F(s) f(t)
"
1 f(t)
e-st f (t) dt
+"
0
2 AF(s) + BG(s) Af(t) + Bg(t)
3 sF(s)  f(+0) f 2 (t)
(1) (n-1) (n)
4 f (t)
snF(s) - sn-1 f (+0) - sn-2 f (+0) -L- f (+0)
t
1
5
F(s)
f (Ä)dÄ
+"
s
0
t Ä
1
6
F(s)
f ()d dÄ
+" +"
s2
0 0
t
7 F1(s)F2(s)
f1(t - Ä) f2 (Ä)dÄ = f1 " f2
+"
0
8  F2 (s) tf(t)
9 ( 1)nF(n)(s) tnf(t)
"
1
10 F(x)dx f (t)
+"
t
s
11 F(s  a) eatf(t)
12 e bsF(s) f(t  b), where f(t) = 0; t < 0
1 t
ëÅ‚ öÅ‚
13 F(cs)
f
íÅ‚
c cłł
1 t
ëÅ‚ öÅ‚
14 F(cs  b)
e(bt) / c f
íÅ‚
c cłł
a
e-st f (t) dt
15 f(t + a) = f(t) periodic signal
+"
0
1 - e-as
a
e-st f (t) dt
16 f(t + a) =  f(t)
+"
0
1 + e-as
F(s)
17 f1(t), the half-wave rectification of f(t) in No. 16.
1 - e-as
as
18 f2(t), the full-wave rectification of f(t) in No. 16.
F(s) coth
2
m
p(s) p(an )
19 ,q(s) = (s - a1)(s - a2 )L(s - am )
eant
"
q(s)
q2 (an )
1
r
(r-n)
p(s) Ć(s) (a)
20 eat
=
"Ćr - n)! (ntn-1 +L
q(s) (s - a)r ( -1)!
n =1
F(s) f(t)
1 sn ´(n)(t) nth derivative of the delta function
d´(t)
2 s
dt
3 1 ´(t)
1
41
s
1
5 t
s2
1 tn-1
6 (n = 12,L)
,
sn (n - 1)!
1
1
7
Ä„t
s
t
8 s 3/2
2
Ä„
2n tn-(1/ 2)
9
s-[n+(1/ 2)] (n = 12L)
, ,
1Å" 3Å" 5L(2n - 1) Ä„
“(k)(k e" 0)
10 tk 1
sk
1
11 eat
s - a
1
12 teat
(s - a)2
1 1
13 (n = 12,L) tn-1eat
,
(s - a)n (n - 1)!
“(k)
(k e" 0)
14
tk-1eat
(s - a)k
1 1
15 (eat - ebt )
(s - a)(s - b) (a - b)
s 1
16 (aeat - bebt )
(s - a)(s - b) (a - b)
1 (b - c)eat + (c - a)ebt + (a - b)ect
17
-
(s - a)(s - b)(s - c) (a - b)(b - c)(c - a)
1
18 e at valid for complex a
(s + a)
1
1
19
(1 - e-at )
s(s + a)
a
1 1
20 (e-at + at -1)
s2 (s + a) a2
îÅ‚
1 1 1 at2 1 Å‚Å‚
21 - t + - e-at śł
s3(s + a) a2 ïÅ‚ a 2 a
ðÅ‚ ûÅ‚
1 1
22 (e-at - e-bt )
(s + a)(s + b) (b - a)
1
1 îÅ‚ 1 Å‚Å‚
23
ïÅ‚1 + (be-at - ae-bt )śł
s(s + a)(s + b)
ab (a - b)
ðÅ‚ ûÅ‚
F(s) f(t)
1 1 îÅ‚ 1 Å‚Å‚
24
(a2e-bt - b2e-at ) + abt - a - bśł
s2 (s + a)(s + b) (ab)2 ïÅ‚(a - b)
ðÅ‚ ûÅ‚
1
îÅ‚
1 a3 - b3 1 (a + b) 1 b a
ëÅ‚ öÅ‚Å‚Å‚
25
ïÅ‚(ab) + t2 - t + íÅ‚ e-at - e-bt
2
s3(s + a)(s + b)
(ab) (a - b) 2 ab (a - b) a2 b2 łłśł
ðÅ‚ ûÅ‚
1 111
26 e-at + e-bt + e-ct
(s + a)(s + b)(s + c) (b - a)(c - a) (a - b)(c - b) (a - c)(b - c)
1 1 1 1 1
27 - e-at - e-bt - e-ct
s(s + a)(s + b)(s + c) abc a(b - a)(c - a) b(a - b)(c - b) c(a - c)(b - c)
ab(ct
Å„Å‚ -1) - ac - bc 1
+ e-at
ôÅ‚
(abc)2 a2 (b - a)(c - a)
1
ôÅ‚
28
òÅ‚
s2 (s + a)(s + b)(s + c)
11
ôÅ‚
+ e-bt + e-ct
ôÅ‚
b2 (a - b)(c - b) c2 (a - c)(b - c)
ół
1 ab + ac + bc 1
Å„Å‚
3
ôÅ‚(abc) [(ab + ac + bc)2 - abc(a + b + c)] - (abc)2 t + 2abc t2
1 ôÅ‚
29
òÅ‚
s3(s + a)(s + b)(s + c)
111
ôÅ‚
- e-at - e-bt - µ-ct
ôÅ‚
a3(b - a)(c - a) b3(a - b)(c - b) c3(a - c)(b - c)
ół
1 1
30
sin at
s2 + a2 a
s
31 cos at
s2 + a2
1 1
32
sinh at
s2 - a2 a
s
33 cosh at
s2 - a2
1
1
34
(1 - cos at)
s(s2 + a2 )
a2
1 1
35 (at - sin at)
s2 (s2 + a2 ) a3
1
1
36
(sin at - at cos at)
(s2 + a2 )2
2a3
s t
37 sin at
(s2 + a2 )2 2a
s2 1
38
(sin at + at cos at)
(s2 + a2 )2 2a
s2 - a2
39 t cos at
(s2 + a2 )2
s cos at - cosbt
40 (a2 `" b2 )
(s2 + a2 )(s2 + b2 ) b2 - a2
1
1
41
eat sin bt
(s - a)2 + b2
b
s - a
42 eat cos bt
(s - a)2 + b2
n
r
1
-e-at
43
"ëÅ‚2nn--r1-1öÅ‚ (-2t)r-1 d [cos(bt)]
[(s + a)2 + b2 ]n
4n-1b2n r=1 ìÅ‚ ÷Å‚ dtr
íÅ‚ Å‚Å‚
F(s) f(t)
n
Å„Å‚ r
ôÅ‚
e-at Å„Å‚
ôÅ‚
r
"ëÅ‚2nn--r1-1öÅ‚ (-2t)r-1 d [a cos(bt) + bsin(bt)]
4n-1b2n òÅ‚ r=1 ìÅ‚ ÷Å‚ dt
ôÅ‚ íÅ‚ Å‚Å‚
ôÅ‚
ół
s ôÅ‚
44
òÅ‚
[(s + a)2 + b2 ]n
n-1
ôÅ‚ r
- r - d üÅ‚
ôÅ‚ - 2b
ìÅ‚ ÷Å‚
"rëÅ‚2nn - 1 2öÅ‚ (-2t)r-1 dtr [sin(bt)]żł
íÅ‚ Å‚Å‚
ôÅ‚
þÅ‚
ół r=1
ëÅ‚ öÅ‚
3a2 at 3 at 3
45
e-at - e(at) / 2 ìÅ‚cos - 3 sin
÷Å‚
s3 + a3 2 2
íÅ‚ Å‚Å‚
4a3
46 sin at cosh at  cos at sinh at
s4 + 4a4
s 1
47
(sin at sinh at)
s4 + 4a4 2a2
1 1
48
(sinh at - sin at)
s4 - a4 2a3
s 1
49
(cosh at - cos at)
s4 - a4 2a2
8a3s2
50 (1 + a2t2) sin at  cos at
(s2 + a2 )3
n
1 s -1öÅ‚ n et d
ëÅ‚
51
Ln (t) = (tne-t )
íÅ‚ Å‚Å‚
s s n! dtn
[Ln(t) is the Laguerre polynomial of degree n]
1 t(n-1) e-at
52
where n is a positive integer
(s + a)n (n - 1)!
