Laplace Tablas

7/23/2019 Laplace Tablas

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Appendix A / Laplace Transform Tables 863

f(t) F(s)

1 Unit impulse d(t) 1

2 Unit step 1(t)

3 t

4

5 tn (n=1, 2, 3, p)

6 eat

7 teat

8

9 tneat (n=1, 2, 3, p)

10 sinvt

11 cosvt

12 sinhvt

13 coshvt

14

15

16

17 1

s(s + a)(s + b)

1

abc1 + 1

a - bAbe-at - ae-btBd

s

(s + a)(s + b)

1

b - aAbe-bt - ae-atB

1

(s + a)(s + b)

1

b - aAe-at - e-btB

1

s(s + a)

1

aA1 - e-atB

s

s2 - v2

v

s2 - v2

s

s2 + v2

v

s2 + v2

n!

(s + a)n+1

1(s + a)n

1(n - 1)!

tn-1e-at

(n = 1, 2, 3, p)

1

(s + a)2

1

s + a

n!

sn+1

1

sntn-1

(n - 1)! (n = 1, 2, 3, p)

1

s2

1

s

Table A1 Laplace Transform Pairs

(continues on next page)


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864 Appendix A / Laplace Transform Tables

18

19

20 eat sinvt

21 eat cosvt

22

23

24

25 1-cosvt

26 vt-sinvt

27 sinvt-vtcosvt

28

29 tcosvt

30

31 s2

As2

+ v

2

B2

1

2v(sinvt + vtcosvt)

s

As2 + v21B As2 + v22B1

v22 - v

21

Acosv1t - cosv2tB Av21 Z v22B

s2 - v2

As2 + v2B2

s

As2 + v2B21

2vtsinvt

2v3

As2 + v2B2

v3

s2As2 + v2B

v2

sAs2 + v2B

(0 6 z 6 1, 0 6 f 6 p2)

v2n

sAs2 + 2zvns + v2nBf = tan-121 - z2

z

1 -1

21 - z2e-zvnt sin Avn21 - z2 t + fB

(0 6 z 6 1, 0 6 f 6 p2)

s

s2 + 2zvns + v2nf = tan-1

21 - z2

z

-1

21 - z2e-zvnt sin Avn21 - z

2 t - fB

v2n

s2 + 2zvns + v2n

vn

21 - z2e-zvnt sinvn21 - z

2 t (0 6 z 6 1)

s + a

(s + a)2 + v2

v

(s + a)2 + v2

1

s2(s + a)

1

a2Aat - 1 + e-atB

1

s(s + a)21

a2A1 - e-at - ate-atB

Table A1 (continued)


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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 l Cf(t)g(t) D =1

2pj3c+jq

c-

jq

F(p)G(s - p)dp

l c3t

0

f1(t - t)f2(t)dt d = F1(s)F2(s)

l cf a1abd = aF(as)

l c 1t

f(t) d = 3q

s

F(s)ds if limt S 0

1

tf(t)exists

l C tnf(t) D = (-1)n dn

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

l Ct2f(t) D =d2

ds2F(s)

l C tf(t) D = -dF(s)

ds

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

l C e-atf(t) D = F(s + a)

3q

0

f(t)dt = lims S 0

F(s) if3q

0

f(t)dtexists

l c 3

t

0f(t)dtd =

F(s)

s

l;c 3p3f(t)(dt)

n d = F(s)sn

+ an

k = 1

1

sn - k + 1c3p3f(t)(dt)

k dt = 0;

l;c 3f(t)dtd =

F(s)

s +

1

sc 3f(t)dtd t = 0;

f(t)(k - 1)

=dk - 1

dtk - 1f(t)

l;c d ndtn

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

k = 1

sn - kf(0; )(k - 1)

l;

cd 2

dt2f(t)

d = s2F(s) - sf(0; ) - f

#

(0; )

l;c ddt

f(t) d = sF(s) - f(0 ; )l Cf1(t) ; f2(t) D = F1(s) ; F2(s)

l CAf(t) D = AF(s)

Table A2 Properties of Laplace Transforms

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