## Definition of Laplace Transform

Let f(t) be a given function which is defined for t ≥ 0. If there exists a function F(s) so that

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

then F(s) is called the Laplace Transform of f(t), and will be denoted by $\mathcal{L} \left\{f(t)\right\}$. Notice the integrator e-st dt where s is a parameter which may be real or complex.

Thus,

$\mathcal{L} \left\{f(t)\right\} = F(s)$

The symbol $\mathcal{L}$ which transform f(t) into F(s) is called the Laplace transform operator.

Laplace transformation is a powerful method of solving linear differential equations. It reduces the problem of solving differential equations into algebraic equations. For more information about the application of Laplace transform in engineering, see this Wikipedia article and this Wolfram article.

## Laplace Transform by Direct Integration

To get the Laplace transform of the given function   $f(t)$,   multiply   $f(t)$   by   $e^{-st}$   and integrate with respect to   $t$   from zero to infinity. In symbol,

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

Example
Find the Laplace transform of f(t) = 1 when t > 0.

Solution

## Table of Laplace Transforms of Elementary Functions

Below are some functions f(t) and their Laplace transforms F(s).

$f(t)$ $F(s) = \mathcal{L} \left\{f(t)\right\}$
$1$ $\dfrac{1}{s}$
$t$ $\dfrac{1}{s^2}$
$t^2$ $\dfrac{2}{s^3}$
$t^n, \, (n = 1, \, 2, \, 3\, ...)$ $\dfrac{n!}{s^{n + 1}}$
$t^a, \, (a \, \text{ is positive})$ $\dfrac{\Gamma(a + 1)}{s^{a + 1}}$
$e^{at}$ $\dfrac{1}{s - a}$
$\sin bt$ $\dfrac{b}{s^2 + b^2}$
$\cos bt$ $\dfrac{s}{s^2 + b^2}$
$\sinh at$ $\dfrac{a}{s^2 - a^2}$
$\cosh at$ $\dfrac{s}{s^2 - a^2}$

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