# §33.8 Continued Fractions

With arguments $\eta,\rho$ suppressed,

 33.8.1 $\frac{\mathop{F_{\ell}\/}\nolimits'}{\mathop{F_{\ell}\/}\nolimits}=S_{\ell+1}-% \cfrac{R_{\ell+1}^{2}}{T_{\ell+1}-\cfrac{R_{\ell+2}^{2}}{T_{\ell+2}-\cdots}}.$

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

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

where

 33.8.3 $\displaystyle a$ $\displaystyle=1+\ell\pm i\eta,$ $\displaystyle b$ $\displaystyle=-\ell\pm i\eta,$ $\displaystyle c$ $\displaystyle=\pm i(1-(\eta/\rho)).$ 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

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 $u=\ifrac{\mathop{F_{\ell}\/}\nolimits'}{\mathop{F_{\ell}\/}\nolimits}$ and $p+iq=\ifrac{\mathop{{H^{+}_{\ell}}\/}\nolimits'}{\mathop{{H^{+}_{\ell}}\/}\nolimits}$, then

 33.8.4 $\displaystyle\mathop{F_{\ell}\/}\nolimits$ $\displaystyle=\pm(q^{-1}(u-p)^{2}+q)^{-1/2},$ $\displaystyle\mathop{F_{\ell}\/}\nolimits'$ $\displaystyle=u\mathop{F_{\ell}\/}\nolimits,$
 33.8.5 $\displaystyle\mathop{G_{\ell}\/}\nolimits$ $\displaystyle=q^{-1}(u-p)\mathop{F_{\ell}\/}\nolimits,$ $\displaystyle\mathop{G_{\ell}\/}\nolimits'$ $\displaystyle=q^{-1}(up-p^{2}-q^{2})\mathop{F_{\ell}\/}\nolimits.$

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).