8.8: A Brief Table of Laplace Transforms

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Contributed by William F. Trench
Andrew G. Cowles Distinguished Professor Emeritus (Mathamatics) at Trinity University

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Table $ \PageIndex{1}$
$ \displaystyle f(t)$	$ \displaystyle F(s)$
1	$ \displaystyle 1\over s$	$ \displaystyle (s > 0)$
$ \displaystyle t^n$	$ \displaystyle n!\over s^{n+1}$	$ \displaystyle (s > 0)$
($ \displaystyle n = \mbox{ integer } > 0$)
$ \displaystyle t^p,\; p > -1$	$ \displaystyle \Gamma (p+1) \over s^{(p+1)}$	$ \displaystyle (s>0)$
$ \displaystyle e^{at}$	$ \displaystyle 1 \over s-a$	$ \displaystyle (s > a)$
$ \displaystyle t^ne^{at}$	$ \displaystyle n! \over (s-a)^{n+1}$	$ \displaystyle (s > 0)$
($ \displaystyle n= \text{ integer } > 0$)
$ \displaystyle \cos \omega t$	$ \displaystyle \frac{s}{s^{2}+\omega ^{2}}$	$ \displaystyle (s > 0)$
$ \displaystyle \sin \omega t$	$ \displaystyle \omega \over s^2+\omega^2$	$ \displaystyle (s > 0)$
$ \displaystyle e^{\lambda t} \cos \omega t$	$ \displaystyle s - \lambda \over (s-\lambda)^2+\omega^2$	$ \displaystyle (s > \lambda)$
$ \displaystyle e^{\lambda t} \sin \omega t$	$ \displaystyle \omega \over (s-\lambda)^2+\omega^2$	$ \displaystyle (s > \lambda)$
$ \displaystyle \cosh bt$	$ \displaystyle s \over s^2-b^2$	$ \displaystyle (s > \|b\|)$
$ \displaystyle \sinh bt$	$ \displaystyle b \over s^2-b^2$	$ \displaystyle (s > \|b\|)$
$ \displaystyle t \cos \omega t$	$ \displaystyle s^2-\omega^2 \over (s^2+\omega^2)^2$	$ \displaystyle (s>0)$
$ \displaystyle t \sin \omega t$	$ \displaystyle 2\omega s \over (s^2+\omega^2)^2$	$ \displaystyle (s>0)$
$ \displaystyle \sin \omega t -\omega t\cos \omega t$	$ \displaystyle 2\omega^3\over (s^2+\omega^2)^2$	$ \displaystyle (s>0)$
$ \displaystyle \omega t - \sin \omega t$	$ \displaystyle \omega^3 \over s^2(s^2+\omega^2)^2$	$ \displaystyle (s>0)$
$ \displaystyle \frac{1}{t}\sin\omega t$	$ \displaystyle \arctan \left({\omega \over s}\right)$	$ \displaystyle (s>0)$
$ \displaystyle e^{at}f(t)$	$ \displaystyle F(s-a)$
$ \displaystyle t^kf(t)$	$ \displaystyle (-1)^{k}F^{(k)}(s)$
$ \displaystyle f(\omega t)$	$ \displaystyle \frac{1}{\omega}F\left(\frac{s}{\omega } \right), \quad \omega >0$
$ \displaystyle u(t-\tau)$	$ \displaystyle e^{-\tau s} \over s$	$ \displaystyle (s>0)$
$ \displaystyle u(t-\tau)f(t-\tau)\, (\tau > 0)$	$ \displaystyle e^{-\tau s}F(s)$
$ \displaystyle \displaystyle {\int^t_o f(\tau)g(t-\tau)\, d\tau}$	$ \displaystyle F(s) \cdot G(s)$
$ \displaystyle \delta(t-a)$	$ \displaystyle e^{-as}$	$ \displaystyle (s>0)$

\( \displaystyle f(t)\)	\( \displaystyle F(s)\)
1	\( \displaystyle 1\over s\)	\( \displaystyle (s > 0)\)
\( \displaystyle t^n\)	\( \displaystyle n!\over s^{n+1}\)	\( \displaystyle (s > 0)\)
(\( \displaystyle n = \mbox{ integer } > 0\))
\( \displaystyle t^p,\; p > -1\)	\( \displaystyle \Gamma (p+1) \over s^{(p+1)}\)	\( \displaystyle (s>0)\)
\( \displaystyle e^{at}\)	\( \displaystyle 1 \over s-a\)	\( \displaystyle (s > a)\)
\( \displaystyle t^ne^{at}\)	\( \displaystyle n! \over (s-a)^{n+1}\)	\( \displaystyle (s > 0)\)
(\( \displaystyle n= \text{ integer } > 0\))
\( \displaystyle \cos \omega t\)	\( \displaystyle \frac{s}{s^{2}+\omega ^{2}}\)	\( \displaystyle (s > 0)\)
\( \displaystyle \sin \omega t\)	\( \displaystyle \omega \over s^2+\omega^2\)	\( \displaystyle (s > 0)\)
\( \displaystyle e^{\lambda t} \cos \omega t\)	\( \displaystyle s - \lambda \over (s-\lambda)^2+\omega^2\)	\( \displaystyle (s > \lambda)\)
\( \displaystyle e^{\lambda t} \sin \omega t\)	\( \displaystyle \omega \over (s-\lambda)^2+\omega^2\)	\( \displaystyle (s > \lambda)\)
\( \displaystyle \cosh bt\)	\( \displaystyle s \over s^2-b^2\)	\( \displaystyle (s > \|b\|)\)
\( \displaystyle \sinh bt\)	\( \displaystyle b \over s^2-b^2\)	\( \displaystyle (s > \|b\|)\)
\( \displaystyle t \cos \omega t\)	\( \displaystyle s^2-\omega^2 \over (s^2+\omega^2)^2\)	\( \displaystyle (s>0)\)
\( \displaystyle t \sin \omega t\)	\( \displaystyle 2\omega s \over (s^2+\omega^2)^2\)	\( \displaystyle (s>0)\)
\( \displaystyle \sin \omega t -\omega t\cos \omega t\)	\( \displaystyle 2\omega^3\over (s^2+\omega^2)^2\)	\( \displaystyle (s>0)\)
\( \displaystyle \omega t - \sin \omega t\)	\( \displaystyle \omega^3 \over s^2(s^2+\omega^2)^2\)	\( \displaystyle (s>0)\)
\( \displaystyle \frac{1}{t}\sin\omega t\)	\( \displaystyle \arctan \left({\omega \over s}\right)\)	\( \displaystyle (s>0)\)
\( \displaystyle e^{at}f(t)\)	\( \displaystyle F(s-a)\)
\( \displaystyle t^kf(t)\)	\( \displaystyle (-1)^{k}F^{(k)}(s)\)
\( \displaystyle f(\omega t)\)	\( \displaystyle \frac{1}{\omega}F\left(\frac{s}{\omega } \right), \quad \omega >0\)
\( \displaystyle u(t-\tau)\)	\( \displaystyle e^{-\tau s} \over s\)	\( \displaystyle (s>0)\)
\( \displaystyle u(t-\tau)f(t-\tau)\, (\tau > 0)\)	\( \displaystyle e^{-\tau s}F(s)\)
\( \displaystyle \displaystyle {\int^t_o f(\tau)g(t-\tau)\, d\tau}\)	\( \displaystyle F(s) \cdot G(s)\)
\( \displaystyle \delta(t-a)\)	\( \displaystyle e^{-as}\)	\( \displaystyle (s>0)\)