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|-
 
|-
 
| align="right" style="padding-right: 1em;" |
 
| align="right" style="padding-right: 1em;" |
| <math> \frac{P(s)}{Q(s)} \\ 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>
 
|-
 
|-
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| <math> \frac1{(s-b)^2-a^2}\ </math>
 
| <math> \frac1{(s-b)^2-a^2}\ </math>
 
| <math> \frac{e^{bt}{sh}\ {at}}a\ </math>
 
| <math> \frac{e^{bt}{sh}\ {at}}a\ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \frac{s-b}{(s-b)^2-a^2} \ </math>
 +
| <math> e^{bt} {ch}\ {at}\ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \frac1{(s-a)(s-b)},\ a \ne b\ </math>
 +
| <math> \frac{e^{bt}-e^{at}}{b-a}\ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \frac{s}{(s-a)(s-b)},\ a \ne b \ </math>
 +
| <math> \frac{be^{bt}-ae^{at}}{b-a}\ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \frac1{(s^2+a^2)^2}\ </math>
 +
| <math> \frac{\sin {at}-at\cos{at}}{2a^3}\ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \frac{s}{(s^2+a^2)^2}\ </math>
 +
| <math> \frac{t\sin {at}}{2a}\ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \frac{s^2}{(s^2+a^2)^2}\ </math>
 +
| <math> \frac{\sin {at}+at\cos {at}}{2a}\ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \frac{s^3}{(s^2+a^2)^2}\ </math>
 +
| <math> \cos {at}-\frac12at\sin {at}\ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \frac{s^2-a^2}{(s^2+a^2)^2}\ </math>
 +
| <math> t\cos {at}\ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \frac{1}{(s^2-a^2)^2}\ </math>
 +
| <math> \frac{at \{ch} {at}-\{sh} {at}}{2a^3}\ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \frac{s}{(s^2-a^2)^2}\ </math>
 +
| <math> \frac{t \{sh} {at}}{2a}\ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \frac{s^2}{(s^2-a^2)^2}\ </math>
 +
| <math> \frac{\{sh} {at}+at \{ch} {at}}{2a}\ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \frac{s^3}{(s^2-a^2)^2}\ </math>
 +
| <math> \{ch} {at}+\frac12at \{sh} {at}\ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \frac{s^2+a^2}{(s^2-a^2)^2}\ </math>
 +
| <math> t\{ch} {at}\ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \frac{1}{(s^2-a^2)^3}\ </math>
 +
| <math> \frac{(3-a^2t^2)\sin {at}-3at\cos {at}}{8a^5}\ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \frac{s}{(s^2-a^2)^3}\ </math>
 +
| <math> \frac{t\sin {at}}-at^2\cos {at}}{8a^3}\ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \frac{s^2}{(s^2-a^2)^3}\ </math>
 +
| <math> \frac{(1+a^2t^2)\sin {at}-at\cos {at}}{8a^3}\ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \frac{s^3}{(s^2-a^2)^3}\ </math>
 +
| <math> \frac{3t\sin {at}+at^2\cos {at}}{}\ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | please continue
 +
| place formula here
 +
|-<math> af_1(s)+bf_2(s)\ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | please continue
 +
| place formula here
 +
|-<math> af_1(s)+bf_2(s)\ </math>
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" | please continue
 
| align="right" style="padding-right: 1em;" | please continue
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|-<math> af_1(s)+bf_2(s)\ </math>
 
|-<math> af_1(s)+bf_2(s)\ </math>
 
|-
 
|-
 
 
|}
 
|}
 
{|
 
{|

Revision as of 20:56, 11 November 2010

There is a problem with the properties listed below. The Laplace transform is a function of a complex variable, denoted by s in all ECE courses. Now below, the Laplace transform appears to be a function of a real variable t. This is seen, for example, by the fact that the function u(t) appears in the table; now u(t) must be a function of a real variable t, because, the statement "t>0" does not make any sense when the variable t is a complex number. Also, one thing that needs to be added is the ROC in the 4th column of the properties table. Please let me know if you need a reference.-pm

