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$$\begin{array}{cc} f(t)&F(s)\\ \hline \frac{1}{2\pi i}\int_{s'-i\infty}^{s'+i\infty}F(s)e^{st}ds&\int_0^\infty f(t)e^{-st}dt\\ \delta(t)&1(s)\\ u(t)&\frac{1}{s}\\ t&\frac{1}{s^2}\\ t^n&\frac{n!}{s^{n+1}}\\ \begin{pmatrix}\sin&\cos\\\sinh&\cosh\end{pmatrix}(\omega t)&\frac{\begin{pmatrix}\omega&s\end{pmatrix}}{s^2\pm \omega^2}\\ u(t-a)f(t-a)&e^{-as}F(s)\\ e^{at}f(t)&F(s-a)\\ (f\star g)(t)=\int_0^t f(t-\tau)g(\tau)d\tau&F(s)G(s)\\ f'(t)&sF(s)-f(0)\\ f^{(n)}(t)&s^nF(s)-s^\downarrow f^{(\uparrow)}(0)\\ \end{array}$$
Paul's Laplace Transforms