Fourier transform of outgoing spherical waves

[AlexBal]

Please, can anyone explain how formula (5) is obtained in J.J. Barton article ''Approximate translation of screened spherical waves" . Phys.Rew. A ,Vol.32,N2, 1985. ?
https://doi.org/10.1103/PhysRevA.32.1019
The same formula are given in the book Pendry J.B. "Low energy electron diffraction. The theory and its application to deformation of surface structure. Academic Press 1974 on the
page 272.
In my derivation, an additional factor is obtained |K|^l / k^{l+1}.

Thanks in advance.

 

[azntoon]

Formula (5) in the article by J.J. Barton is derived using the method of stationary phase to approximate a spherical wave integral. The stationary phase approximation assumes that the integrand is slowly changing in the region of the stationary point, which allows us to replace it with the value of the integrand at the point of stationary phase. This leads to the formula (5), which states that the integral can be approximated asI = \frac{A}{2\pi}\int_0^{2\pi} d\phi e^{i l \phi} f(\phi) \approx \frac{A}{2\pi} f_0 e^{il\phi_0},where A is an amplitude factor, l is the angular momentum quantum number, $\phi$ is the angle of integration, $f(\phi)$ is the integrand, $f_0$ is the value of the integrand at the point of stationary phase $\phi_0$, and $e^{il\phi_0}$ is the phase factor associated with the angular momentum.