| Line 40: | Line 40: | ||
| align="right" style="padding-right: 1em;"|16 || <math>u_{-n}(t) = \underbrace{u(t) *\dots * u(t)}_{n\,\,times}</math> || <math>\frac{1}{s^{n}}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 </math> | | align="right" style="padding-right: 1em;"|16 || <math>u_{-n}(t) = \underbrace{u(t) *\dots * u(t)}_{n\,\,times}</math> || <math>\frac{1}{s^{n}}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 </math> | ||
|} | |} | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
---- | ---- | ||
[[ MegaCollectiveTableTrial1|Back to Collective Table]] | [[ MegaCollectiveTableTrial1|Back to Collective Table]] | ||
Revision as of 04:40, 27 October 2009
Table of Laplace Transform Pairs
| Laplace Transform Pairs | |||
|---|---|---|---|
| Transform Pair | Signal | Transform | ROC |
| 1 | $ \,\!\delta(t) $ | $ 1 $ | $ All\,\, s $ |
| 2 | $ \,\! u(t) $ | $ \frac{1}{s} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $ |
| 3 | $ \,\! -u(-t) $ | $ \frac{1}{s} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace < 0 $ |
| 4 | $ \frac{t^{n-1}}{(n-1)!}u(t) $ | $ \frac{1}{s^{n}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $ |
| 5 | $ -\frac{t^{n-1}}{(n-1)!}u(-t) $ | $ \frac{1}{s^{n}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace < 0 $ |
| 6 | $ \,\!e^{-\alpha t}u(t) $ | $ \frac{1}{s+\alpha} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha $ |
| 7 | $ \,\! -e^{-\alpha t}u(-t) $ | $ \frac{1}{s+\alpha} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace < -\alpha $ |
| 8 | $ \frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(t) $ | $ \frac{1}{(s+\alpha )^{n}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha $ |
| 9 | $ -\frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(-t) $ | $ \frac{1}{(s+\alpha )^{n}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace < -\alpha $ |
| 10 | $ \,\!\delta (t - T) $ | $ \,\! e^{-sT} $ | $ All\,\, s $ |
| 11 | $ \,\![cos( \omega_0 t)]u(t) $ | $ \frac{s}{s^2+\omega_0^{2}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $ |
| 12 | $ \,\![sin( \omega_0 t)]u(t) $ | $ \frac{\omega_0}{s^2+\omega_0^{2}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $ |
| 13 | $ \,\![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 $ |
| 14 | $ \,\![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 $ |
| 15 | $ u_n(t) = \frac{d^{n}\delta (t)}{dt^{n}} $ | $ \,\!s^{n} $ | $ All\,\, s $ |
| 16 | $ u_{-n}(t) = \underbrace{u(t) *\dots * u(t)}_{n\,\,times} $ | $ \frac{1}{s^{n}} $ | $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $ |
