Laplace transform

The Laplace transform is $\hat{f}(p)={\int }_0^{\infty }{e}^{-pt}f(t)dt$ and converts differentiation into multiplication by $p$ plus initial data.

The Laplace transform is a powerful integral transform used in various fields of engineering, physics, and mathematics to analyze and solve differential equations and system dynamics. It converts a function of time, typically denoted as \( f(t) \), which is often defined for \( t \geq 0 \), into a function of a complex variable \( s \), denoted as \( F(s) \).