Sommerfeld identity
The Sommerfeld identity is a mathematical identity, due Arnold Sommerfeld, used in the theory of propagation of waves,
where
is to be taken with positive real part, to ensure the convergence of the integral and its vanishing in the limit and
Here, is the distance from the origin while is the distance from the central axis of a cylinder as in the 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 is the zeroth-order Bessel function of the first kind, better known by the notation in English literature.
This identity is known as the Sommerfeld Identity.
In alternative notation, the Sommerfeld identity can be more easily seen as an expansion of a spherical wave in terms of cylindrically-symmetric waves:
Where
The notation used here is different form that above: is now the distance from the origin and is the radial distance in a cylindrical coordinate system defined as. The physical interpretation is that a spherical wave can be expanded into a summation of cylindrical waves in direction, multiplied by a two-sided plane wave in the direction; see the Jacobi-Anger expansion. The summation has to be taken over all the wavenumbers.
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 but not transforming along the height coordinate.