1 1
53 [1 - e-at - ate-at ]
s(s + a)2 a2
1
1
54
[at - 2 + ate-at + 2e-at ]
s2 (s + a)2
a3
1 1 îÅ‚ 1 Å‚Å‚
ëÅ‚
55
- a2t2 + at + 1öÅ‚ e-at śł
Å‚Å‚
s(s + a)3 a3 ïÅ‚1 íÅ‚ 2
ðÅ‚ ûÅ‚
1 1
56
{e-at + [(a - b)t -1]e-bt}
(s + a)(s + b)2 (a - b)2
1 1 1 îÅ‚ 1 a - 2b Å‚Å‚
57 - e-at - t + e-bt
s(s + a)(s + b)2 ab2 a(a - b)2 ïÅ‚ b(a - b) b2 (a - b)2 śł
ðÅ‚ ûÅ‚
1
11 1 îÅ‚ 1 2(a - b) - b Å‚Å‚
ëÅ‚t öÅ‚
2
58
e-at + - - + t + e-bt
s2 (s + a)(s + b)2 a2 (a - b)2 ab2 íÅ‚ a bÅ‚Å‚ ïÅ‚ b2 (a - b) b3(a - b)2 śł
ðÅ‚ ûÅ‚
Å„Å‚îÅ‚ 1
Å‚Å‚
2c -
ôÅ‚ïÅ‚(c - b)(c - a) t + (c - a)2 a - bb)2 śł e-ct
(c -
ûÅ‚
1 ôÅ‚ðÅ‚
59
òÅ‚
(s + a)(s + b)(s + c)2
ôÅ‚ 11
+ e-at + e-bt
ôÅ‚
(b - a)(c - a)2 (a - b)(c - b)2
ół
11 É
1
60 e-at + sin(Ét - Ć); Ć = tan-1ëÅ‚ öÅ‚
íÅ‚ Å‚Å‚
a2 + É2 É a2 + É2 a
(s + a)(s2 + É2 )
1
1 1 1 a 1
ëÅ‚
61
- sin Ét + cosÉt + e-at öÅ‚
s(s + a)(s2 + É2 ) Å‚Å‚
aÉ2 a2 + É2 íÅ‚ É É2 a
F(s) f(t)
1 1 1
Å„Å‚
t - + e-at
ôÅ‚
aÉ2 a2É2 a2 (a2 + É2 )
1 ôÅ‚
62
òÅ‚
s2 (s + a)(s2 + É2 )
1 a
ôÅ‚
+ cos(Ét + Ć); Ć = tan-1ëÅ‚ öÅ‚
ôÅ‚
íÅ‚ Å‚Å‚
É
É3 a2 + É2
ół
1 1
63 e-at[sin Ét - Ét cosÉt]
[(s + a)2 + É2 ]2 2É3
1 1
64 sinh at
s2 - a2 a
1 1 1
65 sinh at - t
s2 (s2 - a2 ) a3 a2
1 1 1
66
(cosh at -1) - t2
s3(s2 - a2 ) a4 2a2
îÅ‚
a ëÅ‚ öÅ‚Å‚Å‚
1 1 t 3 3
2
67
ïÅ‚e-at - e at - 3 sin at÷łśł
s3 + a3 3a2 ïÅ‚ ìÅ‚cos 2 2
íÅ‚ Å‚Å‚ûÅ‚
śł
ðÅ‚
1 1
68
(sin at cosh at - cos at sinh at)
s4 + 4a4 4a3
1 1
69 (sinh at - sin at)
s4 - a4 2a3
1 1
70 e-at sinh Ét
[(s + a)2 - É2 ] É
Å„Å‚
a 1 (a - b)2 + É2
ôÅ‚ - + e-bt sin (Ét + Ć);
2
s + a
ôÅ‚b + É2 É b2 + É2
71
òÅ‚
s[(s + b)2 + É2 ]
ôÅ‚
ÉÉ
Ć = tan-1ëÅ‚ öÅ‚ + tan-1ëÅ‚ öÅ‚
ôÅ‚
íÅ‚ Å‚Å‚ íÅ‚ - bÅ‚Å‚
b a
ół
Å„Å‚
(a
1 2ab - b)2 + É2
ôÅ‚
[1 + at] - + e-bt sin (Ét + Ć)
b2 + É2 (b2 + É2 )2 É(b2 + É2 )
ôÅ‚
s + a
72
òÅ‚
s2[(s + b)2 + É2 ]
ôÅ‚
ÉÉ
Ć = tan-1ëÅ‚ öÅ‚ + 2 tan-1ëÅ‚ öÅ‚
ôÅ‚
íÅ‚ - bÅ‚Å‚ íÅ‚ b
Å‚Å‚
a
ół
Å„Å‚
a - c 1 (a - b)2 + É2
ôÅ‚ e-ct + e-bt sin (Ét + Ć)
s + a ôÅ‚(c - b)2 + É2 É (c - b)2 + É2
73
òÅ‚
(s + c)[(s + b)2 + É2 ]
ôÅ‚
ÉÉ
Ć = tan-1ëÅ‚ öÅ‚ - tan-1ëÅ‚ öÅ‚
ôÅ‚
íÅ‚ - bÅ‚Å‚ íÅ‚ c - bÅ‚Å‚
a
ół
Å„Å‚ a (c - a)
2
ôÅ‚c(b + É2 ) + c[(b - c)2 + É2 ] e-ct
ôÅ‚
ôÅ‚
s + a ôÅ‚ 1 (a - b)2 + É2
74 - e-bt sin(Ét + Ć)
òÅ‚
s(s + c)[(s + b)2 + É2 ]
É b2 + É2 (b - c)2 + É2
ôÅ‚
ôÅ‚
ôÅ‚
ÉÉ É
Ć = tan-1ëÅ‚ öÅ‚ + tan-1ëÅ‚ öÅ‚ - tan-1ëÅ‚ öÅ‚
ôÅ‚
íÅ‚ Å‚Å‚ íÅ‚ - bÅ‚Å‚ íÅ‚ c - bÅ‚Å‚
b a
ół
s + a
a b - 3a - b a - b 2a - b
îÅ‚3a
75
t + + + t2 + tłł e-bt
s2 (s + b)3
b3 b4 ïÅ‚ b4 2b2 b3 śł
ðÅ‚ ûÅ‚
F(s) f(t)
s + a a - c îÅ‚ a - b c - a a - c Å‚Å‚
76 e-ct + t2 + t + e-bt
(s + c)(s + b)3 (b - c)3 ïÅ‚ 2(c - b) (c - b)2 (c - b)3 śł
ðÅ‚ ûÅ‚
s2 a2 b2 c2
77
e-at + e-bt + e-ct
(s + a)(s + b)(s + c) (b - a)(c - a) (a - b)(c - b) (a - c)(b - c)
s2 a2 îÅ‚ b2 b2 - 2ab Å‚Å‚
78 e-at + t + e-bt
(s + a)(s + b)2 (b - a)2 ïÅ‚(a - b) (a - b)2 śł
ðÅ‚ ûÅ‚
s2 îÅ‚ a2 Å‚Å‚
79
e-at
ïÅ‚2 - 2at + 2 t2 śł
(s + a)3
ðÅ‚ ûÅ‚
a2 É É
s2
80 e-at - sin(Ét + Ć); Ć = tan-1ëÅ‚ öÅ‚
íÅ‚ Å‚Å‚
(a2 + É2 ) a
(s + a)(s2 + É2 )
a2 + É2
Å„Å‚îÅ‚ a2 2aÉ2 Å‚Å‚ É
