Laplace Transform

Definition

Laplace Transformation is an Integral Transformation which transforms a function of the time domain $f(t)$ into a function of the Laplace Frequency to give $F(s)$.

$$\boxed{\mathcal{L}\{f(t)\}={\int }_0^{\infty }f(t)e^{-st}dt=F(s)}$$

where, $t\to$ Time domain, $f(t)\to$ Generalized frequency and $s\to$ Laplace Frequency domain.

Use case

The Laplace Transform is used to solve a particularly hairy Differential Equation. It transforms the D.E into an Algebraic equation in terms of a complex variable $s$. Once the variable is solved for $s$, an Inverse Laplace Transform can be applied to get the solution for $f(t)$.