SOME ELLIPTIC-TYPE INTEGRALS AND THEIR HYPERGEOMETRIC FORMS M.I. QURESHI and NADEEM AHMAD DEPARTMENT OF APPLIED SCIENCES AND HUMANITIES FACULTY OF ENGINEERING AND TECHNOLOGY JAMIA MILLIA ISLAMIA, JAMIA NAGAR, NEW DELHI 110025 (INDIA) E-mail: msq
[email protected] E-mail: nadym
[email protected] M.A. PATHAN CENTER FOR MATHEMATICAL SCIENCES ARUNAPURAM,P.O.,PALA-686574,KERALA (INDIA) E-mail:
[email protected] ABSTRACT Hypergeometric forms of different elliptic-type integrals have been studied due to their applications in certain problems involving computations of the radiation field off axis from a uniform circular disc radiating according to an arbitrary angular distribution law.In this paper we establish Gaussian hypergeometric forms of complete elliptic integral of third kind , Andrews integral, Erd´elyi’s complete elliptic integrals and Wright’s hypergeometric forms of some new elliptic type integrals . A.M.S.(M.O.S.) Subject classification: 33C55, 33E05, 33C65, 33C70. Keywords and Phrases: Elliptic type integral, Generalized Gaussian hypergeometric function , Wright’s generalized hypergeometric function. 1.
INTRODUCTION
The Pochhammer symbol or generalized factorial function or shifted factorial is defined by 1 ; n=0 Γ(a + n) = (1.1) (a)n = Γ(a) a(a + 1)(a + 2).....(a + n − 1); n = 1, 2, 3.... where a 6= 0, −1, −2, ... and the notation Γ stands for Gamma function. (a)−n = where
Γ(a − n) (−1)n = ; Γ(a) (1 − a)n
(1.2)
a 6= ..... − 3, −2, −1, 0, 1, 2, 3..... and n = 1, 2, 3, .....
If m = 1, 2, , 3, 4 . . . and n = 0, 1, 2, 3, . . . , then b+1 b+m−2 b+m−1 b ... (b)mn = mmn m n m n m m n n The notation ∆(N ; b) denotes the array of N parameters given by where N =1,2,3,....... 1
(1.3)
b b+1 b+N −1 N, N ,... N
(α)p+q = (α)p (α + p)q = (α)q (α + q)p A Y
[(aA )]n = (a1 )n (a2 )n ......(aA )n =
(am )n =
m=1
A Y Γ(am + n) Γ(am ) m=1
(1.4) (1.5)
where a1 , a2 , ......, aA , b1 , b2 , ......., bB and z may be real and complex numbers. The generalized hypergeometric function of one variable [14, p.437] is defined as X ∞ a1 , a2 , ....., aA ; (a1 )n (a2 )n .......(aA )n z n F z = A B b1 , b2 , ....., bB ; (b1 )n (b2 )n ......(bB )n n! n=0 or
A FB
X ∞ (aA ); [(aA )]n z n z = (bB ); [(bB )]n n! n=0
(1.6)
where for the sake of convenience (in the contracted notation), (aA ) denotes the array of ’A’ number of parameters given by a1 , a2 , ...., aA . The denominator parameters are neither zero nor negative integers. The numerator parameters may be zero and negative integers. A and B are positive integers or zero.Empty sum is to be interpreted as zero and empty product as unity. Suppose that the numerator parameters are neither zero nor negative integers (otherwise the question of convergence will not arise). If A=B+1 , the series A FB converges for | z |< 1 and diverges for | z |> 1. Wright’s generalized hypergeometric function [18,p.50(1.5.21), p.179(34 iii), p.395(23)], is defined by (α1 , A1 ), . . . , (αp , Ap ); Γ(α1 ) Γ(α2 ) . . . Γ(αp ) z = p Ψq Γ(β1 ) Γ(β2 ) . . . Γ(βq ) (β1 , B1 ), . . . , (βq , Bq ); (α1 , A1 ), . . . , (αp , Ap ); × p Ψ∗q z (1.7) (β1 , B1 ), . . . , (βq , Bq ); ∞ X Γ(α1 + A1 n) Γ(α2 + A2 n) . . . Γ(αp + Ap n) z n Γ(β1 + B1 n) Γ(β2 + B2 n) . . . Γ(βq + Bq n) n! n=0 (1 − α , A ), . . . . . . , (1 − α , A ); 1 1 p p 1,p = Hp,q+1 −z (0, 1), (1 − β1 , B1 ), . . . , (1 − βq , Bq );
