Tablice Laplacea, Matematyka studia, Metody operatorowe w równaniach różniczkowych
[ Pobierz całość w formacie PDF ]Tables of Integral Transforms
619
TABLE B-4
Laplace Transforms
Z
F
(
S
)=
F
(
T
)
exp(−
ST
)
F
(
T
)
DT
0
1
T
N
(
N
=0
,
1
,
2
,
3
, . . .
)
N
!
S
N+1
2
E
AT
1
S
−
A
3 cos
AT
S
S
2
+
A
2
4 sin
AT
A
S
2
+
A
2
5 cosh
AT
S
S
2
−
A
2
6 sinh
AT
A
S
2
−
A
2
7
T
N
E
−AT
(
N
+ 1)
(
S
+
A
)
N+1
8
T
A
(
A >
−1)
(
A
+1)
S
A+1
9
E
AT
cos
BT
S
−
A
(
S
−
A
)
2
+
B
2
10
E
AT
sin
BT
B
(
S
−
A
)
2
+
B
2
11 (
E
AT
−
E
BT
)
A
−
B
(
S
−
A
)(
S
−
B
)
12
1
(
A
−
B
)
(
A E
AT
−
BE
BT
)
S
(
S
−
A
)(
S
−
B
)
13
T
sin
AT
2
AS
(
S
2
+
A
2
)
2
14
T
cos
AT
S
2
−
A
2
(
S
2
+
A
2
)
2
15 sin
AT
sinh
AT
2
SA
2
(
S
4
+ 4
A
4
)
© 2007 by Taylor & Francis Group, LLC
∞
620
INTEGRAL TRANSFORMS and THEIR APPLICATIONS
Z
∞
F
(
T
)
F
(
S
)=
exp(−
ST
)
F
(
T
)
DT
0
16 (sinh
AT
−sin
AT
)
2
A
3
(
S
4
−
A
4
)
17 (cosh
AT
−cos
AT
)
2
A
2
S
(
S
4
−
A
4
)
18
cos
AT
−cos
BT
(
B
2
−
A
2
)
(
A
2
=
B
2
)
S
(
S
2
+
A
2
)(
S
2
+
B
2
)
√
r
S
19
T
√
1
S
r
S
20 2
T
21
T
cosh
AT
(
S
2
+
A
2
)(
S
2
−
A
2
)
−2
22
T
sinh
AT
2
AS
(
S
2
−
A
2
)
−2
23
sin(
AT
)
T
tan
−1
A
S
−
A
T
r
S
exp(−2
√
24
T
−1/2
exp
AS
)
−
A
T
r
A
exp(−2
√
25
T
−3/2
exp
AS
)
26
√
T
(1 +2
AT
)
E
AT
S
(
S
−
A
)
√
S
−
A
27 (1 +
AT
)
E
AT
S
(
S
−
A
)
2
28
√
T
3
(
E
BT
−
E
AT
)
√
S
−
A
−
√
S
−
B
2
29 exp(
A
2
T
)
erf
(
A
√
T
)
√
A
S
(
S
−
A
2
)
30 exp(
A
2
T
)
erfc
(
A
√
T
)
√
√
1
S
(
S
+
A
)
√
√
S
(
S
−
A
2
)
√
31
T
+
A
exp(
A
2
T
)
erf
(
A
T
)
© 2007 by Taylor & Francis Group, LLC
1
Tables of Integral Transforms
621
Z
∞
F
(
T
)
F
(
S
)=
exp(−
ST
)
F
(
T
)
DT
0
32
√
−
A
exp(
A
2
T
)
erfc
(
A
√
T
)
√
1
T
S
+
A
exp(−
AT
)
√
p
1
(
S
+
A
)
33
erf
(
B
−
A
)
T
√
B
−
A
S
+
B
√
√
S
V
34
2
E
I!T
E
−
Z
erfc
(
−
I!T
) (
S
−
I!
)
−1
E
−Z
√
+exp(
Z
)
erfc
(
+
I!T
)
,
q
√
where
=
Z/
2
VT,
=
I!
V
.
