E
Dirac δ-Function
Consider approximating an integral by a sum over strips Δx wide as shown
in figure E.1:
∫ x2
∑
f (x) dx ≃
f (x i )Δx.
(E.1)
x1
i
Consider the function δ(x − x j ) shown in figure E.2,
{ 1/Δx in the jth interval
δ(x − x j ) =
(E.2)
0
all others
Clearly this function has the properties
∑
f (x i )δ(x i − x j )Δx = f (x j )
( E . 3 )
i
and
∑
δ(x i − x j )Δx = 1.
(E.4)
i
In the limit as we pass to an integral form, we might expect (applying (E.1)
to the left-hand sides) that these equations reduce to
∫ x2
f (x)δ(x − x j ) dx = f (x j )
( E . 5 )
x1
and
∫ x2
δ(x − x j ) dx = 1
(E.6)
x1
provided that x 1 < x j < x 2 . Clearly such ‘δ-functions’ can easily be generalized to more dimensions, e.g. three dimensions:
dV = dx dy dz ≡ d
3
r
δ(r − r j ) ≡ δ(x − x j )δ(y − y j )δ(z − z j ). (E.7)
Informally, therefore, we can think of the δ-function as a function that is zero
everywhere except where its argument vanishes – at which point it is infinite
in such a way that its integral has unit area, and equations (E.5) and (E.6)
hold. Do such amazing functions exist? In fact, the informal idea just given
does not define a respectable mathematical function. More properly the use
of the ‘δ-function’ can be justified by introducing the notion of ‘distributions’
377
Dirac δ-Function
Consider approximating an integral by a sum over strips Δx wide as shown
in figure E.1:
∫ x2
∑
f (x) dx ≃
f (x i )Δx.
(E.1)
x1
i
Consider the function δ(x − x j ) shown in figure E.2,
{ 1/Δx in the jth interval
δ(x − x j ) =
(E.2)
0
all others
Clearly this function has the properties
∑
f (x i )δ(x i − x j )Δx = f (x j )
( E . 3 )
i
and
∑
δ(x i − x j )Δx = 1.
(E.4)
i
In the limit as we pass to an integral form, we might expect (applying (E.1)
to the left-hand sides) that these equations reduce to
∫ x2
f (x)δ(x − x j ) dx = f (x j )
( E . 5 )
x1
and
∫ x2
δ(x − x j ) dx = 1
(E.6)
x1
provided that x 1 < x j < x 2 . Clearly such ‘δ-functions’ can easily be generalized to more dimensions, e.g. three dimensions:
dV = dx dy dz ≡ d
3
r
δ(r − r j ) ≡ δ(x − x j )δ(y − y j )δ(z − z j ). (E.7)
Informally, therefore, we can think of the δ-function as a function that is zero
everywhere except where its argument vanishes – at which point it is infinite
in such a way that its integral has unit area, and equations (E.5) and (E.6)
hold. Do such amazing functions exist? In fact, the informal idea just given
does not define a respectable mathematical function. More properly the use
of the ‘δ-function’ can be justified by introducing the notion of ‘distributions’
377