=
∗ p Ψq
X ∞ (α1 )A1 n (α2 )A2 n . . . (αp )Ap n z n (α1 , A1 ), . . . , (αp , Ap ); z = (β1 , B1 ), . . . , (βq , Bq ); (β1 )B1 n (β2 )B2 n . . . (βq )Bq n n! n=0
(1.8)
(1.9)
(1.10)
1,p where Hp,q+1 denotes Fox H-function [18, p.49(15,16)], the coefficients A1 , A2 , . . . , Ap and B1 , B2 , . . . , Bq are positive real numbers. The Fox H-function makes sense when either
δ = (1+B1 +B2 +. . .+Bq )−(A1 +A2 +. . .+Ap ) > 0 and
2
0 < |z| < ∞ ; z 6= 0
or, the equality holds for suitably constrained values of |z| 1 p A2−A2 . . . A−A B1B1 B2B2 . . . BpBp . δ = 0 and 0 < |z| < R = A−A p 1 p Q
p Ψq
Γ(αj ) (α1 , 1), . . . , (αp , 1); j=1 z = Q q (β1 , 1), . . . , (βq , 1); Γ(βj ) j=1
p Fq
α1 , . . . , αp ; z β1 , . . . , β q ;
(1.11)
Appell [17,18] introduced four double hypergeometric series, one of them is given by ∞ X (a)m+n (b)m (c)n xm y n , (d)m+n m!n! m,n=0
F1 [a; b, c; d; x, y] =
max{|x|, |y|} < 1
(1.12)
If Re(M ) > −1 and Re(N ) > −1, then π 2
Z 0
M +1 N +1 Γ Γ 2 2 , sinM θ cosN θdθ = M +N +2 2Γ 2
∞ X
∞ ∞ X X A(p) = A(2p)+ A(2p + 1)
p=0
Z
2a
f (x)dx =
0
1 F0
(1.13)
p=0
2
Ra 0
(1.14)
p=0
if f (2a − x) = f (x) ; if f (2a − x) = −f (x)
f (x)dx;
0
X ∞ a; (a)p z p z = p! −; p=0
= (1 − z)−a ;
An elliptic integral is an integral of the form
R
|z| < 1
(1.15) (1.16)
p R x, P (x) dx, in which
R is a rational function of its arguments and P (x) is a third- or fourth- degree polynomial with distinct zeros. Every elliptic integral can be reduced to a sum of integrals expressible in terms of algebraic, trigonometric, inverse trigonometric, logarithmic and exponential functions (the elementry functions), together with one or more of the following special types of integrals. If P (x) is a polynomial of degree greater than four then integrals may be integrated with the aid of hyper-elliptic functions. Various generalizations of certain families of elliptic-type integrals are studied in a number of earlier works [5,6,11,12,15,16,19 ] on the subject due to their importance for possible applications in certain problems arising in physics and nuclear technology.Hypergeometric forms of different elliptic-type integrals have been studied due to their applications in certain problems involving computations of the radiation field off axis from a uniform circular disc radiating according to an arbitrary angular distribution law.In this paper we stablish Gaussian 3
hypergeometric forms of complete elliptic integral of third kind Π(k, a), Andrews integral, Erd´elyi’s complete elliptic integrals B(k), C(k) and Wright’s hypergeometric forms 2 Ψ∗1 of some new elliptic type integrals in the forms of Z π2 Z π2 sinM θ dθ sinM θ dθ . and (a2 + b2 + c2 − 2ab cosN θ)g (a2 + b2 + c2 − 2ab sinN θ)g 0 0 where ’M ’ is a positive integer. 2.
