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 | $ f(t) = \frac{1}{2 \pi i} \int_{ \gamma - i \infty}^{ \gamma + i \infty} F(s) e^{st}\,ds, $ | ||
| Properties of the Laplace Transform | |||
| Function | Laplace Transform | ROC | |
| $ f(t) \ $ | $ F(s) \ $ | ROC $ R $ | |
| $ af_1(t)+bf_2(t) \ $ | $ aF_1(s)+bF_2(s) \ $ | at least $ R_1 \cap R_2 $ | |
| $ af(at) \ $ | $ F\left( \frac{s}{a} \right) $ | ||
| $ e^{at}f(t) \ $ | $ F(s-a) \ $ | ||
| $ u(t-a) = \begin{cases} f(t-a) & t>a \\ 0 & t<a \end{cases} $ | $ e^{-as}F(s) \ $ | ||
| $ f'(t) \ $ | $ sF(s)-f(0) \ $ | ||
| $ f''(t) \ $ | $ s^2F(s)-sf(0)-f'(0) \ $ | ||
| $ f^{(n)}(t) \ $ | $ s^{n}F(s)-\sum_{k=1}^ns^{n-k}f^{(k)}(0) \ $ | ||
| $ -tf(t) \ $ | $ F'(s) \ $ | ||
| $ t^2f(t) \ $ | $ F''(s) \ $ | ||
| $ (-1)^{(ntn)}f(t) \ $ | $ F^{(n)}(s) \ $ | ||
| $ \int_{0}^{t} f(u) du \ $ | $ \frac{F(s)}s \ $ | ||
| $ \int_{0}^{t}...\int_{0}^{t}f(u)du^n = \int_{0}^{t}\frac{{(t-u)}^{n-1}}{(n-1)!} f(u)du \ $ | $ \frac{F(s)}{s^n} \ $ | ||
| $ \int_{0}^{t}f(u)g(t-u)du \ $ | $ F(s)G(s) \ $ | ||
| $ \frac{f(t)}t \ $ | $ \int_{s}^{\infty}F(u)du \ $ | ||
| $ f(t)=f(t+T) \ $ | $ \frac1{1-e^{-sT}}\int_{0}^{T}e^{-su}f(u)du \ $ | ||
| $ \frac{1}{\sqrt{{\pi}t}}\int_{0}^{\infty}e^{-\frac{u^2}4t}f(u)du $ | $ \frac{F(\sqrt{s})}s \ $ | ||
| $ \int_{0}^{\infty}J_0(2\sqrt{ut})f(u)du \ $ | $ \frac1sF\left(\frac1s\right) \ $ | ||
| $ t^{\frac{n}2}\int_{0}^{\infty}u^{-\frac{n}2}J_n(2\sqrt{ut})f(u)du \ $ | $ \frac1{s^{n+1}}F\left(\frac1s\right) \ $ | ||
| $ \int_{0}^{t}J_0(2\sqrt{u(t-u)})f(u)du \ $ | $ \frac{F(s+\frac1s)}{s^2+1} \ $ | ||
| $ f(t^2) \ $ | $ \frac1{2\sqrt\pi}\int_{0}^{\infty}u^{-\frac32}e^{-\frac{s^2}{4u}}F(u)du \ $ | ||
| $ \int_{0}^{\infty}\frac{t^uf(u)}{\Gamma(u+1)}du \ $ | $ \frac{F(\ln s)}{s\ln s} \ $ | ||
| $ \sum_{k=1}^N \frac{P(\alpha_k)}{Q'(\alpha_k)}e^{\alpha_kt} \ $ | $ \frac{P(s)}{Q(s)} \ $ | ||
| $ 1 \ $ | $ \frac1s \ $ | ||
| $ t \ $ | $ \frac1{s^2} \ $ | ||
| $ \frac{t^{n-1}}{(n-1)!}, \ 0!=1 \ $ | $ \frac1{s^n}, \ n=1,2,3,... \ $ | ||