ôÅ‚ïÅ‚ 2 t - e-at - sin(Ét + Ć);
śł
(a2 + É2 )
ðÅ‚(a + É2 ) (a2 + É2 )2 ûÅ‚
s2 ôÅ‚
81
òÅ‚
(s + a)2 (s2 + É2 )
ôÅ‚
É
Ć = -2 tan-1ëÅ‚ öÅ‚
ôÅ‚
íÅ‚ Å‚Å‚
a
ół
Å„Å‚ a2 b2
ôÅ‚(b - a)(a2 + É2 ) e-at + (a - b)(b2 + É2 ) e-bt
ôÅ‚
s2
82
òÅ‚
(s + a)(s + b)(s2 + É2 )
É îÅ‚ ÉÉ
ôÅ‚
- sin(Ét + Ć); Ć = -ïÅ‚tan-1ëÅ‚ öÅ‚ + tan-1ëÅ‚ öÅ‚Å‚Å‚
ôÅ‚
íÅ‚ Å‚Å‚ íÅ‚ łłśł
ab
ðÅ‚ ûÅ‚
(a2 + É2 )(b2 + É2 )
ół
s2 a É
83 - sin(at) - sin(Ét)
(s2 + a2 )(s2 + É2 ) (É2 - a2 ) (a2 - É2 )
s2 1
84
(sin Ét + Ét cosÉt)
(s2 +É2 )2 2É
Å„Å‚
a2 1 (b2 - É2 )2 + 4b2É2
ôÅ‚ e-at + e-bt sin(Ét + Ć)
ôÅ‚(a - b)2 + É2 É (a - b)2 + É2
s2
85
òÅ‚
(s + a)[(s + b)2 + É2 ]
ôÅ‚
-2bÉ É
Ć = tan-1ëÅ‚ öÅ‚ - tan-1ëÅ‚ öÅ‚
ôÅ‚
íÅ‚ - É2 Å‚Å‚ íÅ‚ a - bÅ‚Å‚
b2
ół
Å„Å‚
a2 îÅ‚ - a)2 + É2 ] + a2 (b - a) Å‚Å‚
a[(b
te-at - 2ïÅ‚ e-at
ôÅ‚
[(b - a)2 + É2 ]2 śł
ðÅ‚ ûÅ‚
ôÅ‚(a - b)2 + É2
ôÅ‚
ôÅ‚
(b2 - É2 )2 + 4b2É2
s2 ôÅ‚
86 + e-bt sin(Ét + Ć)
òÅ‚
(s + a)2[(s + b)2 + É2 ] É[(a - b)2 + É2 ]
ôÅ‚
ôÅ‚
-2bÉ É
ôÅ‚
Ć = tan-1ëÅ‚ öÅ‚ - 2 tan-1ëÅ‚ öÅ‚
ôÅ‚
íÅ‚ - É2 Å‚Å‚ íÅ‚ a - bÅ‚Å‚
b2 -
ôÅ‚
ół
s2 + a b2 + a a a
87
e-bt + t -
s2 (s + b) b2 b b2
s2 + a a a 1
88 t2 - t + [b2 + a - (a + b2 )e-bt ]
s3(s + b) 2b b2 b3
s2 + a a (b2 + a) (c2 + a)
89
+ e-bt - e-ct
s(s + b)(s + c) bc b(b - c) c(b - c)
s2 + a b2 + a c2 + a a a(b + c)
90
e-bt + e-ct + t -
bc
s2 (s + b)(s + c) b2 (c - b) c2 (b - c) b2c2
F(s) f(t)
2
s2 + a b2 + a c2 + a d + a
91 e-bt + e-ct + e-dt
(s + b)(s + c)(s + d) (c - b)(d - b) (b - c)(d - c) (b - d)(c - d)
2
s2 + a a b2 + a c2 + a d + a
92
+ e-bt + e-ct + e-dt
s(s + b)(s + c)(s + d) bcd b(b - c)(d - b) c(b - c)(c - d) d(b - d)(d - c)
Å„Å‚ a a b2 + a
ôÅ‚bcd t - b2c2d (bc + cd + db) + b2 (b - c)(b - d) e-bt
2
s2 + a ôÅ‚
93
òÅ‚
s2 (s + b)(s + c)(s + d) 2
ôÅ‚ c2 + a d + a
+ e-ct + e-dt
ôÅ‚ 2
c2 (c - b)(c - d) d (d - b)(d - c)
ół
s2 + a 1 1
94
(a + É2 )sin Ét - (a - É2 )t cosÉt
(s2 + É2 )2 2É3 2É2
s2 - É2
95 t cosÉt
(s2 + É2 )2
s2 + a
a (a - É2 ) a
96
- t sin Ét - cosÉt
s(s2 + É2 )2
É4 2É3 É4
s(s + a)
îÅ‚
b2 - ab c2 - ac c2 - 2bc + ab Å‚Å‚
97
e-bt + t + e-ct
(s + b)(s + c)2
(c - b)2 ïÅ‚ b - c (b - c)2 śł
ðÅ‚ ûÅ‚
2
Å„Å‚ - ab c2 - ac d - ad
b2
ôÅ‚(c - b)(d - b)2 e-bt + (b - c)(d - c)2 e-ct + (b - d)(c - d) te-dt
s(s + a) ôÅ‚
98
òÅ‚
(s + b)(s + c)(s + d)2 2
ôÅ‚ a(bc - d ) + d(db + dc - 2bc)
+ e-dt
ôÅ‚
(b - d)2 (c - d)2
ół
s2 + a1s + ao b2 - a1b + ao ao a1b - ao
99
e-bt + t +
s2 (s + b) b2 b b2
s2 + a1s + ao a1b - b2 - ao ao a1b - ao b2 - a1b + ao
100
e-bt + t2 + t +
s3(s + b) b3 2b b2 b3
s2 + a1s + ao ao b2 - a1b + ao c2 - a1c + ao
101
+ e-bt + e-ct
s(s + b)(s + c) bc b(b - c) c(c - b)
s2 + a1s + ao ao a1bc - ao (b + c) b2 - a1b + ao c2 - a1c + ao
102
t + + e-bt + e-ct
s2 (s + b)(s + c) bc b2c2 b2 (c - b) c2 (b - c)
2
s2 + a1s + ao b2 - a1b + ao c2 - a1c + ao d - a1d + ao
103
e-bt + e-ct + e-dt
(s + b)(s + c)(s + d) (c - b)(d - b) (b - c)(d - c) (b - d)(c - d)
2
s2 + a1s + ao ao b2 - a1b + ao c2 - a1c + ao d - a1d + ao
104
- e-bt - e-ct - e-dt
