Page 755 - Handbook Of Integral Equations

Supplement 7

Tables of Fourier Sine Transforms

7.1. General Formulas

∞

ˇ
No Original function, f(x) Sine transform, f s (u)= f(x) sin(ux) dx
0
ˇ
ˇ
1 af 1 (x)+ bf 2 (x) af 1s (u)+ bf 2s (u)
u
1
ˇ
2 f(ax), a >0 f s
a a
d 2n
2n
ˇ
3 x f(x), n =1, 2, ... (–1) n f s (u)
du 2n
d 2n+1 ∞
ˇ
n+1
ˇ
4 x 2n+1 f(ax), n =0, 1, ... (–1) 2n+1 f c (u), f c (u)= f(x) cos(xu) dx
du 0
1 u + b u – b
5 f(ax) cos(bx), a, b >0 f s ˇ + F s
2a a a
7.2. Expressions With Power-Law Functions
∞
ˇ
No Original function, f(x) Sine transform, f s (u)= f(x) sin(ux) dx
0
1if 0 < x < a, 1

1 1 – cos(au)
0if a < x u
x if 0 < x <1,

4 2 u
2 2 – x if 1 < x <2, sin u sin
u 2 2
0 if 2 < x
1 π
3
x 2
1
4 , a >0 sin(au) Ci(au) – cos(au) si(au)
a + x
x π –au
5 2 2 , a >0 e
a + x 2
1 π –au
6 2 2 , a >0 2 1 – e
x(a + x ) 2a
a a –au
7 2 2 – 2 2 πe sin(bu)
a +(x – b) a +(x + b)
x + b x – b –au
8 – πe cos(bu)
2
2
a +(x + b) 2 a +(x – b) 2
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 739