Sommerfeld identity
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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 
. The function I0 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
- In(ρ) = Jn(iρ).
This identity is known as the Sommerfeld Identity [Ref.1,Pg.242].
An alternative form is
Where
- kz = (k0 − kρ)1 / 2
[Ref.2,Pg.66].
The physical interpretation is that a spherical wave can be expanded into a summation of cylyndrical waves in ρ direction, multiplied by a plane wave in the z direction. The summation has to be taken over all the wavenumbers kρ.
[edit] 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
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