4.典型函数的拉氏变换F (S)=It f(t)e-s dt(1)指数函数8e-(s+a)t)Lle-"t e(t)] = fo-e-ate-st dt0s+as+a(2)单位阶跃函数8stL[e(t)] = joo e(t)e-s dt = Jo+ e-"t dt1当a=0时e-"(t)=(t)(3)冲激函数LIS(t)]= J 8(t)e-st dt = J S(t)e-s0 dt =1
4.典型函数的拉氏变换 (2)单位阶跃函数 0 1 = − −st e s s 1 = (1)指数函数 L e t e e dt −at −at −st = − 0 [ ( )] 0 1 ( ) + = − − s+a t e s a s + a = 1 − − = 0 L[ (t)] (t)e dt st a 0 e (t) (t) at = = 当 时 − (3)冲激函数 = − − 0 L[ (t)] (t)e dt st = + − 0 − 0 0 (t)e dt s = 1 − + = 0 e dt st ( ) ( ) 0 F S f t e dt st + − − =
$13-2拉普拉斯变换的基本性质F (S) = J+ f(t)e-st dt一线性若LIfi(t)I=F(S),LIf2(t)I=F(S)则L[af.(t)±bf,(t)l =aF(S)±bF,(S)证:[af (t)±bf,(t)le-"dt=af.(t)e-"dt±bf.(t)e-"dt=aF(S)bF(S)U例l:LIUe(t)/ =S(ejar - e-jar )e(t)]例2: L[sin ote(t)] = L[-jo ia0S+0
§13-2 拉普拉斯变换的基本性质 一.线性 [ ( )] ( ) , [ ( )] ( ) 若L f1 t = F1 S L f2 t = F2 S af t bf t e dt −st − [ ( ) ( )] 0 证： 1 2 af t e dt bf t e dt st −st − = − − 0 2 0 1 ( ) ( ) ( ) ( ) = aF1 S bF2 S 例1： L[U(t)] 例2： L[sint (t)] [ ( ) ( )] 1 2 则L af t bf t ] 1 1 [ 2 1 j S j S + j − − = ( ) ( ) = aF1 S bF2 S 2 2 + = S ( ) ( )] 2 1 [ e e t j L j t j t − = − ( ) ( ) 0 F S f t e dt st + − − = S U =
设: LLf(t)I= F(s)导数性质一df (t)1.时域导数性质LI= SF(S)-f(0)dtdf(t)StT80dt = o e-sdf(t) Judv =uv-I vdudt08- Jf(t)(-s)e-" dt-s f(t)=e"f(t)-+sJf(t)e"dt =-f(0-)+ SF(S)=e.1 d例l:LJcos ote(t)]=L[(sin ote(t)lo dt0S0=+の+のOs
二 、导数性质 1. 时域导数性质 ] ( ) (0 ) ( ) [ − = SF S − f dt df t L − − − − = 0 0 ( ) ( ) e dt e df t dt df t st st − − = − − − − 0 ( )( ) 0 e f (t) f t s e dt st st = − f (0 ) + SF(S) − 0 2 2 − + = s s 2 2 + = s s (sin ( ))] 1 1 [cos ( )] [ t t dt d L t t L 例 ： = 设：L[ f (t)] = F(s) udv = uv − vdu + = − − − − 0 ( ) 0 e f (t) s f t e dt st st
例2: L/8(0)=LL6(0)1 =S-15dtdf(t)L- SF(S)-f(0)一dtd'f()) =SISF(S)- f(0)I- (0-)推广：Ldt?=sF(S)-Sf(0-)- (0~)d"f(t)LII= s"F(S) - s"- f(0-)-..- f"- (0-)dt
推广： ] ( ) [ 2 2 dt d f t L [ ( ) (0 )] (0 ) − ' − = S SF S − f − f ( ) (0 ) (0 ) 2 − ' − = S F S − Sf − f ] ( ) [ n n dt d f t L ( ) (0 ) (0 ) −1 − −1 − = − − − n n n S F S S f f ] ( ) (0 ) ( ) [ − = SF S − f dt df t L 例2：L[ (t)] [ (t)] dt d = L 1 1 = = S S
设：LIf()I=F(S2.频域导数性质dF(S)L[-t f(t)] =dsd[ f(n)e"at -Ff()(-)e"dt - Li-f(0)证ds例l: L[te(t)]=例2: Lre(0)-(-1 4()-()例3: LIte-1 =_((S+α)dss+α
例1：L[t (t)] ) 1 ( ds S d = − ) 1 ( 2 S = 2 L[t (t)] n 例 ： ) 1 ( 1) ( ds S d n n n = − ) ! ( +1 = n S n ) 1 ( + = − ds S d 2 ( ) 1 + = S 3 [ ] t L te 例 ： − 2.频域导数性质 dS dF S L t f t ( ) [− ( )] = − − 0 f (t)e dt ds 证：d st = − − − 0 f (t)( t)e dt st = L[−tf (t)] 设：L[ f (t)] = F(S)