The Sommerfeld identity is a mathematical identity, due Arnold Sommerfeld, used in the theory of propagation of waves,
e
i
k
R
R
=
∫
0
∞
I
0
(
λ
r
)
e
−
μ
|
z
|
λ
d
λ
μ
where
μ
=
λ
2
−
k
2
is to be taken with positive real part, to ensure the convergence of the integral and its vanishing in the limit
z
→
±
∞
and
R
2
=
r
2
+
z
2
.
Here,
R
is the distance from the origin while
r
is the distance from the central axis of a cylinder as in the
(
r
,
ϕ
,
z
)
cylindrical coordinate system. Here the notation for Bessel functions follows the German convention, to be consistent with the original notation used by Sommerfeld. The function
I
0
(
z
)
is the zeroth-order Bessel function of the first kind, better known by the notation
I
0
(
z
)
=
J
0
(
i
z
)
in English literature. This identity is known as the Sommerfeld Identity [Ref.1,Pg.242].
In alternative notation, the Sommerfeld identity can be more easily seen as an expansion of a spherical wave in terms of cylindrically-symmetric waves,
e
i
k
0
r
r
=
i
∫
0
∞
d
k
ρ
k
ρ
k
z
J
0
(
k
ρ
ρ
)
e
i
k
z
|
z
|
Where
k
z
=
(
k
0
2
−
k
ρ
2
)
1
/
2
[Ref.2,Pg.66]. The notation used here is different form that above:
r
is now the distance from the origin and
ρ
is the radial distance in a cylindrical coordinate system defined as
(
ρ
,
ϕ
,
z
)
. The physical interpretation is that a spherical wave can be expanded into a summation of cylindrical waves in
ρ
direction, multiplied by a two-sided plane wave in the
z
direction; see the Jacobi-Anger expansion. The summation has to be taken over all the wavenumbers
k
ρ
.
The Sommerfeld identity is closely related to the two-dimensional Fourier transform with cylindrical symmetry, i.e., the Hankel transform. It is found by transforming the spherical wave along the in-plane coordinates (
x
,
y
, or
ρ
,
ϕ
) but not transforming along the height coordinate
z
.
