J. Comp. & Math. Sci. Vol.3 (3), 276-279 (2012)
Some Double Integrals Involving I-Function of one Variable SANDEEP KHARE1 and B. M. L. SRIVASTAVA2 1
Department of Pure &Applied Mathematics, Guru Ghasi Das Central University, Bilaspur, C.G., INDIA 2 Retired Principal, Rewa, M.P., INDIA (Received on: April 15, 2012) ABSTRACT The aim of this paper is to establish some double integrals involving I-function of one variable. Keywords: Double integral, one variable, I-function .
defined by Saxena1 and we will represent here in the following manner:
1. INTRODUCTION The I-function of one variable is
I pi, qi: r [x| [(bj, βj)1, m], [(bji, βji)m + 1, qi] ] = (1/2πω) ∫ θ(s) xs ds L
where ω = √(– 1),
m Π
θ (s) =
j=1
r Σ
i=1
(1)
n
Γ(bj – βjs) Π Γ(1 – aj + αjs) j=1
qi Π
Γ(1 – bji + βjis)
j=m+1
pi Π
Γ(aji – αjis)
,
j=n+1
integral is convergent, when (B>0, A ≤ 0), where Β=
n
pi
m
qi
Σ αj – Σ αji + Σ βj – Σ βji ,
j=1
pi
j=n+1 j=1
qi
A = Σ αji – Σ βji , j=1
(2)
j=m+1
j=1
Journal of Computer and Mathematical Sciences Vol. 3, Issue 3, 30 June, 2012 Pages (248-421)
(3)
277
Sandeep Khare, et al., J. Comp. & Math. Sci. Vol.3 (3), 276-279 (2012)
|arg x| < ½ BĎ&#x20AC;, â&#x2C6;&#x20AC; i â&#x2C6;&#x2C6; (1, 2, â&#x20AC;Ś, r).
(4)
2. FORMULA USED
From MacRobert3, we have
In the present investigation we require the following formulae: From Whitaker and Watson2, we have
2
Ď&#x20AC;
e sinθ cosθ dθ ,
/
(6)
Re(Îą 0, 0.
e cosθ dθ
From Rainville4, we have
,
(5)
Re(ι β 1.
x t x dx t
,
(7)
Re(Ď 0, Ď&#x192; 0.
3. INTEGRALS /
2
/
Ď&#x20AC;
e cosx e siny cosy
"z$2e cosx siny& $2e cosx cosy& 'dxdy .I , , :
Âľ
, Âľ , , , ,Âľ ,â&#x20AC;Ś.. I , " | ', , : Âľ â&#x20AC;Ś.., , Âľ , , , ,Âľ
(8)
provided that Re (Îą + β) > â&#x2C6;&#x2019; 1, Re(Ď&#x192;) > 0, Re(Ď ) > 0, Îť * 0 and Âľ * 0, |arg z| < ½ Ď&#x20AC;Î&#x2019;, where B is given in equation (2).
2
/
Ď&#x20AC;
x t x e cosy
.I , "z$2xe cosy& $2 t x e cosy& 'dxdy , :
Âľ
, Âľ , Âľ , , , ,Âľ ,â&#x20AC;Śâ&#x20AC;Ś. I |â&#x20AC;Śâ&#x20AC;Ś., , Âľ , , , ,Âľ /, ,z - . , : Journal of Computer and Mathematical Sciences Vol. 3, Issue 3, 30 June, 2012 Pages (248-421)
(9)
Sandeep Khare, et al., J. Comp. & Math. Sci. Vol.3 (3), 276-279 (2012)
278
where Re (Îą + β) > â&#x2C6;&#x2019; 1, Re(Ď&#x192;) > 0, Re(Ď ) > 0, Îť * 0 and Âľ * 0, |arg z| < ½ Ď&#x20AC;Î&#x2019;, where B is given in equation (2). Proof: To prove (8), express the I-function of one variable on the left hand side as contour integral with the help of (1) and interchange the order of integration which is justifiable due to given condition, we get I
1 θ s z $ . 2Ď&#x20AC;i #
"
Âľ % $ Âľ$ & cosx % $ Âľ$ & dx' e
." e % $ Âľ$ & siny $ cosy Âľ$ dy'ds Now using the results (5), (6) and interpreting it with the help of (1), we get right hand side of (8). Proceeding on the same lines the results (9) can be established with the help of (5) and (7). 4. PARTICULAR CASES On choosing r = 1 in main integrals, we get following integrals in terms of Hfunction of one variable:
/
/ e cosx e siny cosy
, .H , "z$2e cosx siny& $2e cosx cosy& 'dxdy
Âľ
, Âľ , , , ,Âľ , * , , H , " | ', , Âľ '( , ) , , , Âľ , , , ,Âľ ,
(10)
provided that Re (Îą + β) > â&#x2C6;&#x2019; 1, Re(Ď&#x192;) > 0, Re(Ď ) > 0, Îť * 0 and Âľ * 0, |arg z| < ½ Ď&#x20AC;A , where A is given by n
p
m
q
ÎŁ Îąj â&#x20AC;&#x201C; ÎŁ Îąj + ÎŁ βj â&#x20AC;&#x201C; ÎŁ βj â&#x2030;Ą A > 0.
j=1
j=n+1
j=1
j=m+1
Journal of Computer and Mathematical Sciences Vol. 3, Issue 3, 30 June, 2012 Pages (248-421)
279
Sandeep Khare, et al., J. Comp. & Math. Sci. Vol.3 (3), 276-279 (2012) / x t x e cosy
, .H , "z$2xe cosy& $2 t x e cosy& 'dxdy
Âľ
(expression continue)
, Âľ , Âľ , , , ,Âľ , * , , H "z t/2
|'( , ) , , Âľ , , , ,Âľ , ', , ,
(11)
where Re (Îą + β) > â&#x2C6;&#x2019; 1, Re(Ď&#x192;) > 0, Re(Ď ) > 0, Îť * 0 and Âľ * 0, |arg z| < ½ Ď&#x20AC;A. REFERENCES 1. Saxena, V. P.: Formal Solution of Certain New Pair of Dual Integral Equations Involving H-function, Proc. Nat. Acad. Sci. India, 52(A), III (1982). 2. Whitaker, E. T. and Watson, G. N.: A Course of Modern Analysis, IXth. Edt.
Cambriz, (1952). 3. MacRobert, T. M.: Beta function formulae and integrals involving Efunction, Math. Ann., 142, p. 450-452 (1961). 4. Rainville, E. D.: Special Functions, Macmillan, New York, (1960).
Journal of Computer and Mathematical Sciences Vol. 3, Issue 3, 30 June, 2012 Pages (248-421)