is arbitrary positive integer, and the coefficients An,u (n, u â&#x2030;Ľ 0) are arbitrary constants, real or complex.

From Saran19: 2Îą Î&#x201C;(Îą) Î&#x201C;(Îą) 1 â&#x2C6;Ť (1 + x)Îą â&#x20AC;&#x201C; 1 Pn(x) dx = , â&#x20AC;&#x201C;1 Î&#x201C;(Îą + n + 1) Î&#x201C;(Îą â&#x2C6;&#x2019; n) (2.2.17) provided (Îą â&#x2C6;&#x2019; n) is not a negative integer.

From Erdelyi20:

[ / ] ( )

â&#x2C6;&#x161;Ď&#x20AC; Îą â&#x2C6;&#x2019; Ď&#x192; â&#x20AC;&#x201C; 1 Î&#x201C;(1 + Ď&#x192;)

â&#x2C6;Ť Pν(x)x dx = 0

Î&#x201C;(1 + Ď&#x192;/2 â&#x2C6;&#x2019; ν/2) Î&#x201C;(Ď&#x192;/2 + ν/2 + 3/2)

2.3 ESTABLISHMENT OF DEFINITE AND INDEFINITE INTEGRALS INVOLVING I-FUNCTION OF TWO VARIABLE

%( )

** (sinθ) e

I[$ ,

= (2.2.14)

where Re(Îą + β) > â&#x2C6;&#x2019; 1, Re(Îą) > â&#x2C6;&#x2019; 1/2;

where Re(β + δ) < Re(s) < 1, ι > 0.

From Shrivastava18:

In this section, we shall establish following integrals:

() $!"!# *+ #

( ) (

)

$!# ( / ) ( )

t J (Îąt)J (Îąt) dt

(2.2.15)

(2.2.18)

e x J (x) dx =

,

Re (Ď&#x192;) > â&#x20AC;&#x201C; 1.

where Re(m + n) > 1.

From Mathai17:

!

" , " , ) ( )

(

Ď&#x192;

/ ( ) = &!' ( ) ( ),

"! !, , # , "( , )

1

where Re(1 â&#x2C6;&#x2019; 2u) > 0, n = 0, 1, 2, â&#x20AC;Ś From MacRobert16:

=

[

] dθ

! * / , # ; , ; % , % I& ,' :(;&/ ,'/ :(/ ;&// ,'// :(´´ . # ) ) ) ) ) )

% â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Ś...: , * ,â&#x20AC;Śâ&#x20AC;Ś..:â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Śâ&#x20AC;Ś. $|â&#x20AC;Ś.,â&#x20AC;Śâ&#x20AC;Ś:â&#x20AC;Śâ&#x20AC;Ś..,( / / Âą / ,*):â&#x20AC;Ś,â&#x20AC;Ś. ],

(2.3.1)

provided that h > 0, U > 0, > 0, |argΞ| <

UĎ&#x20AC;, where U and V are given in (1.3.4) and

(1.3.5).

Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411)

395

S. S. Srivastava, et al., J. Comp. & Math. Sci. Vol.5 (4), 390-401 (2014)

Proof:

Proof :

Expressing the I-function of two variables in the integrand as a Mellin-Barnes type integral (1.3.3) and changing the order of the θ-integral and t, u-integral, which is justified due to the absolute convergence of the integrals involved in the process, we have

Expressing the I-function of two variables in the integrand as a Mellin-Barnes type integral (1.3.3) and changing the order of the x-integral and t, u-integral, which is justified due to the absolute convergence of the integrals involved in the process, we have

1 Ď&#x2022; t, u θ t θ u Ξ Ρ 2Ď&#x20AC;i

1 Ď&#x2022; (t, u)θ (t)θ, (u)Ξ- Ρ (2Ď&#x20AC;i) +# +

sinθ e dθ dtdu.

Now evaluate the inner-integral with the help of the formula (2.2.2), we get 1 Ď&#x2022; (t, u)θ (t)θ, (u)Ξ- Ρ (2Ď&#x20AC;i) +# + Î&#x201C; + 2â&#x201E;&#x17D; 0

+ + 1 â&#x2C6;&#x2019; + 1 23 4506 Î&#x201C; + â&#x201E;&#x17D; Î&#x201C; + â&#x201E;&#x17D; 2 2 dtdu.

