Laplace Transforms

Laplace transforms (one-sided)def:laplace-transforms Laplace transform \(\mathcal{L}\): $$\begin{align} \mathcal{L}(y(t)) = Y(s) = \int_{0}^\infty y(t) e^{-s t} dt. \end{align}$$ Inverse Laplace transform \(\mathcal{L}^{-1}\): $$\begin{align} \mathcal{L}^{-1}(Y(s)) = y(t) = \frac{1} {2\pi j}\int_{\sigma-j\infty}^{\sigma+j\infty} Y(s) e^{s t} d s. \end{align}$$