# Coulomb phase shift

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##### 1: 33.25 Approximations
###### §33.25 Approximations
Cody and Hillstrom (1970) provides rational approximations of the phase shift ${\sigma_{0}}\left(\eta\right)=\operatorname{ph}\Gamma\left(1+\mathrm{i}\eta\right)$ (see (33.2.10)) for the ranges $0\leq\eta\leq 2$, $2\leq\eta\leq 4$, and $4\leq\eta\leq\infty$. …
##### 2: 33.13 Complex Variable and Parameters
The quantities $C_{\ell}\left(\eta\right)$, ${\sigma_{\ell}}\left(\eta\right)$, and $R_{\ell}$, given by (33.2.6), (33.2.10), and (33.4.1), respectively, must be defined consistently so that
33.13.1 $C_{\ell}\left(\eta\right)=2^{\ell}e^{\mathrm{i}{\sigma_{\ell}}\left(\eta\right% )-(\pi\eta/2)}\Gamma\left(\ell+1-\mathrm{i}\eta\right)/\Gamma\left(2\ell+2% \right),$
##### 3: 33.2 Definitions and Basic Properties
33.2.9 ${\theta_{\ell}}\left(\eta,\rho\right)=\rho-\eta\ln\left(2\rho\right)-\tfrac{1}% {2}\ell\pi+{\sigma_{\ell}}\left(\eta\right),$
33.2.10 ${\sigma_{\ell}}\left(\eta\right)=\operatorname{ph}\Gamma\left(\ell+1+\mathrm{i% }\eta\right),$
${\sigma_{\ell}}\left(\eta\right)$ is the Coulomb phase shift. … Also, $e^{\mp\mathrm{i}{\sigma_{\ell}}\left(\eta\right)}{H^{\pm}_{\ell}}\left(\eta,% \rho\right)$ are analytic functions of $\eta$ when $-\infty<\eta<\infty$. …
##### 4: 33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta\right|$
${\sigma_{0}}\left(\eta\right)=\eta(\ln\eta-1)+\tfrac{1}{4}\pi+o\left(1\right),$
${\sigma_{0}}\left(\eta\right)=\eta(\ln\left(-\eta\right)-1)-\tfrac{1}{4}\pi+o% \left(1\right),$
##### 7: 5.20 Physical Applications
###### Rutherford Scattering
In nonrelativistic quantum mechanics, collisions between two charged particles are described with the aid of the Coulomb phase shift $\operatorname{ph}\Gamma\left(\ell+1+\mathrm{i}\eta\right)$; see (33.2.10) and Clark (1979). …
##### 8: Bibliography C
• C. W. Clark (1979) Coulomb phase shift. American Journal of Physics 47 (8), pp. 683–684.
• W. J. Cody and K. E. Hillstrom (1970) Chebyshev approximations for the Coulomb phase shift. Math. Comp. 24 (111), pp. 671–677.
• ##### 9: 33.11 Asymptotic Expansions for Large $\rho$
33.11.1 ${H^{\pm}_{\ell}}\left(\eta,\rho\right)\sim e^{\pm\mathrm{i}{\theta_{\ell}}% \left(\eta,\rho\right)}\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}{\left(b% \right)_{k}}}{k!(\pm 2\mathrm{i}\rho)^{k}},$
##### 10: 33.6 Power-Series Expansions in $\rho$
33.6.5 ${H^{\pm}_{\ell}}\left(\eta,\rho\right)=\frac{e^{\pm\mathrm{i}{\theta_{\ell}}% \left(\eta,\rho\right)}}{(2\ell+1)!\Gamma\left(-\ell\pm\mathrm{i}\eta\right)}% \left(\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}}{{\left(2\ell+2\right)_{k}% }k!}(\mp 2\mathrm{i}\rho)^{a+k}\left(\ln\left(\mp 2\mathrm{i}\rho\right)+\psi% \left(a+k\right)-\psi\left(1+k\right)-\psi\left(2\ell+2+k\right)\right)-\sum_{% k=1}^{2\ell+1}\frac{(2\ell+1)!(k-1)!}{(2\ell+1-k)!{\left(1-a\right)_{k}}}(\mp 2% \mathrm{i}\rho)^{a-k}\right),$