33 - Laplace Transforms

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29. Laplace Transforms Contents Page

29.1. Definition of the Laplace Transform . . . . . . . . . . . . .

1020

29.2. Operations for the Laplace Transform . . . . . . . . . . . .

1020

29.3. Table of Laplace Transforms . . . . . . . . . . . . . . . .

1021

29.4. Table of Laplace-Stieltjes Transforms . . . . . . . . . . . .

1029

References . . . . . . . . . . . . . . . . . . . . . . . . . .

1030

1019

29.1. Definition of the Laplace Transform One-dimensional Laplace Transform

j(s) = 4 ~F(t)1 =Jm

29.1.1

0

e-**F[t)dt

29.1.2

lhSBe-WF(t)dt A

exists, then it converges for all s with 9s>so, and the image function is a single valued analytic

a m flm

Deiinition of the Unit Step Function

29.1.3 E+-

98>So.

Two-dimenional Laplace Transform

F(t)is a function of the real variable t and 5 is a complex variable. F(t)is called the original function andf(s) is called the image function. If the integral in 29.1.1 converges for a real s=so, i.e., A 4

{

function of s in the half-plane

u(t)=

0

* 1

O )O)

In the following tables the factor u(t) is to be understood as multiplying the original function F(t).

29.2. Operations for the Laplace Transform'

i*

Original Function F(t)

29.2.1

e-'IF(t)dt

F(t) Inversion Formula

29.2.2

etaj(8)ds

Linearity Property

29.2.3 29.2.4 29.2.5

Image Function f (8)

1 -f (8)

29.2.6

8

29.2.7 Convolution (Faltung) Theorem

29.2.8

r' Fl(2 -r)F2(r)dr=Fl*F2

Jo

-tF(t)

29.2.9 29.2.10

(-l)"t"F(t)

f 1 (df2(8) f' (4

Differentiation

f("'(8)

* Ada ted by permission from R. V. Churchill, Operatiorial mathematics, 2d ed., McGraw-Hill Book Co., Inc., New York, 1958. 1020

N.f.,

1021

LAPLACE TRANSFORMS

Original Function F(t)

Image Function f(s) Integration

29.2.11

j-mf(4dz Linear Transformation

f(s-4

eufF(t)

29.2.12 29.2.13 29.2.14

Translation

29.2.15

F(t-b)u(t-b)

e - b"fs)

(b>O)

Periodic Functions

29.2.16

F(t+a)=F(t)

29.2.17

F(t+a) = -F(t)

Half-Wave Rectification of F(t) in 29.2.17

29.2.18

F(t)

2 (-l)"u(t-na)

'

n-0

Full-Wave Rectification of F(t) in 29.2.17

29.2.19

IF@)I

f ( s ) coth

2

Heaviside Expansion Theorem

29.2.20

P O , q(s)=(s-a,)(s-a,)

. . .

(s--a,)

p(s) a polynomial of degree(s+b) (s+c>

(a, b,c distinct constants) 29.3.15 29.3.16 29.3.17 29.3.18 29.3.19

1 P+a2 S s2+d

U2

S s2--a2

1

s(s2+a2)

- sinh U

ut

cosh ut 1

- (1 -('OS

a2

at)

1 (at-sin ut) a3

29.3.20 29.3.21

ut

cos ut

1

1 s2-

1

- sin U

-

1 (s2+ UZ) 2

1

-

2a3

(sin u t -at cos ut)

LAPLACE TRANSFORMS

1023 F(t)

29.3.22

(s2+a2)

29.3.24

29.3.28 29.3.29

2a

s2-a2 (s2+a2)2

t wsut

S

(s2+a2)(s2+b2)

s+a (S+a)2+b2

29.3.32 29.3.33

cos at)

(a2# b2) b

sin bt

e-at cos bt

3d $+aa 4a3 s4+4a4

sin at cosh at-cos at sinh at 1

29.3.30 29.3.31

- (sin at +at

(s2+a2)

