Page 760 - Handbook Of Integral Equations

∞

ˇ
No Original function, f(x) Sine transform, f s (u)= f(x) sin(ux) dx
0
0 if 0 < u < a,

4 J 0 (ax), a >0 1
√ if a < u
u – a 2
2

sin ν arcsin(u/a)

√
 if 0 < u < a,

a – u
 2 2
ν
5 J ν (ax), a >0, ν > –2  a cos(πν/2)

 ν if a < u,
ξ(u + ξ)
√
2
where ξ = u – a 2

1 arcsin(u/a)if 0 < u < a,
6 J 0 (ax), a >0, ν >0
x π/2 if a < u
 –1
 ν sin ν arcsin(u/a) if 0 < u < a,
1  ν
7 J ν (ax), a >0, ν > –1 a sin(πν/2)
x √ ν if a < u

ν u + u – a
 2 2
0 if 0 < u < a,

 √ π(2a) ν
1
ν
8 x J ν (ax), a >0, –1< ν <
2 if a < u
 1 2 2 ν+1/2
Γ – ν u – a
2

2u
–1 –ax
9 x e J 0 (bx), a >0 arcsin
2
2
2
(u + b) + a + (u – b) + a 2

J 0 (ax) b sinh(bu)K 0 (ab)if 0 < u < a,
–1
10 , a, b >0
2
x + b 2 0 if a < u

xJ 0 (ax) 0 if 0 < u < a,
11 , a, b >0 1 –bu
2
x + b 2 2 πe I 0 (ab)if a < u
√
xJ 2n+1/2 (ax) (–1) sinh(bu)K 2n+1/2 (ab)if 0 < u < a,
n
12 x + b 2 ,
2
a, b >0, n = 0,1,2, ... 0 if a < u
ν
x J ν (ax)
, b ν–1 sinh(bu)K ν (ab)if 0 < u < a,
2
13 x + b 2
a, b >0, –1< ν < 5 0 if a < u
2
x 1–ν J ν (ax)
, 0 if 0 < u < a,
14 x + b 2 1 πb e I ν (ab)if a < u
2
–ν –bu
a, b >0, ν > – 3 2
2
√ 1 a 2
15 J 0 a x , a >0 cos
u 4u
1 √ 2 a 2
16 √ J 1 a x , a >0 sin
x a 4u
√
x ν/2 J ν a x , a ν a 2 πν
17 1 cos –
a >0, –2< ν < 2 u 4u 2
ν ν+1
2
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
Page 744