Inverse Laplace Transform

Reverse of Laplace transform. Denoted as $\mathcal{L}^{-1} \{ F(s) \}$.

$$\mathcal{L}\{ f(t) \} = F(s) \implies \mathcal{L}^{-1} \{ F(s) \} = f(t)$$

Properties

Basic Rule

$$L^{-1}\{F(s+a)\} = e^{-at}f(t)$$

Linearity

$$\mathcal{L}^{-1} \{ cF(s) + G(s) \} = c\mathcal{L}^{-1}\{ F(s) \} + \mathcal{L}^{-1} \{ G(s) \}$$

Time Shift

$$L^{-1}\{e^{-as}F(s)\}=u(t-a)f(t-a)$$

Scaling

For $\alpha \gt 0$.

$$L^{-1}\{F(\alpha s)\} = \frac{1}{\alpha} f\left(\frac{t}{\alpha}\right)$$