TableOfLaplaceTransforms 2009-04-23 11:47:47 2009-04-23 11:47:47 Topic % this is the default PlanetPhysics preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Below are tables of \PMlinkname{Laplace transforms}{LaplaceTransform}; one lists some of the common properties, and the other lists some common examples. \subsubsection*{Properties} \begin{center} \begin{tabular}{|c|c|p{4cm}|c|} \hline\hline Original & Transformed & comment & derivation \\ \hline\hline $af(t)+bg(t)$ & $a\mathcal{L}\{f(t)\}+b\mathcal{L}\{g(t)\}$ & linearity & \\ \hline $f(t)*g(t)$ & $\mathcal{L}\{f(t)\}\mathcal{L}\{g(t)\}$ & convolution property & \PMlinkexternal{here}{http://planetmath.org/encyclopedia/LaplaceTransformOfConvolution}\\ \hline $\displaystyle{\int_a^bf(t,\,x)\,dx}$ & $\displaystyle{\int_a^b\mathcal{L}\{f(t,\,x)\}\,dx}$ & integration with respect to a parametre & \PMlinkid{here}{11521}\\ \hline $\displaystyle{\frac{\partial}{\partial x}f(t,\,x)}$ & $\displaystyle{\frac{\partial}{\partial x}\mathcal{L}\{f(t,\,x)\}}$ & diffentiation with respect to a parameter & \\ \hline $f(\displaystyle{\frac{t}{a}})$ & $aF(as)$ & $\mathcal{L}\{f(t)\} = F(s)$ & \PMlinkexternal{here}{http://planetmath.org/encyclopedia/RulesForLaplaceTransform.html}\\ \hline $e^{at}f(t)$ & $F(s-a)$ & $\mathcal{L}\{f(t)\} = F(s)$ & \PMlinkexternal{here}{http://planetmath.org/encyclopedia/RulesForLaplaceTransform.html}\\ \hline $f(t-a)$ & $e^{-as}F(s)$ & $\mathcal{L}\{f(t)\} = F(s)$ & \PMlinkexternal{here}{http://planetmath.org/encyclopedia/DelayTheorem.html}\\ \hline $t^nf(t)$ & $(-1)^nF^{(n)}(s)$ & $\mathcal{L}\{f(t)\} = F(s)$ & \PMlinkexternal{here}{http://planetmath.org/encyclopedia/LaplaceTransformOfTnft.html}\\ \hline $\displaystyle\frac{f(t)}{t}$ & $\displaystyle\int_s^\infty F(u)\,du$ & $\mathcal{L}\{f(t)\} = F(s)$ & \PMlinkexternal{here}{http://planetmath.org/encyclopedia/LaplaceTransformOfFracftt.html}\\ \hline $\displaystyle{\int_0^tf(u)\,du}$ & $\displaystyle{\frac{F(s)}{s}}$ & $\mathcal{L}\{f(t)\} = F(s)$ & \PMlinkexternal{here}{http://planetmath.org/encyclopedia/LaplaceTransformOfIntegral.html}\\ \hline $f'(t)$ & $sF(s)-\lim_{x\to0+}f(x)$ & $\mathcal{L}\{f(t)\} = F(s)$ & \PMlinkexternal{here}{http://planetmath.org/encyclopedia/LaplaceTransformOfDerivative.html}\\ \hline $f''(t)$ & $s^2F(s)-s\lim_{x\to0+}f'(x)-\lim_{x\to0+}f(x)$ & $\mathcal{L}\{f(t)\} = F(s)$ & \\ \hline \end{tabular} \end{center} \subsubsection*{Examples} \begin{center} \begin{tabular}{|c|c|c|c|c|} \hline\hline $f(t)$ & $\mathcal{L}\{f(t)\}$ & conditions & explanation & derivation \\ \hline\hline $e^{at}$ & $\displaystyle{\frac{1}{s-a}}$ & $s>a$ & & trivial\\ \hline $\cos{at}$ & $\displaystyle{\frac{s}{s^{2}+a^{2}}}$ & $s>0$ & & \PMlinkexternal{here}{http://planetmath.org/encyclopedia/LaplaceTransformOfCosineAndSine.html}\\ \hline $\sin{at}$ & $\displaystyle{\frac{a}{s^{2}+a^{2}}}$ & $s>0$ & & \PMlinkexternal{here}{http://planetmath.org/encyclopedia/LaplaceTransformOfCosineAndSine.html}\\ \hline $\cosh{at}$ & $\displaystyle{\frac{s}{s^{2}-a^{2}}}$ & $s>|a|$ & & \PMlinkexternal{here}{http://planetmath.org/encyclopedia/LaplaceTransformOfCosineAndSine.html}\\ \hline $\sinh{at}$ & $\displaystyle{\frac{a}{s^{2}-a^{2}}}$ & $s>|a|$ & & \PMlinkexternal{here}{http://planetmath.org/encyclopedia/LaplaceTransformOfCosineAndSine.html}\\ \hline $\displaystyle\frac{\sin{t}}{t}$ & $\displaystyle\arctan\frac{1}{s}$ & $s>0$ & See sinc function & \PMlinkexternal{here}{http://planetmath.org/encyclopedia/LaplaceTransformOfIntegralSine.html}\\ \hline $t^r$ & $\displaystyle{\frac{\Gamma(r+1)}{s^{r+1}}}$ & $r>-1,\;\;s>0$ & gamma function $\Gamma$ & \PMlinkexternal{here}{http://planetmath.org/encyclopedia/LaplaceTransformOfPowerFunction.html}\\ \hline $\displaystyle e^{a^2t}\,{\rm erf}\,a\sqrt{t}$ & $\displaystyle\frac{a}{(s\!-\!a^2)\sqrt{s}}$ & $s>a^2$ & See error function & \PMlinkexternal{here}{http://planetmath.org/encyclopedia/UsingConvolutionToFindLaplaceTransform.html}\\ \hline $\displaystyle e^{a^2t}\,{\rm erfc}\,a\sqrt{t}$ & $\displaystyle\frac{1}{(a\!+\!\sqrt{s})\sqrt{s}}$ & $s>0$ & See error function & \PMlinkexternal{here}{http://planetmath.org/encyclopedia/UsingConvolutionToFindLaplaceTransform.html}\\ \hline $\displaystyle\frac{1}{\sqrt{t}}$ & $\displaystyle\sqrt{\frac{\pi}{s}}$ & $s>0$ & & \PMlinkexternal{here}{http://planetmath.org/encyclopedia/LaplaceTransformOfPowerFunction.html}\\ \hline $J_0(at)$ & $\displaystyle\frac{1}{\sqrt{s^2+a^2}}$ & $s>0$ & Bessel function $J_0$ & \PMlinkexternal{here}{http://planetmath.org/encyclopedia/InverseLaplaceTransformOfDerivatives.html}\\ \hline $e^{-t^2}$ & $\displaystyle\frac{\sqrt{\pi}}{2}e^\frac{s^2}{4}\mathrm{erfc}\Big(\frac{s}{2}\Big)$ & $s>0$ & See error function & \PMlinkexternal{here}{http://planetmath.org/encyclopedia/LaplaceTransformOfAGaussianFunction.html}\\ \hline $\ln{t}$ & $\displaystyle-\frac{\gamma+\ln{s}}{s}$ & $s>0$ & Euler'sconstant $\gamma$ & \PMlinkexternal{here}{http://planetmath.org/encyclopedia/LaplaceTransformOfLogarithm.html}\\ \hline $\delta(t)$ & $1$ & & Dirac delta function & \\ \hline \end{tabular} \end{center} \subsubsection*{Rational Functions} \begin{center} \begin{tabular}{|c|c|c|c|c|} \hline\hline $f(t)$ & $\mathcal{L}\{f(t)\}$ & conditions & explanation & derivation \\ \hline\hline 1 & $\displaystyle{1 \over s}$ & & & \\ \hline $t$ & $\displaystyle{1 \over s^2}$ & & &\PMlinkexternal{here}{http://planetmath.org/encyclopedia/LaplaceTransformOfIntegral.html}\\ \hline $\displaystyle{t^{n-1} \over (n-1)!}$ & $\displaystyle{1 \over s^n}$ & & &\PMlinkexternal{here}{http://planetmath.org/encyclopedia/LaplaceTransformOfIntegral.html} \\ \hline $\displaystyle{1 \over t+a}$ & $e^{as} {\rm E}_1(as)$ & $a > 0$ & exponential integral ${\rm E}_1$ & \PMlinkexternal{here}{http://planetmath.org/encyclopedia/Ei.html}\\ \hline $\displaystyle{1 \over (t+a)^2}$ & $\displaystyle{1 \over a}-se^{as}{\rm E}_1(as)$ & $a > 0$ & &\PMlinkexternal{here}{http://planetmath.org/encyclopedia/Ei.html}\\ \hline $\displaystyle{1 \over (t+a)^n}$ & $a^{1-n} e^{as} E_n (as)$ & $a > 0,\;\; n \in \mathbb{N}$ & ? & \\ \hline $L_n(t)$ & $\displaystyle\frac{1}{s}\!\left(\!\frac{s-1}{s}\!\right)^n$ & $s > 0$ & Laguerre polynomial $L_n$ & \\ \hline \end{tabular} \end{center}