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%%% Primary Title: table of Laplace transforms
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\begin{document}
A list of \htmladdnormallink{Laplace transforms}{http://planetphysics.us/encyclopedia/LaplaceTransform.html} is provided in the table below; one lists some of the common properties, and the other lists some common examples. The tables are followed by a subsection outlining the Physics and Engineering areas in which the Laplace transforms are intensely utilized at present. A list of references is also provided in \htmladdnormallink{relation}{http://planetphysics.us/encyclopedia/Bijective.html} to possible \htmladdnormallink{non-commutative}{http://planetphysics.us/encyclopedia/AbelianCategory3.html} or higher dimensional extensions of the classical Laplace transforms (LTs).
\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
$f(t)*g(t)$ & $\mathcal{L}\{f(t)\}\mathcal{L}\{g(t)\}$ & \htmladdnormallink{convolution}{http://planetphysics.us/encyclopedia/AssociatedGroupoidAlgebraRepresentations.html} property & \htmladdnormallink{here}{http://planetmath.org/encyclopedia/LaplaceTransformOfConvolution.html}\\
\hline
$\displaystyle{\int_a^bf(t,\,x)\,dx}$ & $\displaystyle{\int_a^b\mathcal{L}\{f(t,\,x)\}\,dx}$
& integration with respect to a parametre & \htmladdnormallink{here}{http://planetmath.org/encyclopedia/IntegrationOfLaplaceTransformWithRespectToParameter.html}\\
\hline
$\displaystyle{\frac{\partial}{\partial x}f(t,\,x)}$ & $\displaystyle{\frac{\partial}{\partial x}\mathcal{L}\{f(t,\,x)\}}$ & diffentiation with respect to a \htmladdnormallink{parameter}{http://planetphysics.us/encyclopedia/Parameter.html} & \\
\hline
$f(\displaystyle{\frac{t}{a}})$ & $aF(as)$ & $\mathcal{L}\{f(t)\} = F(s)$
& \htmladdnormallink{here}{http://planetmath.org/encyclopedia/RulesForLaplaceTransform.html}\\
\hline
$e^{at}f(t)$ & $F(s-a)$ & $\mathcal{L}\{f(t)\} = F(s)$ & \htmladdnormallink{here}{http://planetmath.org/encyclopedia/RulesForLaplaceTransform.html}\\
\hline
$f(t-a)$ & $e^{-as}F(s)$ & $\mathcal{L}\{f(t)\} = F(s)$ & \htmladdnormallink{here}{http://planetmath.org/encyclopedia/DelayTheorem.html}\\
\hline
$t^nf(t)$ & $(-1)^nF^{(n)}(s)$ & $\mathcal{L}\{f(t)\} = F(s)$ & \htmladdnormallink{here}{http://planetmath.org/encyclopedia/LaplaceTransformOfTnft.html}\\
\hline
$\displaystyle\frac{f(t)}{t}$ & $\displaystyle\int_s^\infty F(u)\,du$ & $\mathcal{L}\{f(t)\} = F(s)$ &
\htmladdnormallink{here}{http://planetmath.org/encyclopedia/LaplaceTransformOfFracftt.html}\\
\hline
$\displaystyle{\int_0^tf(u)\,du}$ & $\displaystyle{\frac{F(s)}{s}}$ & $\mathcal{L}\{f(t)\} = F(s)$ & \htmladdnormallink{here}{http://planetmath.org/encyclopedia/LaplaceTransformOfIntegral.html}\\
\hline
$f'(t)$ & $sF(s)-\lim_{x\to0+}f(x)$ & $\mathcal{L}\{f(t)\} = F(s)$ & \htmladdnormallink{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
$\cos{at}$ & $\displaystyle{\frac{s}{s^{2}+a^{2}}}$ & $s>0$ & & \htmladdnormallink{here}{http://planetmath.org/encyclopedia/LaplaceTransformOfCosineAndSine.html}\\
\hline
$\sin{at}$ & $\displaystyle{\frac{a}{s^{2}+a^{2}}}$ & $s>0$ & & \htmladdnormallink{here}{http://planetmath.org/encyclopedia/LaplaceTransformOfCosineAndSine.html}\\
\hline
$\cosh{at}$ & $\displaystyle{\frac{s}{s^{2}-a^{2}}}$ & $s>|a|$ & & \htmladdnormallink{here}{http://planetmath.org/encyclopedia/LaplaceTransformOfCosineAndSine.html}\\
\hline
$\sinh{at}$ & $\displaystyle{\frac{a}{s^{2}-a^{2}}}$ & $s>|a|$ & & \htmladdnormallink{here}{http://planetmath.org/encyclopedia/LaplaceTransformOfCosineAndSine.html}\\
\hline
$\displaystyle\frac{\sin{t}}{t}$ & $\displaystyle\arctan\frac{1}{s}$ & $s>0$ & See sinc \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} &
\htmladdnormallink{here}{http://planetmath.org/encyclopedia/LaplaceTransformOfIntegralSine.html}\\
\hline
$t^r$ & $\displaystyle{\frac{\Gamma(r+1)}{s^{r+1}}}$ & $r>-1,\;\;s>0$ & \htmladdnormallink{gamma function}{http://planetphysics.us/encyclopedia/GammaFunction.html} $\Gamma$ &
\htmladdnormallink{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 & \htmladdnormallink{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 & \htmladdnormallink{here}{http://planetmath.org/encyclopedia/UsingConvolutionToFindLaplaceTransform.html}\\
