§33.8 Continued Fractions

With arguments $\eta ,\rho$ suppressed,

33.8.1		$$\frac{F_{\mathrm{\ell }}^{\prime }}{F_{\mathrm{\ell }}}=S_{\mathrm{\ell }+1}-\frac{R_{\mathrm{\ell }+1}^2}{T_{\mathrm{\ell }+1}-}\frac{R_{\mathrm{\ell }+2}^2}{T_{\mathrm{\ell }+2}-}\mathrm{\cdots }.$$
ⓘ Symbols: $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $R_{\mathrm{\ell }}$: factor, $S_{\mathrm{\ell }}$: factor and $T_{\mathrm{\ell }}$: 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		$$\frac{H_{\mathrm{\ell }}^{\pm }{}^{\prime }}{H_{\mathrm{\ell }}^{\pm }}=c\pm \frac{\mathrm{i}}{\rho }\frac{ab}{2(\rho -\eta \pm \mathrm{i})+}\frac{(a+1)(b+1)}{2(\rho -\eta \pm 2\mathrm{i})+}\mathrm{\cdots },$$
ⓘ Symbols: $\mathrm{i}$: imaginary unit, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter, $a$: coefficient, $b$: coefficient and $c$: coefficient Referenced by: §33.8 Permalink: http://dlmf.nist.gov/33.8.E2 Encodings: TeX, pMML, png See also: Annotations for §33.8 and Ch.33

where

33.8.3		$a$	$=1+\mathrm{\ell }\pm \mathrm{i}\eta ,$
		$b$	$=-\mathrm{\ell }\pm \mathrm{i}\eta ,$
		$c$	$=\pm \mathrm{i}(1-(\eta /\rho )).$
ⓘ Defines: $a$: coefficient (locally), $b$: coefficient (locally) and $c$: coefficient (locally) Symbols: $\mathrm{i}$: imaginary unit, $\mathrm{\ell }$: 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 $\rho \ne 0$.

If we denote $u=F_{\mathrm{\ell }}^{\prime }/F_{\mathrm{\ell }}$ and $p+\mathrm{i}q=H_{\mathrm{\ell }}^{+}{}^{\prime }/H_{\mathrm{\ell }}^{+}$, then

33.8.4		$F_{\mathrm{\ell }}$	$=\pm {(q^{-1}{(u-p)}^2+q)}^{-1/2},$
		$F_{\mathrm{\ell }}^{\prime }$	$=u{F}_{\mathrm{\ell }},$
ⓘ Symbols: $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: 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		$G_{\mathrm{\ell }}$	$=q^{-1}(u-p)F_{\mathrm{\ell }},$
		$G_{\mathrm{\ell }}^{\prime }$	$=q^{-1}(up-p^2-q^2)F_{\mathrm{\ell }}.$
ⓘ Symbols: $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $u$: ratio, $p$: real part and $q$: imaginary part Permalink: http://dlmf.nist.gov/33.8.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.8 and Ch.33

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).