Sommerfeld identity

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

\frac{{e^{ik r} }} {r} = \int\limits_0^\infty I_0(\lambda r) \frac{{\lambda d \lambda}}{{\mu}}

where

\mu = \sqrt {\lambda ^2 - k^2 }

is to be taken with positive real part, to ensure the convergence of the integral. 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].

An alternative form is

\frac{{e^{ik_0 r} }} {r} = i\int\limits_0^\infty {dk_\rho \frac{{k_\rho }} {{k_z }}J_0 (k_\rho \rho )e^{ik_z \left| z \right|} }

[Ref.2].

[edit] References

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


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