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International Journal of Scientific and Research Publications, Volume 2, Issue 10, October 2012 35
ISSN 2250-3153
∞
∫
0 (c / a)
exp{-(cx + a / x)}dx = 2(c / a)
so the p- integral
1
v
2
K v (2 ac ), (1.2.5)
∞
∫
0
e
2 2
C T
− p−
4 P ρ−1
2 ρ
p dp = 2( ) k (2 ac )
p
CT
and changing 0 F 1 in to Bessel function of the kind the relation
1
( z)
k
2
1 2
( k + 1; − z )
J k (z)= Γ(
k + 1)
0F 1
4
(1.2.6)
(1.2.7)
∞
∫
0
t
λ−1
J
µ
( at)
J
v
( bt)
J
2
λ−1
µ v ⎧1
⎫ ⎧1
⎫
a b Γ⎨
( λ + µ + v + ρ⎬Γ⎨
( λ + µ + v − ρ⎬
⎩2
⎭ ⎩2
⎭
c Γ(
µ + 1) Γ(
v + 1)
ρ ( ct)
dt =
λ+
µ + v
F 4
2 2
⎡1
1
a b ⎤
⎢ ( λ + µ + v + p),
( λ + µ + v − p),
µ + 1, + v + 1; ,
2 2 ⎥
⎣2
2
c c ⎦
(1.2.8)
Where F 4 is fourth type of Appell’s function
1.3 REDUCTION OF F E , F G , F K AND F N INTO HORN’S FUNCTION
Let us consider the integral S. Saran (1955) for F E viz.
F E =
Γ(
γ ) Γ(
γ ) Γ(2
− γ − γ )
2 3
2 3 −γ
− 3
( −t)
2 ( t −1
(2 i)
) γ
2
π
c
∫
F 2 (α 1 ,β 1 , β 2 ,γ 1 , γ 2 + γ 3 -1; cosh x,
cosh y cosh z
+ )
t 1−
t dt (1.3.1)
where
|cosh x|+
coshy
coshz
+
t 1− t
< 1 along the contour.
Puttting β 1 = γ 1 and β 2 = γ 2 + γ 3 -1 in equation (1.3.1) we get
Γ(
γ
2
) Γ(
γ
3)
Γ(2
− γ
2
− γ
3)
2
F E = (2Πi)
coshy
coshz
−
∫ (-t) -γ2 (-t) –γ3 (1-cosh x - t 1− t ) -α1 dt
We can expand
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