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E. Dirac δ-Function
FIGURE E.3
The function (E.10) for finite N .
δ-function in this sense:
(
1
lim
for −∈/2 ≤ x ≤ ∈/2
δ(x) = ∊→0 ∈
(E.8)
0
for |x| > ∈/2
1
∈
δ(x) = lim
(E.9)
∊→0 π x 2 + ∈ 2
1 sin(N x)
δ(x) = lim
.
(E.10)
N →∞ π
x
The first of these is essentially the same as (E.2), and the second is a ‘smoother’
version of the first. The third is sketched in figure E.3: as N tends to infinity, the peak becomes infinitely high and narrow, but it still preserves unit
area.
Usually, under integral signs, δ-functions can be manipulated with no danger of obtaining a mathematically incorrect result. However, care must be
taken when products of two such generalized functions are encountered.
E. Dirac δ-Function
FIGURE E.3
The function (E.10) for finite N .
δ-function in this sense:
(
1
lim
for −∈/2 ≤ x ≤ ∈/2
δ(x) = ∊→0 ∈
(E.8)
0
for |x| > ∈/2
1
∈
δ(x) = lim
(E.9)
∊→0 π x 2 + ∈ 2
1 sin(N x)
δ(x) = lim
.
(E.10)
N →∞ π
x
The first of these is essentially the same as (E.2), and the second is a ‘smoother’
version of the first. The third is sketched in figure E.3: as N tends to infinity, the peak becomes infinitely high and narrow, but it still preserves unit
area.
Usually, under integral signs, δ-functions can be manipulated with no danger of obtaining a mathematically incorrect result. However, care must be
taken when products of two such generalized functions are encountered.