provided that the integral converges. 11Depending on the definition of integral one is using, one may prefer to define the Laplace transform as It suffices that be defined when and can be complex. We will usually denote the Laplace transform of by . Some of the most common Laplace transforms are:
For more particular Laplace transforms, see the table of Laplace
Notice the Laplace transform is a linear transformation. It is worth noting that, if
for some , then is an analytic function in the complex half-plane .
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 (http://planetmath.org/ImageEquation)”.
|Date of creation||2014-03-10 10:50:28|
|Last modified on||2014-03-10 10:50:28|
|Last modified by||pahio (2872)|