Page 753 - Handbook Of Integral Equations

∞

ˇ
No Original function, f(x) Cosine transform, f c (u)= f(x) cos(ux) dx
0
1 √ √ π ab a + b 2 π
2
9 √ sin a x sin b x , a, b >0 sin sin –
x u 2u 4u 4

π u 2 u 2
2
10 sin ax , a >0 cos – sin
8a 4a 4a
√
π Au 2 Bu 2
2 2 exp – sin ϕ– ,
2
2
2
2 1/4
11 exp –ax sin bx , a >0 (A +B ) A +B 2 A +B 2
A =4a, B =4b, ϕ = 1 arctan(b/a)
2
1 – cos(ax) 1 2
12 , a >0 ln 1 – a

x 2 u 2

1 – cos(ax) 1 π(a – u)if u < a,
13 , a >0 2
x 2 0 if u > a
14 x ν–1 cos(ax), a >0, 0 < ν <1 1 Γ(ν) cos 1 πν |u – a| –ν +(u + a) –ν
2 2
1 –1 –ab
cos(ax) 2 πb e cosh(bu)if u < a,
15 , a, b >0 1 –1 –bu
2
x + b 2 πb e cosh(ab) if u > a
2
b 1 1
16 e –bx cos(ax), a, b >0 2 2 + 2 2
2 (a + u) + b (a – u) + b
1 √ π a 2 π
17 √ cos a x sin +
x u 4u 4
1 √ √ π ab a + b 2 π
2
18 √ cos a x cos b x cos sin +
x u 2u 4u 4
2
1 π a + u 2 au
2
19 exp –bx cos(ax), b >0 exp – cosh
2 b 4b 2b

2 π 1 –1 2 1 –1 2
20 cos ax , a >0 cos a u + sin a u
8a 4 4
√
π Au 2 Bu 2
2 2 exp – cos ϕ– ,
2
2
2 1/4
2
21 exp –ax cos bx , a >0 (A +B ) A +B 2 A +B 2
A =4a, B =4b, ϕ = 1 arctan(b/a)
2
6.7. Expressions With Special Functions
∞
ˇ
No Original function, f(x) Cosine transform, f c (u)= f(x) cos(ux) dx
0
1
u
1 Ei(–ax) – arctan
u a
0 if 0 < u < a,

2 Ci(ax) π
– if a < u
2u
1
3 si(ax) – ln u + a , u ≠ a
2u u – a
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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