4典型函数的拉氏变换F(S)=∫+∫(1)et (1)指数函数 L[ e(t)=o e edt==ee -s+a)t 01 0s+ (2)单位阶跃函数1 Le(ol=o c()e tat =rof e-dt =es o I so 当a=0时c"e(t)=68() (3)冲激函数 2L6()=8()e=6()eh=
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 线性F(S)hf(lex孤 + 若印=F(S),L()=E(S) 则Lf()±b2()=aF1(S)±bF2(S) 证:{gf()±b2(O)le=gf()-db()et =1(S)±bF(S) 例1:LU民() U 5方 S 例24smao)1=2y(-1 2jS-j。S+jS2+a
§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 =
、导数性质设:()=F() 1时域导数性质 df (t) 02 =SF(S)-∫(0) dt dde-stdtsgo e= f(o udy =uv -vdu s e"f(x--o f(oes)e"dt oO P =e "f(to +s o f(t)e "dt==f(0 )+SF(S 0 il: Lcos ote(t)=L(sin ote(t)l o dt S 0= PDODOORCNL O S+O5210
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:L8(=146()=s1=1 dt 2 °cS =SF(S(0) 推广:d2f(1=SF(S)f(0)-f(0) 2 =S2F(S)-Sf(0)-f(0) s LId SF(S)-S"/(o.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
2频域导数性质设:L/(O=F(S) oopooLl-tf(tl= dF(S) d HE: o f(t)e-at=o f(ete dt= -tf(oI 例1:Lte()=-()=(2) 例2:1rO1(1yd()=(m) S 例3:L|tea= 1= ds s+a (S+a)2
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)