Laplace transform

The Laplace transform is a powerful tool formulated to solve a wide variety ofinitial-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 Problems ODE's or PDE's Algebra Problems Difficult Very Easy Solutions of Initial-Value Problems    Solutions of Algebra Problems

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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.

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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.