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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