69 Laplace transform table

Examples. Let $a$ be a non-zero real number, $b>0$ and $\alpha>-1$ .

\begin{pmatrix}f(x)&{\cal L}(f)(s)\\ 1&1/s\\ x^{n}&{{n!}\over{s^{n+1}}}\\ e^{ax}&{{1}\over{s-a}}\quad(s>a)\\ x^{\alpha}&{{\Gamma(\alpha+1)}\over{s^{\alpha+1}}}\\ \sin ax&{{a}\over{s^{2}+a^{2}}}\\ \cos ax&{{s}\over{s^{2}+a^{2}}}\\ \delta_{b}(dx)&e^{-bs}\\ H(x-b)&{{e^{-bs}}\over{s}}\end{pmatrix}