The Laplace transform converts a function of time into a function of a complex variable $s$:

$$\mathcal L{f(t)}(s)=\int_0^\infty e^{-st}f(t),dt,$$

whenever the improper integral converges.

The transform is linear:

$$\mathcal L{af+bg}=a\mathcal L{f}+b\mathcal L{g}.$$

Common transforms include

$$\mathcal L{1}=\frac1s,$$

$$\mathcal L{e^{at}}=\frac1{s-a},$$

and

$$\mathcal L{\sin(\omega t)}=\frac{\omega}{s^2+\omega^2}.$$

The exponential factor weights later times and turns many operations in time into algebraic operations in $s$. This makes the Laplace transform useful for linear differential equations and, later, system analysis.