## ABSTRACT

The Laplace transform is a mathematical tool based on integration that has a number of applications. In particular, it can simplify the solving of many differential equations. It is particularly useful when dealing with nonhomogeneous equations in which the forcing functions are not continuous. This makes it a valuable tool for engineers and scientists dealing with “real-world” applications.

This chapter introduces the Laplace transform. It provides both the basic integral definition of the Laplace transform and examples of the computation of the transforms of several basic functions. This leads to the first table of transforms. The linearity of the transform is also discussed, and the first of several important identities (the “first translation identity”) is developed and its use illustrated. The chapter ends with an in depth discussion of when a function is “Laplace transformable” and of some of the general properties of the transformed functions.

It should be noted that the Laplace transform is just one of many “integral transforms” in general use. Conceptually and computationally, it is probably the simplest. Understanding the Laplace transform makes it much easier to pick up the other transforms as needed.