Laplace Transform Catalog
#Math
| $\displaystyle x(t)$ | $\displaystyle X(s)$ | Region of Convergence |
|---|---|---|
| $\displaystyle \delta(t)$ | $\displaystyle 1$ | All $\displaystyle s \in \mathbb{C}$ |
| $\displaystyle u(t)$ | $\displaystyle \frac{1}{s}$ | $\displaystyle \text{Re}(s)>0$ |
| $\displaystyle e^{-at}u(t)$ | $\displaystyle \frac{1}{s+a}$ | $\displaystyle \text{Re}(s)>-a$ |
| $\displaystyle e^{at}u(t)$ | $\displaystyle \frac{1}{s-a}$ | $\displaystyle \text{Re}(s)>a$ |
| $\displaystyle t^{n}u(t)$ | $\displaystyle \frac{n!}{s^{n+1}}$ | $\displaystyle \text{Re}(s)>0$ |
| $\displaystyle \cos({\omega}_{0}t)u(t)$ | $\displaystyle \frac{s}{s^{2}+{\omega}_{0}^{2}}$ | $\displaystyle \text{Re}(s)>0$ |
| $\displaystyle \sin({\omega}_{0}t)u(t)$ | $\displaystyle \frac{{\omega}{0}}{s^{2}+{\omega}{0}^{2}}$ | $\displaystyle \text{Re}(s)>0$ |
| $\displaystyle e^{-at}\cos(\omega t)u(t)$ | $\displaystyle \frac{s+a}{(s+a)^{2}+\omega ^{2}}$ | $\displaystyle \text{Re}(s)>-a$ |
| $\displaystyle e^{-at}\sin(\omega t)u(t)$ | $\displaystyle \frac{\omega}{(s+a)^{2}+\omega ^{2}}$ | $\displaystyle \text{Re}(s)>-a$ |
| $\displaystyle \delta(t-T)$ | $\displaystyle e^{-sT}$ | All $\displaystyle s \in \mathbb{C}$ |
| $\displaystyle u(-t)$ | $\displaystyle -\frac{1}{s}$ | $\displaystyle \text{Re}(s)<0$ |
All of these have a $\displaystyle u(t)$ factor because we assume casual signals