Difference between revisions of "Laplace Transforms Table" - Rhea

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Table of (bidirectional) Laplace Transform Pairs and Properties

Laplace Transform Pairs and Properties
Definition
(bidirectional) Laplace Transform	$F(s)=\int_{-\infty}^\infty f(t) e^{-st}dt, \ s\in {\mathbb C} \$
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)} \$
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	$\,\! u(t)$	$\frac{1}{s}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0$
	$\,\! -u(-t)$	$\frac{1}{s}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace < 0$
	$\frac{t^{n-1}}{(n-1)!}u(t)$	$\frac{1}{s^{n}}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0$
	$-\frac{t^{n-1}}{(n-1)!}u(-t)$	$\frac{1}{s^{n}}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace < 0$
	$\,\!e^{-\alpha t}u(t)$	$\frac{1}{s+\alpha}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha$
	$\,\! -e^{-\alpha t}u(-t)$	$\frac{1}{s+\alpha}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace < -\alpha$
	$\frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(t)$	$\frac{1}{(s+\alpha )^{n}}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha$
	$-\frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(-t)$	$\frac{1}{(s+\alpha )^{n}}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace < -\alpha$
	$\,\!\delta (t - T)$	$\,\! e^{-sT}$	$\text{All}\,\, s\in {\mathbb C}$
	$\,\cos( \omega_0 t)u(t)$	$\frac{s}{s^2+\omega_0^{2}}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0$
	$\, \sin( \omega_0 t)u(t)$	$\frac{\omega_0}{s^2+\omega_0^{2}}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0$
	$\,e^{-\alpha t}\cos( \omega_0 t) u(t)$	$\frac{s+\alpha}{(s+\alpha)^{2}+\omega_0^{2}}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha$
	$\, e^{-\alpha t}\sin( \omega_0 t)u(t)$	$\frac{\omega_0}{(s+\alpha)^{2}+\omega_0^{2}}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha$
	$u_n(t) = \frac{d^{n}\delta (t)}{dt^{n}}$	$\,\!s^{n}$	$All\,\, s$
	$u_{-n}(t) = \underbrace{u(t) \dots u(t)}_{n\,\,times}$	$\frac{1}{s^{n}}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0$	$1 \$	$\frac{1}{s} \$
	$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 = \text{Eular constant}=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 = \text{Eular constant}=0.5772156... \end{array} \$
	$\begin{array}{lcl} - \left (\ln t+\gamma \right ) \\ \gamma = \text{Eular constant}=0.5772156... \end{array} \$	$\frac{\ln s}{s}\$
	$\begin{array}{lcl} \left ( \ln t+\gamma \right )^2-\frac16{\pi}^2 \\ \gamma = \text{Eular constant}=0.5772156... \end{array} \$	$\frac{\ln^2 s}{s}\$
	$t^n\ln t\$	$\frac{\Gamma'(n+1)-\Gamma(n+1)\ln s}{s^{n+1}} \quad n>-1\$
	$\frac{\sin {at}}{t}\$	${Arc}\ {tg}\ (a/s)\$
	$Si(at)\$	$\frac{{Arc}\ {tg}\ (a/s)}{s}\$
	$\frac{e^{-2\sqrt{at}}}{\sqrt{\pi t}} \$	$\frac{e^{a/s}}{\sqrt{s}}\ erfc(\sqrt{a/s})\$
	$\frac{2a}{\sqrt{\pi }}e^{-a^2t^2}\$	$e^{s^2/4a^2}\ erfc(s/2a)\$
	$erf(at)\$	$\frac{e^{s^2/4a^2}\ erfc(s/2a)}{s}\$
	$\frac1{\sqrt{\pi (t+a)}}\$	$\frac{e^{as}erfc\sqrt{as}}{\sqrt{s}}\$
	$\frac1{t+a}\$	$e^{as}Ei(as)\$
	$\frac1{t^2+a^2}\$	$\frac1a \left [ \cos {as} \left \{ \frac{\pi }{2}-Si(as) \right \}-\sin {as}\ Ci(as) \right ]\$
	$\frac{t}{t^2+a^2}\$	$\sin {as} \left \{ \frac{\pi }{2}-Si(as) \right \}+\cos {as}\ Ci(as)\$
	${Arc}\ {tg}(t/a)\$	$\frac{\cos {as} \left \{ \frac{\pi }{2}-Si(as) \right \}-\sin {as}\ Ci(as)}{s}\$
	$\frac12\ln \left (\frac{t^2+a^2}{a^2} \right )\$	$\frac{\sin {as} \left \{ \frac{\pi }{2}-Si(as) \right \}+\cos {as}\ Ci(as)}{s}\$
	$\frac1t \ln \left ( \frac{t^2+a^2}{a^2} \right )\$	$\left [ \frac{\pi}{2}-Si(as) \right ]^2 + Ci^2(as)\$