12

On applying (1.3.3), the result (2.3.1) is established. **

1

â&#x2C6;&#x2019; x 1 +

x P , (x)

( )

I[

]dx

1 â&#x2C6;&#x2019; x 1 + x - P , x dx dtdu.

Now evaluate the inner-integral with the help of the formula (2.2.5), and applying (1.3.3), the definition of the I-function of two variables, we get the result (2.3.2). / ** cos (ux) sin x/2 I[ ]dx

, ; , ; ,

= â&#x2C6;&#x161;(Ď&#x20AC;) I& ,'# :(;& / ,' / :(/%;&//%,'// :(´´ )

%

)

)

)

)

)

# â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Ś...:. # ,*/,â&#x20AC;Śâ&#x20AC;Ś.., # ,* :â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Śâ&#x20AC;Ś.

[ $|â&#x20AC;Ś...,â&#x20AC;Śâ&#x20AC;Ś:

# ,* ,â&#x20AC;Śâ&#x20AC;Ś, # ,* :â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Śâ&#x20AC;Ś.

], (2.3.3)

provided that h > 0, Ď&#x2030;1 > 0 and U > 0, >

0, |argΞ| < UĎ&#x20AC;, where U and V are given in (1.3.4) and (1.3.5).

& ' &!' #!7!'/ [ ] [ ]

& ' &!' [

] [

]

, # ; , ; % , % // &) ,') :(;&/) ,'/) :(/ ;&// ) ,') :(´´

= I [

, , , :â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Śâ&#x20AC;Ś.

â&#x20AC;Ś...,â&#x20AC;Śâ&#x20AC;Ś: , , , ,â&#x20AC;Śâ&#x20AC;Ś..:â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Śâ&#x20AC;Ś.

â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Ś...:â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś.,

|

Proof : ],

(2.3.2) provided that k, m and k - (m+n)/2 are nonnegative integers, δ > 0 and U > 0, >

0, |argΞ| < UĎ&#x20AC;, where U and V are given in (1.3.4) and (1.3.5).

Expressing the I-function of two variables in the integrand as a Mellin-Barnes type integral (1.3.3) and changing the order of the x-integral and t, u-integral, which is justified due to the absolute convergence of the integrals involved in the process, we have

Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411)

396

S. S. Srivastava, et al., J. Comp. & Math. Sci. Vol.5 (4), 390-401 (2014)

1 Ď&#x2022; (t, u)θ (t)θ, (u)Ξ- Ρ (2Ď&#x20AC;i) +# +

x ( # *-) cos ux sin dx dtdu.

2

Now evaluate the inner-integral with the help of the formula given in Bajpai [34]:

3,1

1 â&#x2C6;&#x2019; y 1 + y 1 P

(y) dy

Now evaluate the inner-integral with the help of the formula given in Bajpai33:

=

x # cos ux sin dx 2

and applying (1.3.3), the definition of the Ifunction two variables, we get the result (2.3.4).

= â&#x2C6;&#x161;

( # ) ( # ) #

( # ) ( # )

[

%!

8!9 !# [ 0 1] , # ; , ; % , % . I& ,' :(;&/ ,'/ :(/ ;&// ,'// :(´´ 2! ) ) ) ) ) )

â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Ś...: " $, ,â&#x20AC;Śâ&#x20AC;Ś, ! $, :â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Śâ&#x20AC;Ś. ], |â&#x20AC;Ś...,â&#x20AC;Śâ&#x20AC;Ś: , ,â&#x20AC;Śâ&#x20AC;Ś.., " , :â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Śâ&#x20AC;Ś.

(2.3.4)

provided that k > 0, Ď&#x2030;2 > 0 and U > 0, >

0, |argΞ| < UĎ&#x20AC;, where U and V are given in (1.3.4) and (1.3.5).

%( )

** (sinθ) cosuθ I[$

, ; , ; , )

)

)

)

)

â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Ś...:â&#x20AC;Śâ&#x20AC;Ś.., , , , :â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Śâ&#x20AC;Ś.

