Page 736 - Handbook Of Integral Equations

1 c+i∞ px ˜
˜
No Laplace transform, f(p) Inverse transform, f(x)= e f(p) dp
2πi
c–i∞
√
1 ∞ cos 2 xt
1
17 √ f ˜ √ f(t) dt
p p πx
0
√
1
1 ∞ sin 2 xt
18 √ f ˜ √ f(t) dt
p p p
0 πt
1 ∞ ν √
1
19 2ν+1 f ˜ (x/t) J 2ν 2 xt f(t) dt
p p 0
1 ∞ √
1
20 f ˜ J 0 2 xt f(t) dt
p p
0
1 1 x √
˜
21 f p + J 0 2 xt – t 2 f(t) dt
p p 0
1 a x x – t ν √
˜
22 2ν+1 f p + J 2ν 2 axt – at 2 f(t) dt
p p 0 at
∞ t t 2

√
23 f ˜ p √ exp – f(t) dt
0 2 πx 3 4x
1 √ 1 ∞ t 2
24 √ f ˜ p √ exp – f(t) dt
p πx 0 4x
x 2
1 t t

˜
√

25 f p + p √ exp – f(t) dt
2 π (x – t) 3/2 4(x – t)
0
x √
2
2
26 f ˜ p + a 2 f(x) – a f x – t 2 J 1 (at) dt
0
x
√
2

2
27 f ˜ p – a 2 f(x)+ a f x – t 2 I 1 (at) dt
0
2
f ˜ p + a 2 x √
2
28 J 0 a x – t 2 f(t) dt
2
p + a 2 0
f ˜ p – a 2 x √
2
2
29 I 0 a x – t 2 f(t) dt
p – a 2 0
2
x
√

2
2
30 f ˜ (p + a) – b 2 e –ax f(x)+ be –ax f x – t 2 I 1 (bt) dt
0
∞ x
t–1
˜
31 f(ln p) f(t) dt
0 Γ(t)
1 ∞ x t
˜
32 f(ln p) f(t) dt
p Γ(t +1)
0
˜
2
˜
33 f(p – ia)+ f(p + ia), i = –1 2f(x) cos(ax)
˜
˜
2
34 i f(p – ia) – f(p + ia) , i = –1 2f(x) sin(ax)
˜
df(p)
35 –xf(x)
dp
n ˜
d f(p) n
36 (–x) f(x)
dp n
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
Page 720