Inverse Laplace transform

This is the transformation we do to get back into the continuous time domain from the complex frequency domain.

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

The Formal Definition

Formally, the inverse Laplace transform is defined as a complex contour integral known as the Bromwich integral:

$$f(t) = \frac{1}{2\pi j} \int_{\gamma - j\infty}^{\gamma + j\infty} F(s) e^{st} \, ds$$

here, $s = \gamma + j\omega$ and $\omega$ ranges from $-\infty$ to $+\infty$ $\gamma$ is a chosen real number that is entirely within the ROC of $F(s)$

Practical Solving problems

in practical, never really requires the formal definition to solve inverse laplace transforms. You can use partial fraction decomposition and Laplace Transform Table