§33.19 Power-Series Expansions in $r$

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, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $r$ : real variable, $\epsilon$ : real parameter and $\alpha _{k}$ : term
Referenced by:: §33.18, §33.19
Permalink:: http://dlmf.nist.gov/33.19.E1
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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:: $!$ : $n!$ : factorial, $k$ : 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:: $k$ : 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, $k$ : 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:: $k$ : 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:: $k$ : 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 $r$ , except $r=0$ in the case of (33.19.3).