Modern Control Engineering

Appendix A / Laplace Transform Tables 865

1

2

3

4

5

where

6

7

8

9

10

11

12

13

14

15

16

17

18 lCf(t)g(t)D=

1

2 pj 3

c+jq

c-jq

F(p)G(s-p)dp

lc

3

t

0

f 1 (t-t)f 2 (t)dtd=F 1 (s)F 2 (s)

lcfa

1

a

bd =aF(as)

lc

1

t

f(t)d =

3

q

s

F(s)ds if limtS 0

1

t

f(t) exists

lCtnf(t)D=(-1)n

dn

dsn

F(s) (n=1, 2, 3,p)

lCt^2 f(t)D=

d^2

ds^2

F(s)

lCtf(t)D=-

dF(s)

ds

lCf(t-a)1(t-a)D=e-asF(s) a 0

lCe-atf(t)D=F(s+a)

3

q

0

f(t)dt=limsS 0 F(s) if

3

q

0

f(t)dt exists

lc

3

t

0

f(t)dtd=

F(s)

s

l;c

3

p

3

f(t)(dt)nd=

F(s)

sn

+ a

n

k= 1

1

sn-k+^1

c

3

p

3

f(t)(dt)kd

t= 0 ;

l;c

3

f(t)dtd=

F(s)

s

+

1

s

c

3

f(t)dtd

t= 0 ;

f(t)

(k-1)

=

dk-^1

dtk-^1

f(t)

l;c

dn

dtn

f(t)d =snF(s)- a

n

k= 1

sn-kf( 0 ;)

(k- 1 )

l;c

d^2

dt^2

f(t)d=s^2 F(s)-sf(0 ;)-f

(0 ;)

l;c

d

dt

f(t)d=sF(s)-f(0 ;)

lCf 1 (t);f 2 (t)D=F 1 (s);F 2 (s)

lCAf(t)D=AF(s)

Table A–2 Properties of Laplace Transforms