\hline
$\displaystyle\frac{1}{\sqrt{t}}$ & $\displaystyle\sqrt{\frac{\pi}{s}}$ & $s>0$ & & \htmladdnormallink{here}{http://planetmath.org/encyclopedia/LaplaceTransformOfPowerFunction.html}\\
\hline
$J_0(at)$ & $\displaystyle\frac{1}{\sqrt{s^2+a^2}}$ & $s>0$ & \htmladdnormallink{Bessel function}{http://planetphysics.us/encyclopedia/BesselEquation2.html} $J_0$ & \htmladdnormallink{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 & \htmladdnormallink{here}{http://planetmath.org/encyclopedia/LaplaceTransformOfAGaussianFunction.html}\\
\hline
$\ln{t}$ & $\displaystyle-\frac{\gamma+\ln{s}}{s}$ & $s>0$ & Euler'sconstant $\gamma$ & \htmladdnormallink{here}{http://planetmath.org/encyclopedia/LaplaceTransformOfLogarithm.html}\\
\hline
$\delta(t)$ & $1$ & & Dirac delta function & \\
\hline
$\mathcal{L}\{\delta(t\!-\!a)\}$ & $e^{-as}.$ & & Dirac delta with delay & \\
\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
$t$ & $\displaystyle{1 \over s^2}$ & & &\htmladdnormallink{here}{http://planetmath.org/encyclopedia/LaplaceTransformOfIntegral.html}\\ \hline
$\displaystyle{t^{n-1} \over (n-1)!}$ & $\displaystyle{1 \over s^n}$ & & &\htmladdnormallink{here}{http://planetmath.org/encyclopedia/LaplaceTransformOfIntegral.html} \\ \hline
$\displaystyle{1 \over t+a}$ & $e^{as} {\rm E}_1(as)$ & $a > 0$ & exponential integral ${\rm E}_1$ & \htmladdnormallink{here}{http://planetmath.org/encyclopedia/Ei.html}\\ \hline
$\displaystyle{1 \over (t+a)^2}$ & $\displaystyle{1 \over a}-se^{as}{\rm E}_1(as)$ & $a > 0$ & &\htmladdnormallink{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}
\subsubsection{Applications of Laplace Transforms in Physics, Engineering and Mathematical Biophysics/Theoretical Biology}
Although possibly `less popular' with physicists than the \htmladdnormallink{Fourier transform}{http://planetphysics.us/encyclopedia/FourierTransforms.html},
the Laplace transform has applications both in Astrophysics, Engineering and Mathematical Biophysics.
In Astrophysics the Laplace transform is employed to succesfully `sharpen up' images of distant planets obtained by satellite mounted-telescopes of various kinds without having the disadvantage of \htmladdnormallink{FT}{http://planetphysics.us/encyclopedia/FourierTransforms.html} that may lose fine detail through exponential multiplication ``smoothing'' of partially fuzzy images.
On the other hand, in Engineering applications the Laplace transform is often
employed to calculate the transfer function of an engineered \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} such as
an electrical network or electronic circuit.
In Mathematical Biophysics (and also in Optimal Control theories) both the Laplace and Fourier transforms are employed to model living systems and their components, and also to optimize such models.
\begin{remark}
This contributed topic entry, in addition to the most used, or useful, Laplace transforms, may also contain information on how the convolution of Laplace transforms \htmladdnormallink{work}{http://planetphysics.us/encyclopedia/Work.html}, and also, possibly, higher dimensional generalizations of Laplace transforms, such as 2D Laplace transforms, non-commutative integral generalizations \`a la A. Connes of LTs, etc.
\end{remark}
\textbf{[More to be added...]}
\begin{thebibliography}{9}
\bibitem{AC79}
A. Connes.1979. Sur la th\'eorie noncommutative de l' integration, {\em Lecture Notes in Math.}, Springer-Verlag, Berlin, {\bf 725}: 19-14.
\bibitem{RW97}
A. Ramsay and M. E. Walter, Fourier-Stieltjes algebras of locally compact groupoids,\emph{J. Functional Anal}. \textbf{148}: 314-367 (1997).
\bibitem{PALT2k3}
A. L. T. Paterson, The Fourier-Stieltjes and Fourier algebras for locally
compact groupoids., (2003) \htmladdnormallink{Free PDF file download}{http://aux.planetmath.org/files/objects/10739/AFourierStjelties_LocallyCompactsGds_Harmonic0310138v1.pdf}.
\bibitem{BA91}
B. Aniszczyk. 1991. A rigid Borel space., {\em Proceed. AMS.}, 113 (4):1013-1015.,
\htmladdnormallink{available online}{http://www.jstor.org/pss/2048777}.
\end{thebibliography}
\end{document}