â&#x20AC;Ś...,â&#x20AC;Śâ&#x20AC;Ś: , , , ,â&#x20AC;Śâ&#x20AC;Ś.:â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Śâ&#x20AC;Ś.

[ |

1 Ď&#x2022; (t, u)θ (t)θ (u)Ξ Ρ (2Ď&#x20AC;i)

",!

1 â&#x2C6;&#x2019; y 1 + y ! P (y) dy dtdu.

(2.3.5)

],

%( )

** (sinθ) sinuθ I[$

] dθ

, ; , ; ,

% % = â&#x2C6;&#x161;Ď&#x20AC; sin ( ) I& ,'# :(;& / ,' / :( / ;&// ,'// :(´´ )

)

)

)

)

)

â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Ś...:â&#x20AC;Śâ&#x20AC;Ś.., , , , :â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Śâ&#x20AC;Ś. ],

â&#x20AC;Ś...,â&#x20AC;Śâ&#x20AC;Ś: , , , ,â&#x20AC;Śâ&#x20AC;Ś.:â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Śâ&#x20AC;Ś.

Expressing the I-function of two variables in the integrand as a Mellin-Barnes type integral (1.3.3) and changing the order of the y-integral and t, u-integral, which is justified due to the absolute convergence of the integrals involved in the process, we have

)

provided that Ď > â&#x2C6;&#x2019;1, δ > 0 and U > 0, >

0, |argΞ| < UĎ&#x20AC;, where U and V are given in (1.3.4) and (1.3.5).

[ |

Proof:

] dθ

% % = â&#x2C6;&#x161;Ď&#x20AC; cos ( ) I& ,'# :(;& / ,' / :( / ;&// ,'// :(´´

,

(" ) ( ! )

,

and applying (1.3.3), the definition of the Ifunction of two variables, we get the result (2.3.3).

** 1 â&#x2C6;&#x2019; y 1 + y ! P ",! (y) I[ ( #) ] dy

=

( % &) ( ) (" )

(2.3.6)

provided that Ď > â&#x2C6;&#x2019;1, δ > 0 and U > 0, >

0, |argΞ| < UĎ&#x20AC;, where U and V are given in (1.3.4) and (1.3.5).

Proof: Expressing the I-function of two variables in the integrand as a Mellin-Barnes type integral (1.3.3) and changing the order of the θ-integral and t, u-integral, which is justified

Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411)

397

S. S. Srivastava, et al., J. Comp. & Math. Sci. Vol.5 (4), 390-401 (2014)

due to the absolute convergence of the integrals involved in the process, we have

the integrals involved in the process, we have

1 Ď&#x2022; (t, u)θ (t)θ, (u)Ξ- Ρ (2Ď&#x20AC;i) +# +

1 Ď&#x2022; (t, u)θ (t)θ> (u)Ξ Ρ?

(2Ď&#x20AC;i) =ŕ°­ =ŕ°Ž 6

sinθ -

cosuθ dθ dtdu.

଴

** sin x e 'F ([ :) ] - ´ :.

] dx *´F ,´ [ ´ ] . I[

, ; , ; , )

)

)

: : ´ < < 4 5 (! ´ < -! :

)

Evaluate the inner-integral with the help of the formula (2.2.6) and using multiplication formula for the Gammafunction given in Erdelyi [32, p.4, (11)], we get â&#x2C6;&#x161;Ď&#x20AC;e@A / â&#x2C6;&#x2018; ,B

ŕą&#x152; ๨ ๨ ŕą&#x2018; ๊ ๊ ŕą? ! ŕą&#x2019; ๊ B! ๨

1 . Ď&#x2022; (t, u)θ (t)θ> (u)Ξ Ρ?

(2Ď&#x20AC;i) =ŕ°­ =ŕ°Ž 6 Ď&#x2030; + 2hr + 2ks Ď&#x2030; + 2hr + 2ks + 1 Î&#x201C; + Îťt Î&#x201C; + Îťt 2 2

Ď&#x2030; + 2hr + 2ks Âą m + 1 Î&#x201C; + Îťt 2 dtdu.

% % I& ,'# :(;& / ,' / :(/ ;&// ,'// :(´´ )

ఎ๥

ŕ°Žŕą&#x17E;

dtdu.

