§33.8 Continued Fractions

Keywords:: Coulomb functions: variables $\rho,\eta$ , continued fractions
Referenced by:: §33.23(v)
Permalink:: http://dlmf.nist.gov/33.8

With arguments $\eta,\rho$ suppressed,

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

Symbols:: $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : 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

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

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

Symbols:: $\mathop{{H^{{\pm}}_{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial functions, $\ell$ : 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

where

33.8.3

$a=1+\ell\pm i\eta,$

$b=-\ell\pm i\eta,$

$c=\pm i(1-(\eta/\rho)).$

Symbols:: $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable, $\eta$ : real parameter, $a$ : coefficient, $b$ : coefficient and $c$ : coefficient
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 $u=\ifrac{{\mathop{F_{{\ell}}\/}\nolimits^{{\prime}}}}{\mathop{F_{{\ell}}\/}\nolimits}$ and $p+iq=\ifrac{{\mathop{{H^{{+}}_{{\ell}}}\/}\nolimits^{{\prime}}}}{\mathop{{H^{{+}}_{{\ell}}}\/}\nolimits}$ , then

33.8.4

$\mathop{F_{{\ell}}\/}\nolimits=\pm(q^{{-1}}(u-p)^{2}+q)^{{-1/2}},$

${\mathop{F_{{\ell}}\/}\nolimits^{{\prime}}}=u\mathop{F_{{\ell}}\/}\nolimits,$

Symbols:: $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : 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
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33.8.5

$\mathop{G_{{\ell}}\/}\nolimits=q^{{-1}}(u-p)\mathop{F_{{\ell}}\/}\nolimits,$

${\mathop{G_{{\ell}}\/}\nolimits^{{\prime}}}=q^{{-1}}(up-p^{2}-q^{2})\mathop{F_{{\ell}}\/}\nolimits.$

Symbols:: $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial function, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : regular Coulomb radial function, $\ell$ : 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

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).