Page 749 - Handbook Of Integral Equations

Supplement 6

Tables of Fourier Cosine Transforms

6.1. General Formulas

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

1 sin(au)
0if a < x u
x if 0 < x <1,

4 2 u
2 2 – x if 1 < x <2, cos u sin
u 2 2
0 if 2 < x
1
3 , a >0 – sin(au) si(au) – cos(au) Ci(au)
a + x
π –au
1 e (the integral is understood
4 , a >0 2a
2
a + x 2 in the sense of Cauchy principal value)
1 π sin(au)
5 , a >0
a – x 2 2u
2
a a –au
6 2 2 + 2 2 πe cos(bu)
a +(b + x) a +(b – x)
b + x b – x –au
7 + πe sin(bu)
2
2
a +(b + x) 2 a +(b – x) 2
1 1 –3 au π au
8 , a >0 2 πa exp – √ sin + √
4
a + x 4 2 4 2
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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