Now evaluate the inner-integral with the help of the formula (2.2.3), and applying (1.3.3), the definition of the I-function of two variables, we get the result (2.3.5). **Similarly on applying (2.2.4), the result (2.3.6) is established.

= â&#x2C6;&#x161;Ď&#x20AC;e / â&#x2C6;&#x2018;(,-#

ŕŽ

ŕŽ&#x201C;ŕą&#x2018; :ŕ­˘á&#x2C6;şŕ­ąŕ­§ŕ­Źŕ­śá&#x2C6;ť ŕŽ&#x2018; :ŕ­Ąá&#x2C6;şŕ­ąŕ­§ŕ­Źŕ­śá&#x2C6;ť 67 sin x னାଶŕŽ&#x203A;୲ିଵ e୧୍ଡ଼ ŕ­&#x201D;F ŕ­&#x2022;[ ŕą&#x152; ŕŽ&#x2019;ŕą?] ŕ­&#x2122;F ŕ­&#x161;[ ŕŽ&#x201D;ŕą&#x2019; ]dx8

)

/ / ,02,1 ,02,â&#x20AC;Ś:â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś. â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Ś...:1

[ | ], / Âą ,0):â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Śâ&#x20AC;Ś. â&#x20AC;Ś...,â&#x20AC;Śâ&#x20AC;Ś:â&#x20AC;Śâ&#x20AC;Ś..,(

(2.3.7)

provided that h and k are positive integers, P < Q (or P = Q + 1 and |c| < 1), U´ < V´ (or U´ = V´ + 1 and |d| < 1), no one of the β and δ ´ are zero negative integer and Re (Ď&#x2030;) > 0

and and U > 0, > 0, |argΞ| < UĎ&#x20AC;, where

U and V are given in (1.3.4) and (1.3.5).

Proof of (2.3.7)

Now applying (1.3.3), the value of the integral (2.3.7) is obtained. %

**

x e H x I[$

] dx

, ; , ; ,

% % = â&#x2C6;&#x161;Ď&#x20AC; 2 I& ,'# :(;& / ,' / :(/;&// ,'// :(´´ )

[

%

)

)

)

)

â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Ś...: , * ,â&#x20AC;Śâ&#x20AC;Ś:â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Śâ&#x20AC;Ś. $|â&#x20AC;Ś...,â&#x20AC;Śâ&#x20AC;Ś:â&#x20AC;Śâ&#x20AC;Ś.,( ,*):â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Śâ&#x20AC;Ś. ],

)

(2.3.8)

provided that h > 0, Ď = 0, 1, 2, ... and

U > 0, > 0, |argΞ| < UĎ&#x20AC;, where U and V are given in (1.3.4) and (1.3.5).

Proof:

Expressing the I-function of two variables in the integrand as a Mellin-Barnes type integral (1.3.3) and changing the order of the x-integral and t, u-integral, which is justified due to the absolute convergence of

To establish (2.3.8), expressing the Ifunction of two variables in the integrand as a Mellin-Barnes type integral (1.3.3) and changing the order of the x-integral and t, u-

Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411)

398

S. S. Srivastava, et al., J. Comp. & Math. Sci. Vol.5 (4), 390-401 (2014)

integral, which is justified due to the absolute convergence of the integrals involved in the process, we have 1 Ď&#x2022; (t, u)θ (t)θ, (u)Ξ- Ρ (2Ď&#x20AC;i) +# +

x *- e H x dx dtdu.

Now evaluate the inner-integral with the help of the formula given in Bajpai35: x

e

** y

( ) , ( )

% 9 J y I[$ ] dy

, ; , ; ,

% % = 2: I& ,'# :(;& / ,'/ :( / ;&// ,'// :(´´ )

)

)

)

)

)

% â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Ś...:â&#x20AC;Śâ&#x20AC;Ś..,â&#x20AC;Śâ&#x20AC;Ś:â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Śâ&#x20AC;Ś. [ $|â&#x20AC;Ś...,â&#x20AC;Śâ&#x20AC;Ś:. 7!., /,â&#x20AC;Śâ&#x20AC;Ś..,.7 ., /:â&#x20AC;Ś,â&#x20AC;Śâ&#x20AC;Ś. ],

Now evaluate the inner-integral with the help of the formula given in Bajpai [36]:

y

(2.3.9)

provided that k > 0 and U > 0, >

0, |argΞ| < UĎ&#x20AC;, where U and V are given in (1.3.4) and (1.3.5).