LEGENDRE’S FORM OF SOME WELL KNOWN ELLIPTIC INTEGRALS The incomplete elliptic integral of first kind is defined by Z φ dθ p = F (k, φ) 0 1 − k 2 sin2 θ
The complete elliptic integral of first kind is Z π2 π dθ p = K(k) = K = 2 2 2 0 1 − k sin θ
(2.1)
1 2 , 2 ; k2
1 2 F1
1 ;
π , K(1) = ∞ 2 The incomplete elliptic integral of second kind is Z φp 1 − k 2 sin2 θ dθ = E(k, φ)
(2.2)
K(0) =
(2.3)
0
The complete elliptic integral of second kind is Z π2 p π 1 − k 2 sin2 θ dθ = E(k) = E = 2 0
−1 2 , 2 ; k2
1 2 F1
1
;
(2.4)
π , E(1) = 1, |arg(1 ± k)| < π 2 Such type of integrals were obtained to calculate the perimeter of Ellipse, Sine curve, Cosine curve and Lemniscate of Bernoulli. The above integrals arise in the solution of certain classes of vibration problems.Here φ is known as the amplitude and k is the modulus of elliptic integral.When φ = π2 , we get complete elliptic integral.The values of F(k, φ) and E(k, φ) can be evaluated for given values of k and φ. These values were tabulated by Jahnke and Emde [ 9,10 ] for 0 ≤ φ ≤ π2 and 0 < k < 1. √ 0 The quantity k = 1 − k 2 is called complementary modulus. The incomplete elliptic integral of third kind is defined by Z φ dθ p = Π(k, φ, a) (2.5) 2 2 0 (1 + a2 sin θ) 1 − k 2 sin θ E(0) =
4
where a 6= k, 0, is called characteristic parameter of integral of third kind. The complete elliptic integral of third kind [14], is given by Z 0
π 2
a2
(1 +
dθ p = Π(k, a) sin θ) 1 − k 2 sin2 θ 2
(2.6)
where 0 < k 2 < 1, −∞ < a2 < +∞, a2 6= −1 Another incomplete elliptic integral [14,p.744] is given by φ
Z
sin2 θdθ p
0
1 − k 2 sin2 θ
= D(k, φ)
(2.7)
The complete elliptic integral [14,pp.771-772] is given by π 2
Z
sin2 θdθ p
0
1 − k 2 sin2 θ
= D(k)
π = 4
1 2 , 2 ; k2
3 2 F1
2 ;
(2.8)
In L.C.Andrews [2,p.139(Q.No.13)] the following integral was represented in terms of K(k) and E(k) in the form Z π 2 k2 cos θ dθ √ = √ K(k) − E(k) (2.9) 1− 2 a2 + b2 + c2 − 2ab cos θ k ab 0 4ab where k 2 = (a+b) 2 +c2 In the work of Erd´ elyi, A.et.al[ 7 ,pp.317-318,p.321(25)] elliptic integrals are given as follows: Z φ cos2 θ dθ p = B(k, φ) (2.10) 0 1 − k 2 sin2 θ
Z 0
φ
Z 0
Z 0
π 2
cos2 θ dθ p
1 − k 2 sin2 θ
= B(k)
(sin θ cos θ)2 dθ p 3 = C(k, φ) 1 − k 2 sin2 θ
π 2
(sin θ cos θ)2 dθ p 3 = C(k) 2 2 1 − k sin θ
5
(2.11)
(2.12)
(2.13)
The hypergeometric forms of complete elliptic integrals K(k), E(k) and D(k) are given in the literature on elliptic integrals. To represent the above elliptic integrals, different notations were used by W.W.Bell [ 3 ,p.224] ,L.C.Andrews [2,pp.133-134], Tuma[20,pp.172-174], Marichev[13,pp.174181], Gradshteyn and Ryzhik [8,pp.904-905],Abramowitz and Stegun [1] and Prudnikov et.al.[14,pp.744,771-772].