	$\mathcal{N}(t)\ =\ fonction nulle\$	$0\$
	$\delta(t)\ =\ fonction delta\$	$1\$
	$\delta(t-a)\$	$e^{-as}\$
	$\mu(t-a)\$	$\frac{e^{-as}}{s}\$
	$\frac xa+\frac2{\pi} \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \sin {\frac{n \pi x}{a}} \cos {\frac{n\pi t}{a}}\$	$\frac{{sh}\ sx}{s\ {sh}\ sa}\$
	$\frac4{\pi} \sum_{n=1}^{\infty} \frac{(-1)^n}{2n-1} \sin {\frac{(2n-1) \pi x}{2a}} \sin {\frac{(2n-1)\pi t}{2a}}\$	$\frac{{sh}\ sx}{s\ {ch}\ sa}\$
	$\|fracta+\frac2{\pi} \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \cos {\frac{n \pi x}{a}} \sin {\frac{n\pi t}{a}}\$	$\frac{{ch}\ sx}{s\ {sh}\ as}\$
	$1+\frac4{\pi} \sum_{n=1}^{\infty} \frac{(-1)^n}{2n-1} \cos {\frac{n \pi x}{a}} \cos {\frac{(2n-1)\pi t}{2a}}\$	$\frac{{ch}\ sx}{s\ {ch}\ as}\$
	$\frac {xt}a+\frac{2a}{{\pi}^2 } \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2} \sin {\frac{n \pi x}{a}} \sin {\frac{n\pi t}{a}}\$	$\frac{{sh}\ sx}{s^2\ {sh}\ sa}\$
	$x+\frac{8a}{{\pi}^2 } \sum_{n=1}^{\infty} \frac{(-1)^n}{(2n-1)^2} \sin {\frac{(2n-1) \pi x}{2a}} \cos {\frac{(2n-1) \pi t}{2a}}\$	$\frac{{sh}\ sx}{s^2\ {ch}\ sa}\$
	$\frac{t^2}{2a}+\frac{2a}{{\pi}^2 } \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2} \cos {\frac{n \pi x}{a}} \left ( 1-\cos {\frac{n \pi t}{a}} \right )\$	$\frac{{ch}\ sx}{s^2\ {sh}\ sa}\$
	$t+\frac{8a}{{\pi}^2 } \sum_{n=1}^{\infty} \frac{(-1)^n}{(2n-1)^2} \cos {\frac{(2n-1) \pi x}{2a}} \sin {\frac{(2n-1) \pi t}{2a}}\$	$\frac{{ch}\ sx}{s^3\ {sh}\ sa}\$
	$\frac12(t^2+x^2-a^2)-\frac{16a^2}{{\pi}^3 } \sum_{n=1}^{\infty} \frac{(-1)^n}{(2n-1)^3} \cos {\frac{(2n-1) \pi x}{2a}} \cos {\frac{(2n-1) \pi t}{2a}}\$	$\frac{{ch}\ sx}{s^3\ {ch}\ sa}\$
	$\frac{2 \pi}{a^2} \sum_{n=1}^{\infty} (-1)^nne^{-(2n-1)^2{\pi}^2t/4a^2}\sin {\frac{n \pi x}{a}}\$	$\frac{{ch}\ x\sqrt{s}}{{sh}\ a\sqrt{s}}\$
	$\frac{2 \pi}{a^2} \sum_{n=1}^{\infty} (-1)^nne^{-(2n-1)^2{\pi}^2t/4a^2}\sin {\frac{n \pi x}{a}}\$	$\frac{{ch}\ x\sqrt{s}}{{sh}\ a\sqrt{s}}\$
	$\frac{2 \pi}{a^2} \sum_{n=1}^{\infty} (-1)^nne^{-(2n-1)^2{\pi}^2t/4a^2}\sin {\frac{n \pi x}{a}}\$	$\frac{{sh}\ x\sqrt{s}}{{sh}\ a\sqrt{s}}\$
	$\frac{\pi}{a^2} \sum_{n=1}^{\infty} (-1)^{n-1}(2n-1)e^{-(2n-1)^2{\pi}^2t/4a^2}\cos {\frac{(2n-1) \pi x}{2a}}\$	$\frac{{ch}\ x\sqrt{s}}{{ch}\ a\sqrt{s}}\$
	$\frac{2}{a} \sum_{n=1}^{\infty} (-1)^{n-1}e^{-(2n-1)^2{\pi}^2t/4a^2}\sin {\frac{(2n-1) \pi x}{2a}}\$	$\frac{{sh}\ x\sqrt{s}}{\sqrt{s}{ch}\ a\sqrt{s}}\$
	$\frac1a+\frac2a\sum_{n=1}^{\infty} (-1)^ne^{-n^2{\pi}^2t/a^2}\cos {\frac{n \pi x}{a}}\$	$\frac{{ch}\ x\sqrt{s}}{\sqrt{s}{sh}\ a\sqrt{s}}\$
	$\frac{x}{a}+\frac{2}{\pi} \sum_{n=1}^{\infty} \frac{(-1)^n}{n}e^{-n^2{\pi}^2t/a^2} \sin {\frac{n \pi x}{a}}\$	$\frac{{sh}\ x\sqrt{s}}{s{sh}\ a\sqrt{s}}\$
	$1+\frac4{\pi}\sum_{n=1}^{\infty} \frac{(-1)^n}{2n-1}e^{-(2n-1)^2{\pi}^2t/a^2}\cos {\frac{(2n-1) \pi x}{2a}}\$	$\frac{{ch}\ x\sqrt{s}}{s{ch}\ a\sqrt{s}}\$
	$\frac{xt}{a}+\frac{2a^2}{{\pi}^3}\sum_{n=1}^{\infty} \frac{(-1)^n}{n^3}(1-e^{-n^2{\pi}^2t/a^2})\sin {\frac{n \pi x}{a}}\$	$\frac{{sh}\ x\sqrt{s}}{s^2{sh}\ a\sqrt{s}}\$
	$\frac12(x^2+a^2)+t-\frac{16a^2}{{\pi}^3}\sum_{n=1}^{\infty} \frac{(-1)^n}{(2n-1)^3}e^{-{(2n-1)}^2{\pi}^2t/a^2}\cos {\frac{(2n-1) \pi x}{2a}}\$	$\frac{{ch}\ x\sqrt{s}}{s^2{ch}\ a\sqrt{s}}\$

Go to Relevant Course Page: ECE 301

Go to Relevant Course Page: ECE 438

Go to Relevant Course Page: ECE 538

Back to Collective Table

@@ Line 1: / Line 1: @@
-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
+[[Category:Formulas]]
+[[Category:Laplace transform]]
+[[Category:ECE301]]
+[[Category:ECE438]]
+'''THIS PAGE IS UNDER CONSTRUCTION. PLEASE CHECK BACK LATER.'''
