§ 15-5 Several basis theorems for the Laplace transform
1.The linearity theorem ? ? )()()()(
2121 sFsFtftfL ???Example:
? ?)()())(( 1)( 1 sVLtF i n dsssVIf ????? ???
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ssss
sV
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1)(
)(1)(1 tuetue tt ?? ???? ?? ????? )( tuee
tt
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ss
LsVLt
11
)()( 11
)()( skVtk ??
2,The time differentiation theorem
)()s(sVdt )t(d ??? 0?? )()(s)s(Vsdt )t(d ' ????? 00222 ???
3,Time integration theorem
? ? ?t s sVdxx0 )()(?
4,The sinusoidal function
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??? ?? ?
j
eeLttuL tjtj
2)(s i n
??
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??? ?? ?
2)(co s
tjtj ee
LttuL
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dt
tdLttuL
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?? s i n)(c os
22
11
2
1
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?? ?????? s)jsjs(j
22
11
2
1
??? ?????? s
s)
jsjs(
2222
1
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? ???? s
s
ss
5,Time shift theorem
If a time function is delayed by a time in the time
domain,the result in the frequency domain is a
multiplication by, If we have se ?? )()( sFtf ?
?
? ? )0()()()( ???? ? ??? ? sFetutfL s
Example 1,If f (t) = (a) 5u(t)
(b) 5u(t-5)
(c) 5u(t+5),Find F(s)
Solution:
(a) F(s)=5/s; (b) F(s)=(5/s)e-5s;
ss
edte tsts 5
055 0 ?
?
???
?? ?
? dtetusFc ts? ? ??? 0 )5(5)()(
Example 2,If f (t)=2e-3 t[ u(t+1)- u(t-2)],find F(s).
Solution,f (t) =2e -3 tu(t+1)-2e -3 tu(t-2)
=2e –3 tu(t+1)-2e -6e -3(t-2)u(t-2)
)1(3231232)( 2626 ss esesessF ???? ???????
Example 3,Find F(s)
t
)(tf
0
A
d
f(t)=Au (t)-Au (t-d)
Solution,
)1()( dsds esAesAsAsF ?? ?????
Example 4,Find F(s)
)2(3)2(23)(23)( ????? tuttuttutf
)2(3)2(3)2()2(23)(23 ???????? tutututttu
)2()2(23)(23 ???? tutttu
)e(sess)s(F ss 22222 12 3123123 ?? ??????
Solution:
2
3
)(tf
t
6,Convolution )()()()()()(
122121 tftfsFsFtftf ????
)()(11))(( 1)( 21 sVsVsssssV ???????? ????
)()(1)( 11 tuetssV t??? ????
)()(1)( 22 tuetssV t??? ????
????? dt )()( 20 1 ?? ? ?
? ??? t t deee0 ??????
0
1 )( tee t ????
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??
)( tuee
tt
??
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?? ??
? ? )()()()()( 21211 ttsVSVLt ??? ???? ?
??? ???? dtueue t )()( )(0 ?? ??? ??
? ??? tt dee 0 )( ?????
? ? )(1)( tuee tt ??? ?? ??? ??