| $ \frac{t^{n-1}}{\Gamma(n)} \ $ | $ \frac1{s^n}, \ n>0 \ $ | ||
| $ e^{at}\ $ | $ \frac1{s-a}\ $ | ||
| $ \frac{t^{n-1}e^{at}}{(n-1)!}, \ 0!=1\ $ | $ \frac1{(s-a)^n}, \ n=1,2,3,...\ $ | ||
| $ \frac{t^{n-1}e^{at}}{\Gamma(n)}\ $ | $ \frac1{(s-a)^n}, \ n>0\ $ | ||
| $ \frac{\sin {at}}{a} \ $ | $ \frac1{s^2+a^2}\ $ | ||
| $ \cos {at} \ $ | $ \frac{s}{s^2+a^2} \ $ | ||
| $ \frac{e^{bt}\sin{at}}{a} \ $ | $ \frac1{(s-b)^2+a^2}\ $ | ||
| $ e^{bt}\cos{at}\ $ | $ \frac{s-b}{(s-b)^2+a^2}\ $ | ||
| $ \left(\frac{{sh}\ {at}}{a}\right)\ $ | $ \frac{1}{s^2-a^2} \ $ | ||
| $ {ch}\ {at}\ $ | $ \frac{s}{s^2-a^2}\ $ | ||
| $ \frac{e^{bt}{sh}\ {at}}a\ $ | $ \frac1{(s-b)^2-a^2}\ $ | ||
| $ e^{bt} {ch}\ {at}\ $ | $ \frac{s-b}{(s-b)^2-a^2} \ $ | ||
| $ \frac{e^{bt}-e^{at}}{b-a}\ $ | $ \frac1{(s-a)(s-b)},\ a \ne b\ $ | ||
| $ \frac{be^{bt}-ae^{at}}{b-a}\ $ | $ \frac{s}{(s-a)(s-b)},\ a \ne b \ $ | ||
| $ \frac{\sin {at}-at\cos{at}}{2a^3}\ $ | $ \frac1{(s^2+a^2)^2}\ $ | ||
| $ \frac{t\sin {at}}{2a}\ $ | $ \frac{s}{(s^2+a^2)^2}\ $ | ||
| $ \frac{\sin {at}+at\cos {at}}{2a}\ $ | $ \frac{s^2}{(s^2+a^2)^2}\ $ | ||
| $ \cos {at}-\frac12at\sin {at}\ $ | $ \frac{s^3}{(s^2+a^2)^2}\ $ | ||
| $ t\cos {at}\ $ | $ \frac{s^2-a^2}{(s^2+a^2)^2}\ $ | ||
| $ \frac{at\ {ch}\ {at}-{sh}\ {at}}{2a^3}\ $ | $ \frac{1}{(s^2-a^2)^2}\ $ | ||
| $ \frac{t\ {sh}\ {at}}{2a}\ $ | $ \frac{s}{(s^2-a^2)^2}\ $ | ||
| $ \frac{{sh}\ {at}+at\ {ch}\ {at}}{2a}\ $ | $ \frac{s^2}{(s^2-a^2)^2}\ $ | ||
| $ {ch}\ {at}+\frac12at\ {sh}\ {at} \ $ | $ \frac{s^3}{(s^2-a^2)^2}\ $ | ||
| $ t\ {ch}\ {at}\ $ | $ \frac{s^2+a^2}{(s^2-a^2)^2}\ $ | ||
| $ \frac{(3-a^2t^2)\sin {at}-3at\cos {at}}{8a^5}\ $ | $ \frac{1}{(s^2+a^2)^3}\ $ | ||
| $ \frac{t\sin {at}-at^2\cos {at}}{8a^3}\ $ | $ \frac{s}{(s^2+a^2)^3}\ $ | ||
| $ \frac{(1+a^2t^2)\sin {at}-at\cos {at}}{8a^3}\ $ | $ \frac{s^2}{(s^2+a^2)^3}\ $ | ||
| $ \frac{3t\sin {at}+at^2\cos {at}}{8a}\ $ | $ \frac{s^3}{(s^2+a^2)^3}\ $ | ||
| $ \frac{(3-a^2t^2)\sin {at}+5at\cos {at}}{8a}\ $ | $ \frac{s^4}{(s^2+a^2)^3}\ $ | ||