J y dy = 2

:

(

C!D )

!E C ( )

,

and applying (1.3.3), the definition of the Ifunction, we get the result (2.3.9).

**

1 â&#x2C6;&#x2019; x 1 + ,

and applying (1.3.3), the definition of the Ifunction of two variables, we get the result (2.3.8).

x P

H x dx

= â&#x2C6;&#x161;Ď&#x20AC; 2 ( )

y - J y dy dtdu.

%( ) F ( )

(x)I[$

] dx

( )& ( )& = 2 â&#x2C6;&#x2018; # ( )& ! , # ; , ; % , % I& ,' :(;&/ ,'/ :(/ ;&// ,'// :(´´ ) ) ) ) ) )

[

â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Ś...: , ,â&#x20AC;Śâ&#x20AC;Ś..:â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Śâ&#x20AC;Ś. |â&#x20AC;Ś...,â&#x20AC;Śâ&#x20AC;Ś: ,- , , - ,â&#x20AC;Śâ&#x20AC;Ś,:â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Śâ&#x20AC;Ś. ],

(2.3.10) provided that Re(Ď + δΝ) > â&#x2C6;&#x2019; 1, Re(Ď&#x192; + γΝ) > â&#x2C6;&#x2019; 1, for some suitable constant Îť and

U > 0, > 0, |argΞ| < UĎ&#x20AC;, where U and V are given in (1.3.4) and (1.3.5).

6

, 1 â&#x2C6;&#x2019; x G 1 + x H PI (x) I[K(6 L) ** 06 J

ŕ˛&#x2039; (60L)ŕ°ˇŕ˛&#x152;

] dx

= 2 â&#x2C6;&#x2018; #

( )& ( )& ( )& ! , ; , ; , I& ,'# :(;& / ,' / :(/%;&//%,'// :(´´ ) ) ) ) ) )

(expression continue)

Proof: To establish (2.3.9), expressing the I-function of two variables in the integrand as a Mellin-Barnes type integral (1.3.3) and changing the order of the y-integral and t, uintegral, which is justified due to the absolute convergence of the integrals involved in the process, we have

â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Ś...: , , ,- ,â&#x20AC;Ś..:â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Śâ&#x20AC;Ś. [ ], |â&#x20AC;Ś...,â&#x20AC;Śâ&#x20AC;Ś: ,- ,â&#x20AC;Śâ&#x20AC;Ś:â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Śâ&#x20AC;Ś.

(2.3.11) provided that Re(Ď + δΝ) > â&#x2C6;&#x2019; 1, Re(Ď&#x192; + γΝ) > â&#x2C6;&#x2019; 1, for some suitable constant Îť and

U > 0, > 0, |argΞ| < UĎ&#x20AC;, where U and V are given in (1.3.4) and (1.3.5).

Proof:

1 Ď&#x2022; (t, u)θ (t)θ, (u)Ξ- Ρ (2Ď&#x20AC;i) +# +

In the left-hand side we use the definition of I-function of two variables in the integrand

Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411)

399

S. S. Srivastava, et al., J. Comp. & Math. Sci. Vol.5 (4), 390-401 (2014)

as a Mellin-Barnes type integral (1.3.3) and changing the order of the x-integral and t, uintegral, if permissible, we obtain 1 Ď&#x2022; (t, u)θ (t)θ> (u)Ξ Ρ?

(2Ď&#x20AC;i) =ŕ°­ =ŕ°Ž 6

1 â&#x2C6;&#x2019; x G 1 + x H0 PI 6

,

06

Using (2.2.7), we get

x dx dtdu.