3. HYPERGEOMETRIC FORMS OF REMAINING ELLIPTIC INTEGRALS (2.6),(2.9),(2.11),(2.13) Consider the complete elliptic integral of third kind (2.6) Z
π 2
dθ p sin θ) 1 − k 2 sin2 θ
Π(k, a) = (1 +
0
Z =
π 2
[1 − k 2 sin2 θ]
2
a2 −1 2
[1 − (−a2 sin2 θ)]−1 dθ
0 π 2
Z = 0
X X ∞ ∞ ( 12 )q (k 2 sin2 θ)q (1)p (−a2 sin2 θ)p dθ p! q! p=0 q=0
Z π2 ∞ ∞ X (1)p (−a2 )p X ( 21 )q (k 2 )q sin2p+2q θ dθ = p! q! 0 p=0 q=0 √
∞ ∞ π X (1)p (−a2 )p X ( 12 )q (k 2 )q Γ( 12 + p + q) 2 p=0 p! q! (p + q)! q=0
=
√ = Z
π 2
Π(k, a) = 0
∞ ∞ πΓ( 12 ) XX ( 12 )p+q (1)p ( 12 )q (−a2 )p (k 2 )q 2 (1)p+q p!q! p=0 q=0
dθ π 1 1 2 2 p = F1 ; 1, ; 1; −a , k (3.1) 2 2 2 (1 + a2 sin2 θ) 1 − k 2 sin2 θ
where max {| − a2 |, |k 2 |} < 1 and F1 is Appell’s function of first kind. Next, we consider the integral (2.9) Z π cos θ dθ A.I. = 1 2 2 2 0 [a + b + c − 2ab cos θ] 2 =√
1 a2 + b2 + c2
Z π 1− 0
2ab cos θ a2 + b2 + c2
6
−1 2 cos θ dθ
=√
∞ 2ab p Z π X ( 12 )p ( a2 +b 1 2 +c2 ) cosp+1 θ dθ p! a2 + b2 + c2 p=0 0
Now using the series identity (1.14), we get 1 A.I. = √ 2 a + b2 + c2
+
X ∞ 2ab 2p Z ( 21 )2p ( a2 +b 2 +c2 ) 2p!
p=0
∞ 2ab 2p+1 Z X ( 12 )2p+1 ( a2 +b 2 +c2 )
(2p + 1)!
p=0
π
cos2p+1 θ dθ
0 π
cos2p+2 θ dθ
0
Now using the property of definite integral (1.15), we get Z π2 ∞ X ( 12 )2p+1 (2ab)2p+1 2 cos2p+2 θ dθ A.I. = √ a2 + b2 + c2 p=0 (2)2p (a2 + b2 + c2 )2p+1 0 Now using the integral (1.13), we get A.I. =
∞ 2ab 2p X ( 23 )p ( 54 )p ( a2 +b π ab 2 +c2 ) 2 (a2 + b2 + c2 ) 32 p=0 (2)p p!