+<center><font size= 4>
+'''[[Collective_Table_of_Formulas|Collective Table of Formulas]]'''
+</font size>
+Table of (bidirectional) [[Laplace_transform|Laplace Transform]] Pairs and Properties
+(used in [[ECE301]] and [[ECE438]])
+</center>
+----
 {|
 |-
-! style="background: rgb(228, 188, 126) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | Laplace Transform Pairs and Properties
+! style="background: rgb(228, 188, 126) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | [[Laplace_transform|Laplace Transform]] Pairs and Properties
 |-
 ! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" | Definition
 |-
-| align="right" style="padding-right: 1em;" | Laplace Transform
+| align="right" style="padding-right: 1em;" | (bidirectional) [[Laplace_transform|Laplace Transform]]
 |<math>F(s)=\int_{-\infty}^\infty f(t) e^{-st}dt, \ s\in {\mathbb C} \ </math>
-|-
-| align="right" style="padding-right: 1em;" | Inverse Laplace Transform
-| add formula here
 |-
 ! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" | Properties of the Laplace Transform
+|-
+| style="padding-right: 1em;" |
+| Function
+| Laplace Transform
+| ROC
 |-
 | align="right" style="padding-right: 1em;" |
-|function <math> F(s) \ </math>
+|<math>f(t) \ </math>
-| Laplace transform <math>f(t) \ </math>
+| <math>F(s) \ </math>
 | ROC <math>R</math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> aF_1(s)+bF_2(s) \ </math>
 | <math> af_1(t)+bf_2(t) \ </math>
+| <math> aF_1(s)+bF_2(s) \ </math>
 | at least <math>R_1 \cap R_2 </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> F\left( \frac{s}{a} \right) </math>
 | <math> af(at) \ </math>
+|<math> F\left( \frac{s}{a} \right) </math>
 |-
 | align="right" style="padding-right: 1em;" |
+| <math> e^{at}f(t) \ </math>
 | <math> F(s-a) \ </math>
-|<math> e^{at}f(t) \ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> e^{-as}F(s) \ </math>
 | <math> u(t-a) =
 \begin{cases}
 f(t-a)  & t>a \\
 & t<a
 \end{cases} </math>
+| <math> e^{-as}F(s) \ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> sF(s)-f(0) \ </math>
 | <math> f'(t) \ </math>
+| <math> sF(s)-f(0) \ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> s^2F(s)-sf(0)-f'(0) \ </math>
 | <math> f''(t) \ </math>
+| <math> s^2F(s)-sf(0)-f'(0) \ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> s^{n}F(s)-\sum_{k=1}^ns^{n-k}f^{(k)}(0) \ </math>
 | <math> f^{(n)}(t) \ </math>
+| <math> s^{n}F(s)-\sum_{k=1}^ns^{n-k}f^{(k)}(0) \ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> F'(s) \ </math>
 | <math> -tf(t) \ </math>
+| <math> F'(s) \ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math>F''(s) \ </math>
 | <math> t^2f(t) \ </math>
+| <math>F''(s) \ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> F^{(n)}(s) \ </math>
 | <math> (-1)^{(ntn)}f(t) \ </math>
+| <math> F^{(n)}(s) \ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{F(s)}s \ </math>
 | <math> \int_{0}^{t} f(u) du \ </math>
+| <math> \frac{F(s)}s \ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <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> \frac{F(s)}{s^n} \ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math>F(s)G(s) \ </math>
 | <math> \int_{0}^{t}f(u)g(t-u)du \ </math>
+| <math>F(s)G(s) \ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \int_{s}^{\infty}F(u)du \ </math>
 | <math> \frac{f(t)}t \ </math>
+| <math> \int_{s}^{\infty}F(u)du \ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac1{1-e^{-sT}}\int_{0}^{T}e^{-su}f(u)du \ </math>
 | <math> f(t)=f(t+T) \ </math>
+| <math> \frac1{1-e^{-sT}}\int_{0}^{T}e^{-su}f(u)du \ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <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{F(\sqrt{s})}s \ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac1sF\left(\frac1s\right) \ </math>
 | <math> \int_{0}^{\infty}J_0(2\sqrt{ut})f(u)du \ </math>
+| <math> \frac1sF\left(\frac1s\right) \ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac1{s^{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> \frac1{s^{n+1}}F\left(\frac1s\right) \ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <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> \frac{F(s+\frac1s)}{s^2+1} \ </math>
 |-
 | 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> f(t^2) \ </math>
+| <math> \frac1{2\sqrt\pi}\int_{0}^{\infty}u^{-\frac32}e^{-\frac{s^2}{4u}}F(u)du \ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{F(\ln s)}{s\ln s} \ </math>
 | <math> \int_{0}^{\infty}\frac{t^uf(u)}{\Gamma(u+1)}du \ </math>
+| <math> \frac{F(\ln s)}{s\ln s} \ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{P(s)}{Q(s)} \ </math>
 | <math> \sum_{k=1}^N \frac{P(\alpha_k)}{Q'(\alpha_k)}e^{\alpha_kt} \ </math>
+| <math> \frac{P(s)}{Q(s)} \ </math>
 |-
+| align="right" style="padding-right: 1em;" | please continue
+| place formula here
+|-<math> af_1(s)+bf_2(s)\ </math>
+|-
+|}
+{|
+! colspan="4" style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" | Laplace Transform Pairs
+!
+!
+|-
+| style="padding-right: 1em;" |
+| Signal
+| Laplace Transform
+| ROC
+|-
+| align="left" style="padding-right: 1em;" | unit impulse/Dirac delta
+| <math>\,\!\delta(t)</math>
+| <span class="texhtml">1</span>
+| <math>\text{All}\, s \in {\mathbb C}</math>
+|-
+| align="right" style="padding-right: 1em;" | unit step function
+| <math>\,\! u(t)</math>
+| <math>\frac{1}{s}</math>
+| <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math>\,\! -u(-t)</math>
+| <math>\frac{1}{s}</math>
