8.8 A Brief Table of Laplace Transforms

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