ABSTRACT
“Convolution” is an operation involving two functions that turns out to be very useful in a number of applications. There are two particular reasons for introducing it here. First of all, convolution provides a way to deal with inverse transforms of fairly arbitrary products of functions. Secondly, it is a major element in some relatively simple formulas for solving a number of differential equations.
This chapter begins with the basic formula for the convolution of two functions and a brief discussion of some issues regarding notation. The basic computations are illustrated, and simple, yet useful, identities are derived for the convolution. Next is the derivation of the main identity of the chapter: the equality of the Laplace transform of the convolution of two functions and the product of the corresponding transformed functions. Equivalently, the inverse transform of a product is the corresponding convolution. Its use is then demonstrated in the computation of some inverse transforms and, more importantly, in finding a general formula for the solutions of an important class of nonhomogeneous differential equations (Duhamel’s principle).
