# Laplace transform

Let $f(t)$ be a function defined on the interval  $[0,\,\infty)$. The Laplace transform of $f(t)$ is the function $F(s)$ defined by

 $F(s)\,=\,\int_{0}^{\infty}e^{-st}f(t)\,dt,$

provided that the integral converges. 11Depending on the definition of integral one is using, one may prefer to define the Laplace transform as $\lim_{x\to 0+}\int_{x}^{\infty}e^{-st}f(t)\,dt$ It suffices that $f$ be defined when $t>0$ and $s$ can be complex. We will usually denote the Laplace transform of $f$ by $\mathcal{L}\{f\}$. Some of the most common Laplace transforms are:

1. 1.

$\displaystyle\mathcal{L}\{e^{at}\}\,=\,\frac{1}{s-a},\;\;s>a$

2. 2.

$\displaystyle\mathcal{L}\{\cos(bt)\}\,=\,\frac{s}{s^{2}+b^{2}},\;\;s>0$

3. 3.

$\displaystyle\mathcal{L}\{\sin(bt)\}\,=\,\frac{b}{s^{2}+b^{2}},\;\;s>0$

4. 4.

$\displaystyle\mathcal{L}\{t^{n}\}\,=\,\frac{\Gamma(n+1)}{s^{n+1}},\;\;s>0,\;n>% -1.$

5. 5.

$\displaystyle\mathcal{L}\{f^{\prime}\}\,=\,s\mathcal{L}\{f\}-\lim_{x\to 0+}f(x)$

For more particular Laplace transforms, see the table of Laplace transforms.

Notice the Laplace transform is a linear transformation. It is worth noting that, if

 $\int_{0}^{\infty}e^{-st}|f(t)|\,dt<\infty$

for some  $s\in\mathbb{R}$, then $\mathcal{L}\{f\}$ is an analytic function in the complex half-plane $\{z\mid\;\Re z>s\}$.

Much like the Fourier transform, the Laplace transform has a convolution. However, the form of the convolution used is different.

 $\mathcal{L}\{f*g\}=\mathcal{L}\{f\}\mathcal{L}\{g\}$

where

 $(f*g)(t)=\int_{0}^{t}f(t-s)g(s)\,ds$

and

 $\mathcal{L}\{fg\}(s)=\int_{c-i\infty}^{c+i\infty}\mathcal{L}\{f\}(z)\mathcal{L% }\{g\}(s-z)\,dz$

The most popular usage of the Laplace transform is to solve initial value problems by taking the Laplace transform of both sides of an ordinary differential equation; see the entry “image equation (http://planetmath.org/ImageEquation)”.

Title Laplace transform LaplaceTransform 2014-03-10 10:50:28 2014-03-10 10:50:28 rspuzio (6075) pahio (2872) 26 rspuzio (2872) Definition msc 44A10 DiscreteFourierTransform UsingLaplaceTransformToInitialValueProblems UsingLaplaceTransformToSolveHeatEquation