The Laplace Transform

The definition of the Laplace Transform that we will use is called a "one-sided" (or unilateral) Laplace Transform and is given by:

The Laplace Transform seems, at first, to be a fairly abstract and esoteric concept.  In practice, it allows one to (more) easily solve a huge variety of problems that involve linear systems, particularly differential equations.  It allows for compact representation of systems (via the "Transfer Function"), it simplifies evaluation of the convolution integral, and it turns problems involving differential equations into algebraic problems.  As indicated by the quotes in the animation above (from some students at Swarthmore College), it almost magically simplifies problems that otherwise are very difficult to solve.

There are a few things to note about the Laplace Transform.

Before we show how the Laplace Transform is useful, we need to lay some groundwork.  We start by finding the Laplace Transform of some functions and from there move on to finding properties of the Laplace Transform.  With tables of the Laplace Transform of Functions and Properties of the Laplace Transform it becomes possible to find the Laplace Transform of almost any function of interest without resorting to the integral shown above.   Applications of the Laplace Transform are discussed next - mostly the use of the Laplace Transform to solve differential equations.  Finally, the inverse Laplace Transform is covered (though this is a large enough topic that it has its own page elsewhere).