+| <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace < 0 </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math>\frac{t^{n-1}}{(n-1)!}u(t)</math>
+| <math>\frac{1}{s^{n}}</math>
+| <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math>-\frac{t^{n-1}}{(n-1)!}u(-t)</math>
+| <math>\frac{1}{s^{n}}</math>
+| <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace  < 0 </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math>\,\!e^{-\alpha t}u(t)</math>
+| <math>\frac{1}{s+\alpha}</math>
+| <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math>\,\! -e^{-\alpha t}u(-t)</math>
+| <math>\frac{1}{s+\alpha}</math>
+| <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace  < -\alpha </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math>\frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(t)</math>
+| <math>\frac{1}{(s+\alpha )^{n}}</math>
+| <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math>-\frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(-t)</math>
+| <math>\frac{1}{(s+\alpha )^{n}}</math>
+| <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace < -\alpha </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math>\,\!\delta (t - T)</math>
+| <math>\,\! e^{-sT}</math>
+| <math>\text{All}\,\, s\in {\mathbb C}</math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math>\,\cos( \omega_0 t)u(t)</math>
+| <math>\frac{s}{s^2+\omega_0^{2}}</math>
+| <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math>\, \sin( \omega_0 t)u(t)</math>
+| <math>\frac{\omega_0}{s^2+\omega_0^{2}}</math>
+| <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math>\,e^{-\alpha t}\cos( \omega_0 t) u(t)</math>
+| <math>\frac{s+\alpha}{(s+\alpha)^{2}+\omega_0^{2}}</math>
+| <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math>\, e^{-\alpha t}\sin( \omega_0 t)u(t)</math>
+| <math>\frac{\omega_0}{(s+\alpha)^{2}+\omega_0^{2}}</math>
+| <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math>u_n(t) = \frac{d^{n}\delta (t)}{dt^{n}}</math>
+| <math>\,\!s^{n}</math>
+| <math>All\,\, s</math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <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;" |
-| <math> \frac1s \ </math>
 | <math> 1 \ </math>
+| <math> \frac{1}{s} \ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac1{s^2} \ </math>
 | <math> t \ </math>
+| <math> \frac1{s^2} \ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac1{s^n}, \    n=1,2,3,... \ </math>
 | <math> \frac{t^{n-1}}{(n-1)!},  \ 0!=1 \ </math>
+| <math> \frac1{s^n}, \    n=1,2,3,... \ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac1{s^n}, \ n>0 \ </math>
 | <math> \frac{t^{n-1}}{\Gamma(n)} \ </math>
+| <math> \frac1{s^n}, \ n>0 \ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac1{s-a}\ </math>
 | <math> e^{at}\ </math>
+| <math> \frac1{s-a}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac1{(s-a)^n}, \ n=1,2,3,...\ </math>
 | <math> \frac{t^{n-1}e^{at}}{(n-1)!}, \ 0!=1\ </math>
+| <math> \frac1{(s-a)^n}, \ n=1,2,3,...\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac1{(s-a)^n}, \ n>0\ </math>
 | <math> \frac{t^{n-1}e^{at}}{\Gamma(n)}\ </math>
+| <math> \frac1{(s-a)^n}, \ n>0\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac1{s^2+a^2}\ </math>
 | <math> \frac{\sin {at}}{a} \ </math>
+| <math> \frac1{s^2+a^2}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{s}{s^2+a^2} \ </math>
 | <math> \cos {at} \ </math>
+| <math> \frac{s}{s^2+a^2} \ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac1{(s-b)^2+a^2}\ </math>
 | <math> \frac{e^{bt}\sin{at}}{a} \ </math>
+| <math> \frac1{(s-b)^2+a^2}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{s-b}{(s-b)^2+a^2}\ </math>
 | <math> e^{bt}\cos{at}\ </math>
+| <math> \frac{s-b}{(s-b)^2+a^2}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{1}{s^2-a^2} \ </math>
 | <math> \left(\frac{{sh}\ {at}}{a}\right)\ </math>
+| <math> \frac{1}{s^2-a^2} \ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{s}{s^2-a^2}\ </math>
 | <math> {ch}\ {at}\ </math>
+| <math> \frac{s}{s^2-a^2}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac1{(s-b)^2-a^2}\ </math>
 | <math> \frac{e^{bt}{sh}\ {at}}a\ </math>
+| <math> \frac1{(s-b)^2-a^2}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{s-b}{(s-b)^2-a^2} \ </math>
 | <math> e^{bt} {ch}\ {at}\ </math>
+| <math> \frac{s-b}{(s-b)^2-a^2} \ </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>
+| <math> \frac1{(s-a)(s-b)},\ a \ne b\ </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>
+| <math> \frac{s}{(s-a)(s-b)},\ a \ne b \ </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>
+| <math> \frac1{(s^2+a^2)^2}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{s}{(s^2+a^2)^2}\ </math>
 | <math> \frac{t\sin {at}}{2a}\ </math>
+| <math> \frac{s}{(s^2+a^2)^2}\ </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>
+| <math> \frac{s^2}{(s^2+a^2)^2}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{s^3}{(s^2+a^2)^2}\ </math>
 | <math> \cos {at}-\frac12at\sin {at}\ </math>
+| <math> \frac{s^3}{(s^2+a^2)^2}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{s^2-a^2}{(s^2+a^2)^2}\ </math>
 | <math> t\cos {at}\ </math>
+| <math> \frac{s^2-a^2}{(s^2+a^2)^2}\ </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>
+| <math> \frac{1}{(s^2-a^2)^2}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{s}{(s^2-a^2)^2}\ </math>
 | <math> \frac{t\ {sh}\ {at}}{2a}\ </math>
+| <math> \frac{s}{(s^2-a^2)^2}\ </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>
+| <math> \frac{s^2}{(s^2-a^2)^2}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{s^3}{(s^2-a^2)^2}\ </math>