s(s + b)(s + c)(s + d) bcd b(c - b)(d - b) c(b - c)(d - c) d(b - d)(c - d)
s2 + a1s + ao ao b2 - a1b + ao b2 - ao
105
- t e-bt + e-bt
s(s + b)2 b2 b b2
s2 + a1s + ao
ao a1b - 2ao b2 - a1b + ao 2ao - a1b
106
t + + t µ-bt + e-bt
s2 (s + b)2
b2 b3 b2 b3
s2 + a1s + ao b2 - a1b + ao c2 - a1c + ao c2 - 2bc + a1b - ao
107
e-bt + t e-ct + e-ct
(s + b)(s + c)2 (c - b)2 (b - c) (b - c)2
Å„Å‚ d3
b3 c3
ôÅ‚(b - c)(d - b)2 e-bt + (c - b)(d - c)2 e-ct + (d - b)(c - d) t e-dt
s3 ôÅ‚
108
òÅ‚
(s + b)(s + c)(s + d)2 2 2
ôÅ‚ d [d - 2d(b + c) + 3bc]
+ e-dt
ôÅ‚
(b - d)2 (c - d)2
ół
©1999 CRC Press LLC
F(s) f(t)
Å„Å‚ b3 c3
ôÅ‚(b - c)(d - b)( f - b)2 e-bt + (c - b)(d - c)( f - c)2 e-ct
ôÅ‚
ôÅ‚
3 3
d f
ôÅ‚
+ e-dt + t e- ft
ôÅ‚
(d - b)(c - d)( f - d)2 ( f - b)(c - f )(d - f )
s3 ôÅ‚
109
(s + b)(s + c)(s + d)(s + f )2 òÅ‚ îÅ‚ 3 f 2
ôÅ‚
+
ïÅ‚(b - f )(c - f )(d - f )
ôÅ‚
ðÅ‚
ôÅ‚
ôÅ‚
3
f [(b - f )(c - f ) + (b - f )(d - f ) + (c - f )(d - f )]
)( Å‚Å‚
ôÅ‚
+ µ-dt
ôÅ‚
(b - f )2 (c - f )2 (d - f )2 śł
ûÅ‚
ół
s3 b3 b2 (3c - b) c3 c2 (3b - c)
110
- t e-bt + e-bt - t e-ct + e-ct
(s + b)2 (s + c)2 (c - b)2 (c - b)3 (b - c)2 (b - c)3
3
Å„Å‚
d b3
ôÅ‚- e-dt + t e-bt
(b - d)2 (c - d)2 (c - b)2 (b - d)
ôÅ‚
ôÅ‚
ôÅ‚
îÅ‚ Å‚Å‚
s3 ôÅ‚ 3b2 b3(c + 2d - 3b) c3
111 + e-bt + t e-ct
(s + d)(s + b)2 (s + c)2 òÅ‚ ïÅ‚(c - b)2 (d - b) + (c - b)3(d - b)2 śł (b - c)2 (c - d)
ðÅ‚ ûÅ‚
ôÅ‚
ôÅ‚
ôÅ‚
îÅ‚ 3c2 c3(b + 2d - 3c) Å‚Å‚
+ + e-ct
ôÅ‚
ïÅ‚(b +
ôÅ‚ - c)2 (d - c) (b - c)3(d - c)2 śł
ðÅ‚ ûÅ‚
ół
Å„Å‚
b3 c3
ôÅ‚ e-bt + e-ct
ôÅ‚(b - c)(b2 + É2 ) (c - b)(c2 + É2 )
ôÅ‚
ôÅ‚
s3 É2
112
- sin(Ét + Ć)
òÅ‚
(s + b)(s + c)(s2 + É2 )
(b2 + É2 )(c2 + É2 )
ôÅ‚
ôÅ‚
ôÅ‚
c É
Ć = tan-1ëÅ‚ öÅ‚ - tan-1ëÅ‚ öÅ‚
ôÅ‚
íÅ‚ Å‚Å‚ íÅ‚ Å‚Å‚
É b
ół
Å„Å‚ b3 c3
ôÅ‚(b - c)(d - b)(b2 + É2 ) e-bt + (c - b)(d - c)(c2 + É2 ) e-ct
ôÅ‚
ôÅ‚
d3
ôÅ‚
+ e-dt
2
ôÅ‚
(d - b)(c - d)(d + É2 )
s3 ôÅ‚
113
òÅ‚
(s + b)(s + c)(s + d)(s2 + É2 )
É2
ôÅ‚
- cos(Ét - Ć)
ôÅ‚
2
(b2 + É2 )(c2 + É2 )(d + É2 )
ôÅ‚
ôÅ‚
ôÅ‚ É É É
Ć = tan-1ëÅ‚ öÅ‚ + tan-1ëÅ‚ öÅ‚ + tan-1ëÅ‚ öÅ‚
ôÅ‚
íÅ‚ Å‚Å‚ íÅ‚ Å‚Å‚ íÅ‚ Å‚Å‚
b c d
d
ół
Å„Å‚ b3 b2 (b2 + 3É2 ) É2
ôÅ‚- b2 + É2 t e-bt + (b2 + É2 )2 e-bt - (b2 + É2 ) sin(Ét + Ć)
s3 ôÅ‚
114
òÅ‚
(s + b)2 (s2 + É2 )
ôÅ‚ b É
Ć = tan-1ëÅ‚ öÅ‚ - tan-1ëÅ‚ öÅ‚
ôÅ‚
íÅ‚ Å‚Å‚ íÅ‚ Å‚Å‚
É b
ół
s3
115 cos(Ét) cosh(Ét)
s4 + 4É4
s3
1
116
[cosh(Ét) + cos(Ét)]
2
s4 -É4
F(s) f(t)
Å„Å‚ ao ao (b + c) - a1bc -b3 + a2b2 - a1b + ao
t - + e-bt
ôÅ‚
bc b2c2 b2 (c - b)
s3 + a2s2 + a1s + ao ôÅ‚
117
òÅ‚
s2 (s + b)(s + c)
ôÅ‚ -c3 + a2c2 - a1c + ao
+ e-ct
ôÅ‚
c2 (b - c)
ół
Å„Å‚ -b3 + a2b2 - a1b + ao -c3 + a2c2 - a1c + ao
ao
- e-bt - e-ct
ôÅ‚
s3 + a2s2 + a1s + ao ôÅ‚bcd b(c - b)(d - b) c(b - c)(d - c)
118
òÅ‚
s(s + b)(s + c)(s + d) 3 2
ôÅ‚ -d + a2d - a1d + ao
- e-dt
ôÅ‚
d(b - d)(c - d)
ół
Å„Å‚ ao îÅ‚ a1 ao (bc + bd + cd) -b3 + a2b2 - a1b + ao
Å‚Å‚
ôÅ‚bcd t + ïÅ‚ bcd - b2c2d 2 śł + b2 (c - b)(d - b) µ-bt
ðÅ‚ ûÅ‚
ôÅ‚
s3 + a2s2 + a1s + ao
119
òÅ‚
2
s2 (s + b)(s + c)(s + d)
