eFunda: Laplace Transforms
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Introduction

The Laplace transform is a powerful tool formulated to solve a wide variety of initial-value problems. The strategy is to transform the difficult differential equations into simple algebra problems where solutions can be easily obtained. One then applies the Inverse Laplace transform to retrieve the solutions of the original problems. This can be illustrated as follows:

 Initial-Value ProblemsODE's or PDE's
 Algebra Problems

Difficult

Very Easy
 Solutions ofInitial-Value Problems

 Solutions ofAlgebra Problems

Definition of the Laplace Transform

For a function defined on , its Laplace transform is denoted as obtained by the following integral:

where is real and is called the Laplace Transform Operator.

 Conditions for the Existence of a Laplace Transform of f(t) = F(s) 1) is piecewise continuous on . 2) is of exponential order as . That is, there exist real constants , , and such that for all . Note that conditions 1 and 2 are sufficient, but not necessary, for to exist.

Definition of the Inverse Laplace Transform

If the Laplace transform of is , then the we say that the Inverse Laplace Transform of is . Or,

where is called the Inverse Laplace Transform Operator.

 Conditions for the Existence of an Inverse Laplace Transform of F(s) = f(t) 1) . 2) is finite.