
1488 INTEGRAL TRANSFORMS
28.4 Tables of Fourier Sine Transforms
28.4.1 G eneral Formulas
No. Original function, f(x)
Sine transform,
ˇ
f
s
(u) =
Z
∞
0
f(x) sin(ux) dx
1 af
1
(x) + bf
2
(x)
a
ˇ
f
1s
(u) + b
ˇ
f
2s
(u)
2 f(ax), a > 0
1
a
ˇ
f
s
u
a
3
x
2n
f(x), n = 1, 2, . . .
(−1)
n
d
2n
du
2n
ˇ
f
s
(u)
4
x
2n+1
f(ax), n = 0, 1, . . .
(−1)
n+1
d
2n+1
du
2n+1
ˇ
f
c
(u),
ˇ
f
c
(u) =
Z
∞
0
f(x) cos(xu) dx
5 f(ax) cos(bx), a, b > 0
1
2a
h
ˇ
f
s
u + b
a
+ F
s
u −b
a
i
28.4.2 Expressions with Power-Law Functions
No. Original function, f(x)
Sine transform,
ˇ
f
s
(u) =
Z
∞
0
f(x) sin(ux) dx
1
n
1 if 0 < x < a,
0 if a < x
1
u
1 − cos(au)
2
(
x if 0 < x < 1,
2 −x if 1 < x < 2,
0 if 2 < x
4
u
2
sin u sin
2
u
2
3
1
x
π
2
4
1
a + x
, a > 0
sin(au) Ci(au) − cos(au) si(