PlanetMath: Laplace transform

PlanetMath

(more info)

Math for the people, by the people.

Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS

Login

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (211)
Orphanage
Unclass'd (1)
Unproven (472)
Corrections (36)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Owner confidence rating: High

Entry average rating: No information on entry rating

Laplace transform

(Definition)

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

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

provided that the integral converges. ¹ It suffices that

be defined when

and

can be complex. We will usually denote the Laplace transform of

by $\mathcal{L}\{f\}$ . Some of the most common Laplace transforms are:

$\mathcal{L}\{e^{at}\}\,=\,\frac{1}{s-a}\,,\,s>a$
$\mathcal{L}\{\cos(bt)\}\,=\,\frac{s}{s^{2}+b^{2}}\,,\,s>0$
$\mathcal{L}\{\sin(bt)\}\,=\,\frac{b}{s^{2}+b^{2}}\,,\,s>0$
$\mathcal{L}\{t^{n}\}\,=\,\frac{\Gamma(n+1)}{s^{n+1}}\,,\,s>0, n>-1.$
$\mathcal{L}\{f'\} = s \mathcal{L}\{f\}-\lim_{x \to 0+}f(x)$

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

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

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.

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

where

$\displaystyle (f*g) (t) = \int_0^t f(t-s) g(s) \, ds$

and

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

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.

Footnotes

...¹: Depending on the definition of integral one is using, one may prefer to define the Laplace transform as $\lim_{x \to 0+} \int_{x}^{\infty}e^{-st} f(t)\,dt$

"Laplace transform" is owned by rspuzio. [ full author list (3) | owner history (1) ]

(view preamble)

See Also: discrete Fourier transform

Attachments:: Mellin's inverse formula (Result) by pahio
Laplace transform of a Gaussian function (Application) by perucho
existence of Laplace transform (Theorem) by rspuzio
inverse Laplace transform of derivatives (Derivation) by pahio
delay theorem (Theorem) by pahio
Laplace transform of sine integral (Example) by pahio
table of Laplace transforms (Feature) by CWoo

Log in to rate this entry.
(view current ratings)

Cross-references: ordinary differential equation, sides, initial value problems, convolution, Fourier transform, analytic function, linear transformation, complex, converges, integral, interval, function
There are 18 references to this entry.

This is version 20 of Laplace transform, born on 2003-06-11, modified 2008-05-07.
Object id is 4343, canonical name is LaplaceTransform.
Accessed 18965 times total.

Classification:

44A10 (Integral transforms, operational calculus :: Laplace transform)

Pending Errata and Addenda

None.[ View all 11 ]

Discussion

forum policy

Interact

post | correct | update request | add derivation | add example | add (any)