33 Coulomb Functions Variables $r,\epsilon$ 33.18 Limiting Forms for Large $\ell$ 33.20 Expansions for Small $|\epsilon|$

§33.19 Power-Series Expansions in

Notes:: See Seaton (2002a, Eqs. 15–17, 31–48).
Keywords:: Coulomb functions: variables $r,\epsilon$
Referenced by:: §33.23(ii), §33.23(iii), §33.23(iii)
Permalink:: http://dlmf.nist.gov/33.19

33.19.1 $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)=r^{{\ell+1}}\sum_{{k=0}}^{% \infty}\alpha_{k}r^{k},$

Symbols:: $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : regular Coulomb function, : nonnegative integer, $\ell$ : nonnegative integer, : real variable, $\epsilon$ : real parameter and $\alpha_{k}$ : term
Referenced by:: §33.18, §33.19
Permalink:: http://dlmf.nist.gov/33.19.E1
Encodings:: TeX, pMML, png

where

33.19.2

$\alpha_{0}=2^{{\ell+1}}/(2\ell+1)!,$

$\alpha_{1}=-\alpha_{0}/(\ell+1),$

$k(k+2\ell+1)\alpha_{k}+2\alpha_{{k-1}}+\epsilon\alpha_{{k-2}}=0,$ $k=2,3,\dots$ .

Symbols:: : : factorial, : nonnegative integer, $\ell$ : nonnegative integer, $\epsilon$ : real parameter and $\alpha_{k}$ : term
Permalink:: http://dlmf.nist.gov/33.19.E2
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33.19.3 $2\pi\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)={\sum_{{k=0}}^{{2\ell}% }\frac{(2\ell-k)!\gamma_{k}}{k!}(2r)^{{k-\ell}}-\sum_{{k=0}}^{\infty}\delta_{k% }r^{{k+\ell+1}}}-A(\epsilon,\ell)\left(2\mathop{\ln\/}\nolimits|2r/\kappa|+% \realpart{\mathop{\psi\/}\nolimits\!\left(\ell+1+\kappa\right)}+\realpart{% \mathop{\psi\/}\nolimits\!\left(-\ell+\kappa\right)}\right){\mathop{f\/}% \nolimits\!\left(\epsilon,\ell;r\right),}$ $r\neq 0$ .

Here $\kappa$ is defined by (33.14.6), $A(\epsilon,\ell)$ is defined by (33.14.11) or (33.14.12), $\gamma_{0}=1$ , $\gamma_{1}=1$ , and

33.19.4 $\gamma_{k}-\gamma_{{k-1}}+\tfrac{1}{4}(k-1)(k-2\ell-2)\epsilon\gamma_{{k-2}}=0,$ $k=2,3,\dots$ .

Symbols:: : nonnegative integer, $\ell$ : nonnegative integer, $\epsilon$ : real parameter and $\gamma_{k}$ : term
Permalink:: http://dlmf.nist.gov/33.19.E4
Encodings:: TeX, pMML, png

Also,

33.19.5

$\delta_{0}=\left(\beta_{{2\ell+1}}-2(\mathop{\psi\/}\nolimits\!\left(2\ell+2% \right)+\mathop{\psi\/}\nolimits\!\left(1\right))A(\epsilon,\ell)\right)\alpha% _{0},$

$\delta_{1}=\left(\beta_{{2\ell+2}}-2(\mathop{\psi\/}\nolimits\!\left(2\ell+3% \right)+\mathop{\psi\/}\nolimits\!\left(2\right))A(\epsilon,\ell)\right)\alpha% _{1},$

Symbols:: $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, : nonnegative integer, $\ell$ : nonnegative integer, $\epsilon$ : real parameter, $A(\epsilon,\ell)$ : function, $\alpha_{k}$ : term, $\delta_{k}$ : term and $\beta_{k}$ : term
Permalink:: http://dlmf.nist.gov/33.19.E5
Encodings:: TeX, TeX, pMML, pMML, png, png

33.19.6 $k(k+2\ell+1)\delta_{k}+2\delta_{{k-1}}+\epsilon\delta_{{k-2}}+2(2k+2\ell+1)A(% \epsilon,\ell)\alpha_{k}=0,$ $k=2,3,\dots$ ,

Symbols:: : nonnegative integer, $\ell$ : nonnegative integer, $\epsilon$ : real parameter, $A(\epsilon,\ell)$ : function, $\alpha_{k}$ : term and $\delta_{k}$ : term
Permalink:: http://dlmf.nist.gov/33.19.E6
Encodings:: TeX, pMML, png

with $\beta_{0}=\beta_{1}=0$ , and

33.19.7 $\beta_{k}-\beta_{{k-1}}+\tfrac{1}{4}(k-1)(k-2\ell-2)\epsilon\beta_{{k-2}}+% \tfrac{1}{2}(k-1)\epsilon\gamma_{{k-2}}=0,$ $k=2,3,\dots$ .

Symbols:: : nonnegative integer, $\ell$ : nonnegative integer, $\epsilon$ : real parameter, $\gamma_{k}$ : term and $\beta_{k}$ : term
Permalink:: http://dlmf.nist.gov/33.19.E7
Encodings:: TeX, pMML, png

The expansions (33.19.1) and (33.19.3) converge for all finite values of , except in the case of (33.19.3).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 33.18 Limiting Forms for Large $\ell$ 33.20 Expansions for Small $|\epsilon|$