 | <math> {ch}\ {at}+\frac12at\ {sh}\ {at} \ </math>
+| <math> \frac{s^3}{(s^2-a^2)^2}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{s^2+a^2}{(s^2-a^2)^2}\ </math>
 | <math> t\ {ch}\ {at}\ </math>
+| <math> \frac{s^2+a^2}{(s^2-a^2)^2}\ </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>
+| <math> \frac{1}{(s^2+a^2)^3}\ </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>
+| <math> \frac{s}{(s^2+a^2)^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>
+| <math> \frac{s^2}{(s^2+a^2)^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}}{8a}\ </math>
+| <math> \frac{s^3}{(s^2+a^2)^3}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{s^4}{(s^2+a^2)^3}\ </math>
 | <math> \frac{(3-a^2t^2)\sin {at}+5at\cos {at}}{8a}\ </math>
+| <math> \frac{s^4}{(s^2+a^2)^3}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{s^5}{(s^2+a^2)^3}\ </math>
 | <math> \frac{(8-a^2t^2)\cos {at}-7at\sin {at}}{8}\ </math>
+| <math> \frac{s^5}{(s^2+a^2)^3}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{3s^2-a^2}{(s^2+a^2)^3}\ </math>
 | <math> \frac{t^2\sin {at}}{2a}\ </math>
+| <math> \frac{3s^2-a^2}{(s^2+a^2)^3}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{s^3-3a^2s}{(s^2+a^2)^3}\ </math>
 | <math> \frac12t^2\cos {at}\ </math>
+| <math> \frac{s^3-3a^2s}{(s^2+a^2)^3}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{s^4-6a^2s^2+a^4}{(s^2+a^2)^4}\ </math>
 | <math> \frac16t^3\cos {at}\ </math>
+| <math> \frac{s^4-6a^2s^2+a^4}{(s^2+a^2)^4}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{s^3-a^2s}{(s^2+a^2)^4}\ </math>
 | <math> \frac{t^3\sin {at}}{24a}\ </math>
+| <math> \frac{s^3-a^2s}{(s^2+a^2)^4}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{1}{(s^2-a^2)^3}\ </math>
 | <math> \frac{3+a^2t^2\ {sh}\ {at}-3at\ {ch}\ {at}}{8a^5}\ </math>
+| <math> \frac{1}{(s^2-a^2)^3}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{s}{(s^2-a^2)^3}\ </math>
 | <math> \frac{at^2\ {ch}\ {at}-t\ {sh}\ {at}}{8a^3}\ </math>
+| <math> \frac{s}{(s^2-a^2)^3}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{s^2}{(s^2-a^2)^3}\ </math>
 | <math> \frac{at\ {ch}\ {at}+(a^2t^2-1)\ {sh}\ {at}}{8a^3}\ </math>
+| <math> \frac{s^2}{(s^2-a^2)^3}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{s^3}{(s^2-a^2)^3}\ </math>
 | <math> \frac{3t\ {sh}\ {at}+at^2\ {ch}\ {at}}{8a}\ </math>
+| <math> \frac{s^3}{(s^2-a^2)^3}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{s^4}{(s^2-a^2)^3}\ </math>
 | <math> \frac{(3+a^2t^2)\ {sh}\ {at}+5at\ {ch}\ {at}}{8a}\ </math>
+| <math> \frac{s^4}{(s^2-a^2)^3}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{s^5}{(s^2-a^2)^3}\ </math>
 | <math> \frac{(8+a^2t^2)\ {ch}\ {at}+7at\ {sh}\ {at}}{8}\ </math>
+| <math> \frac{s^5}{(s^2-a^2)^3}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{3s^2+a^2}{(s^2-a^2)^3}\ </math>
 | <math> \frac{t^2\ {sh}\ {at}}{2a}\ </math>
+| <math> \frac{3s^2+a^2}{(s^2-a^2)^3}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{s^3+3a^2s}{(s^2-a^2)^3}\ </math>
 | <math> \frac12t^2\ {ch}\ {at}\ </math>
+| <math> \frac{s^3+3a^2s}{(s^2-a^2)^3}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{s^4+6a^2s^2+a^4}{(s^2-a^2)^4}\ </math>
 | <math> \frac16t^3\ {ch}\ {at}\ </math>
+| <math> \frac{s^4+6a^2s^2+a^4}{(s^2-a^2)^4}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{s^3+a^2s}{(s^2-a^2)^4}\ </math>
 | <math> \frac{t^3\ {sh}\ {at}}{24a}\ </math>
+| <math> \frac{s^3+a^2s}{(s^2-a^2)^4}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{1}{s^3+a^3}\ </math>
 | <math> \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 \}\ </math>
+| <math> \frac{1}{s^3+a^3}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{s}{s^3+a^3}\ </math>
 | <math> \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 \}\ </math>
+| <math> \frac{s}{s^3+a^3}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{s^2}{s^3+a^3}\ </math>
 | <math> \frac13 \left \{ e^{-at}+ 2e^{at/2} \cos {\frac{\sqrt{3}at}{2}} \right \}\ </math>
+| <math> \frac{s^2}{s^3+a^3}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{1}{s^3-a^3}\ </math>
 | <math> \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 \}\ </math>
+| <math> \frac{1}{s^3-a^3}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{s}{s^3-a^3}\ </math>
 | <math> \frac{e^{-at/2}}{3a} \left \{ \sqrt{3}\sin {\frac{\sqrt{3}at}{2}}-\cos {\frac{\sqrt{3}at}{2}}+e^{3at/2} \right \}\ </math>
+| <math> \frac{s}{s^3-a^3}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{s^2}{s^3-a^3}\ </math>
 | <math> \frac13 \left \{ e^{at}+ 2e^{-at/2} \cos {\frac{\sqrt{3}at}{2}} \right \}\ </math>
+| <math> \frac{s^2}{s^3-a^3}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{1}{s^4+4a^4}\ </math>
 | <math> \frac1{4a^3} \left (\sin {at}\ {ch}\ {at}-\cos {at}\ {sh}\ {at} \right )\ </math>
+| <math> \frac{1}{s^4+4a^4}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
+| <math> \frac{\sin {at}\ {sh}\ {at}}{2a^2}\ </math>
 | <math> \frac{s}{s^4+4a^4}\ </math>
-| <math> \frac{\sin {at}\ {sh}\ {at}}{2a^2}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
+| <math> \frac1{2a} \left (\sin {at}\ {ch}\ {at}+\cos {at}\ {sh}\ {at} \right )\ </math>
 | <math> \frac{s^2}{s^4+4a^4}\ </math>
-| <math> \frac1{2a} \left (\sin {at}\ {ch}\ {at}+\cos {at}\ {sh}\ {at} \right )\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{s^3}{s^4+4a^4}\ </math>
 | <math> \cos {at}\ {ch}\ {at}\ </math>
+| <math> \frac{s^3}{s^4+4a^4}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{1}{s^4-a^4}\ </math>
 | <math> \frac1{2a^3} \left (\ {sh}\ {at}-\sin {at} \right )\ </math>
