拉氏变换习题解答
习题一
1．求下列函数的拉氏变换,并用查表的方法来验证结果.
(1)
()
sin
2
t
ft=
； (2)
()
2t
f
te
−
= ； (3)
(
)
2
f
tt
=
； (4)
()
sin cos
f
tt= t；
(5)
()
sinh
f
t= kt； (6)
()
cosh
f
tk= t； (7)
(
)
2
cos
f
t= t; (10)
()
2
cos
f
tt= .
解 (1) &
()
ii
22
00
()
sin
22i
tt
st
st
teee
f
t e dt dt
−
−
+∞ +∞
−
−
==
⎡⎤
⎣⎦
∫∫
ii
() ()
22
0
1
()
2i
st st
eedt
+∞
−− −+
=−
∫
ii
() ()
22
00
1
ii
2i
22
||
st st
ee
ss
−− −+
+
∞+∞
⎡
⎤
⎢
⎥
=−
⎢
⎥
⎢
⎥
−+ −−
⎣
⎦
ii
11 1 1
22
ii
ii
2i 2i
22
22
ss
ss
ss
⎡⎤
+−+
⎢⎥
=−=
⎢⎥
⎛⎞⎛
⎢⎥
−+
−+
⎜⎟⎜
⎣⎦
⎝⎠⎝
⎞
⎟
⎠
()
2
2
1
2
2
Re 0
1
41
4
s
s
s
== >
+
+
(2) &
()
2(2)
00
tst st
f
teedte
+∞ +∞
−− −+
==
⎡⎤
⎣⎦
∫∫
dt
()
(2)
0
1
Re 2
(2) 2
|
st
e
s
ss
+∞
−+
=
=>
−+ +
−
(3) &
()
22
0
00
1
2
|
st
st st
e
f
ttedtt tedt
ss
−
+∞ +∞
+∞
−−
==−+
⎡⎤
⎣⎦
∫∫
22
0
0
22
|
st st
te e dt
ss
+∞
+∞
−−
=− +
∫
()
32
0
22
Re 0
|
st
t
es
ss
+∞
−
=
=− = >
(4) &
()
00
1
sin cos sin 2
2
st st
f
t t te dt te dt
+∞ +∞
−−
==⎡⎤
⎣⎦
∫∫
(2i) (2i)
0
1
4i
st st
ee dt
+∞
−− −+
⎡⎤
=−
⎣⎦
∫
()()
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
+−
−
−−
=
+∞
+−
+∞
−−
i2i2i4
1
||
0
i)2(
0
i)2(
s
e
s
e
tsts
()
0Re
4
1
i2
1
i2
1
i4
1
2
>
+
=
⎟
⎠
⎞
⎜
⎝
⎛
+
−
−
= s
s
ss
(5) &
()
00
sinh
2
kt kt
st st
ee
f
tktedt edt
−
+∞ +∞
−−
−
==
⎡⎤
⎣⎦
∫∫
(
)
() ()
00
1
2
skt skt
edt edt
+∞ +∞
−− −+
=−
∫∫
() ()
00
1
2( ) ( )
||
skt skt
ee
sk sk
+∞ +∞
−− −+
⎛⎞
⎜⎟
=−
⎜⎟
−− −+
⎝⎠
()
22
11 1
Re max{ , }
2
k
sk
sk sk s k
⎛⎞
=−= >−
⎜⎟
−+ −
⎝⎠
k
- 1 -
