Cmjv04i01p0053

J. Comp. & Math. Sci. Vol.4 (1), 53-55 (2013)

On Some Double Integrals Involving H-function and Some Commonly Used Functions R. P. KUSHWAHA1 and PINKEY SIKARWAR2 1

Department of Mathematics Jaypee Polytechnic & Training Centre, Rewa, M. P., INDIA. 2 Department of Mathematics & Computer Science Govt. P. G. College, Shahdol, M.P., INDIA. (Received on: February 1, 2013) ABSTRACT The aim of this paper is to obtain some double integrals involving Fox’s H-function.

Keywords: Fractional integration, H-function of one variable, Gamma and Beta functions, Eulerian integrals, Mellin-Barnes contour integrals.

1. INTRODUCTION H-function of one variable which is introduced by Fox1, will be represented as follows: (a , α )

m, n s j j 1, p Hp, q [x| (bj, βj)1, q ] = (1/2πi) ∫ θ(s) x ds (1)

aj (j = 1, …, q) are complex numbers. L is a suitable contour of Barnes type such that poles of Γ(bj – β js) (j = 1, …, m) lie to the right and poles of Γ(1 – aj + αjs) (j = 1, …, n) to the left of L. These assumptions for the H-function will be adhered to through out this paper.

L

According to Braakasma

where i = √(– 1), n m Π Γ(bj – β js)Π Γ(1 – aj + αjs) j=1

j=1

θ (s) = q

Π

p

j=m+1

Γ(1 – bj + β js) Π

Γ(aj – αjs)

j=n+1

x is not equal to zero and an empty product is interpreted as unity; p, q, m, n are integers satisfying 1 ≤ m ≤ q, 0 ≤ n ≤ p, αj (j = 1, …., p), β j (j = 1, …, q) are positive numbers and

m, n α (aj, αj)1, p Hp, q [x| (bj, βj)1, q ] = O (|x| ) for small x, p

q

where Σ αj – Σ β j ≤ 0 and α = min R(bh/β h) j=1 j=1 (h = 1, .., k) and m, n (aj, αj)1, p ] = O (|x|β) for large x, Hp, q [x| (b , β ) j

j 1, q

where

Journal of Computer and Mathematical Sciences Vol. 4, Issue 1, 28 February, 2013 Pages (1-79)

54

R. P. Kushwaha, et al., J. Comp. & Math. Sci. Vol.4 (1), 53-55 (2013)

n

p

m

Re (α + β ) > −1.

q

Σ αj – Σ αj + Σ β j – Σ β j ≡ A > 0, j=1

j=n+1

p

j=1

(2)

j=m+1

From MacRobert3, we have

q

π /2

Σ αj – Σ β j < 0 j=1

∫e

j=1

i (α + β )θ

(sin θ ) α −1 (cos θ ) β −1 dθ =

0

e πiα / 2 Γ(α )Γ( β ) Γ(α + β )

Re (α ) > 0, Re ( β ) > 0.

|arg x| < ½ Aπ and β = max R[(aj – 1)/αj] (j = 1, .., n)

(4)

From Rainville4, we have 2. FORMULA USED t

In the present investigation we require the following formula: From Whitaker and Watson2, we have π /2

∫

2α + β +1

π

0

e i (α − β )θ (cosθ ) α + β dθ =

πΓ(α + β + 1) α + β +1 2 Γ(α + 1)Γ( β + 1)

(3) π /2π /2

∫ ∫ 0

2α + β +1

π

0

∫x

ρ −1

(t − x ) σ −1 dx = t ρ +σ −1

0

Γ ( ρ )Γ (σ ) Γ( ρ + σ )

(5)

Re ( ρ ) > 0, Re (σ ) > 0.

3. DOUBLE INTEGRALS In this section we will establish the following double integrals:

e i (α − β ) x (cos x)α + β e i ( ρ +σ ) y (sin y )σ −1 (cos y ) ρ −1

× H pm,,qn [ z (2e i ( x + y ) cos x sin y ) λ (2e i ( y − x ) cos x cos y ) µ ]dxdy

=

π.e

iπσ 2

iπλ 2

ze m,n+3 H [ p+3,q+3 2α +β +1 2λ+µ

(−α −β ,λ+µ ),(1−σ ,λ ),(1−ρ ,µ ),(a j ,α j )1, p (bj ,β j )1, q ,(1−σ −ρ ,λ+µ),(−α ,λ),(−β ,µ )

]

(6)

provided that Re (α + β) > −1, Re(σ) > 0, Re(ρ) > 0, λ ≥ 0 and µ ≥ 0 |arg z| < ½ πA, where A is given in equation (2). t π /2

∫∫ 0

0

2α + β +1

π

x ρ −1 (t − x) σ −1 e i (α − β ) y (cos y ) α + β

× H pm,,qn [ z (2 xe iy cos y ) λ (2(t − x)e −iy cos y ) µ ]dxdy

=

πt ρ +σ −1 α + β +1

2

H pm+,n3+,q3+3 [ z(t / 2) λ +µ

( −α −β ,λ +µ ),(1−σ ,λ ),(1− ρ ,µ ),( a j ,α j )1, p (b j , β j )1, q ,(1−σ −ρ ,λ +µ ),( −α ,λ ),( − β , µ )

],

Journal of Computer and Mathematical Sciences Vol. 4, Issue 1, 28 February, 2013 Pages (1-79)

(7)

R. P. Kushwaha, et al., J. Comp. & Math. Sci. Vol.4 (1), 53-55 (2013)

where Re (α + β) > − 1, Re(σ) > 0, Re(ρ) > 0, given in equation (2).

Proof: To prove (6), express the H-function of one variable on the left hand side as

I=

55

λ ≥ 0 and µ ≥ 0 , |arg z| < ½ πA , where A is contour integral and interchange the order of integration which is justifiable due to given condition, we get

1 θ ( s) z s 2πi ∫L

π /2

×[ ∫ 0

2(α +λs)+( β +µs)+1

π

eix[(α +λs)−( β +µs) (cosx)[(α +λs)+(β +µs)] dx

π /2

×[ ∫ eiy[(σ +λs)+(ρ+µs)] (siny)(σ +λs)−1 (cosy)(ρ+µs)−1 dy]ds 0

Now using the results (3), (4) and interpreting it with the help of (1), we get right hand side of (6). Preceding on the same lines the integral (7) can be established with the help of the results given in section 2.

REFERENCES 1. Fox, C.: The G and H-functions as symmetrical Fourier kernels Trans Amer.

Math. Soc. 98, p. 395-429 (1961).

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, NewYork, (1960).

Journal of Computer and Mathematical Sciences Vol. 4, Issue 1, 28 February, 2013 Pages (1-79)