# Applied Mathematics/Laplace Transforms

The Laplace transform is an integral transform which is widely used in physics and engineering. Laplace transform is denoted as $\displaystyle\mathcal{L} \left\{f(t)\right\}$.

The Laplace transform is named after mathematician and astronomer Pierre-Simon Laplace.

## Definition

For a function f(t), using Napier's constant"e" and complex number "s", the Laplace transform F(s) is defined as follow:

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

The parameter s is a complex number:

$s = \sigma + i \omega, \,$ with real numbers σ and ω.

This $F(s)$ is the Laplace transform of f(t).

## Examples of Laplace transform

Examples of Laplace transform
function result of Laplace transform
$C$ (constant) $\frac{C}{s}$
$t$ $\frac{1}{s^2}$
$t^n$ (n is natural number) $\frac{n!}{s^{n+1}}$
$\frac{t^{n-1}}{(n-1)!}$ $\frac{1}{s^{n}}$
$e^{at}$ $\frac{1}{s-a}$
$e^{-at}$ $\frac{1}{s+a}$
${\rm cos}\ \omega t$ $\frac{s}{s^2 + {\omega}^2}$
${\rm sin}\ \omega t$ $\frac{\omega}{s^2 + {\omega}^2}$
$\frac{t^{n-1}}{\Gamma(n)}$ $\frac{1}{s^{n}}$ (n>0)
$\delta (t-a)$ (Delta function) $e^{-as}$
$H(t-a)$ (Heaviside function) $\frac {e^{-as}}{s}$

## Examples of calculation

(1)Suppose $f(t)=C$ (C = constant)
$\int_0^{\infty} e^{-st} C \,dt$
$=\frac{C}{s}$
$=F(s)$

(2)Suppose $f(t)=e^{-at}$
$\int_0^{\infty} e^{-st} \cdot e^{-at} \,dt$
$=\int_0^{\infty} e^{-(s+a)t} \,dt$
$=\left[\frac{-e^{-(s+a)t}}{s+a}\right]_0^\infty$
$=\frac{1}{s+a}$