248
6.8
CHAP. 6 Laplace Transforms
Laplace Transform: General Formulas
Formula
#
F(s) 5 l{ f (t)} 5
Name, Comments
Sec.
`
e2stf (t) dt
Definition of Transform
0
6.1
f (t) 5 l21{F(s)}
Inverse Transform
l{af (t) 1 bg(t)} 5 al{ f (t)} 1 bl{g(t)}
Linearity
6.1
s-Shifting
(First Shifting Theorem)
6.1
l{eatf (t)} 5 F(s 2 a)
l21{F(s 2 a)} 5 eatf (t)
l( f r ) 5 sl( f ) 2 f (0)
l( f s ) 5 s 2l( f ) 2 sf (0) 2 f r (0)
l( f
n
(n)
) 5 s l( f ) 2 s
Differentiation of Function
(n21)
f (0) 2 Á
6.2
Á 2 f (n21)(0)
le
t
# f (t) dtf 5 1s l( f )
Integration of Function
0
t
( f * g)(t) 5
# f (t)g(t 2 t) dt
0
t
5
# f (t 2 t)g(t) dt
Convolution
6.5
t-Shifting
(Second Shifting Theorem)
6.3
0
l( f * g) 5 l( f )l(g)
l{ f (t 2 a) u(t 2 a)} 5 e2asF(s)
˛
21
l
{e2asF (s)} 5 f (t 2 a) u(t 2 a)
l{tf (t)} 5 2F r (s)
le
f (t)
l( f ) 5
t
f 5
#
Differentiation of Transform
6.6
`
s ) d|
s
F( |
Integration of Transform
s
1
1 2 e2ps
p
#e
0
2st
f (t) dt
f Periodic with Period p
6.4
Project
16
SEC. 6.9 Table of Laplace Transforms
6.9
249
Table of Laplace Transforms
For more extensive tables, see Ref. [A9] in Appendix 1.
F (s) 5 l{ f (t)}
f (t)
˛
1
2
3
4
5
6
7
8
9
1>s
1>s 2
1>s n
1> 1s
1>s 3>2
1>s a
(n 5 1, 2, Á )
(a . 0)
1
s2a
1
n
(s 2 a)
1
k
teat
(n 5 1, 2, Á )
1
t n21eat
(n 2 1)!
(k . 0)
1 k21 at
t
e
G(k)
(s 2 a)
11
12
13
14
15
16
17
18
1
(s 2 a)(s 2 b)
s
(s 2 a)(s 2 b)
1
2
t 6.1
eat
(s 2 a)2
1
10
1
t
t n21>(n 2 1)!
1> 1pt
21t> p
t a21>G(a)
Sec.
(a Þ b)
(a Þ b)
t 6.1
1
(eat 2 ebt)
a2b
1
(aeat 2 bebt)
a2b
1
sin vt
v
2
s 1v
s
cos vt
s 2 1 v2
1
1
sinh at
a
s2 2 a2
s
cosh at
s2 2 a2
1
2
2
2
2
(s 2 a) 1 v
s2a
t 6.1
1 at
e sinh vt
v
eat cos vt
(s 2 a) 1 v
19
20
1
1
s(s 2 1 v2)
v2
1
1
s 2(s 2 1 v2)
v3
(1 2 cos vt)
x 6.2
(vt 2 sin vt)
(continued )
250
CHAP. 6 Laplace Transforms
Table of Laplace Transforms (continued )
F (s) 5 l{ f (t)}
f (t)
˛
1
21
22
23
24
1
2
(sin vt 2 vt cos vt)
2v3
t
sin vt
2v
2 2
(s 1 v )
s
(s 2 1 v2) 2
s2
2
2 2
2
2
2
2
(a 2 Þ b 2)
(s 1 a )(s 1 b )
1
26
27
28
29
30
31
34
35
36
37
4k 3
1
s 4 1 4k 4
1
2k 2
1
s4 2 k 4
s
2k 3
1
s4 2 k 4
2k 2
s 1 4k
s
1s 2 a 2 1s 2 b
1
1s 1 a 1s 1 b
1
(sin kt cos kt 2 cos kt sinh kt)
sin kt sinh kt
(sinh kt 2 sin kt)
(cosh kt 2 cos kt)
1
2 2pt 3
1pt
3>2
(s 2 2 a 2)k
(ebt 2 eat)
e2(a1b)t>2I0 a
1
s
(s 2 a)
1
(cos at 2 cos bt)
a2b
tb
2
I 5.5
J 5.4
J0(at)
2s 1 a 2
2
eat(1 1 2at)
k21>2
(k . 0)
1p t
a b
G(k) 2a
Ik21>2(at)
I 5.5
e2as>s
e2as
u(t 2 a)
d(t 2 a)
6.3
6.4
1 2k>s
e
s
1 2k>s
e
1s
J0(2 1kt)
J 5.4
1
38
s
39
b 2 a2
4
32
33
1
2
1
4
t 6.6
1
(sin vt 1 vt cos vt)
2v
(s 1 v )
s
25
Sec.
3>2
1
1pt
1
ek>s
e2k1s
1pk
(k . 0)
cos 2 1kt
sinh 2 1kt
k
2 2pt
2
3
e2k
>4t
(continued )