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 formula_3 and
Here, formula_5 is the distance from the origin while formula_6 is the distance from the central axis of a cylinder as in the formula_7 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 formula_8 is the zeroth-order Bessel function of the first kind, better known by the notation formula_9 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: formula_6 is now the distance from the origin and formula_13 is the radial distance in a cylindrical coordinate system defined as formula_14. The physical interpretation is that a spherical wave can be expanded into a summation of cylindrical waves in formula_13 direction, multiplied by a two-sided plane wave in the formula_16 direction; see the Jacobi-Anger expansion. The summation has to be taken over all the wavenumbers formula_17.
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 (formula_18,formula_19, or formula_13, formula_21) but not transforming along the height coordinate formula_16. 