1550078481-Ordinary_Differential_Equations__Roberts_

268 Ordinary Differential Equations

where Eis a small positive constant. A graph of one function d, (x) is shown

in Figure 5. 7. Notice that for all E > 0, the total impulse of d, ( x) is

J

Id, = oo d. ( x) dx = 1• -^1 dx = 1.

• oo O E

1

£

y

y = d 8 (x)

Figure 5. 7 Graph of the Impulse Step Function d< ( x)

x

We would like to define an ideali zed impulse function o(x) to be the limit as
E ---+ o+ of the impulse step functions d< ( X) and we would like for the integral
of the idealized step function 0 ( x ) to be the limit as E ---+ o+ of Id,. Thus, we
define

{

oo,

o(x) = lim d€(x) =

<->O+ O

'

x=O

Notice that o ( x) is not a function in the usual sense but is a "generalized

function." The function o(x) is called the Dirac delta function in honor
of the British physicist Paul Adrien Maurice Dirac (1902-1982), who won
the Nobel prize in 19 33. Laurent Schwartz developed a mathematical theory
called distribution theory in the early 1950s. Within the framework of this
theory, the Dirac delta function is an acceptable generalized function and is
integrable with

(1)

J

oo o(x) dx = li m f

00

d€(x) dx = l.

-oo <->O+ .f _oo

The first integral of equation (1) is not the Riemann integral with which

we are familiar but the generali zed integral of distribution theory. Because

of the property of the Dirac delta function exhibited in equation (1), the

"function" o ( x) is often referred to as the unit impulse function. Following

the notation developed earlier for translating functions, we have

{

oo,

o(x - c) =

0,

X=C