ôÅ‚ -c3 + a2c2 - a1c + ao -d3 + a2d - a1d + ao dt
+ e-ct + e-
ôÅ‚
2
c2 (b - c)(d - c) d (b - d)(c - d)
)
ół
Å„Å‚ -b3 + a2b2 - a1b + ao -c3 + a2c2 - a1c + ao
e-bt + e-ct
ôÅ‚
(c - b)(d - b)( f - b) (b - c)(d - c)( f - c)
s3 + a2s2 + a1s + ao ôÅ‚
120
òÅ‚
(s + b)(s + c)(s + d)(s + f ) 2 3 2
ôÅ‚ -d3 + a2d - a1d + ao - f + a2 f - a1 f + ao ft
+ e-dt + e-
ôÅ‚
(b - d)(c - d)( f - d) (b - f )(c - f )(d - f )
-
ół
Å„Å‚ -b3 + a2b2 - a1b + ao -c3 + a2c2 - a1c + ao
ao
ôÅ‚bcdf - b(c - b)(d - b)( f - b) e-bt - c(b - c)(d - c)( f - c) e-ct
s3 + a2s2 + a1s + ao ôÅ‚
121
òÅ‚
s(s + b)(s + c)(s + d)(s + f ) 2 3 2
ôÅ‚ -d3 + a2d - a1d + ao - f + a2 f - a1 f + ao ft
a
- e-dt - e-
ôÅ‚
d(b - d)(c - d)( f - d) f (b - f )(c - f )(d - f )
ół
Å„Å‚ -b3 + a2b2 - a1b + ao -c3 + a2c2 - a1c + ao
e-bt + e-ct
ôÅ‚
ôÅ‚(c - b)(d - b)( f - b)(g - b) (b - c)(d - c)( f - c)(g - c)
ôÅ‚
2 3 2
-
s3 + a2s2 + a1s + ao ôÅ‚ -d3 + a2d - a1d + ao - f + a2 f - a1 f + ao ft
122 + e-dt + e-
òÅ‚
(b - d)(c - d)( f - d)(g - d) (b - f )(c - f )(d - f )(g - f )
(s + b)(s + c)(s + d)(s + f )(s + g)
ôÅ‚
ôÅ‚
ôÅ‚ -g3 + a2g2 - a1g + ao
+ e-gt
ôÅ‚
(b - g)(c - g)(d - g)( f - g)
ół
Å„Å‚ -b3 + a2b2 - a1b + ao -c3 + a2c2 - a1c + ao
e-bt + e-ct
ôÅ‚
(c - b)(d - b)2 (b - c)(d - c)2
ôÅ‚
ôÅ‚
2
ôÅ‚ -d3 + a2d - a1d + ao
+ te-dt
s3 + a2s2 + a1s + ao ôÅ‚
123
(b - d)(c - d)
òÅ‚
(s + b)(s + c)(s + d)2
ôÅ‚
2
ôÅ‚
ao (2d - b - c) + a1(bc - d )
ôÅ‚
2 2
+ a2d(db + dc - 2bc) + d (d - 2db - 2dc + 3bc)
+
ôÅ‚
+ e-dt
ôÅ‚
(b - d)2 (c - d)2
ół
3 2
Å„Å‚ -b3 + a2b2 - a1b + ao -c + a2c - a1c + ao
ao
ôÅ‚bcd 2 - b(c - b)(d - b)2 e-bt - c(b - c)(d - c) 2 e-ct
ôÅ‚
ôÅ‚
3 2 2
s3 + a2s2 + a1s + ao ôÅ‚ -d + a2d - a1d+ a 3d - 2a2d + a1 -
o
124 - te-dt - e-dt
òÅ‚
d(b - d)(c - d) d (b - d)(c - d)
s(s + b)(s + c)(s + d)2
ôÅ‚
ôÅ‚
2
(-d3 + a2d - a1d + ao )[(b - d)(c - d) - d(b - d) - d(c - d)]
ôÅ‚
- e-dt
ôÅ‚ 2
d (b - d)2 (c - d)2
ół
F(s) f(t)
Å„Å‚ -b3 + a2b2 - a1b + ao -c3 + a2c2 - a1c + ao
e-bt + e-ct
ôÅ‚
(c - b)(d - b)( f - b)2 (b - c)(d - c)( f - c)2
ôÅ‚
ôÅ‚
2 3 2
ôÅ‚ -d3 + a2d - a1d + ao - f + a2 f - a1 f + ao ft
+ e-dt + te-
s3 + a2s2 + a1s + ao ôÅ‚
125
(b f
òÅ‚ - d)(c - d)( f - d)2 (b - f )(c - f )(d - f )
(s + b)(s + c)(s + d)(s + f )2 ôÅ‚
3 2
ôÅ‚
(- f + a2 f - a1 f + ao )[(b - f )(c - f )
ôÅ‚
2
3 f - 2a2 f + a1 ft - f )(d - f ) + (c - f )(d - f )]
+ (b
ôÅ‚
+ e- - e- ft
ôÅ‚
(b - f )(c - f )(d - f ) (b - f )2 (c - f )2 (d - f )2
ół
ół
1
s
eat (1 + 2at)
126
(s - a)3 / 2 Ä„ t
1
(ebt - eat )
127
s - a - s - b
2 Ä„ t3
2
1
1
- aea terfc(a t )
128
Ä„ t
s + a
2
1
s
129 + aea terf (a t )
Ä„ t
s - a2
a t
2 2
1
s 2a
130 - e-a t e d
+"
Ä„ t Ä„ 0
s + a2
1 2
1
131
ea terf (a t )
s (s - a2 ) a
a t
2 2
1 2
132 e-a t e dĄ
+"
s (s + a2 ) a Ä„ 0
b2 - a2 22
133
ea t[b - a erf (a t )] - beb t erfc(b t )
(s - a2 )(b + s )
1
2
134
ea terfc(a t )
s ( s + a)
1
1
135 e-aterf ( b - a t )
(s + a) s + b
b - a
22
b2 - a2 b
136 ea t îÅ‚ erf (a t ) -1Å‚Å‚ + eb terfc(b t )
ïÅ‚ śł
a
s (s - a2 )( s + b) ðÅ‚ ûÅ‚
n!
