Page 738 - 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 1 2
15 x
p 3 2
1 1 –ax
16 e + ax – 1
2
p (p + a) a 2
1 1 –ax –bx
17 a – b + be – ae
p(p + a)(p + b) ab(a – b)
1 1
18 2 1 – e –ax – axe –ax
p(p + a) a 2
1 (c – b)e –ax +(a – c)e –bx +(b – a)e –cx
19
(p + a)(p + b)(p + c) (a – b)(b – c)(c – a)
p a(b – c)e –ax + b(c – a)e –bx + c(a – b)e –cx
20
(p + a)(p + b)(p + c) (a – b)(b – c)(c – a)
2
2
2
p 2 a (c – b)e –ax + b (a – c)e –bx + c (b – a)e –cx
21
(p + a)(p + b)(p + c) (a – b)(b – c)(c – a)
1 1 –ax –bx –bx
22 2 2 e – e +(a – b)xe
(p + a)(p + b) (a – b)
p 1 –ax –bx
23 2 –ae +[a + b(b – a)x e
(p + a)(p + b) (a – b) 2
p 2 1 2 –ax 2 –bx
24 2 a e + b(b – 2a – b x + abx)e
(p + a)(p + b) 2 (a – b)
1 1 2 –ax
25 3 2 x e
(p + a)
p
1
26 3 x 1 – ax e –ax
(p + a) 2
p 2 1 2 2 –ax
27 1 – 2ax + a x e
2
(p + a) 3
1 1
28 1 – cos(ax)
2
p(p + a ) a 2
2
1 1 –bx b
29 2 2 1 – e cos(ax)+ sin(ax)
2
p (p + b) + a a + b 2 a
1 1 a
30 2 2 e –ax + sin(bx) – cos(bx)
2
(p + a)(p + b ) a + b 2 b
p 1
31 2 2 –ae –ax + a cos(bx)+ b sin(bx)
(p + a)(p + b ) a + b 2
2
p 2 1 2 –ax 2
32 a e – ab sin(bx)+ b cos(bx)
(p + a)(p + b ) a + b 2
2
2
2
1 –ax 1 ax/2 √
1 e – e cos(kx) – 3 sin(kx) ,
33 3 3 3a 2 √ 3a 2
p + a 1
k = a 3
2
1 1 √
p – e –ax + e ax/2 cos(kx)+ 3 sin(kx) ,
34 3 3 3a 3a
p + a 1 √
k = a 3
2
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 722