2 2ab 2 F1 1 2 2 2 2 ; a2 + b2 + c2 0 [a + b + c − 2ab cos θ] 2 (3.2) Similarly, we obtain the following hypergeometric forms of Erd´elyi’s complete elliptic integrals (2.11) and (2.13)
Z
π
cos θdθ
5 4, 4;
3
π ab = 2 (a2 + b2 + c2 ) 32
π 2
Z B(k) = 0
Z C(k) = 0
π 2
cos2 θdθ
π p = 2 F1 2 4 1 − k 2 sin θ
1
1 2 , 2 ; k2 2 ;
(sin θ cos θ)2 dθ π p 3 = 2 F1 16 2 1 − k 2 sin θ
3 2 , 2 ; k2
(3.3)
3
3 ;
(3.4)
4. MORE ELLIPTIC TYPE INTEGRALS AND THEIR HYPERGEOMETRIC FORMS On the same parallel lines of the derivations discussed in section 3, we obtain the following hypergeometric forms of some elliptic type integrals Z π2 sin2m θ dθ cos2m θ dθ = 2n 2 2 (a + b + c2 − 2ab cos2n θ)g (a2 + b2 + c2 − 2ab sin θ)g 0 0 1 π ( 21 )m 2ab ∗ (g, 1), ( 2 + m, n); = (4.1) 2 Ψ1 2 (a2 + b2 + c2 )g (1)m (1 + m, n) ; a2 + b2 + c2 Z
π 2
7
When ’n’ is positive integer, we get π 2
Z 0
sin2m θ dθ = (a2 + b2 + c2 − 2ab sin2n θ)g
π ( 21 )m = 2 (a2 + b2 + c2 )g (1)m
n+1 Fn
π 2
Z
(a2
0
+
b2
cos2m θ dθ + c2 − 2ab cos2n θ)g
g, ∆(n; 12 + m); 2ab ∆(n; 1 + m) ; a2 + b2 + c2
(4.2)
Proofs of (4.1) and (4.2) : Left hand side of (4.1) can be written as L1 =
0
sin2m θ dθ 2n g 2ab θ] [1 − ( a2 +b 2 +c2 ) sin
1 2 (a + b2 + c2 )g
Z
1 (a2 + b2 + c2 )g
∞ X (g)k 2ab k! a2 + b2 + c2
= =
π 2
π 2
Z
1 (a2 + b2 + c2 )g
=
1 (a2 + b2 + c2 )g
sin2m θ [1 −
0
2ab 2 a + b2 + c2 k Z π 2
k=0
(g)k k!
2ab a2 + b2 + c2
sin2n θ]−g dθ
sin2m+2nk θ dθ
0
k=0
∞ X
k
Γ( 2m+2nk+1 ) Γ( 12 ) 2 ) 2 Γ( 2m+2nk+2 2
k ∞ Γ(m + 21 ) Γ( 12 ) X (g)k (m + 21 )nk 2ab 1 = 2 (a2 + b2 + c2 )g Γ(m + 1) k! a2 + b2 + c2 (m + 1)nk k=0
=
∞ X π ( 21 )m (g)k (m + 21 )nk 2 2 2 g 2 (a + b + c ) (1)m (m + 1)nk k=0
2ab a2 +b2 +c2
k
k!
which gives the right hand side of equation (4.1). When ’n’ is positive integer and using (1.3), we get L1 =
π ( 12 )m 2 2 (a + b2 + c2 )g (1)m
n+1 Fn
g, ∆(n; 12 + m); 2ab ∆(n; 1 + m) ; a2 + b2 + c2
which is the right hand side of equation (4.2)
Z π2 sin2m+1 θ dθ cos2m+1 θ dθ = 2n+1 (a2 + b2 + c2 − 2ab cos2n+1 θ)g (a2 + b2 + c2 − 2ab sin θ)g 0 0 2n+1 (1)m 2ab ∗ (g, 1), (m + 1, 2 ); = 2 (4.3) 2 Ψ1 a2 + b2 + c2 (a + b2 + c2 )g ( 32 )m (m + 23 , 2n+1 2 ) ; Z
π 2
8
When ’n’ is positive integer, we get π 2
Z
sin2m+1 θ dθ = (a2 + b2 + c2 − 2ab sin2n+1 θ)g
0
(1)m = 2 2 (a + b + c2 )g ( 32 )m
2n+3 F2n+2
Z