35
1
2
E
−AB
erfc
B
−2
AT
2
√
T
E
−B(S+A
2
)
2
+exp(
AB
)
erfc
B
+ 2
AT
2
√
T
Z
T
sin
X
X
1
S
cot
−1
(
S
)
36
SI
(
T
)=
DX
0
Z
∞
cos
X
X
−
1
37
CI
(
T
)=−
DX
2
S
log(1+
S
2
)
T
Z
∞
E
−X
X
1
S
log(1 +
S
)
38
−
EI
(−
T
)=
DX
T
39
J
0
(
AT
)
(
S
2
+
A
2
)
−
2
40
I
0
(
AT
)
(
S
2
−
A
2
)
−
2
41
T
−1
exp(−
AT
)
, A >
0
(
)(
S
+
A
)
−
√
T
2
A
V
J
V
(
AT
) (
S
2
+
A
2
)
−
(
V+
2
)
,
Re
V >
−
2
42
V
+
1
2
43
T
−1
J
V
(
AT
)
V
(
S
2
+A
2
+S
)
V
,
Re
V >
−
2
A
V
44
J
0
(
A
√
T
)
1
S
exp
−
A
2
4
S
© 2007 by Taylor & Francis Group, LLC
1
√
622
INTEGRAL TRANSFORMS and THEIR APPLICATIONS
Z
∞
F
(
T
)
F
(
S
)=
exp(−
ST
)
F
(
T
)
DT
0
2
A
V
√
45
T
V/2
J
V
(
A
T
)
S
−(V+1)
exp
−
A
2
4S
,
Re
V >
−
2
46
√
A
T
exp
−
A
2
4
T
exp(−
A
√
S
)
, A >
0
2
T
47
√
T
exp
−
A
2
4
T
√
S
exp(−
A
√
S
)
, A
≥0
−
A
2
T
2
4
√
A
S
2
A
2
S
A
48 exp
exp
erfc
, A >
0
49 (
T
2
−
A
2
)
−
2
H
(
T
−
A
)
K
0
(
AS
)
, A >
0
50
(
T
−
A
)
exp(−
AS
)
, A
≥0
51
H
(
T
−
A
)
1
S
exp(−
AS
)
, A
≥0
52
′
(
T
−
A
)
S E
−AS
, A
≥0
53
(N)
(
T
−
A
)
S
N
exp(−
AS
)
54
|sin
AT
|
,
(
A >
0)
A
(
S
2
+
A
2
)
coth
S
2
A
55
√
T
cos(2
√
AT
)
√
S
exp
−
A
S
56
√
T
sin(2
√
AT
)
√
1
S
exp
−
A
S
S
57
√
A
cosh(2
√
AT
)
√
S
exp
A
S
58
√
A
sinh(2
√
AT
)
√
1
S
exp
A
S
S
59
erf
T
2
A
1
S
exp(
A
2
S
2
)
erfc
(
AS
)
, A >
0
© 2007 by Taylor & Francis Group, LLC
Tables of Integral Transforms
623
Z
∞
F
(
S
)=
F
(
T
)
exp(−
ST
)
F
(
T
)
DT
0
60
erfc
√
1
S
exp(−
A
√
S
)
, A
≥0
2
T
r
√
61
E
−
A
4T
−
A erfc
√
√
1
S
exp(−
A
S
)
, A
≥0
2
T
S
√
B
exp(−
B
√
S
)
62
E
A(B+AT)
erfc
A
T
+
√
√
√
S
+
A
)
, A
≥0
2
T
S
(
63
J
0
A
√
T
2
−
!
2
H
(
T
−
!
) (
S
2
+
A
2
)
−
2
exp
−
!
√
S
2
+
A
2
64
1
T
(
E
BT
−
E
AT
)
log
S
−
A
S
−
B
65
{
(
T
+
A
)}
−
2
√
S
exp(
AS
)
erfc
(
√
AS
)
, A >
0
1
T
sin(2
A
√
√
66
T
)
erf
S
√
√
√
A
2
S
√
67
T
exp(−2
A
T
)
, A
≥0
S
exp
erfc
S
√
Z
T
cos
U
√
1
2
S
1
S
1+
S
2
1
2
68
C
(
T
)=
DU
√
1+
S
2
+
2
U
0
√
Z
T
sin
U
√
1
2
S
1
−
S
1+
S
2
1
2
69
S
(
T
)=
DU
√
U
1+
S
2
2
0
X
∞
√
√
70
I
(
T
)=1 + 2
exp(−
N
2
T
) (
S
tanh
S
)
−1
N=1
71
T
M
+
−1
E
(M)
,
(±
AT
)
M!S
−
(S
∓A)
M+1
72
1+2AT
√
T
S+A
S
√
S
© 2007 by Taylor & Francis Group, LLC
A
4
T
A
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