Å„Å‚
ôÅ‚(2n)! Ä„t H2n ( t )
ôÅ‚
(1 - s)n
137
òÅ‚
n
sn+(1/ 2)
ôÅ‚ îÅ‚ 2 Å‚Å‚
d
x2
ôÅ‚ ïÅ‚H (t) = Hermite polynomial = ex dxn (e- )śł
n
ðÅ‚ ûÅ‚
ół
n!
(1 - s)n
138 - H2n+1( t )
sn+(3 / 2) Ä„ (2n + 1)!
-at
Å„Å‚
s + 2a ôÅ‚ae [I1(at) + Io (at)]
139
- 1
òÅ‚
-n
s
ôÅ‚ [In (t) = j Jn ( jt) where Jn is Bessel's function of the first kind]
ół
1
a - b
140
e-(1/ 2)(a+b)t Io ëÅ‚ töÅ‚
íÅ‚ Å‚Å‚
s + a s + b 2
F(s) f(t)
k-(1/ 2)
“(k)
t a - b
ëÅ‚ öÅ‚
141 (k e" 0)
Ä„ e-(1/ 2)(a+b)t Ik-(1/ 2)ëÅ‚ töÅ‚
íÅ‚ Å‚Å‚
(s + a)k (s + b)k íÅ‚ - bÅ‚Å‚ 2
a
1
îÅ‚ a - b a - b Å‚Å‚
142
t e-(1/ 2)(a+b)t ïÅ‚Io ëÅ‚ töÅ‚ + I1ëÅ‚ töłśł
(s + a)1/ 2 (s + b)3 / 2 íÅ‚ Å‚Å‚ íÅ‚ Å‚Å‚ûÅ‚
22
ðÅ‚
s + 2a - s 1
143
e-at I1(at)
t
s + 2a + s
k a
(a - b)k (k > 0) e-(1/ 2)(a+b)t Ik - b
ëÅ‚
144 töÅ‚
íÅ‚ Å‚Å‚
t 2
( s + a + s + b )2k
11
( s + a + s )-2v
145
e-(1/ 2)(at)Iv ëÅ‚ atöÅ‚
íÅ‚ Å‚Å‚
av 2
s s + a
1
146 Jo(at)
s2 + a2
( s2 + a2 - s)v
147 av Jv (at)
(v >-1)
s2 + a2
k-(1/ 2)
1 Ä„ t
ëÅ‚ öÅ‚
148 (k > 0) Jk-(1/ 2)(at)
(s2 + a2 )k 2a
“(k)íÅ‚ Å‚Å‚
kak
149
Jk (at)
( s2 + a2 - s)k (k > 0)
t
(s - s2 - a2 )v
150
av Iv (at)
(v >-1)
s2 - a2
k-(1/ 2)
1
Ä„ t
ëÅ‚ öÅ‚
151 (k > 0)
Ik-(1/ 2)(at)
2a
(s2 - a2 )k “(k)íÅ‚ Å‚Å‚
y
1 2
"
2
152 erf( t ); erf(y) the error function = e-u du
+"
s s + 1
Ä„ o
1
153
Jo (at) ; Bessel function of 1st kind, zero order
s2 + a2
1
J1(at)
154
; J1 is the Bessel function of 1st kind, 1st order
s2 + a2 + s at
1
JN (at)
N
155
;N = 12,3L, JN is the Bessel function of 1st kind, Nth order
, ,
aN t
[ s2 + a2 + s]N
t
1
JN (au)
N
156
du;N = 123L, JN is the Bessel function of 1st kind, Nth order
, , ,
+"
aN o u
s[ s2 + a2 + s]N
1
1
157
J1(at); J1 is the Bessel function of 1st kind, 1st order
s2 + a2 ( s2 + a2 + s)a
1
1
158
JN (at) ;N = 12,3,L, JN is the Bessel function of 1st kind, Nth order
,
aN
s2 + a2 [ s2 + a2 + s]N
1
159 Io (at) ; Io is the modified Bessel function of 1st kind, zero order
s2 - a2
e-ks Å„Å‚0 when 0 < t < k
160
Sk (t) =
òÅ‚1 when t > k
s
ół
F(s) f(t)
Å„Å‚0 when 0 < t < k
e-ks
161
òÅ‚t - k when t > k
s2
ół
when 0 < t < k
Å„Å‚0
e-ks ôÅ‚(t - k)µ-1
162
(µ > 0)
òÅ‚
when t > k
sµ
ôÅ‚ “(µ)
ół
Å„Å‚1 when 0 < t < k
1 - e-ks
163
òÅ‚0 when t > k
s
ół
1
1 + coth ks Å„Å‚n when
1
164 2 S(k,t) =
òÅ‚
=
(n -1)k < t < n k (n = 1,2,L)
ół
s(1 - e-ks ) 2s
0 when 0 < t < k
Å„Å‚
1
ôÅ‚1+
165
Sk (t) = a + a2 +L+ an-1
òÅ‚
s(e+ks - a)
ôÅ‚
when nk < t < (n +1)k (n = 1,2,L)
ół
Å„Å‚
M(2k,t) = (-1)n-1
ôÅ‚
ôÅ‚
1
166
when 2k(n -1) < t < 2nk
tanh ks òÅ‚
s
ôÅ‚
(n = 12,L)
,
ôÅ‚
ół
Å„Å‚ 1 1 1- (-1)n
1 (
ôÅ‚2 Mk,t) + 2 = 2
167
òÅ‚
s(1 + e-ks )
ôÅ‚
when (n -1)k < t < nk
ół
Å„Å‚
1 ôÅ‚H(2k,t) [H(2k,t) = k + (r - k)(-1)n where t = 2kn + r;
168 tanh ks
òÅ‚
s2
ôÅ‚ 0 d" r d" 2k; n = 012,L]
, ,
ół
Å„Å‚2S(2k,t + k) - 2 = 2(n -1)
