拉普拉斯变换的性质

设 L [ f ( t ) ] = F ( s ) \mathscr{L}[f(t)] = F(s) L[f(t)]=F(s)， L [ g ( t ) ] = G ( s ) \mathscr{L}[g(t)] = G(s) L[g(t)]=G(s) 则 F [ α f ( t ) + β g ( t ) ] = α F ( s ) + β G ( s ) , \mathscr{F}[\alpha f(t) + \beta g(t)] = \alpha F(s) + \beta G(s), F[αf(t)+βg(t)]=αF(s)+βG(s),

F − 1 [ α F ( s ) + β G ( s ) ] = α f ( t ) + β g ( t ) ] . \mathscr{F}^{-1} [\alpha F(s) + \beta G(s)]= \alpha f(t) + \beta g(t)]. F−1[αF(s)+βG(s)]=αf(t)+βg(t)].

设 L [ f ( t ) ] = F ( s ) \mathscr{L}[f(t)] = F(s) L[f(t)]=F(s)，则对任一常数 a > 0 a>0 a>0有 L [ f ( a t ) ] = 1 a F ( s a ) . \mathscr{L}[f(at)] = \frac{1}{a}F(\frac{s}{a}). L[f(at)]=a1​F(as​).

设 L [ f ( t ) ] = F ( s ) 设\mathscr{L}[f(t)] = F(s) 设L[f(t)]=F(s)，则有 L [ f ′ ( t ) ] = s F ( s ) − f ( 0 ) ; \mathscr{L}[f'(t)] = sF(s) - f(0); L[f′(t)]=sF(s)−f(0); 一般地，有 L [ f ( n ) ] = s n F ( s ) − s n − 1 f ( 0 ) − s n − 2 f ′ ( 0 ) − ⋅ ⋅ ⋅ − f ( n − 1 ) ( 0 ) , \mathscr{L}[f^{(n)}] = s^nF(s) - s^{n-1}f(0) - s^{n-2}f'(0) - ···-f^{(n-1)}(0), L[f(n)]=snF(s)−sn−1f(0)−sn−2f′(0)−⋅⋅⋅−f(n−1)(0), 其中， f ( k ) ( 0 ) 应 理 解 为 lim ⁡ t → 0 + f ( k ) ( t ) f^{(k)}(0)应理解为\lim_{t \to 0^+}f^{(k)}(t) f(k)(0)应理解为limt→0+​f(k)(t)

设 L [ f ( t ) ] = F ( s ) 设\mathscr{L}[f(t)] = F(s) 设L[f(t)]=F(s)，则有 F ′ ( s ) = − L [ t f ( t ) ] ; F'(s) =- \mathscr{L}[tf(t)] ; F′(s)=−L[tf(t)]; 一般地，有 F ( n ) ( s ) = ( − 1 ) n L [ t n f ( t ) ] . F^{(n)}(s) = (-1)^n \mathscr{L}[t^nf(t)]. F(n)(s)=(−1)nL[tnf(t)].

设 L [ f ( t ) ] = F ( s ) 设\mathscr{L}[f(t)] = F(s) 设L[f(t)]=F(s)，则有 L [ ∫ 0 t f ( t ) ] d t = 1 s F ( s ) ; \mathscr{L}[\int_{0}^{t}f(t)]dt = \frac{1}{s}F(s); L[∫0t​f(t)]dt=s1​F(s); 一般地，有 L [ ∫ 0 t d t ∫ 0 t d t ⋅ ⋅ ⋅ ∫ 0 t f ( t ) d t ] = 1 s n F ( s ) . \mathscr{L} [\int_{0}^{t}dt\int_{0}^{t}dt···\int_{0}^{t}f(t)dt]= \frac{1}{s^n}F(s). L[∫0t​dt∫0t​dt⋅⋅⋅∫0t​f(t)dt]=sn1​F(s).

设 L [ f ( t ) ] = F ( s ) 设\mathscr{L}[f(t)] = F(s) 设L[f(t)]=F(s)，则有 ∫ s ∞ F ( s ) d s = L [ f ( t ) t ] , \int_{s}^{\infty}F(s)ds = \mathscr{L}[\frac{f(t)}{t}], ∫s∞​F(s)ds=L[tf(t)​], 一般地，有 ∫ 0 t d t ∫ 0 t d t ⋅ ⋅ ⋅ ∫ 0 t f ( t ) d t = L [ f ( t ) t n ] . \int_{0}^{t}dt\int_{0}^{t}dt···\int_{0}^{t}f(t)dt=\mathscr{L}[ \frac{f(t)}{t^n}]. ∫0t​dt∫0t​dt⋅⋅⋅∫0t​f(t)dt=L[tnf(t)​].

设 L [ f ( t ) ] = F ( s ) ， 当 t < 0 时 f ( t ) = 0 ， 则 对 任 意 非 负 实 数 τ 有 设\mathscr{L}[f(t)] = F(s)，当t