8.4: A Brief Table of Laplace Transforms

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

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