1 Ď&#x2022; (t, u)θ (t)θ (u)Ξ Ρ (2Ď&#x20AC;i)

1 Ď&#x2022; (t, u)θ (t)θ (u)Ξ Ρ (2Ď&#x20AC;i) 3

Î&#x201C; Ď&#x192; â&#x2C6;&#x2019; Îłt + 1 Î&#x201C; Ď + δt + 1 ஡ା஢ାá&#x2C6;şŕŹżŕŽ&#x201C;ାŕŽ&#x201D;á&#x2C6;ťŕ­˛ ି୬,ŕŽ&#x2018;ାŕŽ&#x2019;ା୬ାଵ,஡ାŕŽ&#x201D;୲ାଵ; 2 ଷF ଶ [ŕŽ&#x2018;ାଵ,஡ା஢ାá&#x2C6;şŕŹżŕŽ&#x201C;ାŕŽ&#x201D;á&#x2C6;ťŕ­˛ŕŹžŕŹś; 1]4 Î&#x201C; Ď + Ď&#x192; â&#x2C6;&#x2019; Îłt + δt + 2

dtdu.

Arranging the parameters, we get the required result. ** The proof of the integral (2.3.11) would run parallel to what we have obtained above.

** sin (2n + 1)θ sinθ I[

, ; , ; ,

] dθ

= â&#x2C6;&#x161;Ď&#x20AC; I& ,'# :(;& / ,' / :(/%;&//%,'// :(´´ )

[

% F

)

)

)

)

)

â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Ś...:. ,*/,â&#x20AC;Ś..,( ,*):â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Śâ&#x20AC;Ś. #

$|â&#x20AC;Ś...,â&#x20AC;Śâ&#x20AC;Ś: ,* ,â&#x20AC;Śâ&#x20AC;Ś.,(

,*):â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Śâ&#x20AC;Ś. ],

(2.3.12) provided that Re(3 â&#x2C6;&#x2019; 2u) > 0, n = 0, 1, 2, â&#x20AC;Ś,

h > 0 and U > 0, > 0, |argΞ| < UĎ&#x20AC;, where U and V are given in (1.3.4) and (1.3.5).

( / ) ** cos nθ sin θ/2 I[ ] dθ

, ; , ; ,

= â&#x2C6;&#x161;Ď&#x20AC; I& ,'# :(;& / ,' / :(/%;&//%,'// :(´´ )

%

)

)

)

)

Proof: To establish (2.3.12), expressing the Ifunction of two variables in the integrand as a Mellin-Barnes type integral (1.3.3) and changing the order of the θ-integral and t, uintegral, which is justified due to the absolute convergence of the integrals involved in the process, we have

)

# â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Ś...:. ,*/,â&#x20AC;Ś..,( ,*):â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Śâ&#x20AC;Ś.

[ $|â&#x20AC;Ś...,â&#x20AC;Śâ&#x20AC;Ś: ,* ,â&#x20AC;Śâ&#x20AC;Ś.,( ,*):â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Śâ&#x20AC;Ś. ], (2.3.13)

provided that Re(1 â&#x2C6;&#x2019; 2u) > 0, n = 0, 1, 2, â&#x20AC;Ś,

h > 0 and U > 0, > 0, |argΞ| < UĎ&#x20AC;, where U and V are given in (1.3.4) and (1.3.5).

sin 2n + 1 θ sinθ dθ dtdu.

Now evaluate the inner-integral with the help of the formula (2.2.11), we get 1 Ď&#x2022; (t, u)θ (t)θ (u)Ξ Ρ (2Ď&#x20AC;i) 3 Î&#x201C; â&#x2C6;&#x2019; u + ht Î&#x201C; u + n â&#x2C6;&#x2019; ht dtdu.

2 Î&#x201C;(u â&#x2C6;&#x2019; ht)Î&#x201C;(2 â&#x2C6;&#x2019; u + n + ht)

â&#x2C6;&#x161;Ď&#x20AC;

On applying (1.3.3), the integral (2.3.12) is established. ** The integral (2.3.13) is established on applying the same procedure as above and using the result (2.2.12) in place of (2.2.11). ** / (cosθ) e ( ) I[ (5

/

=

( )

)6 ) /

â&#x2C6;&#x2018; ,

] dθ

, ; , ; , I7 ,8 :/;7 ,8 :/ ;7 ,8 :/´´

/ â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Ś...: , ,â&#x20AC;Śâ&#x20AC;Ś,( , / ):â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Śâ&#x20AC;Ś.

|â&#x20AC;Ś...,â&#x20AC;Śâ&#x20AC;Ś: , , , / ,â&#x20AC;Śâ&#x20AC;Ś.:â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Śâ&#x20AC;Ś. ],(2.3.14)

