Sommerfeld identity

The Sommerfeld identity is a mathematical identity, due Arnold Sommerfeld, used in the theory of propagation of waves,



\frac {R} = \int\limits_0^\infty I_0(\lambda r) e^{ - \mu \left| z \right| } \frac $$

where

\mu = \sqrt {\lambda ^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 \rightarrow \pm \infty $$ 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,\phi,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(iz)$$ 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:

\frac {r} = i\int\limits_0^\infty {dk_\rho  \frac J_0 (k_\rho \rho )e^{ik_z \left| z \right|} } $$ Where

k_z=(k_0^2-k_\rho^2)^{1/2} $$ The notation used here is different form that above: $$r$$ is now the distance from the origin and $$\rho$$ is the radial distance in a cylindrical coordinate system defined as $$(\rho,\phi,z)$$. The physical interpretation is that a spherical wave can be expanded into a summation of cylindrical waves in $$\rho$$ direction, multiplied by a two-sided plane wave in the $$z$$ direction; see the Jacobi-Anger expansion. The summation has to be taken over all the wavenumbers $$k_\rho$$.

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 $$\rho$$, $$\phi$$) but not transforming along the height coordinate $$z$$.