1
ôÅ‚
169
òÅ‚
ssinh ks
ôÅ‚ when (2n - 3)k < t < (2n -1)k (t > 0)
ół
Å„Å‚M(2k,t + 3k) +1 = 1+ (-1)n
1
ôÅ‚
170
òÅ‚
s cosh ks
ôÅ‚
when (2n - 3)k < t <(2n -1)k (t > 0)
ół
Å„Å‚2S(2k,t) -1 = 2n -1
1 ôÅ‚
171
coth ks
òÅ‚
s
ôÅ‚ when 2k(n -1) < t < 2kn
ół
k Ä„ s
172
coth sin kt
s2 + k2 2k
1 Å„Å‚sin t when (2n - 2)Ä„ < t < (2n -1)Ä„
173
òÅ‚0 when (2n -1)Ä„ < t < 2nÄ„
(s2 + 1)(1 - e-Ä„s )
ół
1
174
e-k / s Jo (2 kt )
s
1 1
175 e-k / s cos2 kt
s Ä„t
1 1
176 ek / s cosh2 kt
s Ä„t
1
1
177 sin 2 kt
e-k / s
s3 / 2 Ä„k
1
1
sinh 2 kt
178 ek / s
s3 / 2 Ä„k
F(s) f(t)
(µ-1) / 2
1 t
ëÅ‚ öÅ‚
179
e-k / s (µ > 0) Jµ-1(2 kt )
íÅ‚ Å‚Å‚
sµ k
(µ-1) / 2
1
t
ëÅ‚ öÅ‚
180 ek / s (µ > 0)
Iµ-1(2 kt )
sµ íÅ‚ Å‚Å‚
k
ëÅ‚ öÅ‚
k k2
181 expìÅ‚-
e-k s (k > 0) ÷Å‚
2 Ä„t3 íÅ‚ 4t Å‚Å‚
1 ëÅ‚ öÅ‚
k
182 e-k s (k e" 0)
erfcìÅ‚ ÷Å‚
s
íÅ‚ Å‚Å‚
2 t
2
ëÅ‚ öÅ‚
1 1 k
183
e-k s (k e" 0) expìÅ‚-
÷Å‚
s Ä„ t íÅ‚ 4t Å‚Å‚
ëÅ‚ öÅ‚ ëÅ‚ öÅ‚
t k2 k
184
2 expìÅ‚- - k erfcìÅ‚ ÷Å‚
s-3 / 2e-k s (k e" 0)
÷Å‚
Ä„ íÅ‚ 4t Å‚Å‚ íÅ‚ Å‚Å‚
2 t
ae-k s (k e" 0) 2 ëÅ‚ öÅ‚ ëÅ‚ öÅ‚
k
185
-eak ea t erfcìÅ‚a t + + erfcìÅ‚ k ÷Å‚
÷Å‚
s(a + s )
íÅ‚ 2 t 2 t
Å‚Å‚ íÅ‚ Å‚Å‚
-k s
2 ëÅ‚ öÅ‚
e k
186 (k e" 0) eak ea t erfcìÅ‚a t +
÷Å‚
s (a + s ) íÅ‚ 2 t Å‚Å‚
-k s(s+a)
e when 0 < t < k
Å„Å‚0
187
òÅ‚e-(1/ 2)at
1
s(s + a)
Io ( a t2 - k2 ) when t > k
2
ół
when 0 < t < k
e-k s2 +a2 Å„Å‚0
188
òÅ‚J (a t2 - k ) when t > k
2
(s2 + a2 )
o
ół
-k s2 -a2
e when 0 < t < k
Å„Å‚0
189
òÅ‚I (a t2 - k2 ) when t > k
(s2 - a2 )
o
ół
-k( s2 +a2 -s)
e
190 (k e" 0)
Jo (a t2 + 2kt )
(s2 + a2 )
when 0 < t < k
Å„Å‚0
ôÅ‚
ak
191
òÅ‚
e-ks - e-k s2 +a2 J1(a t2 - k2 ) when t > k
ôÅ‚
t2 - k2
ół
when 0 < t < k
Å„Å‚0
ôÅ‚
ak
192
òÅ‚
e-k s2 +a2 - e-ks
I1(a t2 - k2 ) when t > k
ôÅ‚
t2 - k2
ół
av e-k s2 -a2
v
when 0 < t < k
Å„Å‚0
ëÅ‚
(1/ 2)v
(s2 + a2) s2 + a2 + söÅ‚ ôÅ‚ t - k
193
òÅ‚
íÅ‚ Å‚Å‚
Jv (a t2 - k2 ) when t > k
( )
ôÅ‚
t + k
ół
(v >-1)
1
194
log s “2 (1) - log t [“2 (1) = -0.5772]
s
Å„Å‚ logt üÅ‚
1 “2 (k)
k-1
195
t
òÅ‚[“(k)]2 “(k) żł
sk log s (k > 0)
ół þÅ‚
log s
196
eat[log a - Ei(-at)]
s - a(a > 0)
F(s) f(t)
log s
197
cost Si(t) - sin t Ci(t)
s2 + 1
s log s
198
-sin t Si(t) - cost Ci(t)
s2 + 1
1
ëÅ‚- t
öÅ‚
199 log(1 + ks) (k > 0) -Ei
íÅ‚ Å‚Å‚
s k
s - a 1
200
log (ebt - eat )
s - b t
1 t
201
log(1 + k2s2 ) -2CiëÅ‚ öÅ‚
íÅ‚ Å‚Å‚
s k
1
202 log(s2 + a2 ) (a > 0) 2 log a - 2Ci(at)
s
1 2
203
log(s2 + a2 ) (a > 0) [at log a + sin at - at Ci(at)]
s2 a
s2 + a2 2
204
log (1 - cos at)
s2 t
2
s2 - a2
205 (1 - cosh at)
log
t
s2
k 1
206
arctan sin kt
s t
1 k
207 Si(kt)
arctan
s s
ëÅ‚ öÅ‚
t2
2 1
208
expìÅ‚-
ek s2 erfc(ks) (k > 0)
÷Å‚
2
k Ä„ íÅ‚ 4k Å‚Å‚
2
1 t
ëÅ‚ öÅ‚
209 ek s2 erfc(ks) (k > 0) erf
íÅ‚ Å‚Å‚
s 2k
k
210
ekserfc( ks ) (k > 0)
Ä„ t(t + k)
1
Å„Å‚0 when 0 < t < k
