Page 750 - Handbook Of Integral Equations

∞
ˇ
No Original function, f(x) Cosine transform, f c (u)= f(x) cos(ux) dx
0
1 π ae –bu – be –au
9 2 2 2 2 , a, b >0
2
2
(a + x )(b + x ) 2 ab(a – b )
x 2m π ∂ n √
√
10 (x + a) n+1 , (–1) n+m n a 1/ m –u a
e
2
n, m =1, 2, ... ; n +1 > m ≥ 0 2n! ∂a
1 π
11 √
x 2u
1

√ if 0 < x < a, π
12 x 2 C(au), C(u) is the Fresnel integral
0 if a < x 2u
0 if 0 < x < a,

1 π
13 1 – 2C(au) , C(u) is the Fresnel integral
√ if a < x 2u
x
0 if 0 < x < a,

14 1 π cos(au) – sin(au)
√ if a < x 2u
x – a
1
15 √ K 0 (au)
2
a + x 2
1

√ if 0 < x < a, π
16 a – x 2 2 J 0 (au)
2
0 if a < x
–ν 1 ν–1
17 x , 0 < ν <1 sin πν Γ(1 – ν)u
2
6.3. Expressions With Exponential Functions
∞

ˇ
No Original function, f(x) Cosine transform, f c (u)= f(x) cos(ux) dx
0
a
1 e –ax
a + u 2
2
2
1 b + u 2
–ax
–bx
1

2 e – e ln
x 2 a + u 2
2
√ –ax 1 √ 2 2 –3/4 3 u
3 xe π (a + u ) cos arctan
2 2 a
2
1 –ax π a +(a + u ) 1/2
2 1/2

4 √ e
x 2 a + u 2
2
a n+1 n! k 2k 2k
u
n –ax
5 x e , n =1, 2, ... 2 2 n+1 (–1) C n+1
(a + u ) a
0≤2k≤n+1
∂ n 1
k n u √ ,
n
e
6 x n–1/2 –ax , n =1, 2, ... ∂a r r – a
√
2
2
where r = a + u , k n =(–1) n π/2
u

2 –ν/2
2
7 x ν–1 –ax Γ(ν)(a + u ) cos ν arctan
e
a
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
Page 734