29.3.26 29.3.27

1

S2

29.3.23

29.3.25

t 5 sin at

S

sin at sinh at 2u2 1 s4-a4 S -

1

- (sinh at -sin at)

2a5

1 (cosh at-cos at)

s4-a4

2a2

8a3s2 (s2+u2)3

(l+a2t2)sin at-at cos at

29.3.34

22

29.3.35 29.3.36 --aea't 1 &-t

29.3.37 29.3.38 29.3.39

29.3.40

&

s+a2 1

m3-aZ>

erfc a&

1024

LAPLACE TRANSFORMS

f (8)

F (0

29.3.41

7

29.3.42 29.3.43 29.3.44 29.3.45

ea2t[b--a erf a a - b e b * t erfc bJ?

7

1 &(&+a> 1 (st-4J6+b

b2--a2 &s-a2) (++b)

7

(1-5)"

29.3.46

22

p+*

(1-4"

29.3.47

22

Sn++

9 29.3.49

1

9

&T-(aJGZ

10 9

9 29.3.53

(a-b)'

( m a +JS+bY

(k>O) 1 e-*%()at) a.

29.3.55

1

J92+a2

9

9

6,lO

1025

LAPLACE TRANSFORMS

f(8) 29.3.58

29.3.59

JW-S)~ (k>O)

(

@=a

(8 2-J

29.3.60

F (0

'

1

(y>

-1)

(k>o)

(sa-az)k

kat

t Jt(at)

9

a'I,(at>

9

J1; ("-> k-tlr-t(at) r(k) 2a

29.3.61

1 e-). 8

u(t-k)

29.3.62

21 e-6

(t-k)U(t -k)

29.3.63

1 e-6

1 -e-Ra

29.3.64

29.3.65

(P>O)

B

S

1 1 +coth )ks s (1 -e-k)= 2s

L E

u(t-k)

(t-k)'-l

r (PI

2 u(t-nk)

p-

29.3.66

1

s(e6 -a)

C a"-'u(t-nk) n-1

I f-;-

Lh

k

m

0

29.3.67

29.3.68

29.3.69

1

-

8

tanh ks

1

s(l+e-6) 1

- tanh ks SO

29.3.70

1

s sinh ks

u(t)+2 f: (-l)%(t-!hk) n-1

q

n-

(-l)"u(t-nk)

tu(t ) +2

2

2 n -0 u[t-(2n+l)k]

1

s cosh ks

2

I#-

1

O

3k

2 (-l)"u[t-(%+l)k]

n-0

3L

Zk

b/

hn

5 (- 1) "(t -2nk)u(t-2nk) 111

k

h * -

2

29.3.71

6

h

u(t)-u(t-k)

n-0

6,10

k

Zk

0

I Y

ah

6k

oh

kul 0

I

3k

5,

7k

1026

LAPLACE TRANSFORMS

f (4

F(t)

-1 coth k8

29.3.72

u(t) +2

8

5 u(t-2nk) n-1

p-0

k

T8

f+p coth %

29.3.73

#F/\r\/.

Isinkt1

-

0

1 (sz+l)(1 -e-")

29.3.74

0

k

C (-1)"u(t-n~)

n-0

k

sin t

'!UL 0

29.3.75

1 J -e 8

29.3.76

ze

29.3.77

1 ; &e

1 "

9

1 cash 2&

29.3.79

F aea

1

asin

2@

1 q sinh 2&

*

(i)q (i)9

29.3.80

2e

29.3.81

eg &P

(r>O)

29.3.82

e -k fi

(k>O)

k exp (-g) 2J?rt8

(k2 0)

k erfc 2 4

(cc>O)

I t

-1e

29.3.83

-kfi

8

29.3.84

7 81

(k20)

e-k*

1 -e-&

29.3.85

(krO)

8t

29.3.86

29.3.87

1

-

(n=O, 1,2, . . .; k2O)

81+P e-"J;

E' 8 2

29.3.88

'Bee page 11.

