1550078481-Ordinary_Differential_Equations__Roberts_

The Laplace Trans] orm Method 251

DEFINITION Convolution of f(x) and g(x)

The convolution of f(x) and g(x) is

(1) j(x) * g(x) =fox j(x - ~)g(~) d~.

Making the change of variable ry = x - ~ in the integral appearing in (1), we

see that

f(x) * g(x) =fox f(x - ~)g(~) d~ = -1° J(ry)g(x - ry) dry

=fox g(x - ry)f(ry) dry= g(x) * J(x).

Hence, we have shown that the convolution operator is commutative. In-
deed, the convolution operator has many of the same properties as ordinary
multiplication. For instance,

and

f(x) * (g1(x) + g2(x)) = f(x) * g1(x) + f(x) * g2(x),

f(x) * (g(x) * h(x)) = (f(x) * g(x)) * h(x),

f(x)*O=O.

Consequently, the convolution operator may be thought of as a "generali zed
multiplication" operator. However , the convolution operator does not have
some of the properties of ordinary multiplication. For example, it is not true
for all functions f(x) that f(x) * 1 = f(x).

Suppose that f(x) and g(x) both have a Laplace transform for s >a. That

is , suppose

.C[f(x) ] = fo

00

f(x)e-sx dx and .C[g(x)] = fo

00

g(x)e-sx dx

both exist for s > a. By definition,

provided both integrals exist. The domain of integration, which is the region
above the positive x-axis and below the half-line~ = x, x ;::: 0, and the order