Page 756 - Handbook Of Integral Equations

∞
ˇ
No Original function, f(x) Sine transform, f s (u)= f(x) sin(ux) dx
0
n–2
x πue –au (2n – k – 4)! k
9 2 2 n , a >0, n =1, 2, ... (2au)
(x + a ) 2 2n–2 (n – 1)! a 2n–3 k!(n – k – 2)!
k=0
x 2m+1 π ∂ n
m –u a
10 (x + a) n+1 , (–1) n+m n a e √
2
n, m =0, 1, ... ;0 ≤ m ≤ n 2n! ∂a
1 π
11 √
x 2u
1 √
12 √ 2πu
x x
2 –3/2
2
13 x(a + x ) uK 0 (au)
√ 1/2
2
2
a + x – a π –au
14 √ e
2
a + x 2 2u
–ν

15 x , 0 < ν <2 cos 1 2 πν Γ(1 – ν)u ν–1
7.3. Expressions With Exponential Functions
∞

ˇ
No Original function, f(x) Sine transform, f s (u)= f(x) sin(ux) dx
0
u
–ax
1 e , a >0
a + u 2
2
[n/2]
u
a n+1 k 2k+1
n –ax
2k+1
2 x e , a >0, n =1, 2, ... n! (–1) C n+1
a + u 2 a
2
k=0
1 –ax u
3 e , a >0 arctan
x a
√
√ –ax π 3 u
2
2 –3/4
4 xe , a >0 (a + u ) sin arctan
2 2 a
√
1 –ax π a + u – a) 1/2
2
2
5 √ e , a >0 √
x 2 a + u 2
2
1 –ax √ √
2
2
6 √ e , a >0 2π a + u – a) 1/2
x x
√
n 2 2 1/2
π ∂ a + u – a
n
7 x n–1/2 –ax , a >0, n =1, 2, ... (–1) √
e
2 ∂a n a + u 2
2
u
2 –ν/2
2
8 x ν–1 –ax , a >0, ν > –1 Γ(ν)(a + u ) sin ν arctan
e
a
2
u
u
9 x –2 e –ax – e –bx , a, b >0 u ln u + b 2 + b arctan – a arctan
2
2 u + a 2 b a
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
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