# §33.8 Continued Fractions

With arguments $\eta,\rho$ suppressed,

 33.8.1 $\frac{F_{\ell}'}{F_{\ell}}=S_{\ell+1}-\cfrac{R_{\ell+1}^{2}}{T_{\ell+1}-\cfrac% {R_{\ell+2}^{2}}{T_{\ell+2}-\cdots}}.$ ⓘ Symbols: $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$: regular Coulomb radial function, $\ell$: nonnegative integer, $R_{\ell}$: factor, $S_{\ell}$: factor and $T_{\ell}$: factor Referenced by: §33.8, §33.8 Permalink: http://dlmf.nist.gov/33.8.E1 Encodings: TeX, pMML, png See also: Annotations for §33.8 and Ch.33

For $R$, $S$, and $T$ see (33.4.1).

 33.8.2 $\frac{{H^{\pm}_{\ell}}'}{{H^{\pm}_{\ell}}}=c\pm\frac{\mathrm{i}}{\rho}\cfrac{% ab}{2(\rho-\eta\pm\mathrm{i})+\cfrac{(a+1)(b+1)}{2(\rho-\eta\pm 2\mathrm{i})+% \cdots}},$

where

 33.8.3 $\displaystyle a$ $\displaystyle=1+\ell\pm\mathrm{i}\eta,$ $\displaystyle b$ $\displaystyle=-\ell\pm\mathrm{i}\eta,$ $\displaystyle c$ $\displaystyle=\pm\mathrm{i}(1-(\eta/\rho)).$ ⓘ Defines: $a$: coefficient (locally), $b$: coefficient (locally) and $c$: coefficient (locally) Symbols: $\ell$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter Referenced by: §33.11 Permalink: http://dlmf.nist.gov/33.8.E3 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §33.8 and Ch.33

The continued fraction (33.8.1) converges for all finite values of $\rho$, and (33.8.2) converges for all $\rho\neq 0$.

If we denote $u=\ifrac{F_{\ell}'}{F_{\ell}}$ and $p+\mathrm{i}q=\ifrac{{H^{+}_{\ell}}'}{{H^{+}_{\ell}}}$, then

 33.8.4 $\displaystyle F_{\ell}$ $\displaystyle=\pm(q^{-1}(u-p)^{2}+q)^{-1/2},$ $\displaystyle F_{\ell}'$ $\displaystyle=uF_{\ell},$ ⓘ Symbols: $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$: regular Coulomb radial function, $\ell$: nonnegative integer, $u$: ratio, $p$: real part and $q$: imaginary part Referenced by: §33.8 Permalink: http://dlmf.nist.gov/33.8.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.8 and Ch.33
 33.8.5 $\displaystyle G_{\ell}$ $\displaystyle=q^{-1}(u-p)F_{\ell},$ $\displaystyle G_{\ell}'$ $\displaystyle=q^{-1}(up-p^{2}-q^{2})F_{\ell}.$

The ambiguous sign in (33.8.4) has to agree with that of the final denominator in (33.8.1) when the continued fraction has converged to the required precision. For proofs and further information see Barnett et al. (1974) and Barnett (1996).