+| <math> \frac{1}{s^4-a^4}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
+| <math> \frac1{2a^2} \left (\ {ch}\ {at}-\cos {at} \right )\ </math>
 | <math> \frac{s}{s^4-a^4}\ </math>
-| <math> \frac1{2a^2} \left (\ {ch}\ {at}-\cos {at} \right )\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
+| <math> \frac1{2a} \left (\ {sh}\ {at}+\sin {at} \right )\ </math>
 | <math> \frac{s^2}{s^4-a^4}\ </math>
-| <math> \frac1{2a} \left (\ {sh}\ {at}+\sin {at} \right )\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{s^3}{s^4-a^4}\ </math>
 | <math> \frac12 \left (\ {ch}\ {at}+\cos {at} \right )\ </math>
+| <math> \frac{s^3}{s^4-a^4}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac1{\sqrt{s+a}+\sqrt{s+b}}\ </math>
 | <math> \frac{e^{-bt}-e^{-at}}{2(b-a)\sqrt{\pi t^3}}\ </math>
+| <math> \frac1{\sqrt{s+a}+\sqrt{s+b}}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac1{s\sqrt{s+a}}\ </math>
 | <math> \frac{erf\ \sqrt{at}}{\sqrt{a}}\ </math>
+| <math> \frac1{s\sqrt{s+a}}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac1{\sqrt{s}(s-a)}\ </math>
 | <math> \frac{e^{at}\ {erf}\ \sqrt{at}}{\sqrt{a}}\ </math>
+| <math> \frac1{\sqrt{s}(s-a)}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac1{\sqrt{s-a}+b}\ </math>
 | <math> e^{at} \left \{\frac1{\sqrt{\pi t}}-be^{b^{2}t}\ {erfc}\ (b\sqrt{t}) \right \}\ </math>
+| <math> \frac1{\sqrt{s-a}+b}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac1{\sqrt{s^2+a^2}}\ </math>
 | <math> J_0(at)\ </math>
+| <math> \frac1{\sqrt{s^2+a^2}}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac1{\sqrt{s^2-a^2}}\ </math>
 | <math> I_0(at)\ </math>
+| <math> \frac1{\sqrt{s^2-a^2}}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
+| <math> a^nJ_n(at)\ </math>
 | <math> \frac{{\left (\sqrt{s^2+a^2}-s \right )}^n}{\sqrt{s^2+a^2}},\quad n>-1 \ </math>
-| <math> a^nJ_n(at)\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{{\left (s- \sqrt{s^2-a^2} \right )}^n}{\sqrt{s^2-a^2}},\quad n>-1 \ </math>
 | <math> a^nI_n(at)\ </math>
+| <math> \frac{{\left (s- \sqrt{s^2-a^2} \right )}^n}{\sqrt{s^2-a^2}},\quad n>-1 \ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{e^{-b\sqrt{s^2+a^2}}}{}} \ </math>
 | <math> J_0(a\sqrt{t(t+2b)})\ </math>
+| <math> \frac{e^{b \left (s- \sqrt{s^2+a^2} \right )}}{\sqrt{s^2+a^2}} \ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{e^{b \left (s- \sqrt{s^2+a^2} \right )}}{\sqrt{s^2+a^2}} \ </math>
+| <math> \begin{cases}
-| <math> J_0(a\sqrt{t(t+2b)})\ </math>
+J_0(a\sqrt{t^2-b^2})   & t>b \\
+                      &t<b
+\end{cases}
+\ </math>
+| <math> \frac{e^{-b\sqrt{s^2+a^2}}}{\sqrt{s^2+a^2}} \ </math>
 |-
-| align="right" style="padding-right: 1em;" | please continue
+| align="right" style="padding-right: 1em;" |
-| place formula here
+| <math> tJ_0(at)\ </math>
-|-<math> af_1(s)+bf_2(s)\ </math>
+| <math> \frac1{(s^2+a^2)^{3/2}}\ </math>
 |-
-| align="right" style="padding-right: 1em;" | please continue
+| align="right" style="padding-right: 1em;" |
-| place formula here
+| <math> J_0(at)-atJ_1(at)\ </math>
-|-<math> af_1(s)+bf_2(s)\ </math>
+| <math> \frac{s^2}{(s^2+a^2)^{3/2}}\ </math>
 |-
-| align="right" style="padding-right: 1em;" | please continue
+| align="right" style="padding-right: 1em;" |
-| place formula here
+| <math> \frac{tI_1(at)}{a}\ </math>
-|-<math> af_1(s)+bf_2(s)\ </math>
+| <math> \frac1{(s^2-a^2)^{3/2}}\ </math>
 |-
-| align="right" style="padding-right: 1em;" | please continue
+| align="right" style="padding-right: 1em;" |
-| place formula here
+| <math> I_0(at)+atI_1(at)\ </math>
-|-<math> af_1(s)+bf_2(s)\ </math>
+| <math> \frac{s}{(s^2+a^2)^{3/2}}\ </math>
 |-
-| align="right" style="padding-right: 1em;" | please continue
+| align="right" style="padding-right: 1em;" |
-| place formula here
+| <math> f(t)=n,\ n \leqq t\ <n+1,\ n=0,1,2,... \ </math>
-|-<math> af_1(s)+bf_2(s)\ </math>
+| <math> \frac1{s(e^s-1)}\  =\  \frac{e^{-s}}{s(1-e^{-s})}\ </math>
 |-
-| align="right" style="padding-right: 1em;" | please continue
+| align="right" style="padding-right: 1em;" |
-| place formula here
+| <math> f(t)= \sum_{k=1}^{[t]} r^k\ </math>
-|-<math> af_1(s)+bf_2(s)\ </math>
+| <math> \frac1{s(e^s-r)}\  =\  \frac{e^{-s}}{s(1-re^{-s})}\ </math>
 |-
-| align="right" style="padding-right: 1em;" | please continue
+| align="right" style="padding-right: 1em;" |
-| place formula here
+| <math> f(t)= r^n,\ n\leqq t<n+1,\ n=0,1,2,...\ </math>
-|-<math> af_1(s)+bf_2(s)\ </math>
+| <math> \frac{s^s-1}{s(e^s-r)}\  =\  \frac{1-e^{-s}}{s(1-re^{-s})}\ </math>
 |-
-| align="right" style="padding-right: 1em;" | please continue
+| align="right" style="padding-right: 1em;" |
-| place formula here
+| <math> \frac{\cos {2\sqrt{at}}}{\sqrt{ \pi t}}\ </math>
-|-<math> af_1(s)+bf_2(s)\ </math>
+| <math> \frac{s^{-a/s}}{\sqrt{s}}\ </math>
 |-
-| align="right" style="padding-right: 1em;" | please continue
+| align="right" style="padding-right: 1em;" |
-| place formula here
+| <math> \frac{\sin {2\sqrt{at}}}{\sqrt{ \pi a}}\ </math>
-|-<math> af_1(s)+bf_2(s)\ </math>
+| <math> \frac{e^{-a/s}}{s^{3/2}}\ </math>
 |-
-| align="right" style="padding-right: 1em;" | please continue
+| align="right" style="padding-right: 1em;" |
-| place formula here
+| <math> \left ( \frac{t}{a} \right )^{n/2}J_n(2\sqrt{at})\ </math>
-|-<math> af_1(s)+bf_2(s)\ </math>
+| <math> \frac{e^{-a/s}}{s^n+1} \quad    n>-1 \ </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>
-|-
-|}
-{|
-! colspan="4" style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" | Laplace Transform Pairs
-!
-!