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) \ $ at least $ R_1 \cap R_2 $
$ 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{s^{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 \ $
$ \frac1{s^n}, \ n=1,2,3,... \ $ $ \frac{t^{n-1}}{(n-1)!}, \ 0!=1 \ $
$ \frac1{s^n}, \ n>0 \ $ $ \frac{t^{n-1}}{\Gamma(n)} \ $
$ \frac1{s-a}\ $ $ e^{at}\ $
$ \frac1{(s-a)^n}, \ n=1,2,3,...\ $ $ \frac{t^{n-1}e^{at}}{(n-1)!}, \ 0!=1\ $
$ \frac1{(s-a)^n}, \ n>0\ $ $ \frac{t^{n-1}e^{at}}{\Gamma(n)}\ $
$ \frac1{s^2+a^2}\ $ $ \frac{\sin {at}}{a} \ $
$ \frac{s}{s^2+a^2} \ $ $ \cos {at} \ $
$ \frac1{(s-b)^2+a^2}\ $ $ \frac{e^{bt}\sin{at}}{a} \ $
$ \frac{s-b}{(s-b)^2+a^2}\ $ $ e^{bt}\cos{at}\ $
$ \frac{1}{s^2-a^2} \ $ $ \left(\frac{{sh}\ {at}}{a}\right)\ $
$ \frac{s}{s^2-a^2}\ $ $ {ch}\ {at}\ $
$ \frac1{(s-b)^2-a^2}\ $ $ \frac{e^{bt}{sh}\ {at}}a\ $
$ \frac{s-b}{(s-b)^2-a^2} \ $ $ e^{bt} {ch}\ {at}\ $
$ \frac1{(s-a)(s-b)},\ a \ne b\ $ $ \frac{e^{bt}-e^{at}}{b-a}\ $
$ \frac{s}{(s-a)(s-b)},\ a \ne b \ $ $ \frac{be^{bt}-ae^{at}}{b-a}\ $
$ \frac1{(s^2+a^2)^2}\ $ $ \frac{\sin {at}-at\cos{at}}{2a^3}\ $
$ \frac{s}{(s^2+a^2)^2}\ $ $ \frac{t\sin {at}}{2a}\ $
$ \frac{s^2}{(s^2+a^2)^2}\ $ $ \frac{\sin {at}+at\cos {at}}{2a}\ $
$ \frac{s^3}{(s^2+a^2)^2}\ $ $ \cos {at}-\frac12at\sin {at}\ $
$ \frac{s^2-a^2}{(s^2+a^2)^2}\ $ $ t\cos {at}\ $
$ \frac{1}{(s^2-a^2)^2}\ $ $ \frac{at \{ch} {at}-\{sh} {at}}{2a^3}\ $
$ \frac{s}{(s^2-a^2)^2}\ $ $ \frac{t \{sh} {at}}{2a}\ $
$ \frac{s^2}{(s^2-a^2)^2}\ $ $ \frac{\{sh} {at}+at \{ch} {at}}{2a}\ $
$ \frac{s^3}{(s^2-a^2)^2}\ $ $ \{ch} {at}+\frac12at \{sh} {at}\ $
$ \frac{s^2+a^2}{(s^2-a^2)^2}\ $ $ t\{ch} {at}\ $
$ \frac{1}{(s^2-a^2)^3}\ $ $ \frac{(3-a^2t^2)\sin {at}-3at\cos {at}}{8a^5}\ $
$ \frac{s}{(s^2-a^2)^3}\ $ $ \frac{t\sin {at}}-at^2\cos {at}}{8a^3}\ $
$ \frac{s^2}{(s^2-a^2)^3}\ $ $ \frac{(1+a^2t^2)\sin {at}-at\cos {at}}{8a^3}\ $
$ \frac{s^3}{(s^2-a^2)^3}\ $ $ \frac{3t\sin {at}+at^2\cos {at}}{}\ $
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 $

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