33 Coulomb Functions Variables $\rho,\eta$ 33.7 Integral Representations 33.9 Expansions in Series of Bessel Functions

§33.8 Continued Fractions

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Keywords:: Coulomb functions: variables $\rho,\eta$ , continued fractions
Referenced by:: §33.23(v)
Permalink:: http://dlmf.nist.gov/33.8
See also:: Annotations for Ch.33

With arguments $\eta,\rho$ suppressed,

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

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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 , , and see (33.4.1).

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

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Symbols:: $\mathrm{i}$ : imaginary unit, ${H^{\NVar{\pm}}_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial functions, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable, $\eta$ : real parameter, : coefficient, : coefficient and : 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

$a=1+\ell\pm\mathrm{i}\eta,$

$b=-\ell\pm\mathrm{i}\eta,$

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

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Defines:: : coefficient (locally), : coefficient (locally) and : coefficient (locally)
Symbols:: $\mathrm{i}$ : imaginary unit, $\ell$ : 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

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

$F_{\ell}'=uF_{\ell},$

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Symbols:: $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $\ell$ : nonnegative integer, : ratio, : real part and : 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

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

$G_{\ell}'=q^{-1}(up-p^{2}-q^{2})F_{\ell}.$

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Symbols:: $G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial function, $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $\ell$ : nonnegative integer, : ratio, : real part and : 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).

33.7 Integral Representations 33.9 Expansions in Series of Bessel Functions

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