π 2
(a2
0
+
b2
cos2m+1 θ dθ + c2 − 2ab cos2n+1 θ)g
2 ∆(2; g), ∆(2n + 1; 1 + m); 2ab 1 3 ; a2 + b2 + c2 2 , ∆(2n + 1; 2 + m)
π abg ( 23 )m+n 2(a2 + b2 + c2 )g+1 (2)m+n 2 ∆(2; g + 1), ∆(2n + 1; 23 + m + n); 2ab × 2n+3 F2n+2 3 ; a2 + b2 + c2 2 , ∆(2n + 1; 2 + m + n) +
(4.4)
Proofs of (4.3) and (4.4) : Left hand side of (4.3) can be written as L2 =
1 2 (a + b2 + c2 )g
1 = 2 (a + b2 + c2 )g =
(a2
=
1 + b2 + c2 )g
1 (a2 + b2 + c2 )g
π 2
Z
π 2
sin2m+1 θ dθ 2n+1 g 2ab [1 − ( a2 +b θ] 2 +c2 ) sin
0
2ab sin2n+1 θ]−g dθ sin θ [1 − a2 + b2 + c2 0 k Z π ∞ X 2 (g)k 2ab sin2m+2nk+k+1 θ dθ 2 2 2 k! a + b + c 0
Z
2m+1
k=0
k ∞ X Γ( 2m+2+2nk+k ) Γ( 21 ) (g)k 2ab 2 2m+3+2nk+k 2 2 2 k! a + b + c 2 Γ( ) 2 k=0
=
1 2 (a + b2 + c2 )g
∞ 1 X (g)k Γ(m + 1 + ( 2n+1 2 )k) Γ( 2 ) 2 Γ(m + 23 + ( 2n+1 2 )k) k=0
2ab a2 +b2 +c2
k!
=
2
(a2
Γ( 12 ) Γ(m + 1) 1 2 2 g +b +c ) Γ(m + 32 )
∞ X
(g)k (m + 1)( 2n+1 )k
k=0
(m + 23 )( 2n+1 )k
k
2ab
k
a2 +b2 +c2
2
k!
2
which yields the right hand side of equation (4.3). When ’n’ is positive integer, applying the identity (1.14), we get 1 L2 = 2 (a + b2 + c2 )g
+
2k Z π ∞ X 2 2ab (g)2k sin2m+4nk+2k+1 θ dθ 2 2 2 (2k)! a + b + c 0
k=0
2k+1 Z π ∞ X 2 1 (g)2k+1 2ab sin2m+4nk+2n+2k+2 θ dθ 2 2 2 g 2 2 2 (a + b + c ) (2k + 1)! a + b + c 0 k=0
9
1 = 2 (a + b2 + c2 )g +
2k ∞ 2m+4nk+2k+2 X ( g2 )k ( g+1 ) Γ( 12 ) 2ab 2 )k Γ( 2 a2 + b2 + c2 ( 12 )k ( 22 )k 2 Γ( 2m+4nk+2k+3 ) 2 k=0
1 (g)1 2 2 2 g (a + b + c ) (1)1
2ab 2 a + b2 + c2
X ∞ k=0
2k g+2 2m+4nk+2n+2k+3 ( g+1 ) Γ( 21 ) 2ab 2 )k ( 2 )k Γ( 2 a2 + b2 + c2 ( 22 )k ( 23 )k 2 Γ( 2m+4nk+2n+2k+4 ) 2
=
∞ g+1 g 1 Γ(m + 1) X ( 2 )k ( 2 )k (m + 1)(2n+1)k 3 1 2 (a2 + b2 + c2 )g Γ(m + 2 ) ( 2 )k (m + 32 )(2n+1)k k=0
2ab a2 +b2 +c2
2k
k!
+
∞ g+2 (m + n + 23 )(2n+1)k Γ(m + n + 23 ) X ( g+1 abg 2 )k ( 2 )k (a2 + b2 + c2 )g+1 Γ(m + n + 2) (m + n + 2)(2n+1)k ( 32 )k
2ab a2 +b2 +c2
k=0
which yields the right hand side of equation (4.4). A corresponding hypergeometric forms of the elliptic-type integrals over the intervals [0, π] and [0, 2π] may be deduced by using the properties of definite integrals. Acknowledgement The author M.A.Pathan would like to thank the Department of Science and Technology, Government of India, for the financial assistance for this work under project number SR/S4/MS:794/12.