[

provided that h is a positive number and Re(m + n) > 1 and U > 0, > 0, |argΞ| <

UĎ&#x20AC;, where U and V are given in (1.3.4) and

(1.3.5). ** 1 â&#x2C6;&#x2019; x P (x) I[

9 :

] dx

Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411)

S. S. Srivastava, et al., J. Comp. & Math. Sci. Vol.5 (4), 390-401 (2014) =

(

) ( )

, ; , ; ,

I7 ,8 :/;7 ,8 :/ ;7 ,8 :/´´

â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Ś...:â&#x20AC;Śâ&#x20AC;Ś.,

/ , ,

/ , :â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Śâ&#x20AC;Ś.

[ |â&#x20AC;Ś...,â&#x20AC;Śâ&#x20AC;Ś: / , , / , ,â&#x20AC;Śâ&#x20AC;Ś:â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Śâ&#x20AC;Ś.

], (2.3.15)

provided that Re(2Îą + δ) > 0, Re(2Îą â&#x2C6;&#x2019; δ) >

0 and U > 0, > 0, |argΞ| < UĎ&#x20AC;, where U and V are given in (1.3.4) and (1.3.5).

; ** e x J (x) I[ ] dx

=

! " #

5 , ; , ; ,

( / ) 7 ,8 :/;7 ,8 :/ ;7 ,8 :/´´

I

(

/

[

) â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Ś...: / , ,â&#x20AC;Ś..:â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Śâ&#x20AC;Ś. |â&#x20AC;Ś...,â&#x20AC;Śâ&#x20AC;Ś:

, ,â&#x20AC;Ś..,

, :â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Śâ&#x20AC;Ś. ],

(2.3.16) provided that Re(Îą + β) > â&#x2C6;&#x2019; 1, Re(Îą) > â&#x2C6;&#x2019; 1/2

and U > 0, > 0, |argΞ| < UĎ&#x20AC;, where U and V are given in (1.3.4) and (1.3.5). ;

t J (Îąt)J (Îąt) I[

=

$ â&#x2C6;&#x161;

To establish (2.3.18), express the Ifunction of two variables in the integrand by its equivalent contour integral (1.3.3), changing the order of the θ-integral and t, uintegral, we get 1 Ď&#x2022; (t, u)θ (t)θ (u)Ξ Ρ (2Ď&#x20AC;i)

â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Ś...: / , , , ,â&#x20AC;Śâ&#x20AC;Ś.: ], | ! $

$ ! â&#x20AC;Ś...,â&#x20AC;Śâ&#x20AC;Ś: , ,â&#x20AC;Śâ&#x20AC;Ś., ,

[

Proof of (2.3.18):

sin θ cos θ dθ dtdu.

, ; , ; ,

The proof of the formulae (2.3.15) to (2.3.17) can be developed by proceeding on similar lines with the help of the result (2.2.8), (2.2.14) and (2.2.15) respectively in the place of result (2.2.13) and a relation Ď&#x20AC; Î&#x201C; z Î&#x201C; 1 â&#x2C6;&#x2019; z = . sinĎ&#x20AC;z

] dt

I7 ,8 :/;7 ,8 :/ ;7 ,8 :/´´

400

$

,â&#x20AC;Śâ&#x20AC;Ś..:â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Śâ&#x20AC;Ś.

! $ ! $ , , , :â&#x20AC;Śâ&#x20AC;Ś,â&#x20AC;Śâ&#x20AC;Ś.

],

(2.3.17)

provided that Re(β + δ) < Re(s) < 1, ι > 0

and U > 0, > 0, |argΞ| < UĎ&#x20AC;, where U and V are given in (1.3.4) and (1.3.5).

Proof: Equation (2.3.14) can be proved on using the contour integral (1.3.3), changing the order of the θ-integral and t, u-integral, which is justified under the conditions stated with the result, evaluating the inner integral with the help of (2.2.13) and using a relation Ď&#x20AC; 2Ď&#x20AC;i Î&#x201C; z Î&#x201C; 1 â&#x2C6;&#x2019; z = = ; , sinĎ&#x20AC;z e â&#x2C6;&#x2019; e ; and applying the definition (1.3.1).

