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|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | <math> f(s-a) \</math> | + | | <math> f(s-a) \ </math> |
| − | |<math> e^{at}F(t) \</math> | + | |<math> e^{at}F(t) \ </math> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | <math> e^{-as}f(s) \</math> | + | | <math> e^{-as}f(s) \ </math> |
| <math> u(t-a) = | | <math> u(t-a) = | ||
\begin{cases} | \begin{cases} | ||
| Line 41: | Line 41: | ||
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | <math> sf(s)-F(0) \</math> | + | | <math> sf(s)-F(0) \ </math> |
| <math> F'(t) \ </math> | | <math> F'(t) \ </math> | ||
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | <math> s^2f(s)-sF(0)-F'(0) \</math> | + | | <math> s^2f(s)-sF(0)-F'(0) \ </math> |
| − | | <math> F''(t)\</math> | + | | <math> F''(t) \ </math> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | <math> s^{n}f(s)-\sum_{k=1}^ns^{n-k}F^{(k)}(0)\</math> | + | | <math> s^{n}f(s)-\sum_{k=1}^ns^{n-k}F^{(k)}(0) \ </math> |
| − | | <math> F^{(n)}(t)\</math> | + | | <math> F^{(n)}(t) \ </math> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | <math> f'(s)\</math> | + | | <math> f'(s) \ </math> |
| − | | <math> -tF(t)\</math> | + | | <math> -tF(t) \ </math> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | <math>f''(s)\ </math> | + | | <math>f''(s) \ </math> |
| − | | <math> t^2F(t)\ </math> | + | | <math> t^2F(t) \ </math> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | <math> f^{(n)}(s)\</math> | + | | <math> f^{(n)}(s) \ </math> |
| − | | <math> (-1)^{(ntn)}F(t)\</math> | + | | <math> (-1)^{(ntn)}F(t) \ </math> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | <math> \frac{f(s)}s\</math> | + | | <math> \frac{f(s)}s \ </math> |
| − | | <math> \int_{0}^{t} F(u) du \</math> | + | | <math> \int_{0}^{t} F(u) du \ </math> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | <math> \frac{f(s)}{s^n} \</math> | + | | <math> \frac{f(s)}{s^n} \ </math> |
| − | | <math> \int_{0}^{t}...\int_{0}^{t}F(u)du^n = \int_{0}^{t}\frac{{(t-u)}^{n-1}}{(n-1)!} F(u)du\</math> | + | | <math> \int_{0}^{t}...\int_{0}^{t}F(u)du^n = \int_{0}^{t}\frac{{(t-u)}^{n-1}}{(n-1)!} F(u)du \ </math> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | <math>f(s)g(s)\</math> | + | | <math>f(s)g(s) \ </math> |
| − | | <math> \int_{0}^{t}F(u)G(t-u)du\ </math> | + | | <math> \int_{0}^{t}F(u)G(t-u)du \ </math> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | <math> \int_{s}^{\infty}f(u)du \</math> | + | | <math> \int_{s}^{\infty}f(u)du \ </math> |
| − | | <math> \frac{F(t)}t \</math> | + | | <math> \frac{F(t)}t \ </math> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | <math> \frac1{1-e^{-sT}}\int_{0}^{T}e^{-su}F(u)du \</math> | + | | <math> \frac1{1-e^{-sT}}\int_{0}^{T}e^{-su}F(u)du \ </math> |
| − | | <math> F(t)=F(t+T)\</math> | + | | <math> F(t)=F(t+T) \ </math> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | <math> \frac{f(\sqrt{s})}s\</math> | + | | <math> \frac{f(\sqrt{s})}s \ </math> |
| <math> \frac{1}{\sqrt{{\pi}t}}\int_{0}^{\infty}e^{-\frac{u^2}4t}F(u)du </math> | | <math> \frac{1}{\sqrt{{\pi}t}}\int_{0}^{\infty}e^{-\frac{u^2}4t}F(u)du </math> | ||
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | <math> \frac1sf\left(\frac1s\right)\</math> | + | | <math> \frac1sf\left(\frac1s\right) \ </math> |
| − | | <math> \int_{0}^{\infty}J_0(2\sqrt{ut})F(u)du\</math> | + | | <math> \int_{0}^{\infty}J_0(2\sqrt{ut})F(u)du \ </math> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | <math> \frac1{g^{n+1}}f\left(\frac1s\right)\</math> | + | | <math> \frac1{g^{n+1}}f\left(\frac1s\right) \ </math> |
