Laplace transform

Definition

Let $f : [0, \infty) \to \mathbb{C}$ be a function that is integrable on each interval $[0, T]$ with $T > 0$. The Laplace transform of $f$ is the function $F = \mathcal{L}\{f\}$ given by $$F(s) = \int_0^\infty f(t)\, e^{-s t}\, dt = \lim_{T \to \infty} \int_0^T f(t)\, e^{-s t}\, dt,$$ for each complex number $s$ for which the limit exists.

Remarks

Let $f$ be of exponential order: there are constants $M, a$ with $|f(t)| \le M e^{a t}$ for all $t \ge 0$. Then the integral converges absolutely for $\operatorname{Re} s > a$. Some transforms: $\mathcal{L}\{1\}(s) = 1/s$ and $\mathcal{L}\{e^{a t}\}(s) = 1/(s - a)$ for $\operatorname{Re} s > \operatorname{Re} a$, and $\mathcal{L}\{\sin \omega t\}(s) = \omega / (s^2 + \omega^2)$ for $\operatorname{Re} s > 0$ and real $\omega$. The transform is linear: $\mathcal{L}\{c f + g\} = c \mathcal{L}\{f\} + \mathcal{L}\{g\}$ where both sides converge. It turns derivatives into multiplication by $s$. See the Laplace transform of a derivative.

Explanations

L1 human

The Laplace transform turns a function of time into a function of a complex variable $s$. It weights the function by $e^{-st}$, which decays fast, and adds up the result over all positive times. Its main use is to turn linear differential equations into algebraic equations.