Page 742 - 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 x
6 √ 2
p p π
1
x
7 √ (b – a) –1/2 –ax erf (b – a) 1/2 1/2
e
(p + a) p + b
1 1 ax √
8 √ √ e erf ax
p (p – a) a
1 –3/2 ax √ –1 –1/2 1/2
9 a e erf ax – 2a π x
p 3/2 (p – a)
1 2 √
10 √ π –1/2 –1/2 – ae a x erfc a x
x
p + a
a 2
√
11 √ 1 – e a x erfc a x
p p + a
1 2 √
12 √ e a x erfc a x
p + a p
1 2 1/2 ax √
13 √ √ 2 1 – √ (ax) +(1 – 2ax)e erf ax – 1
p + a π
1 1 1 ax √ 2 √
14 √ √ 2 + 2x – e erfc ax – √ x
p p + a a a πa
1 2
√
x
15 √ √ 2 2π –1/2 1/2 – 2axe a x erfc a x
p p + a
1 2 √ 2 √
2
2
16 √ 3 √ (a x +1) x – ax(2a x +3)e a x erfc a x
p + a π
2 n n–1/2
17 p –n–1/2 , n =1, 2, ... √ x
1 ⋅ 3 ... (2n – 1) π
2 n n–1/2 –ax
18 (p + a) –n–1/2 √ x e
1 ⋅ 3 ... (2n – 1) π
1
19 2 2 J 0 (ax)
p + a
1
20 I 0 (ax)
p – a 2
2
1
1 2 1/2
1

21 exp – ax J 0 (b – a x
2
p + ap + b 2 4
1
1/2
2
2
22 p + a – p √ sin(ax)
2πx 3
√
1 2 2 1/2 2
23 p + a + p √ cos(ax)
2
p + a 2 πx
√
1 2 2 1/2 2
24 2 2 p – a + p √ cosh(ax)
p – a πx
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 726