Laplace transform


Let f(t) be a functionMathworldPlanetmath defined on the interval  [0,). The Laplace transformDlmfMathworldPlanetmath of f(t) is the function F(s) defined by

F(s)=0e-stf(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 limx0+xe-stf(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 {f}. Some of the most common Laplace transforms are:

  1. 1.

    {eat}=1s-a,s>a

  2. 2.

    {cos(bt)}=ss2+b2,s>0

  3. 3.

    {sin(bt)}=bs2+b2,s>0

  4. 4.

    {tn}=Γ(n+1)sn+1,s>0,n>-1.

  5. 5.

    {f}=s{f}-limx0+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

0e-st|f(t)|𝑑t<

for some  s, then {f} is an analytic function in the complex half-plane {zz>s}.

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

{f*g}={f}{g}

where

(f*g)(t)=0tf(t-s)g(s)𝑑s

and

{fg}(s)=c-ic+i{f}(z){g}(s-z)𝑑z

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

Title Laplace transform
Canonical name LaplaceTransform
Date of creation 2014-03-10 10:50:28
Last modified on 2014-03-10 10:50:28
Owner rspuzio (6075)
Last modified by pahio (2872)
Numerical id 26
Author rspuzio (2872)
Entry type Definition
Classification msc 44A10
Related topic DiscreteFourierTransform
Related topic UsingLaplaceTransformToInitialValueProblems
Related topic UsingLaplaceTransformToSolveHeatEquation