Laplace transform

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

$$F(s)={\int }_0^{\mathrm{\infty }}{e}^{-st}f(t)𝑑t,$$

provided that the integral converges. It suffices that $f$ be defined when $t>0$ and $s$ can be complex. We willusually denote the Laplace transform of $f$ by $ℒ\{f\}$. Someof the most common Laplace transforms are:

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

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

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

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

5. $ℒ\{f^{\prime }\}=sℒ\{f\}-\underset{x\to 0+}{lim}f(x)$

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

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

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

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

$$ℒ\{f*g\}=ℒ\{f\}ℒ\{g\}$$

where

$$(f*g)(t)={\int }_0^{t}f(t-s)g(s)𝑑s$$

and

$$ℒ\{fg\}(s)={\int }_{c-i\mathrm{\infty }}^{c+i\mathrm{\infty }}ℒ\{f\}(z)ℒ\{g\}(s-z)𝑑z$$

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