| $ \frac{(8-a^2t^2)\cos {at}-7at\sin {at}}{8}\ $ | $ \frac{s^5}{(s^2+a^2)^3}\ $ | ||
| $ \frac{t^2\sin {at}}{2a}\ $ | $ \frac{3s^2-a^2}{(s^2+a^2)^3}\ $ | ||
| $ \frac12t^2\cos {at}\ $ | $ \frac{s^3-3a^2s}{(s^2+a^2)^3}\ $ | ||
| $ \frac16t^3\cos {at}\ $ | $ \frac{s^4-6a^2s^2+a^4}{(s^2+a^2)^4}\ $ | ||
| $ \frac{t^3\sin {at}}{24a}\ $ | $ \frac{s^3-a^2s}{(s^2+a^2)^4}\ $ | ||
| $ \frac{3+a^2t^2\ {sh}\ {at}-3at\ {ch}\ {at}}{8a^5}\ $ | $ \frac{1}{(s^2-a^2)^3}\ $ | ||
| $ \frac{at^2\ {ch}\ {at}-t\ {sh}\ {at}}{8a^3}\ $ | $ \frac{s}{(s^2-a^2)^3}\ $ | ||
| $ \frac{at\ {ch}\ {at}+(a^2t^2-1)\ {sh}\ {at}}{8a^3}\ $ | $ \frac{s^2}{(s^2-a^2)^3}\ $ | ||
| $ \frac{3t\ {sh}\ {at}+at^2\ {ch}\ {at}}{8a}\ $ | $ \frac{s^3}{(s^2-a^2)^3}\ $ | ||
| $ \frac{(3+a^2t^2)\ {sh}\ {at}+5at\ {ch}\ {at}}{8a}\ $ | $ \frac{s^4}{(s^2-a^2)^3}\ $ | ||
| $ \frac{(8+a^2t^2)\ {ch}\ {at}+7at\ {sh}\ {at}}{8}\ $ | $ \frac{s^5}{(s^2-a^2)^3}\ $ | ||
| $ \frac{t^2\ {sh}\ {at}}{2a}\ $ | $ \frac{3s^2+a^2}{(s^2-a^2)^3}\ $ | ||
| $ \frac12t^2\ {ch}\ {at}\ $ | $ \frac{s^3+3a^2s}{(s^2-a^2)^3}\ $ | ||
| $ \frac16t^3\ {ch}\ {at}\ $ | $ \frac{s^4+6a^2s^2+a^4}{(s^2-a^2)^4}\ $ | ||
| $ \frac{t^3\ {sh}\ {at}}{24a}\ $ | $ \frac{s^3+a^2s}{(s^2-a^2)^4}\ $ | ||
| $ \frac{e^{at/2}}{3a^2} \left \{ \sqrt{3} \sin {\frac{\sqrt{3}at}{2}}-\cos {\frac{\sqrt{3}at}{2}}+e^{-3at/2} \right \}\ $ | $ \frac{1}{s^3+a^3}\ $ | ||
| $ \frac{e^{at/2}}{3a^2} \left \{ \cos {\frac{\sqrt{3}at}{2}}+ \sqrt{3}\sin {\frac{\sqrt{3}at}{2}}-e^{-3at/2} \right \}\ $ | $ \frac{s}{s^3+a^3}\ $ | ||
| $ \frac13 \left \{ e^{-at}+ 2e^{at/2} \cos {\frac{\sqrt{3}at}{2}} \right \}\ $ | $ \frac{s^2}{s^3+a^3}\ $ | ||
| $ \frac{e^{-at/2}}{3a^2} \left \{e^{3at/2}- \cos {\frac{\sqrt{3}at}{2}}- \sqrt{3}\sin {\frac{\sqrt{3}at}{2}} \right \}\ $ | $ \frac{1}{s^3-a^3}\ $ | ||
| $ \frac{e^{-at/2}}{3a} \left \{ \sqrt{3}\sin {\frac{\sqrt{3}at}{2}}-\cos {\frac{\sqrt{3}at}{2}}+e^{3at/2} \right \}\ $ | $ \frac{s}{s^3-a^3}\ $ | ||
| $ \frac13 \left \{ e^{at}+ 2e^{-at/2} \cos {\frac{\sqrt{3}at}{2}} \right \}\ $ | $ \frac{s^2}{s^3-a^3}\ $ | ||
