教学内容释疑解难 第8章 1.若F1(S)=L[f1()],F2(s)=L[f2(1),则 L[f1()*f2()=F1(s)F2(s) 证:∵【<0时,f1(1)=f2(1)=0 f1(1)*f2(t)=f1()2(-)dr f()(-)dr+∫f()/(-ldr f(r)f2(t-r)dt LL(*2(]=If(Of(-r)dr]e"dr L f(rl f2(t-r)e-dt]dr SS(If2(t-r)e"da f(r)e-r f2(t-r)e-s(end(t-t)]dr F2(s)f(re dt F(SF,(s) 2.设L[f()=F(s),证明 (a)I'[F(bs=f()(b>0) (b)L[f(at-b)(at-b)=-F(-)ea(a>0,b>0) 并由此性质计算Lsin(ot+q)(ot+)(o>0,<0) E:(a):L[G]= fGedt=l bfG)e b d() =bF(sb) L[F(b)=f() (b)I[(at-b)u(at-b)]=b f(at-b)e-dt lb - f(at-b)e a e a d(at-b教学内容释疑解难 第 8 章 1. 若 )( = L ， L ，则 1 sF )]([ 1 tf 2 sF )( = )]([ 2 tf L )()()]()([ . 1 2 21 =∗ sFsFtftf 证： Q 时， t < 0 0)()( ， 1 2 tftf == ∴ d)()()()( τττ 2 0 1 2 =∗ 1 − ∫ tfftftf t d)()(d)()( ττττττ 2 21 0 = 1 +− − ∫ ∫ +∞ tff tff t t d)()( τττ 2 0 = 1 − ∫ ∞+ tff ∴ L 1 ∗ 2 tftf )]()([ = tff t stde]d)()([ 2 0 0 1 − +∞ +∞ − ∫ ∫ τττ τ d]de)()[( ττ 0 0 1 2 ttff −st +∞ +∞ ∫ ∫ = − τ ττ τ d]de)()[( 0 1 2 ttff −st +∞ +∞ ∫ ∫ = − τ τ ττ τ τ d)]d(e)([e)( )( 0 0 = 1 2 − − −− +∞ +∞ − ∫ ∫ f tf t s ts ττ τ de)()( 0 12 s fsF − +∞ ∫ = )()( . 21 = sFsF 2. 设 L = sFtf )()]([ , 证明: (a) L-1 )0( )( 1 )]([ = b > b t f b bsF ; (b) L )0 ,0( e)( 1 )]()([ =−− >> − ba a s F a batubatf s a b . 并由此性质计算 L ω ϕ ωtut ++ ϕ ω > ϕ < )0 ,0( )]()[sin( . 证：(a) Q L )d(e)(de)()]([ 0 0 b t b t bft b t f b t f b t sb st ∞+ − − ∞+ ∫ ∫ = = = sbbF )( , ∴ L-1 )( 1 )]([ b t f b bsF = ; (b) L tbatfbatubatf st a b de)()]()([ − +∞ ∫ −=−− )d(ee)( 1 )( batf bat a a sb bat a s a = b − − ∞+ −−− ∫ )(e 1 a s F a s a b − =