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


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.


  2. 2.


  3. 3.


  4. 4.


  5. 5.


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


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.






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 (”.

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