-|-
-| style="padding-right: 1em;" | notes
-| Signal
-| Laplace Transform
-| ROC
-|-
-| align="left" style="padding-right: 1em;" | unit impulse/Dirac delta
-| <math>\,\!\delta(t)</math>
-| <span class="texhtml">1</span>
-| <math>\text{All}\, s \in {\mathbb C}</math>
-|-
-| align="right" style="padding-right: 1em;" | unit step function
-| <math>\,\! u(t)</math>
-| <math>\frac{1}{s}</math>
-| <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math>\,\! -u(-t)</math>
+| <math> \frac{e^{-a^2/4t}}{\sqrt{ \pi t}}\ </math>
-| <math>\frac{1}{s}</math>
+| <math> \frac{e^{-a\sqrt{s}}}{\sqrt{s}}\ </math>
-| <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace < 0 </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math>\frac{t^{n-1}}{(n-1)!}u(t)</math>
+| <math> \frac{a}{2\sqrt{ \pi t^3}}e^{-a^2/4t}\ </math>
-| <math>\frac{1}{s^{n}}</math>
+| <math> e^{-a\sqrt{s}}\ </math>
-| <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math>-\frac{t^{n-1}}{(n-1)!}u(-t)</math>
+| <math> erf(a/2\sqrt{t})\ </math>
-| <math>\frac{1}{s^{n}}</math>
+| <math> \frac{1-e^{-a\sqrt{s}}}{s}\ </math>
-| <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace  < 0 </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math>\,\!e^{-\alpha t}u(t)</math>
+| <math> erfc(a/2\sqrt{t})\ </math>
-| <math>\frac{1}{s+\alpha}</math>
+| <math> \frac{e^{-a\sqrt{s}}}{s}\ </math>
-| <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math>\,\! -e^{-\alpha t}u(-t)</math>
+| <math> e^{b(bt+a)}erfc \left ( b\sqrt{t}+\frac{a}{2\sqrt{t}} \right )\ </math>
-| <math>\frac{1}{s+\alpha}</math>
+| <math> \frac{e^{-a\sqrt{s}}}{\sqrt{s}(\sqrt{s}+b)}\ </math>
-| <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace  < -\alpha </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math>\frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(t)</math>
+| <math> \frac1{\sqrt{\pi t}a^{2n+1}}\int_{0}^{\infty}u^ne^{-u^2/4a^2t}J_{2n}(2\sqrt{u})du \ </math>
-| <math>\frac{1}{(s+\alpha )^{n}}</math>
+| <math> \frac{e^{-a\sqrt{s}}}{s^{n+1}} \quad n>-1\ </math>
-| <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math>-\frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(-t)</math>
+| <math> \frac{e^{-bt}-e^{-at}}{t}\ </math>
-| <math>\frac{1}{(s+\alpha )^{n}}</math>
+| <math> \ln \left ( \frac{s+a}{s+b} \right )\ </math>
-| <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace < -\alpha </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math>\,\!\delta (t - T)</math>
+| <math> Ci(at)\ </math>
-| <math>\,\! e^{-sT}</math>
+| <math> \frac{\ln [(s^2+a^2)/a^2]}{2s}\ </math>
-| <math>\text{All}\,\, s\in {\mathbb C}</math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math>\,\cos( \omega_0 t)u(t)</math>
+| <math> Ei(at)\ </math>
-| <math>\frac{s}{s^2+\omega_0^{2}}</math>
+| <math> \frac{\ln [(s+a)/a]}{s}\ </math>
-| <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math>\, \sin( \omega_0 t)u(t)</math>
+| <math> \ln t\ </math>
-| <math>\frac{\omega_0}{s^2+\omega_0^{2}}</math>
+| <math> \begin{array}{lcl} -\frac{(\gamma+\ln s)}{s} \\
-| <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 </math>
+\gamma = \text{Eular constant}=0.5772156...
+\end{array} \ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math>\,e^{-\alpha t}\cos( \omega_0 t) u(t)</math>
+| <math> \frac{2(\cos {at}-\cos {bt})}{t}\ </math>
-| <math>\frac{s+\alpha}{(s+\alpha)^{2}+\omega_0^{2}}</math>
+| <math> \ln \left ( \frac{s^2+a^2}{s^2+b^2} \right )\ </math>
-| <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math>\, e^{-\alpha t}\sin( \omega_0 t)u(t)</math>
+| <math> \ln^2 t\ </math>
-| <math>\frac{\omega_0}{(s+\alpha)^{2}+\omega_0^{2}}</math>
+| <math> \begin{array}{lcl} \frac{{\pi}^2}{6s}+\frac{ \left (\gamma+\ln s \right )^2}{s} \\
-| <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha </math>
+\gamma = \text{Eular constant}=0.5772156...
+\end{array} \ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math>u_n(t) = \frac{d^{n}\delta (t)}{dt^{n}}</math>
+| <math> \begin{array}{lcl} - \left (\ln t+\gamma \right ) \\
-| <math>\,\!s^{n}</math>
+\gamma = \text{Eular constant}=0.5772156...
-| <math>All\,\, s</math>
+\end{array} \ </math>
+| <math> \frac{\ln s}{s}\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math>u_{-n}(t) = \underbrace{u(t) *\dots * u(t)}_{n\,\,times}</math>
+| <math> \begin{array}{lcl} \left ( \ln t+\gamma \right )^2-\frac16{\pi}^2 \\
-| <math>\frac{1}{s^{n}}</math>
+\gamma = \text{Eular constant}=0.5772156...
-| <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace  > 0 </math>
+\end{array} \ </math>
-|}
+| <math> \frac{\ln^2 s}{s}\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> t^n\ln t\ </math>
+| <math> \frac{\Gamma'(n+1)-\Gamma(n+1)\ln s}{s^{n+1}} \quad n>-1\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \frac{\sin {at}}{t}\ </math>
+| <math> {Arc}\ {tg}\ (a/s)\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> Si(at)\ </math>
+| <math> \frac{{Arc}\ {tg}\ (a/s)}{s}\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \frac{e^{-2\sqrt{at}}}{\sqrt{\pi t}} \ </math>
+| <math> \frac{e^{a/s}}{\sqrt{s}}\ erfc(\sqrt{a/s})\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \frac{2a}{\sqrt{\pi }}e^{-a^2t^2}\ </math>
+| <math> e^{s^2/4a^2}\ erfc(s/2a)\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> erf(at)\ </math>
+| <math> \frac{e^{s^2/4a^2}\ erfc(s/2a)}{s}\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \frac1{\sqrt{\pi (t+a)}}\ </math>
+| <math> \frac{e^{as}erfc\sqrt{as}}{\sqrt{s}}\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \frac1{t+a}\ </math>
+| <math> e^{as}Ei(as)\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \frac1{t^2+a^2}\ </math>
+| <math> \frac1a \left [ \cos {as} \left \{ \frac{\pi }{2}-Si(as) \right \}-\sin {as}\ Ci(as) \right ]\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \frac{t}{t^2+a^2}\ </math>