| − | | <math> t^{\frac{n}2}\int_{0}^{\infty}u^{-\frac{n}2}J_n(2\sqrt{ut})F(u)du \</math> | + | | <math> t^{\frac{n}2}\int_{0}^{\infty}u^{-\frac{n}2}J_n(2\sqrt{ut})F(u)du \ </math> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | <math> \frac{f(s+\frac1s)}{s^2+1}\</math> | + | | <math> \frac{f(s+\frac1s)}{s^2+1} \ </math> |
| − | | <math> \int_{0}^{t}J_0(2\sqrt{u(t-u)})F(u)du\</math> | + | | <math> \int_{0}^{t}J_0(2\sqrt{u(t-u)})F(u)du \ </math> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| <math> \frac1{2\sqrt\pi}\int_{0}^{\infty}u^{-\frac32}e^{-\frac{s^2}{4u}}f(u)du\</math> | | <math> \frac1{2\sqrt\pi}\int_{0}^{\infty}u^{-\frac32}e^{-\frac{s^2}{4u}}f(u)du\</math> | ||
| − | | <math> F(t^2)\</math> | + | | <math> F(t^2) \ </math> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | <math> \frac{f(\ln s)}{s\ln s}\</math> | + | | <math> \frac{f(\ln s)}{s\ln s} \ </math> |
| − | | <math> \int_{0}^{\infty}\frac{t^uF(u)}{\Gamma(u+1)}du\</math> | + | | <math> \int_{0}^{\infty}\frac{t^uF(u)}{\Gamma(u+1)}du \ </math> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | <math> \frac{P(s)}{Q(s)}\</math> | + | | <math> \frac{P(s)}{Q(s)} \ </math> |
| − | | <math> \sum_{k=1}^N \frac{P(\alpha_k)}{Q'(\alpha_k)}e^{\alpha_kt}\</math> | + | | <math> \sum_{k=1}^N \frac{P(\alpha_k)}{Q'(\alpha_k)}e^{\alpha_kt} \ </math> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | <math> \frac1s\</math> | + | | <math> \frac1s \ </math> |
| − | | <math> 1\</math> | + | | <math> 1 \ </math> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | <math> \frac1{s^2}\</math> | + | | <math> \frac1{s^2} \ </math> |
| − | | <math> t(s)\</math> | + | | <math> t(s) \ </math> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | <math> \frac1{s^n} n=1,2,3,...\</math> | + | | <math> \frac1{s^n} n=1,2,3,... \ </math> |
| − | | <math> \frac{t^{n-1}}{(n-1)!}, 0!=1\</math> | + | | <math> \frac{t^{n-1}}{(n-1)!}, 0!=1 \ </math> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | please continue | | align="right" style="padding-right: 1em;" | please continue | ||
Revision as of 15:59, 8 November 2010
IT SEEMS LIKE THE VARIABLES s AND t WERE INTERCHANGED BELOW.
| Laplace Transform Pairs and Properties | |||
|---|---|---|---|
| Definition | |||
| Laplace Transform | $ F(s)=\int_{-\infty}^\infty f(t) e^{-st}dt, \ s\in {\mathbb C} \ $ | ||
| Inverse Laplace Transform | add formula here | ||
| Properties of the Laplace Transform | |||
| function $ f(s) \ $ | Laplace transform $ F(t) \ $ | ROC $ R $ | |
| $ af_1(s)+bf_2(s) \ $ | $ aF_1(t)+bF_2(t) \ $ | ||
| $ f\left( \frac{s}{a} \right) $ | $ 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)-\sum_{k=1}^ns^{n-k}F^{(k)}(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 \ $ | ||
| $ \frac1{1-e^{-sT}}\int_{0}^{T}e^{-su}F(u)du \ $ | $ F(t)=F(t+T) \ $ | ||
| $ \frac{f(\sqrt{s})}s \ $ | $ \frac{1}{\sqrt{{\pi}t}}\int_{0}^{\infty}e^{-\frac{u^2}4t}F(u)du $ | ||
| $ \frac1sf\left(\frac1s\right) \ $ | $ \int_{0}^{\infty}J_0(2\sqrt{ut})F(u)du \ $ | ||
| $ \frac1{g^{n+1}}f\left(\frac1s\right) \ $ | $ t^{\frac{n}2}\int_{0}^{\infty}u^{-\frac{n}2}J_n(2\sqrt{ut})F(u)du \ $ | ||
| $ \frac{f(s+\frac1s)}{s^2+1} \ $ | $ \int_{0}^{t}J_0(2\sqrt{u(t-u)})F(u)du \ $ | ||
| $ \frac1{2\sqrt\pi}\int_{0}^{\infty}u^{-\frac32}e^{-\frac{s^2}{4u}}f(u)du\ $ | $ F(t^2) \ $ | ||
| $ \frac{f(\ln s)}{s\ln s} \ $ | $ \int_{0}^{\infty}\frac{t^uF(u)}{\Gamma(u+1)}du \ $ | ||
| $ \frac{P(s)}{Q(s)} \ $ | $ \sum_{k=1}^N \frac{P(\alpha_k)}{Q'(\alpha_k)}e^{\alpha_kt} \ $ | ||
| $ \frac1s \ $ | $ 1 \ $ | ||
| $ \frac1{s^2} \ $ | $ t(s) \ $ | ||
| $ \frac1{s^n} n=1,2,3,... \ $ | $ \frac{t^{n-1}}{(n-1)!}, 0!=1 \ $ | ||
| 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 | ||
| 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 $ | |||
