§33.19 Power-Series Expansions in $r$

33.19.1		$$f(ϵ,\mathrm{\ell };r)=r^{\mathrm{\ell }+1}\sum _{k=0}^{\mathrm{\infty }}{\alpha }_k{r}^k,$$
ⓘ Symbols: $f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter and ${\alpha }_k$: term Referenced by: §33.18, §33.19 Permalink: http://dlmf.nist.gov/33.19.E1 Encodings: TeX, pMML, png See also: Annotations for §33.19 and Ch.33

where

33.19.2		${\alpha }_0$	$=2^{\mathrm{\ell }+1}/(2\mathrm{\ell }+1)!,$
		${\alpha }_1$	$=-{\alpha }_0/(\mathrm{\ell }+1),$
		$k(k+2\mathrm{\ell }+1){\alpha }_k+2{\alpha }_{k-1}+ϵ{\alpha }_{k-2}$	$=0,$
	$k=2,3,\mathrm{\dots }$.
ⓘ Defines: ${\alpha }_k$: term (locally) Symbols: $!$: factorial (as in $n!$), $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer and $ϵ$: real parameter Permalink: http://dlmf.nist.gov/33.19.E2 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §33.19 and Ch.33

33.19.3		$$2\pi h(ϵ,\mathrm{\ell };r)=\begin{array}{l}\sum _{k=0}^{2\mathrm{\ell }}\frac{(2\mathrm{\ell }-k)!{\gamma }_k}{k!}{(2r)}^{k-\mathrm{\ell }}-\sum _{k=0}^{\mathrm{\infty }}{\delta }_k{r}^{k+\mathrm{\ell }+1}\\ -A(ϵ,\mathrm{\ell })\left(2\ln |2r/\kappa |+\mathrm{\Re }\psi \left(\mathrm{\ell }+1+\kappa \right)+\mathrm{\Re }\psi \left(-\mathrm{\ell }+\kappa \right)\right)f(ϵ,\mathrm{\ell };r),\end{array}$$
$r\ne 0$.
ⓘ Symbols: $f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter, $\kappa$: quantity, $A(ϵ,\mathrm{\ell })$: function, ${\gamma }_k$: term and ${\delta }_k$: term Referenced by: §33.19 Permalink: http://dlmf.nist.gov/33.19.E3 Encodings: TeX, pMML, png See also: Annotations for §33.19 and Ch.33

Here $\kappa$ is defined by (33.14.6), $A(ϵ,\mathrm{\ell })$ is defined by (33.14.11) or (33.14.12), ${\gamma }_0=1$, ${\gamma }_1=1$, and

33.19.4		$${\gamma }_k-{\gamma }_{k-1}+\frac{1}{4}(k-1)(k-2\mathrm{\ell }-2)ϵ{\gamma }_{k-2}=0,$$
$k=2,3,\mathrm{\dots }$.
ⓘ Symbols: $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $ϵ$: real parameter and ${\gamma }_k$: term Permalink: http://dlmf.nist.gov/33.19.E4 Encodings: TeX, pMML, png See also: Annotations for §33.19 and Ch.33

Also,

33.19.5		${\delta }_0$	$=\left({\beta }_{2\mathrm{\ell }+1}-2(\psi \left(2\mathrm{\ell }+2\right)+\psi \left(1\right))A(ϵ,\mathrm{\ell })\right){\alpha }_0,$
		${\delta }_1$	$=\left({\beta }_{2\mathrm{\ell }+2}-2(\psi \left(2\mathrm{\ell }+3\right)+\psi \left(2\right))A(ϵ,\mathrm{\ell })\right){\alpha }_1,$
ⓘ Defines: ${\delta }_k$: term (locally) Symbols: $\psi \left(z\right)$: psi (or digamma) function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $ϵ$: real parameter, $A(ϵ,\mathrm{\ell })$: function, ${\alpha }_k$: term and ${\beta }_k$: term Permalink: http://dlmf.nist.gov/33.19.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.19 and Ch.33

33.19.6		$$k(k+2\mathrm{\ell }+1){\delta }_k+2{\delta }_{k-1}+ϵ{\delta }_{k-2}+2(2k+2\mathrm{\ell }+1)A(ϵ,\mathrm{\ell }){\alpha }_k=0,$$
$k=2,3,\mathrm{\dots }$,
ⓘ Symbols: $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $ϵ$: real parameter, $A(ϵ,\mathrm{\ell })$: function, ${\alpha }_k$: term and ${\delta }_k$: term Permalink: http://dlmf.nist.gov/33.19.E6 Encodings: TeX, pMML, png See also: Annotations for §33.19 and Ch.33

with ${\beta }_0={\beta }_1=0$, and

33.19.7		$${\beta }_k-{\beta }_{k-1}+\frac{1}{4}(k-1)(k-2\mathrm{\ell }-2)ϵ{\beta }_{k-2}+\frac{1}{2}(k-1)ϵ{\gamma }_{k-2}=0,$$
$k=2,3,\mathrm{\dots }$.
ⓘ Symbols: $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $ϵ$: real parameter, ${\gamma }_k$: term and ${\beta }_k$: term Permalink: http://dlmf.nist.gov/33.19.E7 Encodings: TeX, pMML, png See also: Annotations for §33.19 and Ch.33

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).