+| <math>  \sin {as} \left \{ \frac{\pi }{2}-Si(as) \right \}+\cos {as}\ Ci(as)\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> {Arc}\ {tg}(t/a)\ </math>
+| <math> \frac{\cos {as} \left \{ \frac{\pi }{2}-Si(as) \right \}-\sin {as}\ Ci(as)}{s}\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \frac12\ln \left (\frac{t^2+a^2}{a^2} \right )\ </math>
+| <math> \frac{\sin {as} \left \{ \frac{\pi }{2}-Si(as) \right \}+\cos {as}\ Ci(as)}{s}\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \frac1t \ln \left ( \frac{t^2+a^2}{a^2} \right )\ </math>
+| <math> \left [ \frac{\pi}{2}-Si(as) \right ]^2 + Ci^2(as)\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \mathcal{N}(t)\ =\ fonction nulle\ </math>
+| <math> 0\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \delta(t)\ =\ fonction delta\ </math>
+| <math> 1\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \delta(t-a)\ </math>
+| <math> e^{-as}\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \mu(t-a)\ </math>
+| <math> \frac{e^{-as}}{s}\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \frac xa+\frac2{\pi} \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \sin {\frac{n \pi  x}{a}} \cos {\frac{n\pi t}{a}}\ </math>
+| <math> \frac{{sh}\ sx}{s\ {sh}\ sa}\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \frac4{\pi} \sum_{n=1}^{\infty} \frac{(-1)^n}{2n-1} \sin {\frac{(2n-1) \pi  x}{2a}} \sin {\frac{(2n-1)\pi t}{2a}}\ </math>
+| <math> \frac{{sh}\ sx}{s\ {ch}\ sa}\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> |fracta+\frac2{\pi} \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \cos {\frac{n \pi  x}{a}} \sin {\frac{n\pi t}{a}}\ </math>
+| <math> \frac{{ch}\ sx}{s\ {sh}\ as}\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> 1+\frac4{\pi} \sum_{n=1}^{\infty} \frac{(-1)^n}{2n-1} \cos {\frac{n \pi  x}{a}} \cos {\frac{(2n-1)\pi t}{2a}}\ </math>
+| <math> \frac{{ch}\ sx}{s\ {ch}\ as}\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \frac {xt}a+\frac{2a}{{\pi}^2 } \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2} \sin {\frac{n \pi  x}{a}} \sin {\frac{n\pi t}{a}}\ </math>
+| <math> \frac{{sh}\ sx}{s^2\ {sh}\ sa}\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> x+\frac{8a}{{\pi}^2 } \sum_{n=1}^{\infty} \frac{(-1)^n}{(2n-1)^2} \sin {\frac{(2n-1) \pi  x}{2a}} \cos {\frac{(2n-1) \pi t}{2a}}\ </math>
+| <math> \frac{{sh}\ sx}{s^2\ {ch}\ sa}\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \frac{t^2}{2a}+\frac{2a}{{\pi}^2 } \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2} \cos {\frac{n \pi  x}{a}} \left ( 1-\cos {\frac{n \pi t}{a}} \right )\ </math>
+| <math> \frac{{ch}\ sx}{s^2\ {sh}\ sa}\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> t+\frac{8a}{{\pi}^2 } \sum_{n=1}^{\infty} \frac{(-1)^n}{(2n-1)^2} \cos {\frac{(2n-1) \pi  x}{2a}} \sin {\frac{(2n-1) \pi t}{2a}}\ </math>
+| <math> \frac{{ch}\ sx}{s^3\ {sh}\ sa}\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \frac12(t^2+x^2-a^2)-\frac{16a^2}{{\pi}^3 } \sum_{n=1}^{\infty} \frac{(-1)^n}{(2n-1)^3} \cos {\frac{(2n-1) \pi  x}{2a}} \cos {\frac{(2n-1) \pi t}{2a}}\ </math>
+| <math> \frac{{ch}\ sx}{s^3\ {ch}\ sa}\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \frac{2 \pi}{a^2} \sum_{n=1}^{\infty} (-1)^nne^{-(2n-1)^2{\pi}^2t/4a^2}\sin {\frac{n \pi  x}{a}}\ </math>
+| <math> \frac{{ch}\ x\sqrt{s}}{{sh}\ a\sqrt{s}}\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \frac{2 \pi}{a^2} \sum_{n=1}^{\infty} (-1)^nne^{-(2n-1)^2{\pi}^2t/4a^2}\sin {\frac{n \pi  x}{a}}\ </math>
+| <math> \frac{{ch}\ x\sqrt{s}}{{sh}\ a\sqrt{s}}\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \frac{2 \pi}{a^2} \sum_{n=1}^{\infty} (-1)^nne^{-(2n-1)^2{\pi}^2t/4a^2}\sin {\frac{n \pi  x}{a}}\ </math>
+| <math> \frac{{sh}\ x\sqrt{s}}{{sh}\ a\sqrt{s}}\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \frac{\pi}{a^2} \sum_{n=1}^{\infty} (-1)^{n-1}(2n-1)e^{-(2n-1)^2{\pi}^2t/4a^2}\cos {\frac{(2n-1) \pi  x}{2a}}\ </math>
+| <math> \frac{{ch}\ x\sqrt{s}}{{ch}\ a\sqrt{s}}\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \frac{2}{a} \sum_{n=1}^{\infty} (-1)^{n-1}e^{-(2n-1)^2{\pi}^2t/4a^2}\sin {\frac{(2n-1) \pi  x}{2a}}\ </math>
+| <math> \frac{{sh}\ x\sqrt{s}}{\sqrt{s}{ch}\ a\sqrt{s}}\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \frac1a+\frac2a\sum_{n=1}^{\infty} (-1)^ne^{-n^2{\pi}^2t/a^2}\cos {\frac{n \pi  x}{a}}\ </math>
+| <math> \frac{{ch}\ x\sqrt{s}}{\sqrt{s}{sh}\ a\sqrt{s}}\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \frac{x}{a}+\frac{2}{\pi} \sum_{n=1}^{\infty} \frac{(-1)^n}{n}e^{-n^2{\pi}^2t/a^2} \sin {\frac{n \pi x}{a}}\ </math>
+| <math> \frac{{sh}\ x\sqrt{s}}{s{sh}\ a\sqrt{s}}\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> 1+\frac4{\pi}\sum_{n=1}^{\infty} \frac{(-1)^n}{2n-1}e^{-(2n-1)^2{\pi}^2t/a^2}\cos {\frac{(2n-1) \pi  x}{2a}}\ </math>
+| <math> \frac{{ch}\ x\sqrt{s}}{s{ch}\ a\sqrt{s}}\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \frac{xt}{a}+\frac{2a^2}{{\pi}^3}\sum_{n=1}^{\infty} \frac{(-1)^n}{n^3}(1-e^{-n^2{\pi}^2t/a^2})\sin {\frac{n \pi  x}{a}}\ </math>
+| <math> \frac{{sh}\ x\sqrt{s}}{s^2{sh}\ a\sqrt{s}}\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \frac12(x^2+a^2)+t-\frac{16a^2}{{\pi}^3}\sum_{n=1}^{\infty} \frac{(-1)^n}{(2n-1)^3}e^{-{(2n-1)}^2{\pi}^2t/a^2}\cos {\frac{(2n-1) \pi  x}{2a}}\ </math>
+| <math> \frac{{ch}\ x\sqrt{s}}{s^2{ch}\ a\sqrt{s}}\ </math>
+|-
+|}
 ----
+[[ECE301|Go to Relevant Course Page: ECE 301]]
+[[ECE438|Go to Relevant Course Page: ECE 438]]
+[[ECE538|Go to Relevant Course Page: ECE 538]]
 [[Collective Table of Formulas|Back to Collective Table]]
 [[Category:Formulas]]

Difference between revisions of "Laplace Transforms Table" - Rhea

Latest revision as of 12:58, 22 May 2015

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