| 线性定理 - 齐次性 | L [ a f ( t ) ] = a F ( s ) L[af(t)]=aF(s) L[af(t)]=aF(s) |
| 线性定理 - 叠加性 | L ( f 1 ( t ) ± f 2 ( t ) ) = F 1 ( s ) ± F 2 ( s ) L(f_1(t)\pm f_2(t))=F_1(s)\pm F_2(s) L(f1(t)±f2(t))=F1(s)±F2(s) |
| 微分定理 - 一阶导 | L [ d f ( t ) d t ] = s F ( s ) − f ( 0 ) L[\frac{df(t)}{dt}]=sF(s)-f(0) L[dtdf(t)]=sF(s)−f(0) |
| 微分定理 - 二阶导 | L [ d 2 f ( t ) d t 2 ] = s 2 F ( s ) − s f ( 0 ) − f ′ ( 0 ) L[\frac{d^2f(t)}{dt^2}]=s^2F(s)-sf(0)-f'(0) L[dt2d2f(t)]=s2F(s)−sf(0)−f′(0) |
| 微分定理 - n阶导 | L [ d n f ( t ) d t n ] = s n F ( s ) − ∑ k = 1 n s n − k f k − 1 ( 0 ) L[\frac{d^n f(t)}{dt^n}]=s^nF(s)-\sum_{k=1}^{n}s^{n-k}f^{k-1}(0) L[dtndnf(t)]=snF(s)−∑k=1nsn−kfk−1(0) |
| 微分定理 | L [ t f ( t ) ] = − d d s F ( s ) L[tf(t)]=-\frac{d}{ds}F(s) L[tf(t)]=−dsdF(s) |
| 积分定理 - 一阶导 | L [ ∫ f ( t ) d t ] = F ( s ) s + [ ∫ f ( t ) d t ] t = 0 s L[\int f(t)dt]=\frac{F(s)}{s}+\frac{[\int f(t)dt]_{t=0}}{s} L[∫f(t)dt]=sF(s)+s[∫f(t)dt]t=0 |
| 积分定理 - 二阶导 | L [ ∬ f ( t ) ( d t ) 2 ] = F ( s ) s 2 + [ ∫ f ( t ) d t ] t = 0 s 2 + [ ∬ f ( t ) ( d t ) 2 ] t = 0 s L[\iint f(t)(dt)^2]=\frac{F(s)}{s^2}+\frac{[\int f(t)dt]_{t=0}}{s^2}+\frac{[\iint f(t)(dt)^2]_{t=0}}{s} L[∬f(t)(dt)2]=s2F(s)+s2[∫f(t)dt]t=0+s[∬f(t)(dt)2]t=0 |
| 积分定理 - n阶导 | L [ ∫ … ∫ ⏞ n f ( t ) ( d t ) n ] = F ( s ) s n + ∑ k = 1 n [ ∫ … ∫ ⏞ k f ( t ) ( d t ) k ] t = 0 s n − k + 1 L[\overbrace{\int \dotso \int}^{n}f(t)(dt)^n]=\frac{F(s)}{s^n}+\sum_{k=1}^n\frac{[\overbrace{\int \dotso \int}^{k}f(t)(dt)^k]_{t=0}}{s^{n-k+1}} L[∫…∫ nf(t)(dt)n]=snF(s)+∑k=1nsn−k+1[∫…∫ kf(t)(dt)k]t=0 |
| 延迟定理 | L [ f ( t − T ) 1 ( t − T ) ] = e − T s F ( s ) L[f(t-T)1(t-T)]=e^{-Ts}F(s) L[f(t−T)1(t−T)]=e−TsF(s) |
| 衰减定理 | L [ f ( t ) e − a t ] = F ( s + a ) L[f(t)e^{-at}]=F(s+a) L[f(t)e−at]=F(s+a) |
| 终值定理 | lim t → ∞ f ( t ) = lim s → 0 s F ( s ) \lim\limits_{t \to \infty}f(t)=\lim\limits_{s \to 0}sF(s) t→∞limf(t)=s→0limsF(s) |
| 初值定理 | lim t → 0 f ( t ) = lim s → ∞ s F ( s ) \lim\limits_{t \to 0}f(t)=\lim\limits_{s \to \infty}sF(s) t→0limf(t)=s→∞limsF(s) |
| 卷积定理 | L [ ∫ 0 t f 1 ( t − τ ) f 2 ( τ ) d τ ] = F 1 ( s ) F 2 ( s ) L[\int_{0}^{t}f_1(t-\tau)f_2(\tau)d\tau]=F_1(s)F_2(s) L[∫0tf1(t−τ)f2(τ)dτ]=F1(s)F2(s) |
| 尺度定理 | L [ f ( a t ) ] = 1 ∣ a ∣ f ( s a ) L[f(at)]=\frac{1}{\vert a\vert}f(\frac{s}{a}) L[f(at)]=∣a∣1f(as) |