Now evaluating the inner integral with the help of the result (2.2.1) and finally interpreting in view of (1.3.3), the integral (2.3.18) is obtained. The proof of the integrals (2.3.19) to (2.3.21) would run parallel to what we have obtained above.

Proof of (2.3.22): Replace the express the I-function of two variables in the integrand by its equivalent contour integral (1.3.3), changing the order of the θ-integral and t, u-integral, evaluate the inner integral with the help of (2.2.17) and finally interpret it with (1.3.3), to get (2.3.22).

The result (2.3.23) can be established exactly on the same lines in view of (2.2.18).

Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411)

401

S. S. Srivastava, et al., J. Comp. & Math. Sci. Vol.5 (4), 390-401 (2014)

REFERENCES 1. Ronghe, A. K.: Some Weber - Schaf heiflin type integrals involving Ifunction, Vijnana Parishad Anusandhan Patrika, Vol. 37, No.1, p.23-27 (1994).

2. Saxena, R. K. and Singh, V.: Integration of generalized H-function with respect to their parameters, Vijnana Parishad Anusandhan Patrika, Vol. 36, No.1, p.55-62 (1993).

3. Sharma, C.K. and Tiwari, D.K.: Integrals involving a general class of polynomials Lauricella functions and the multivariable I-function, Vijnana Parishad Anusandhan Patrika,Vol. 40, No.2, p.77-87 April (1997).

4. Goyal, A. and Agrawal, R. D.: Integral involving the product of I- function of two variables, Journal of M.A.C.T. Bhopal, Vol. No. 28, p.147-156 (1995).

5. Mohan, R. and Bhargava, M.: Finite integrals involving the multivariable Ifunction and a general class of polynomials, Acta Ciencia Indica, Vol. XXIV (m) No.3, p. 211-214 (1998).

6. Srivastava, Renu and Srivastava, S. S.: Integration of Certain Products Involving I-function and Double Hypergeometric Function, Vikram Mathematical Journal, Vol. 17, p. 74-82 (1997).

7. Srivastava, Renu: Some finite double integral formulae involving I-function, Vijnana Parishad Anusandhan Patrika, Vol. 46, No.02, p.127-137 April (2002).

8. Shailesh Jaloree, Anil Goyal and R. D. Agrawal. Fractional Integral Formulae involving the product of a general class of polynomials and I-function of two variables II., Jnanabha, Vol. 30, p. 7579 (2000).

9. Sharma C. K. and Mishra, P. L.: On the I-function of two variables and its certain properties, ACI, 17, 1-4 (1991).

10. Erdelyi, A.: Higher Transcendental Functions, Vol.II, McGraw-Hill, New York, (1953).

11. Nielsen, N.: Handbuch der Theorie der gamma function, Leipzig, (1906).

12. Kuipers, L.: Integral Transforms in the theory of Jacobi Polynomials & Generalised Legender Associated Functions (First Part), Proc. Kon. Ned. Ak. V. Wet. Amsterdam, Ser. A. 62, pp.

148-152 (1959).

13. Mishra Sadhna: Integrals involving Exponential function, Generalized Hypergeometric series and Fox's HFunction and Fourier series for products of Generalized Hypergeometric functions, J. Indian Acad. Math. Vol. 12, No. 1 (1990).

14. Erdelyi, A. Tables of Integral Transform, Vol.II, McGraw-Hill, New York (1953).

15. MacRobert, T. M.: Fourier series for the E-function, Math. Z., 76, 79-82,(1961).

16. MacRobert, T. M.: Function of a Complex Variable, Fifth ed., Macmillan, London, (1962).

17. Mathai, A. M. and Saxena, R. K.: Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences, Springer-Verlag, Berlin, Heidelberg and New York, (1973).

18. Shrivastava, H. M.: A contour integral involving Fox's H-function, Indian J. Math. 14, 1 â&#x20AC;&#x201C; 6 (1972).

19. Saran, N.: Special Functions, Pragati Prakashan, Meerut, (1989).

20. Erdelyi, A. H. T. F.: Vol.1, McGrawHill, NewYork, (1953).

21. Rainville, E. D.: Special Functions, Macmillan, NewYork, (1960).

Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411)