Bootstrap

常用拉普拉斯变换

  • 基本性质
性质公式表示
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=1nsnkfk1(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=1nsnk+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(tT)1(tT)]=eTsF(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)eat]=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) tlimf(t)=s0limsF(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) t0limf(t)=slimsF(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(tτ)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)]=a1f(as)
  • 常用函数的变换
时间函数变换后时间函数变换后
δ ( t ) \delta(t) δ(t)1 1 − e − a t 1-e^{-at} 1eat a s ( s + a ) \frac{a}{s(s+a)} s(s+a)a
δ T ( t ) = ∑ n = 0 ∞ δ ( t − n T ) \delta_T(t)=\sum_{n=0}^\infty\delta(t-nT) δT(t)=n=0δ(tnT) 1 1 − e − T s \frac{1}{1-e^{-Ts}} 1eTs1 e − a t − e − b t e^{-at}-e^{-bt} eatebt b − a ( s + a ) ( s + b ) \frac{b-a}{(s+a)(s+b)} (s+a)(s+b)ba
1 ( t ) 1(t) 1(t) 1 s \frac{1}{s} s1 sin ⁡ ω t \sin \omega t sinωt ω s 2 + ω 2 \frac{\omega}{s^2+\omega^2} s2+ω2ω
t t t 1 s 2 \frac{1}{s^2} s21 cos ⁡ ω t \cos \omega t cosωt s s 2 + ω 2 \frac{s}{s^2+\omega^2} s2+ω2s
t 2 2 \frac{t^2}{2} 2t2 1 s 3 \frac{1}{s^3} s31 e − a t sin ⁡ ω t e^{-at}\sin \omega t eatsinωt ω ( s + a ) 2 + ω 2 \frac{\omega}{(s+a)^2+\omega^2} (s+a)2+ω2ω
t n n \frac{t^n}{n} ntn 1 s n + 1 \frac{1}{s^{n+1}} sn+11 e − a t cos ⁡ ω t e^{-at}\cos \omega t eatcosωt s + a ( s + a ) 2 + ω 2 \frac{s+a}{(s+a)^2+\omega^2} (s+a)2+ω2s+a
e − a t e^{-at} eat 1 s + a \frac{1}{s+a} s+a1 a t / T a^{t/T} at/T 1 s − ( 1 / t ) ln ⁡ a \frac{1}{s-(1/t)\ln a} s(1/t)lna1
t e − a t te^{-at} teat 1 ( s + a ) 2 \frac{1}{(s+a)^2} (s+a)21
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