Engineering Tables/Laplace Transform Table

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  Time Domain Laplace Domain
x(t) = \mathcal{L}^{-1}\left\{ X(s) \right\} X(s) = \mathcal{L} \left\{ x(t) \right\}
1  \frac{1}{2\pi j} \int_{\sigma-j\infty}^{\sigma+j\infty} X(s)e^{st}ds  \int_{-\infty}^\infty x(t)e^{-st}dt
2  \delta (t) \,  1 \,
3  \delta (t-a)\,  e^{-as}\,
4  u(t) \,  \frac{1}{s}
5  u(t-a)\,  \frac{e^{-as}}{s}
6  t u(t) \,  \frac{1}{s^2}
7  t^nu(t) \,  \frac{n!}{s^{n+1}}
8  \frac{1}{\sqrt{\pi t}}u(t)  \frac{1}{\sqrt{s}}
9  e^{at}u(t) \,  \frac{1}{s-a}
10  t^n e^{at}u(t) \,  \frac{n!}{(s-a)^{n+1}}
11   \cos (\omega t) u(t) \,  \frac{s}{s^2+\omega^2}
12   \sin (\omega t) u(t) \,  \frac{\omega}{s^2+\omega^2}
13   \cosh (\omega t) u(t) \,  \frac{s}{s^2-\omega^2}
14  \sinh (\omega t) u(t) \,  \frac{\omega}{s^2-\omega^2}
15  e^{at}  \cos (\omega t) u(t) \,  \frac{s-a}{(s-a)^2+\omega^2}
16  e^{at} \sin (\omega t) u(t) \,  \frac{\omega}{(s-a)^2+\omega^2}
17  \frac{1}{2\omega^3}(\sin \omega t-\omega t \cos \omega t)  \frac{1}{(s^2+\omega^2)^2}
18  \frac{t}{2\omega} \sin \omega t  \frac{s}{(s^2+\omega^2)^2}
19  \frac{1}{2\omega}(\sin \omega t+\omega t \cos \omega t)  \frac{s^2}{(s^2+\omega^2)^2}