(n=O, 1,2, . . .; k>O)

e -kfi

e-& a

377

z

1 2 ,alze g

g

2"

1 G cos 2&

29.3.78

1 -E

r

JO(2@)

8

1

ek

Zk

(k20)

J,-,(2@)

9

4+*(2*)

9

-L exp G t

4

7

(-g)

exp ( - 3 - k

erfc %k= 2 4 i erfc k 2$

k

(4t)tn inerfc -

7

24

(-:)

(3

22

exP 2 " V Hn 1 - exp

&

7

(-E)-~~Fvz~ . (

22

erfc a4+-

7

*

1027 29.3.89

9

9

9 29.3.94

e - k ( w - 8 )

(k20)

JW

9 1

- In

29.3.98

S

-?-In

s

t(r=.57721 56649 . . . Euler's constant)

tk- 1

29.3.99

$1.s

(k>O)

r(k) irLW-1. tl

6

29.3.100

In s -

@>O>

eo'[ln u+E1(ut)]

5

In s -

cos t Si (t)--sin t Ci (t,

5

s In s

-sin t Si (t)-cos t Ci (t)

5

s--a

29.3.101

sa+ 1

29.3.102 29.3.103

ss+1 1

-In (l+ks) S

(k>O) 1 (e-b:-e-a'

29.3.104

S+U In s+b

29.3.105

1S In (l+k2s2)

29.3.106

1

-In ($+az) S

t

(k>O)

(u>O)

-2 ci

1

(i)

2 In u-2 Ci (ut)

5

1028 29.3.107

LAPLACE TRANSFORMS

1 In (sa+aa) S2

5

(a>O) 2 t

29.3.108

h-#+aa Sa

- (l--cos at)

29.3.109

In-sa-a' 82

2t (l-cosh

29.3.110

k arctan -

29.3.111

- arctan -

71 sin kt

8

1

k

8

8

29.3.112

ekS8'erfc ks

29.3.113

18 ekar'erfc ks

at)

Si (kt)

(k>O) (k>O)

7

hap(-$)

7

t erfB

7

zsin 1 2k4

7

-1

29.3.114 29.3.115 29.3.116

61 err erfc f i k

29.3.117

29.3.118

(k20)

G i !

k

erfc G

-2k&

&i e 1

u(t-k)

29.3.119

9 7 7 p

29.3.120

9

+P(-;)

9

p1 x p ( - 5 )

29.3.121

1 ehK1(ks) 8

(k>O)

29.3.122 29.3.123

9

29.3.124

fi'e-**IO(ks)

(k>O)

29.3.125

e-&Il(ks)

(k>O)

9

1 Jt (2k-tt)

k-t TkJ-

[u(t) -u( t -2k)

I

[~(t)-~(t-2k)]

1029

LAPLACE TRANSFORMS

f(8) 29.3.126

eaaEl(us)

29.3.127

--se"El(us) U

(u>O)

1

(u>O)

29.3.128 u'-"e"E,(as) 29.3.129

F(t)

(u>O;n=O,

5

1 t+a

5

-

1,2,. . .) 5

k - ~ i ( s ) ] cos s+ci(s) sin s

5

1

1 -

(t+4" 1

t2+1

29.4. Table of Laplace-Stieltjes Transforms

W)

ds) 29.4.1

1-

29.4.2

e-"

W)

e-''d+(t)

(k>O)

u(t-k)

5 u(t-nk)

29.4.3 29.4.4 29.4.5

n=O

1 l-te-" 1

sinh ks

2 (-l)"u(t--nk)

n -0

(k>O)

29.4.6

n =O

n-0

29.4.7

tanh ks

29.4.8

1 sinh (ks+u)

(k>O)

u(t)+2

(k>o)

29.4.9 29.4.10

sinh (hs+b) sinh (ks+u)

(Oh
33 - Laplace Transforms

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