| Laplace Transform Pairs and Properties | |||||
|---|---|---|---|---|---|
| Definition | |||||
| Laplace Transform | $ X(s)=\int_{-\infty}^\infty x(t) e^{-st}dt, \ s\in {\mathbb C} \ $ | ||||
| Inverse Laplace Transform | add formula here | ||||
| Properties of the Laplace Transform | |||||
| f(s) | F(t) | ||||
| $ af_1(s)+bf_2(s) $ | $ aF_1(t)+bF_2(t) $ | ||||
| $ f(s/a) $ | $ aF(at) $ | ||||
| $ f(s-a) $ | $ e^{at}F(t) $ | ||||
| $ e^{-as}f(s) $ | $ u(t-a) = \begin{cases} F(t-a) & t>a \\ 0 & t<a \end{cases} $ | ||||
| $ sf(s)-F(0) $ | $ F'(t) $ | ||||
| $ s^2f(s)-sF(0)-F'(0) $ | $ F''(t) $ | ||||
| $ s^{n}f(s)-s^{n-1}F(0)-s^{n-2}F'(0)-...-F^{n-1}(0) $ | $ F^{(n)}(t) $ | ||||
| $ f'(s) $ | $ -tF(t) $ | ||||
| $ f''(s) $ | $ t^2F(t) $ | ||||
| $ f^{(n)}(s) $ | $ (-1)^{(ntn)}F(t) $ | ||||
| $ \frac{f(s)}s $ | $ \int_{0}^{t} F(u) du $ | ||||
| $ \frac{f(s)}{s^n} $ | $ \int_{0}^{t}...\int_{0}^{t}F(u)du^n = \int_{0}^{t}\frac{{(t-u)}^{n-1}}{(n-1)!} F(u)du $ | ||||
| $ f(s)g(s) $ | $ \int_{0}^{t}F(u)G(t-u)du $ | ||||
| $ \int_{s}^{\infty}f(u)du $ | $ \frac{F(t)}t $ | ||||
| please continue | place formula here | ||||
| please continue | place formula here | ||||
| please continue | place formula here | ||||
| please continue | place formula here | ||||
| please continue | place formula here | ||||
| Laplace Transform Pairs | |||||
| notes | Signal | Laplace Transform | ROC | ||
| unit impulse/Dirac delta | $ \,\!\delta(t) $ | 1 | $ \text{All}\, s \in {\mathbb C} $ | ||
| unit step function | $ \,\! u(t) $ | $ \frac{1}{s} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $ | ||
| $ \,\! -u(-t) $ | $ \frac{1}{s} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace < 0 $ | |||
| $ \frac{t^{n-1}}{(n-1)!}u(t) $ | $ \frac{1}{s^{n}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $ | |||
| $ -\frac{t^{n-1}}{(n-1)!}u(-t) $ | $ \frac{1}{s^{n}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace < 0 $ | |||
| $ \,\!e^{-\alpha t}u(t) $ | $ \frac{1}{s+\alpha} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha $ | |||
| $ \,\! -e^{-\alpha t}u(-t) $ | $ \frac{1}{s+\alpha} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace < -\alpha $ | |||
| $ \frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(t) $ | $ \frac{1}{(s+\alpha )^{n}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha $ | |||
| $ -\frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(-t) $ | $ \frac{1}{(s+\alpha )^{n}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace < -\alpha $ | |||
| $ \,\!\delta (t - T) $ | $ \,\! e^{-sT} $ | $ \text{All}\,\, s\in {\mathbb C} $ | |||
| $ \,\cos( \omega_0 t)u(t) $ | $ \frac{s}{s^2+\omega_0^{2}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $ | |||
| $ \, \sin( \omega_0 t)u(t) $ | $ \frac{\omega_0}{s^2+\omega_0^{2}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $ | |||
| $ \,e^{-\alpha t}\cos( \omega_0 t) u(t) $ | $ \frac{s+\alpha}{(s+\alpha)^{2}+\omega_0^{2}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha $ | |||
| $ \, e^{-\alpha t}\sin( \omega_0 t)u(t) $ | $ \frac{\omega_0}{(s+\alpha)^{2}+\omega_0^{2}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha $ | |||
| $ u_n(t) = \frac{d^{n}\delta (t)}{dt^{n}} $ | $ \,\!s^{n} $ | $ All\,\, s $ | |||
| $ u_{-n}(t) = \underbrace{u(t) *\dots * u(t)}_{n\,\,times} $ | $ \frac{1}{s^{n}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $ | |||