| $ \frac1{4a^3} \left (\sin {at}\ {ch}\ {at}-\cos {at}\ {sh}\ {at} \right )\ $ | $ \frac{1}{s^4+4a^4}\ $ | ||
| $ \frac{\sin {at}\ {sh}\ {at}}{2a^2}\ $ | $ \frac{s}{s^4+4a^4}\ $ | ||
| $ \frac1{2a} \left (\sin {at}\ {ch}\ {at}+\cos {at}\ {sh}\ {at} \right )\ $ | $ \frac{s^2}{s^4+4a^4}\ $ | ||
| $ \cos {at}\ {ch}\ {at}\ $ | $ \frac{s^3}{s^4+4a^4}\ $ | ||
| $ \frac1{2a^3} \left (\ {sh}\ {at}-\sin {at} \right )\ $ | $ \frac{1}{s^4-a^4}\ $ | ||
| $ \frac1{2a^2} \left (\ {ch}\ {at}-\cos {at} \right )\ $ | $ \frac{s}{s^4-a^4}\ $ | ||
| $ \frac1{2a} \left (\ {sh}\ {at}+\sin {at} \right )\ $ | $ \frac{s^2}{s^4-a^4}\ $ | ||
| $ \frac12 \left (\ {ch}\ {at}+\cos {at} \right )\ $ | $ \frac{s^3}{s^4-a^4}\ $ | ||
| $ \frac{e^{-bt}-e^{-at}}{2(b-a)\sqrt{\pi t^3}}\ $ | $ \frac1{\sqrt{s+a}+\sqrt{s+b}}\ $ | ||
| $ \frac{erf\ \sqrt{at}}{\sqrt{a}}\ $ | $ \frac1{s\sqrt{s+a}}\ $ | ||
| $ \frac{e^{at}\ {erf}\ \sqrt{at}}{\sqrt{a}}\ $ | $ \frac1{\sqrt{s}(s-a)}\ $ | ||
| $ e^{at} \left \{\frac1{\sqrt{\pi t}}-be^{b^{2}t}\ {erfc}\ (b\sqrt{t}) \right \}\ $ | $ \frac1{\sqrt{s-a}+b}\ $ | ||
| $ J_0(at)\ $ | $ \frac1{\sqrt{s^2+a^2}}\ $ | ||
| $ I_0(at)\ $ | $ \frac1{\sqrt{s^2-a^2}}\ $ | ||
| $ a^nJ_n(at)\ $ | $ \frac{{\left (\sqrt{s^2+a^2}-s \right )}^n}{\sqrt{s^2+a^2}},\quad n>-1 \ $ | ||
| $ a^nI_n(at)\ $ | $ \frac{{\left (s- \sqrt{s^2-a^2} \right )}^n}{\sqrt{s^2-a^2}},\quad n>-1 \ $ | ||
| $ J_0(a\sqrt{t(t+2b)})\ $ | $ \frac{e^{b \left (s- \sqrt{s^2+a^2} \right )}}{\sqrt{s^2+a^2}} \ $ | ||
| $ \begin{cases} J_0(a\sqrt{t^2-b^2}) & t>b \\ 0 &t<b \end{cases} \ $ | $ \frac{e^{-b\sqrt{s^2+a^2}}}{\sqrt{s^2+a^2}} \ $ | ||
| $ tJ_0(at)\ $ | $ \frac1{(s^2+a^2)^{3/2}}\ $ | ||
| $ J_0(at)-atJ_1(at)\ $ | $ \frac{s^2}{(s^2+a^2)^{3/2}}\ $ | ||
| $ \frac{tI_1(at)}{a}\ $ | $ \frac1{(s^2-a^2)^{3/2}}\ $ | ||
| $ I_0(at)+atI_1(at)\ $ | $ \frac{s}{(s^2+a^2)^{3/2}}\ $ | ||
| $ f(t)=n,\ n \leqq t\ <n+1,\ n=0,1,2,... \ $ | $ \frac1{s(e^s-1)}\ =\ \frac{e^{-s}}{s(1-e^{-s})}\ $ | ||
| $ f(t)= \sum_{k=1}^{[t]} r^k\ $ | $ \frac1{s(e^s-r)}\ =\ \frac{e^{-s}}{s(1-re^{-s})}\ $ | ||
