Page 759 - Handbook Of Integral Equations

∞
ˇ
No Original function, f(x) Sine transform, f s (u)= f(x) sin(ux) dx
0
1 1
1 2 (u +2a)ln |u +2a| + (u – 2a)ln |u – 2a|
9 sin (ax), a >0 4 1 4
x 2 – u ln u
2
2
1 π u + b 2 bu
2
10 exp –ax sin(bx), a >0 exp – sinh
2 a 4a 2a
0 if 0 < u < a – b,

1 π
11 sin(ax) sin(bx), a ≥ b >0 4 if a – b < u < a + b,
x
0 if a + b < u
√
a π a √

12 sin , a >0 √ J 1 2 au
x 2 u
1 π √ √ √
a
13 √ sin , a >0 sin 2 au – cos 2 au +exp –2 au
x x 8u

π a 2
√ √ –3/2
14 exp –a x sin a x , a >0 a u exp –
8 2u

 0 if 0 < u < a,
cos(ax) 1
15 , a >0 4 π if u = a,
x  1 π if a < u
2
π(u + a) –ν – sign(u – a)|u – a| –ν
16 x ν–1 cos(ax), a >0, |ν| <1 1
4Γ(1 – ν) cos πν
2
1 –ab
x cos(ax) – πe sinh(bu)if u < a,
17 , a, b >0 1 2 –bu
2
x + b 2 πe cosh(ab) if u > a
2
1 – cos(ax) u 2 2 a
18 , a >0 u – a + u + a
x 2 2 ln u 2 2 ln u – a
1 √ π a 2 π
19 √ cos a x cos +
x u 4u 4
1 √ √ π ab a + b 2 π
2
20 √ cos a x cos b x , a, b >0 cos cos +
x u 2u 4u 4
7.7. Expressions With Special Functions
∞

ˇ
No Original function, f(x) Sine transform, f s (u)= f(x) sin(ux) dx
0
1 u 2
1 erfc(ax), a >0 1 – exp –
u 4a 2
1 2
2 ci(ax), a >0 – ln 1 – u

2u a 2
0 if 0 < u < a,

3 si(ax), a >0 1 –1
– πu if a < u
2
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 743