§33.16 Connection Formulas

33.16.1		$$F_{\mathrm{\ell }}(\eta ,\rho )=\frac{(2\mathrm{\ell }+1)!C_{\mathrm{\ell }}\left(\eta \right)}{{(-2\eta )}^{\mathrm{\ell }+1}}f(1/{\eta }^2,\mathrm{\ell };-\eta \rho ),$$
ⓘ Symbols: $C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $!$: factorial (as in $n!$), $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter Referenced by: §33.18 Permalink: http://dlmf.nist.gov/33.16.E1 Encodings: TeX, pMML, png See also: Annotations for §33.16(i), §33.16 and Ch.33

33.16.2		$$G_{\mathrm{\ell }}(\eta ,\rho )=\frac{\pi {(-2\eta )}^{\mathrm{\ell }}}{(2\mathrm{\ell }+1)!C_{\mathrm{\ell }}\left(\eta \right)}h(1/{\eta }^2,\mathrm{\ell };-\eta \rho ),$$
ⓘ Symbols: $C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter Referenced by: §33.18 Permalink: http://dlmf.nist.gov/33.16.E2 Encodings: TeX, pMML, png See also: Annotations for §33.16(i), §33.16 and Ch.33

where $C_{\mathrm{\ell }}\left(\eta \right)$ is given by (33.2.5) or (33.2.6).

When $ϵ>0$ denote

33.16.3		$$\tau ={ϵ}^{1/2}(>0),$$
ⓘ Defines: $\tau$: parameter (locally) Symbols: $ϵ$: real parameter Referenced by: §33.16(iv) Permalink: http://dlmf.nist.gov/33.16.E3 Encodings: TeX, pMML, png See also: Annotations for §33.16(ii), §33.16 and Ch.33

and again define $A(ϵ,\mathrm{\ell })$ by (33.14.11) or (33.14.12). Then for $r>0$

33.16.4		$$f(ϵ,\mathrm{\ell };r)={\left(\frac{2}{\pi \tau }\frac{1-{\mathrm{e}}^{-2\pi /\tau }}{A(ϵ,\mathrm{\ell })}\right)}^{1/2}{F}_{\mathrm{\ell }}(-1/\tau ,\tau r),$$
ⓘ Symbols: $f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter, $A(ϵ,\mathrm{\ell })$: function and $\tau$: parameter Referenced by: §33.16(iv), item 1 Permalink: http://dlmf.nist.gov/33.16.E4 Encodings: TeX, pMML, png See also: Annotations for §33.16(ii), §33.16 and Ch.33

33.16.5		$$h(ϵ,\mathrm{\ell };r)={\left(\frac{2}{\pi \tau }\frac{A(ϵ,\mathrm{\ell })}{1-{\mathrm{e}}^{-2\pi /\tau }}\right)}^{1/2}{G}_{\mathrm{\ell }}(-1/\tau ,\tau r).$$
ⓘ Symbols: $h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter, $A(ϵ,\mathrm{\ell })$: function and $\tau$: parameter Permalink: http://dlmf.nist.gov/33.16.E5 Encodings: TeX, pMML, png See also: Annotations for §33.16(ii), §33.16 and Ch.33

Alternatively, for

33.16.6		$f(ϵ,\mathrm{\ell };r)$	$={(-1)}^{\mathrm{\ell }+1}{\left({\displaystyle \frac{2}{\pi \tau }}{\displaystyle \frac{{\mathrm{e}}^{2\pi /\tau }-1}{A(ϵ,\mathrm{\ell })}}\right)}^{1/2}{F}_{\mathrm{\ell }}(1/\tau ,-\tau r),$
ⓘ Symbols: $f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter, $A(ϵ,\mathrm{\ell })$: function and $\tau$: parameter Permalink: http://dlmf.nist.gov/33.16.E6 Encodings: TeX, pMML, png See also: Annotations for §33.16(ii), §33.16 and Ch.33
33.16.7		$h(ϵ,\mathrm{\ell };r)$	$={(-1)}^{\mathrm{\ell }}{\left({\displaystyle \frac{2}{\pi \tau }}{\displaystyle \frac{A(ϵ,\mathrm{\ell })}{{\mathrm{e}}^{2\pi /\tau }-1}}\right)}^{1/2}{G}_{\mathrm{\ell }}(1/\tau ,-\tau r).$
ⓘ Symbols: $h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter, $A(ϵ,\mathrm{\ell })$: function and $\tau$: parameter Referenced by: §33.16(iv), item 1 Permalink: http://dlmf.nist.gov/33.16.E7 Encodings: TeX, pMML, png See also: Annotations for §33.16(ii), §33.16 and Ch.33

When denote

33.16.8		$$\nu =1/{(-ϵ)}^{1/2}(>0),$$
ⓘ Defines: $\nu$: parameter (locally) Symbols: $ϵ$: real parameter Referenced by: §33.16(v) Permalink: http://dlmf.nist.gov/33.16.E8 Encodings: TeX, pMML, png See also: Annotations for §33.16(iii), §33.16 and Ch.33

