Sommerfeld Identity

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

:$\frac = \int\limits_0^\infty I_0(\lambda r) e^ \frac$

where
:$\mu = \sqrt$
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:
:$\frac = i\int\limits_0^\infty$
Where
:$k_z=(k_0^2-k_\rho^2)^$
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 $z$.

Notes

References

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Mathematical identities
Wave mechanics

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