Sommerfeld identity

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

:$frac$e^{ik R} {R} = intlimits_0^infty I_0(lambda r) e^{ - mu left| z ight| } fraclambda d lambdamu

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 $z\; ightarrow\; pm\; infty$ 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. The function $I\_0$ is a Bessel function. Here the notation for Bessel functions follows the German convention, to be consistent with the original notation used by Sommerfeld. In English literature it is more common to use

:$I\_n(\; ho)=J\_n(i\; ho)$.

This identity is known as the Sommerfeld Identity [Ref.1,Pg.242] .

An alternative form is

:$frac$e^{ik_0 r} {r} = iintlimits_0^infty {dk_ ho frack_ ho k_z J_0 (k_ ho ho )e^{ik_z left| z ight } Where:$k\_z=(k\_0^2-k\_\; ho^2)^\{1/2\}$

[Ref.2,Pg.66] . The notation used here is different form that above: $r$ is now the distance from the origin and $ho$ is the axial distance in a cylindrical system defined as $(\; ho,phi,z)$.

The physical interpretation is that a spherical wave can be expanded into a summationof cylindrical waves in $ho$ direction, multiplied by a plane wave in the $z$ direction.The summation has to be taken over all the wavenumbers $k\_\; ho$.

References

# Sommerfeld, A.,"Partial Differential Equations in Physics",Academic Press,New York,1964
# Chew, W.C.,"Waves and Fields in Inhomogenous Media",Van Nostrand Reinhold,New York,1990