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The Deltafunction as the Limit of Some Special Functions

​
function
Bessel
Airy
Hermite
Dirichlet
Fejér
sigmoid
closure
n
1
Another Demonstration gives some representations for the Dirac deltafunction as the limit of elementary functions (see the related links below).This Demonstration illustrates several additional limiting relations involving special functions and the deltafunction:
Bessel:
δ(x)=
lim
n∞
n
J
n
(n(x+1))
, (
n≠integer).
Airy:
δ(x)=
lim
n∞
nAi(nx)
.
Hermite: The
th
m
derivative of the deltafunction is given by
(m)
δ
(x)=
lim
n∞
m
(-1)
π
-
2
n
2
x
m-1
n
H
m
(nx)
. Shown is
δ'(x)
with
m=1
.
Dirichlet kernel:
Ш(x)=
lim
n∞
1
2π
sinn+
1
2
x
sin
x
2
, where the delta-comb or shah function is defined by
Ш(x)=
∞
∑
n=-∞
δ(x-2nπ)
. This is a periodic extension of the deltafunction.
Fejér kernel:
Ш(x)=
lim
n∞
1
2πn
2
sin
nx
2
sin
x
2
, with the same periodicity as the Dirichlet kernel.
Sigmoid: The derivative of the sigmoid function:
δ(x)=
lim
n∞
d
dx
1
1-
-nx
.
Closure: An orthonormal set of functions {
ϕ
n
(x)
} obeys the closure relation
lim
n∞
n
∑
k=0
*
ϕ
k
(x)
ϕ
k
(x')=δ(x-x')
. This is shown for orthonormalized Hermite functions
ϕ
n
(x)=
-1/2
(
n
2
n!
π
)
-
2
x
2
H
n
(x)
with
x'=1
.
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