33 Coulomb FunctionsVariables $\rho ,\eta $33.10 Limiting Forms for Large $\rho $ or Large $\left|\eta \right|$33.12 Asymptotic Expansions for Large $\eta $

For large $\rho $, with $\mathrm{\ell}$ and $\eta $ fixed,

33.11.1 | $${H}_{\mathrm{\ell}}^{\pm}(\eta ,\rho )\sim {\mathrm{e}}^{\pm \mathrm{i}{\theta}_{\mathrm{\ell}}(\eta ,\rho )}\sum _{k=0}^{\mathrm{\infty}}\frac{{\left(a\right)}_{k}{\left(b\right)}_{k}}{k!{(\pm 2\mathrm{i}\rho )}^{k}},$$ | ||

where ${\theta}_{\mathrm{\ell}}(\eta ,\rho )$ is defined by (33.2.9), and $a$ and $b$ are defined by (33.8.3).

An equivalent formulation is given by

33.11.2 | ${F}_{\mathrm{\ell}}(\eta ,\rho )$ | $=g(\eta ,\rho )\mathrm{cos}{\theta}_{\mathrm{\ell}}+f(\eta ,\rho )\mathrm{sin}{\theta}_{\mathrm{\ell}},$ | ||

${G}_{\mathrm{\ell}}(\eta ,\rho )$ | $=f(\eta ,\rho )\mathrm{cos}{\theta}_{\mathrm{\ell}}-g(\eta ,\rho )\mathrm{sin}{\theta}_{\mathrm{\ell}},$ | |||

33.11.3 | ${F}_{\mathrm{\ell}}^{\prime}(\eta ,\rho )$ | $=\widehat{g}(\eta ,\rho )\mathrm{cos}{\theta}_{\mathrm{\ell}}+\widehat{f}(\eta ,\rho )\mathrm{sin}{\theta}_{\mathrm{\ell}},$ | ||

${G}_{\mathrm{\ell}}^{\prime}(\eta ,\rho )$ | $=\widehat{f}(\eta ,\rho )\mathrm{cos}{\theta}_{\mathrm{\ell}}-\widehat{g}(\eta ,\rho )\mathrm{sin}{\theta}_{\mathrm{\ell}},$ | |||

33.11.4 | $${H}_{\mathrm{\ell}}^{\pm}(\eta ,\rho )={\mathrm{e}}^{\pm \mathrm{i}{\theta}_{\mathrm{\ell}}}(f(\eta ,\rho )\pm \mathrm{i}g(\eta ,\rho )),$$ | ||

where

33.11.5 | $f(\eta ,\rho )$ | $\sim {\displaystyle \sum _{k=0}^{\mathrm{\infty}}}{f}_{k},$ | ||

$g(\eta ,\rho )$ | $\sim {\displaystyle \sum _{k=0}^{\mathrm{\infty}}}{g}_{k},$ | |||

33.11.6 | $\widehat{f}(\eta ,\rho )$ | $\sim {\displaystyle \sum _{k=0}^{\mathrm{\infty}}}{\widehat{f}}_{k},$ | ||

$\widehat{g}(\eta ,\rho )$ | $\sim {\displaystyle \sum _{k=0}^{\mathrm{\infty}}}{\widehat{g}}_{k},$ | |||

33.11.7 | $$g(\eta ,\rho )\widehat{f}(\eta ,\rho )-f(\eta ,\rho )\widehat{g}(\eta ,\rho )=1.$$ | ||

Here ${f}_{0}=1$, ${g}_{0}=0$, ${\widehat{f}}_{0}=0$, ${\widehat{g}}_{0}=1-(\eta /\rho )$, and for $k=0,1,2,\mathrm{\dots}$,

33.11.8 | ${f}_{k+1}$ | $={\lambda}_{k}{f}_{k}-{\mu}_{k}{g}_{k},$ | ||

${g}_{k+1}$ | $={\lambda}_{k}{g}_{k}+{\mu}_{k}{f}_{k},$ | |||

${\widehat{f}}_{k+1}$ | $={\lambda}_{k}{\widehat{f}}_{k}-{\mu}_{k}{\widehat{g}}_{k}-({f}_{k+1}/\rho ),$ | |||

${\widehat{g}}_{k+1}$ | $={\lambda}_{k}{\widehat{g}}_{k}+{\mu}_{k}{\widehat{f}}_{k}-({g}_{k+1}/\rho ),$ | |||

where

33.11.9 | ${\lambda}_{k}$ | $={\displaystyle \frac{(2k+1)\eta}{(2k+2)\rho}},$ | ||

${\mu}_{k}$ | $={\displaystyle \frac{\mathrm{\ell}(\mathrm{\ell}+1)-k(k+1)+{\eta}^{2}}{(2k+2)\rho}}.$ | |||