2805 Laplace Transform Table

Properties and Rules

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learn in this session $$ $$\begin{aligned} &f(t) &&F(s)=\int_{0^-}^\infty f(t)e^{-st}dt &&\text{(Definition)}\\ &af(t)+bg(t) &&aF(s)+bG(s) &&\text{(Linearity)}\\ &e^{zt}f(t) &&F(s-z) &&\text{($s$-shift)} \end{aligned}$$ $$

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Will learn in a future session $$ $$\begin{aligned} &f'(t) &&sF(s)-f(0^-)\\ &f''(t) &&s^2F(s)-sf(0^-)-f'(0^-)\\ &f^{(n)}(t) &&s^nF(s)-s^{n-1}f(0^-)-\cdots-f^{(n-1)}(0^-)\\ &tf(t) &&-F'(s)\\ &t^nf(t) &&(-1)^nF^{(n)}(s)\\ &u(t-a)f(t-a) &&e^{-as}F(s) &&\text{($t$-translation)}\\ &u(t-a)f(t) &&e^{-as}\mathcal{L}(f(t+a)) &&\text{($t$-translation)}\\ &(f*g)(t)=\int_{0^-}^{t^+}f(t-\tau)g(\tau)d\tau &&F(s)G(s)\\ &\int_{0^-}^{t^+}f(\tau)d\tau &&\frac{F(s)}{s} &&\text{(integration rule)}\\ \end{aligned}$$ $$

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Interesting, but not included in this course $$\begin{aligned} &\frac{f(t)}{t} &&\int_s^\infty F(\sigma)d\sigma \end{aligned}$$

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Function Table

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learn in this session $$\begin{aligned} &1 &&1/s &&\text{Re}(s)>0\\ &e^{at} &&1/(s-a) &&\text{Re}(s)>a\\ &t &&1/s^2 &&\text{Re}(s)>0\\ &t^n &&n!/s^{n+1} &&\text{Re}(s)>0\\ &\cos(\omega t) &&s/(s^2+\omega^2) &&\text{Re}(s)>0\\ &\sin(\omega t) &&\omega/(s^2+\omega^2) &&\text{Re}(s)>0\\ &e^{zt}\cos(\omega t) &&(s-z)/((s-z)^2+\omega^2) &&\text{Re}(s)>\text{Re}(z)\\ &e^{zt}\sin(\omega t) &&\omega/((s-z)^2+\omega^2) &&\text{Re}(s)>\text{Re}(z)\\ &\delta(t) &&1 &&\text{all }s\\ &\delta(t-a) &&e^{-as} &&\text{all }s\\ &\cosh(kt)=\frac{e^{kt}+e^{-kt}}{2} &&s/(s^2-k^2) &&\text{Re}(s)>k\\ &\sinh(kt)=\frac{e^{kt}-e^{-kt}}{2} &&k/(s^2-k^2) &&\text{Re}(s)>k \end{aligned}$$

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Will learn in a future session $$\begin{aligned} &\frac{1}{2\omega^3}(\sin(\omega t)-\omega t\cos(\omega t)) &&\frac{1}{(s^2+\omega^2)^2} &&\text{Re}(s)>0\\ &\frac{t}{2\omega}\sin(\omega t) &&\frac{s}{(s^2+\omega^2)^2} &&\text{Re}(s)>0\\ &\frac{1}{2\omega}(\sin(\omega t)+\omega t\cos(\omega t)) &&\frac{s^2}{(s^2+\omega^2)^2} &&\text{Re}(s)>0\\ &u(t-a) &&e^{-as}/s &&\text{Re}(s)>0\\ &t^ne^{at} &&n!/(s-a)^{n+1} &&\text{Re}(s)>a \end{aligned}$$

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Interesting, but not included in this course $$\begin{aligned} &\frac{1}{\sqrt{\pi t}} &&\frac{1}{\sqrt{s}} &&\text{Re}(s)>0\\ &t^a &&\frac{\Gamma(a+1)}{s^{a+1}} &&\text{Re}(s)>0 \end{aligned}$$

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