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

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

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

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

- 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”.

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## Attached Articles

Laplace transform of power function by pahio

Laplace transform of cosine and sine by pahio

Laplace transform of convolution by pahio

Laplace transform of derivative by pahio

Laplace transform of logarithm by pahio

rules for Laplace transform by pahio

integration of Laplace transform with respect to parameter by pahio

image equation by pahio

differentiation of Laplace transform with respect to parameter by pahio

## Corrections

only need t>0 ; specify s by Mathprof ✓

typo by yark ✓

Laplace transform by perucho ✓

Laplace transform by perucho ✓

## Comments

## Bilateral Transform

Should the bilateral laplace transform (x=-oo,oo rather than x=0-,oo)be referenced here?

## Re: Bilateral Transform

I would think that the best way to deal with the bilateral transform would be to add it as an attachments to this entry. While one is at it, it might also not be bad to add attachments about other variations on the theme such as the Laplace-Mellin transform and the Laplace-Stieltjes transform. Since each of these variants behaves somewhat differently and enough can be said about each one to warrant an entry, I think that adding attachments, I would think that would be the best way to go.

While I might get around to doing this one day, I have a lot else to do first, so don't hold your breath waiting. However, if you would like to add and attachment about the bilateral Laplace transform, please go right ahead.