| $ f(t)= r^n,\ n\leqq t<n+1,\ n=0,1,2,...\ $ | $ \frac{s^s-1}{s(e^s-r)}\ =\ \frac{1-e^{-s}}{s(1-re^{-s})}\ $ | ||
| $ \frac{\cos {2\sqrt{at}}}{\sqrt{ \pi t}}\ $ | $ \frac{s^{-a/s}}{\sqrt{s}}\ $ | ||
| $ \frac{\sin {2\sqrt{at}}}{\sqrt{ \pi a}}\ $ | $ \frac{e^{-a/s}}{s^{3/2}}\ $ | ||
| $ \left ( \frac{t}{a} \right )^{n/2}J_n(2\sqrt{at})\ $ | $ \frac{e^{-a/s}}{s^n+1} \quad n>-1 \ $ | ||
| $ \frac{e^{-a^2/4t}}{\sqrt{ \pi t}}\ $ | $ \frac{e^{-a\sqrt{s}}}{\sqrt{s}}\ $ | ||
| $ \frac{a}{2\sqrt{ \pi t^3}}e^{-a^2/4t}\ $ | $ e^{-a\sqrt{s}}\ $ | ||
| $ erf(a/2\sqrt{t})\ $ | $ \frac{1-e^{-a\sqrt{s}}}{s}\ $ | ||
| $ erfc(a/2\sqrt{t})\ $ | $ \frac{e^{-a\sqrt{s}}}{s}\ $ | ||
| $ e^{b(bt+a)}erfc \left ( b\sqrt{t}+\frac{a}{2\sqrt{t}} \right )\ $ | $ \frac{e^{-a\sqrt{s}}}{\sqrt{s}(\sqrt{s}+b)}\ $ | ||
| $ \frac1{\sqrt{\pi t}a^{2n+1}}\int_{0}^{\infty}u^ne^{-u^2/4a^2t}J_{2n}(2\sqrt{u})du \ $ | $ \frac{e^{-a\sqrt{s}}}{s^{n+1}} \quad n>-1\ $ | ||
| $ \frac{e^{-bt}-e^{-at}}{t}\ $ | $ \ln \left ( \frac{s+a}{s+b} \right )\ $ | ||
| $ Ci(at)\ $ | $ \frac{\ln [(s^2+a^2)/a^2]}{2s}\ $ | ||
| $ Ei(at)\ $ | $ \frac{\ln [(s+a)/a]}{s}\ $ | ||
| $ \ln t\ $ | $ \begin{array}{lcl} -\frac{(\gamma+\ln s)}{s} \\ \gamma = constant Euler=0.5772156... \end{array} \ $ | ||
| $ \frac{2(\cos {at}-\cos {bt})}{t}\ $ | $ \ln \left ( \frac{s^2+a^2}{s^2+b^2} \right )\ $ | ||
| $ \ln^2 t\ $ | $ \begin{array}{lcl} \frac{{\pi}^2}{6s}+\frac{ \left (\gamma+\ln s \right )^2}{s} \\ \gamma = constant Euler=0.5772156... \end{array} \ $ | ||
| $ \begin{array}{lcl} - \left (\ln t+\gamma \right ) \\ \gamma = constant Euler=0.5772156... \end{array} \ $ | $ \frac{\ln s}{s}\ $ | ||
| $ \begin{array}{lcl} \left ( \ln t+\gamma \right )^2-\frac16{\pi}^2 \\ \gamma = constant Euler=0.5772156... \end{array} \ $ | $ \frac{\ln^2 s}{s}\ $ | ||
| please continue | place formula here | ||
| please continue | place formula here | ||
| please continue | place formula here | ||
| Laplace Transform Pairs | |||||
|---|---|---|---|---|---|
| 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 $ | |||