211 erfc( ks)
òÅ‚(Ä„t)-1/ 2
s when t > k
ół
1
1
212 ekserfc( ks ) (k > 0)
Ä„(t + k)
s
ëÅ‚ öÅ‚
k 1
213
erf sin (2k t )
ìÅ‚ ÷Å‚
Ä„t
íÅ‚ s Å‚Å‚
2 ëÅ‚ öÅ‚ 1
1 k
e-2k t
214
ek / serfcìÅ‚ ÷Å‚
Ä„ t
s íÅ‚ s Å‚Å‚
1
215 - easEi(-as)
; (a > 0)
t + a
1 1
216
+ seasEi(-as) ; (a > 0)
a (t + a)2
Ä„ 1
îÅ‚
217 - Si(s)Å‚Å‚ cos s +Ci(s)sin s
ïÅ‚ śł
2 t2 + 1
ðÅ‚ ûÅ‚
F(s) f(t)
when 0 < t < k [Kn(t) is Bessel function of the
Å„Å‚0
218 Ko(ks)
òÅ‚(t2 - k2 )-1/ 2 when t > k second kind of imaginary argument]
ół
ëÅ‚ öÅ‚
1 k2
219
Ko (k s ) expìÅ‚-
÷Å‚
2t íÅ‚ 4t Å‚Å‚
1 1
220 eksK1(ks) t (t + 2k)
s k
2
1
ëÅ‚ öÅ‚
1 k
221 K1(k s )
expìÅ‚-
÷Å‚
s k íÅ‚ 4t Å‚Å‚
2
1 k
222 Ko (2 2kt )
ek / sKo ëÅ‚ öÅ‚
íÅ‚ Å‚Å‚
s Ä„t
s
Å„Å‚ - t)]-1/ 2 when 0 < t < 2k
[t(2k
223 Ä„e ksIo(ks)
òÅ‚
when t > 2k
ół0
k
Å„Å‚ - t
when 0 < t < 2k
ôÅ‚
224
e-ks I1(ks)
Ä„k t(2k
òÅ‚ - t)
ôÅ‚0
when t > 2k
ół
"
2 u[t - (2k + 1)a]
"
k=0
1
225
ssinh (as)
"
2 (-1)k u(t - 2k -1)
"
k=0
1
226
s cosh s
"
u(t) + 2 (-1)k u(t - ak)
"
k=1
square wave
1 asöÅ‚
ëÅ‚
227 tanh
2
íÅ‚ Å‚Å‚
s
"
u(t - ak)
"
k=0
stepped function
1 asöÅ‚
ëÅ‚1 + coth
228
íÅ‚ 2 Å‚Å‚
2s
F(s) f(t)
"
mt - ma u(t - ka)
"
k=1
saw - tooth function
m ma as
ëÅ‚coth - 1öÅ‚
229 -
íÅ‚ Å‚Å‚
s2 2s 2
"
îÅ‚ Å‚Å‚
1
ïÅ‚t + 2 (-1)k (t - ka) Å" u(t - ka)śł
"
a
ïÅ‚ śł
ðÅ‚ k=1 ûÅ‚
triangular wave
1 as
ëÅ‚ öÅ‚
230 tanh
Å‚Å‚
s2 íÅ‚ 2
"
(-1)k u(t - k)
"
k=0
1
231
s(1 + e-s )
"
îÅ‚ Ä„öÅ‚Å‚Å‚ Ä„
öÅ‚
ïÅ‚sin aëÅ‚t - k a łłśł Å" uëÅ‚t - k a Å‚Å‚
"
íÅ‚ íÅ‚
ðÅ‚ ûÅ‚
k=0
half - wave rectification of sine wave
a
232
Ä„
- s
a
(s2 + a2 )(1- e )
"
îÅ‚ Ä„
ëÅ‚t öÅ‚Å‚Å‚ Ä„ öÅ‚
sin (at) Å" u(t) + 2 a - k Å" uëÅ‚t - k
[ ]
ïÅ‚sin íÅ‚ a łłśł íÅ‚ a Å‚Å‚
"
ðÅ‚ ûÅ‚
k=1
full - wave rectification of sine wave
îÅ‚ a Å‚Å‚ Ä„söÅ‚
ëÅ‚
233
coth
ïÅ‚(s2 śł
íÅ‚ a Å‚Å‚
+ a2 ) 2
ðÅ‚ ûÅ‚
u(t - a)
1
234
e-as
s
F(s) f(t)
u(t - a) - u(t - b)
1
235
(e-as - e-bs )
s
m Å"(t - a) Å" u(t - a)
m
236
e-as
s2
mt Å" u(t - a)
or
ma + m(t
[ - a) Å" u(t - a)
]
ma m
îÅ‚ Å‚Å‚e-as
237
+
ïÅ‚ śł
s s2
ðÅ‚ ûÅ‚
(t - a)2 Å" u(t - a)
2
238
e-as
s3
t2 Å" u(t - a)
îÅ‚ 2 2a a2 Å‚Å‚
239
+ + e-as
ïÅ‚ śł
s3 s2 s
ðÅ‚ ûÅ‚
mt Å" u(t) - m(t - a) Å" u(t - a)
m m
240 - e-as
s2 s2
mt - 2m(t - a) Å"u(t - a) + m(t - 2a) Å" u(t - 2a)
m 2m m
241 - e-as + e-2as
s2 s2 s2
mt -[ - a) Å" u(t - a)
ma + m(t
]
m ma m
ëÅ‚ öÅ‚
242 - + e-as
s2 íÅ‚ s s2 Å‚Å‚
F(s) f(t)
05t2 for 0 d" t < 1
.
1 - 0.5(t - 2)2 for 0 d" t < 2
1 for 2 d" t
(1 - e-s )2
243
s3
05t2 for 0 d" t < 1
.
075- (t -15)2 for 1 d" t < 2
. .
05(t - 3)2 for 2 d" t < 3
.
3
îÅ‚ - e-s ) Å‚Å‚
(1
244
0 for 3 < t
ïÅ‚ śł
s
ðÅ‚ ûÅ‚
(ebt - 1) Å" u(t) - (ebt -1) Å" u(t - a) + Ke-b(t-a) Å" u(t - a)
b
+ (eba -1)
s(s - b) where K = (eba -1)
245 b
îÅ‚ Å‚Å‚
s +
ïÅ‚ śł
1
eba -1 -as
-
ïÅ‚ śłe
s + b s(s - b)
ïÅ‚ śł
ðÅ‚ ûÅ‚