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

8 N. Elezović et al.
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For the Fourier cosine transform, we similarly find that
F c (S(r))(x) =
=
=
∫ ∞
0
∫ ∞
0
∫ ∞
= π 4
= π 2
0
∫ ∞
cos(xr)S(r)dr
( ∫ 1 ∞
cos(xr)
r 0
(∫ ∞
t
e t − 1
0
∫ ∞
x
0
)
t sin(rt)
e t − 1 dt dr
sin(rt) cos(xr)
dr
r
t
[1 + sgn(t − x)]dt
e t − 1
)
dt
t
dt
e t − 1
(x > 0). (35)
II. In order to obtain the Fourier sine and the Fourier cosine transforms of S μ+1 (r), we first
set ν = 1/2 and ν =−1/2 in the Sonine–Schafheitlin formula (see, for example, [18, p. 401,
Equation 13.4 (2)]):
∫ ∞
0
t −λ a μ Ɣ (μ + ν − λ + 1/2)
J μ (at)J ν (bt)dt =
2 λ b μ−λ+1 Ɣ(μ + 1)Ɣ (λ − μ + ν + 1/2)
( μ + ν − λ + 1
· 2F 1 , μ − ν − λ + 1 )
; μ + 1; a2
2
2
b 2
( )
R(μ + ν + 1) >R(λ) > −1; 0