5.4.1 The Diracδ-function
The defining relation for the Diracδ-function on the real line is such that for all test functions
f(a test function belongs to a dense set ofC∞functions onRwith compact support, often
called the Schwartz space), we have
∫+∞−∞dxδ(x−y)f(x) =f(y) (5.47)
It follows immediately thatδ(x) has support only atx= 0, so thatδ(x) = 0 unlessx= 0,
so thatxδ(x) = 0 for allx, and that
∫+∞−∞dxδ(x−y) = 1 (5.48)
for ally. The Diracδ-function may be viewed as a limit of a sequence of continuous or
smooth functionsδn(x), which converge in the sense that their pairing against any function
in the Schwartz space converges. Examples of such sequences are
δn(x) =
{0 |x|> 1 /(2n)
n |x|< 1 /(2n)
δn(x) =
n
√
π
e−n
(^2) x 2
δn(x) =
n
π
1
1 +n^2 x^2
(5.49)
Derivatives ofδ(x) may be defined by using the rule that the derivative and the integral
commute with one another. Since the test functions have compactsupport, one may always
integrate by parts under the integral without producing boundary terms. The defining
relation of the derivativeδ′(x−y) =∂xδ(x−y) is by
∫+∞−∞dxδ′(x−y)f(x) =−f′(y) (5.50)
As a result, we have for example,
xδ′(x) =−δ(x) (5.51)
Diracδ-functions may be multiplied under certain restricted conditions, essentially when
their supports are appropriately weighed. For example, we have the following integral for-
mulas,
∫+∞−∞