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 ${\displaystyle z\rightarrow \pm \infty }$ and

Here, ${\displaystyle R}$ is the distance from the origin while ${\displaystyle r}$ is the distance from the central axis of a cylinder as in the ${\displaystyle (r,\phi ,z)}$ 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 ${\displaystyle I_{0}(z)}$ is the zeroth-order Bessel function of the first kind, better known by the notation ${\displaystyle I_{0}(z)=J_{0}(iz)}$ in English literature. This identity is known as the Sommerfeld Identity [Ref.1,Pg.242].

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