Sommerfeld identity
The Sommerfeld identity is a mathematical identity, due Arnold Sommerfeld, used in the theory of propagation of waves, e i k R R = ∫ 0 ∞ I 0 ( λ r ) e − μ | z | λ d λ μ {\displaystyle {\frac {e^{ikR}}{R}}=\int \limits _{0}^{\infty }I_{0}(\lambda r)e^{-\mu \left|z\right|}{\frac {\lambda d\lambda }{\mu }}} where μ = λ 2 − k 2 {\displaystyle \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 → ± ∞ {\displaystyle z\rightarrow \pm \infty } and R 2 = r 2 + z 2 {\displaystyle R^{2}=r^{2}+z^{2}} . Here, R {\displaystyle R} is the distance from the origin while r {\displaystyle r} is the distance from the central axis of a cylinder as in the ( r , ϕ , z ) {\displaystyle (r,\phi ,z)} cylindrical coordinate system.