# §33.6 Power-Series Expansions in $\rho$

 33.6.1 $F_{\ell}\left(\eta,\rho\right)=C_{\ell}\left(\eta\right)\sum_{k=\ell+1}^{% \infty}A_{k}^{\ell}(\eta)\rho^{k},$
 33.6.2 $F_{\ell}'\left(\eta,\rho\right)=C_{\ell}\left(\eta\right)\sum_{k=\ell+1}^{% \infty}kA_{k}^{\ell}(\eta)\rho^{k-1},$

where $A_{\ell+1}^{\ell}=1$, $A_{\ell+2}^{\ell}=\eta/(\ell+1)$, and

 33.6.3 $(k+\ell)(k-\ell-1)A_{k}^{\ell}=2\eta A_{k-1}^{\ell}-A_{k-2}^{\ell},$ $k=\ell+3,\ell+4,\dots$, ⓘ Symbols: $k$: nonnegative integer, $\ell$: nonnegative integer, $\eta$: real parameter and $A_{\ell+1}^{\ell}$: coefficient A&S Ref: 14.1.6 Permalink: http://dlmf.nist.gov/33.6.E3 Encodings: TeX, pMML, png See also: Annotations for §33.6 and Ch.33

or in terms of the hypergeometric function (§§15.1, 15.2(i)),

 33.6.4 $A_{k}^{\ell}(\eta)=\dfrac{(-\mathrm{i})^{k-\ell-1}}{(k-\ell-1)!}\*{{}_{2}F_{1}% }\left(\ell+1-k,\ell+1-\mathrm{i}\eta;2\ell+2;2\right).$
 33.6.5 ${H^{\pm}_{\ell}}\left(\eta,\rho\right)=\frac{e^{\pm\mathrm{i}{\theta_{\ell}}% \left(\eta,\rho\right)}}{(2\ell+1)!\Gamma\left(-\ell\pm\mathrm{i}\eta\right)}% \left(\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}}{{\left(2\ell+2\right)_{k}% }k!}(\mp 2\mathrm{i}\rho)^{a+k}\left(\ln\left(\mp 2\mathrm{i}\rho\right)+\psi% \left(a+k\right)-\psi\left(1+k\right)-\psi\left(2\ell+2+k\right)\right)-\sum_{% k=1}^{2\ell+1}\frac{(2\ell+1)!(k-1)!}{(2\ell+1-k)!{\left(1-a\right)_{k}}}(\mp 2% \mathrm{i}\rho)^{a-k}\right),$ ⓘ Symbols: ${\theta_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)$: phase of Coulomb functions, $\Gamma\left(\NVar{z}\right)$: gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $\psi\left(\NVar{z}\right)$: psi (or digamma) function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, ${H^{\NVar{\pm}}_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)$: irregular Coulomb radial functions, $\ln\NVar{z}$: principal branch of logarithm function, $k$: nonnegative integer, $\ell$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter and $a$: variable A&S Ref: 14.1.14 (equivalent) Referenced by: §33.13, §33.6, §33.6, Erratum (V1.0.19) for Equation (33.6.5) Permalink: http://dlmf.nist.gov/33.6.E5 Encodings: TeX, pMML, png Errata (effective with 1.0.19): On the right-hand side the factor in the denominator was incorrectly written as $\Gamma\left(-\ell+\mathrm{i}\eta\right)$. This has been corrected to $\Gamma\left(-\ell\pm\mathrm{i}\eta\right)$. Reported 2018-05-15 by Ian Thompson See also: Annotations for §33.6 and Ch.33

where $a=1+\ell\pm\mathrm{i}\eta$ and $\psi\left(x\right)=\Gamma'\left(x\right)/\Gamma\left(x\right)$5.2(i)).

The series (33.6.1), (33.6.2), and (33.6.5) converge for all finite values of $\rho$. Corresponding expansions for ${H^{\pm}_{\ell}}'\left(\eta,\rho\right)$ can be obtained by combining (33.6.5) with (33.4.3) or (33.4.4).