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
is the distance from the central axis of a cylinder as in the
Here the notation for Bessel functions follows the German convention, to be consistent with the original notation used by Sommerfeld.
is the zeroth-order Bessel function of the first kind, better known by the notation
[1] In alternative notation, the Sommerfeld identity can be more easily seen as an expansion of a spherical wave in terms of cylindrically-symmetric waves:[2] Where The notation used here is different form that above:
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 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
[3] This mathematical physics-related article is a stub.