§4.3*拉普拉斯变换 拉氏变换及存在定理 l、拉氏变换C(d0()=0当7<0则/(kd<易 此时:F/(]上k"e“h 而:10k=0d 20
§ 4.3* 拉普拉斯变换 一、拉氏变换及存在定理 、拉氏变换 ò f ( )t dt 难对f ( )t 要求苛刻 ¥ -¥ 1 : 0, f t = 0当t < 0则 f t e bt dt < ¥易 [ ( ) ] ( ) ò ¥ -¥ - - - F f t e = f t e e dt bt bt iwt 此时: ( ) ( ) ò ¥ -¥ - + = f t e dt b iw t ( ) [ ( ) ] ò ¥ -¥ - - = w p b b w f t e F f t e e d t t i t 2 1 而:
若记p=B+10(0)中1/1则d 则f[小]=∫f(k"dt 拉氏变换 f()=∫FDh-F拉氏逆变换 2Ti +10 2、存在条件:)(及导数除有限个第一类间断点外连续 2)/()sM,MB20是增长指数
b w ( ) [ ( ) ] w b p i F p F f t e dp id t = + = = 若记 - ,则 则 [ ] ( ) ( ) [ ] [ ]的拉氏逆变换 拉氏变换 F p e dp F p i f t F p f t e dt i i pt pt = - = ò ò - + - b w b w w 2p 1 0 2、存在条件 :(1)f (t)及导数除有限个第一类 间断点外连续 (2) f (t) £ Meb0 t , Mb0 ³ 0 b0是增长指数
讲义p221 例0)pje"e"d 1 (pay|∞ 设a=a1+ia2 p-a (B-a1) cos(@-a2t-isin(@ p-a Rep> rea p-a
( ) [ ] ò ¥ - = 0 : 1 L e e e dt 例 at at pt ( ) 0 1 2 1 e a a ia p a p a t = + - = - - - ¥ 设 讲义p221 ( ) { [ ( ) ( ) ]} - - ¥ - - - - = - 0 2 2 1 cos sin 1 e a t i a t p a a t w w b p a p a ,Re Re 1 > - =
i[]=e" dt=ee " dt=, Rep>o Il"dt=ide te Rep>0 P丿p l6=r(+ Rep>o
(2) [ ] = ? k L t [ ] ,Re 0 1 0 0 0 0 = = = > ò ò ¥ - ¥ - p p L t e dt e e dt pt t pt [ ] ò ò ¥ - ¥ - = = 0 0 1 pt pt L t te dt tde ,Re 0 1 1 2 0 0 = > ÷ ÷ ø ö ç ç è æ - × ú û ù ê ë é = - ò ¥ - ¥ - p p p te e dt pt pt [ ] ( ) , Re 0 ! 1 1 1 > + = = + + p p k p k L t k k k G
性质(对照傅氏变换) 1线性:(+5)=a([()+B[/( 记L[/()=F(p) 2延迟:Lpwy()=F(p-n) 3位移:L/(-r)=eF 4相似:/()-12a>0-270 5微分:(=p()-p/(0) L(o=PF(P)-f(o
二、性质(对照傅氏变换) L[ f (t) f (t)] L[f (t)] L[f (t)] 1 2 1 2 1.线性 : a + b = a + b 记L[f (t)] = F(p) [ ( )] ( ) 0 0 2. : L e f t F p p p t 延迟 = - L[f (t )] e F[p] pt t - 3.位移: - = [ ( )] , 0 1 4. : ÷ > ø ö ç è æ = a a p F a 相似 L f at ( ) 5. : [ ( )] ( ) (0) 1 L f t p F p p f n n n- 微分 = - ( ) ( ) 0 (0) -2 -1 - ¢ - - n n p f K f 即 L[f ¢(t)] = pF(p)- f (0)
L”(=p()-/0)-(0) 6积分4/kr=1( 7卷积:L[(0)*f2(=Lf()L[f f(5()=j((-axr 例:0m=4“ 1 2iLp-认kP+hRep>0
[ ( )] ( ) (0) (0) 2 L f ¢¢ t = p F p - pf - f ¢ ( ) L[f ( )t ] p L f d t 1 6. : 0 ú = û ù ê ë é ò 积分 t t L[f (t) f (t)] L[f (t)] L[f (t)] 1 2 1 2 7.卷积 : * = × ( ) ( ) ( ) ( ) ò * = - t f t f t f f t d 0 1 2 1 2 t t t ( ) [ ] ú û ù ê ë é - = - i e e L kt L ikt ikt 2 例: 1 sin ,Re 0 1 1 2 1 > ú û ù ê ë é + - - = p i p ik p ik
+2 (2)L sin t 2丌 3 in e p p2+1 3) COS ht]- l d sin kt k dt k +k
2 2 p k k + = ( ) ú û ù ê ë é ÷ ø ö ç è æ - 3 2 2 sin p L t [ ] 1 1 sin 2 3 2 3 2 + = = × - - p e L t e p p p p ( ) [ ] ú û ù ê ë é = kt dt d k L kt L sin 1 3 cos 2 2 1 p k k p k + = ×
(4)知F(p)= 求L[F()=? L(=|- p2+1p2+1 L. L cost* cost]=[cost cost -tdr cos(2τ-t)+ coSta
( ) ( ) ( ) [ ( )] ? 1 4 1 2 2 2 = + = - L F p p p 已知F p 求 [ ( )] ú û ù ê ë é + × + = - - 1 1 2 2 1 1 p p p p L F p L [ ] ( ) ò = × * = - - t L L t t t d 0 1 cos cos cost cos t t [ ( ) ] ò = - + t t t d 0 cos 2 cos 2 1 t t
a=2τ-t osada +-t cost t cost+tsin t
ò- = - = × + t t t d t cost 2 1 cos 4 1 2 a a a t (t cost tsin t) 2 1 = +
原函数存在定理 若F单值在05ag:≤27中当p→OF()→ 则(1∑→全平面奇点 拉氏反演及展开定理 h 例:F(p) 求f() ch l 单值,奇点是单极点
三、原函数存在定理 若F(p)单值,在0 £ arg z £ 2p中当p ® ¥ F(p)® 0 ( ) =å [ [ ] ] ® k k p t k f t F p e p 则 res k 全平面奇点 拉氏反演及展开定理 ( ) f ( )t l a p p x a p 例 F p 求 ch ch : = 单值,奇点是单极点