References [1] Abramowitz, M. and Stegun, I.A. Handbook of Mathematical Functions, with Formulas, Graphs and Mathematical Tables , National Bureau of Standards Washington, D.C., 1964, Reprinted by Dover Publication , New York , 1972. [2] Andrews, L.C., Special functions For Engineers and Applied Mathematician, MacMillan Co., New York , 1985. [3] Bell, W.W. Special Functions for Scientists and Engineers , D. Van Nostrand Company Ltd., London, 1968. [4] Byrd, P.F. and Friedman, M.D., Handbook of Elliptic Integrals for Engineers and Scientists, Springer - Verlag, Heidelberg , 1971. [5] Carlson, B. C., Normal elliptic integrals of the first and second kinds, Duke. Math. J. 31, (1964), 405-419. [6] Carlson, B. C., Some series and bounds for incomplete elliptical integrals., J. Math. and Phys. 40, (1961), 125-134. [7] Erd´ elyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G., Higher transcendental functions (Bateman Manuscript Project) Vol. II, McGrawHill Book Co. Inc. New York, Toronato and London, 1953.
10
k!
2k
[8] Gradshteyn, I.S., and Ryzhik, I.M., Tables of Integrals, Series and Products, Fourth Edition, Academic Press, New York, 1965. [9] Jahnke, E., Emde, F., Tables with formulas and curves of functions, Dover Publication, New York, (1945), 73-84. ..
[10] Jahnke, E., Emde, F. and Losch, F., Tables of Higher Functions, Edition, McGraw-Hill, New York, 1960.
6th
[11] Jeffrey, A., Hand book of Mathematical formulas and integrals second edition , Academic Press, San Francisco, New York, London and Tokyo, 2000. [12] Pathan, M.A., On unified elliptic-type integrals, Proceedings of the international conference on Mathematics and its applications (ICMA: 2004), April 5-7, 2004, Kuwait foundation for the advancement of Sciences (KFAS), Kuwait University, Department of Mathematics and Computer Science, 375-387. [13] Marichev, O.I., Hand book of Integral Transforms of Higher Transcendental functions, Theory and algorithmic tables , Translated by L.W. Longdon, Ellis Horwood Limited, Chichester, Halsted Press, John Wiley and Sons, New York, Brisbane, Chichester and Toronto, 1983. [14] Prudnikov, A.P., Brychkov, Yu.A. and Marichev, O.I., Integrals and series Vol-3: More Special Functions, Nauka, Moscow, (1986). Translated from the Russian by G.G.Gould, Gordon and Breach Science Publishers, New York, Philadelphia, London, Paris, Montreux, Tokyo, Melbourne, 1990. [15] Saxena, R.K. and Pathan, M.A., Asymptotic Formulas for Unified Elliptictype Integrals, Demonstratio Mathematica , 36(3), (2003), 580-589. [16] Spiegel, M.R., Schaum’s outline series : Mathematical hand book of formula and tables , McGraw-Hill Publishing Company, New York, San Francisco, London, Singapore and Tokyo, International edition, 1990. [17] Srivastava, H.M. and Karlsson, P.W., Multiple Gaussian hypergeometric series, Halsted press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1985. [18] Srivastava, H.M. and Manocha, H.L., A treatise on generating functions, Halsted press (Ellis Horwood Ltd., Chichester), John Wiley and Sons, New York, Chicago, Brisbane and Toronto, 1984. [19] Srivastava, H.M. and Siddiqi, R.N. A Unified Representation of Certain Families of Elliptic-type Integrals related to Radiation Field problems , Radiat. Phys. Chem. , 46(3), (1995), 303-315. [20] Tuma, J.J., Engineering Mathematics Hand book, Definition, Theorem, Formulas and Tables , Third edition, McGraw-Hill book Company, New York, San francisco, London, Singapore and Tokyo, 1987.
11
