Integral representations and integral transforms of some families of ...

10 N. Elezović et al.
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and
F c (S μ+1 (r))(x) =
√ π · 2
1/2−μ
∫ ∞
t μ+1/2
cos(xr)S μ+1 (r)dr =
0
Ɣ(μ + 1) x e t − 1 c(μ; x,t)dt
(x > 0; μ>0), (47)
∫ ∞
where (1)
s (μ; x,t), (2)
s (μ; x,t) and c (μ; x,t) are defined by (44), (45) and (46), respectively.
In a similar manner, we find from the integral representation (11) that
and
F s
(
S
(α,0)
μ (r;{k 2/α }) ) (x) =
∫ ∞
0
sin(xr)S μ
(α,0) (r;{k 2/α })dr
∫ ∞
t μ−1/2 (∫ ∞
)
r 1/2−μ J
Ɣ(μ) 0 e t μ−1/2 (rt) sin(xr)dr dt
− 1 0
( ∫ x
t μ−1/2
Ɣ(μ) 0 e t − 1 (1) s (μ; x,t)dt
t μ−1/2
)
(μ; x,t)dt
√ π · 2
3/2−μ
=
√ π · 2
3/2−μ
=
+
∫ ∞
x
e t − 1 (2) s
)
(
x>0; μ> 1 2
F c
(
S
(α,0)
μ (r;{k 2/α }) ) (x) =
∫ ∞
0
cos(xr)S μ
(α,0) (r;{k 2/α })dr
√ π · 2
3/2−μ
=
Ɣ(μ)
∫ ∞
t μ−1/2
x
(
x>0; μ> 1 2
(48)
e t − 1 c(μ; x,t)dt
)
, (49)
where (1)
s (μ; x,t), (2)
s (μ; x,t) and c (μ; x,t) are defined, as before, by (44), (45) and (46),
respectively.
III. In the evaluation of the Fourier sine and the Fourier cosine transforms of the general
Mathieu Series S μ
(α,β) (r;{k 2/α }), we shall make use of the following integral formulas proven
earlier by Miller and Srivastava [9, pp. 225 and 226]:
⎧
∫ ∞
⎨ (1)
sin(2ax) 1 F 2 (α; β,γ;−b 2 x 2 s (α, β, γ ; a,b) (0