33.16.9		${\zeta }_{\mathrm{\ell }}(\nu ,r)$	$=W_{\nu ,\mathrm{\ell }+\frac{1}{2}}\left(2r/\nu \right),$
		${\xi }_{\mathrm{\ell }}(\nu ,r)$	$=\mathrm{\Re }\left({\mathrm{e}}^{\mathrm{i}\pi \nu }{W}_{-\nu ,\mathrm{\ell }+\frac{1}{2}}\left({\mathrm{e}}^{\mathrm{i}\pi }2r/\nu \right)\right),$
ⓘ Defines: ${\zeta }_{\mathrm{\ell }}(\nu ,r)$: function (locally) and ${\xi }_{\mathrm{\ell }}(\nu ,r)$: function (locally) Symbols: $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\Re }$: real part, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $\nu$: parameter Referenced by: §33.16(v), §33.21(i) Permalink: http://dlmf.nist.gov/33.16.E9 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.16(iii), §33.16 and Ch.33

and again define $A(ϵ,\mathrm{\ell })$ by (33.14.11) or (33.14.12). Then for $r>0$

33.16.10		$f(ϵ,\mathrm{\ell };r)$	$={(-1)}^{\mathrm{\ell }}{\nu }^{\mathrm{\ell }+1}\left(-{\displaystyle \frac{\cos \left(\pi \nu \right){\zeta }_{\mathrm{\ell }}(\nu ,r)}{\mathrm{\Gamma }\left(\mathrm{\ell }+1+\nu \right)}}+{\displaystyle \frac{\sin \left(\pi \nu \right)\mathrm{\Gamma }\left(\nu -\mathrm{\ell }\right){\xi }_{\mathrm{\ell }}(\nu ,r)}{\pi }}\right),$
ⓘ Symbols: $f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter, $\nu$: parameter, ${\zeta }_{\mathrm{\ell }}(\nu ,r)$: function and ${\xi }_{\mathrm{\ell }}(\nu ,r)$: function Referenced by: §33.16(v), item 2 Permalink: http://dlmf.nist.gov/33.16.E10 Encodings: TeX, pMML, png See also: Annotations for §33.16(iii), §33.16 and Ch.33
33.16.11		$h(ϵ,\mathrm{\ell };r)$	$={(-1)}^{\mathrm{\ell }}{\nu }^{\mathrm{\ell }+1}A(ϵ,\mathrm{\ell })\left({\displaystyle \frac{\sin \left(\pi \nu \right){\zeta }_{\mathrm{\ell }}(\nu ,r)}{\mathrm{\Gamma }\left(\mathrm{\ell }+1+\nu \right)}}+{\displaystyle \frac{\cos \left(\pi \nu \right)\mathrm{\Gamma }\left(\nu -\mathrm{\ell }\right){\xi }_{\mathrm{\ell }}(\nu ,r)}{\pi }}\right).$
ⓘ Symbols: $h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter, $A(ϵ,\mathrm{\ell })$: function, $\nu$: parameter, ${\zeta }_{\mathrm{\ell }}(\nu ,r)$: function and ${\xi }_{\mathrm{\ell }}(\nu ,r)$: function Permalink: http://dlmf.nist.gov/33.16.E11 Encodings: TeX, pMML, png See also: Annotations for §33.16(iii), §33.16 and Ch.33

Alternatively, for

33.16.12		$$f(ϵ,\mathrm{\ell };r)=\frac{{(-1)}^{\mathrm{\ell }}{\nu }^{\mathrm{\ell }+1}}{\pi }\left(-\frac{\pi {\xi }_{\mathrm{\ell }}(-\nu ,r)}{\mathrm{\Gamma }\left(\mathrm{\ell }+1+\nu \right)}+\sin \left(\pi \nu \right)\cos \left(\pi \nu \right)\mathrm{\Gamma }\left(\nu -\mathrm{\ell }\right){\zeta }_{\mathrm{\ell }}(-\nu ,r)\right),$$
ⓘ Symbols: $f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter, $\nu$: parameter, ${\zeta }_{\mathrm{\ell }}(\nu ,r)$: function and ${\xi }_{\mathrm{\ell }}(\nu ,r)$: function Permalink: http://dlmf.nist.gov/33.16.E12 Encodings: TeX, pMML, png See also: Annotations for §33.16(iii), §33.16 and Ch.33

33.16.13		$$h(ϵ,\mathrm{\ell };r)={(-1)}^{\mathrm{\ell }}{\nu }^{\mathrm{\ell }+1}A(ϵ,\mathrm{\ell })\mathrm{\Gamma }\left(\nu -\mathrm{\ell }\right){\zeta }_{\mathrm{\ell }}(-\nu ,r)/\pi .$$
ⓘ Symbols: $h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter, $A(ϵ,\mathrm{\ell })$: function, $\nu$: parameter and ${\zeta }_{\mathrm{\ell }}(\nu ,r)$: function Referenced by: §33.16(v), item 2 Permalink: http://dlmf.nist.gov/33.16.E13 Encodings: TeX, pMML, png See also: Annotations for §33.16(iii), §33.16 and Ch.33

When $ϵ>0$, again denote $\tau$ by (33.16.3). Then for $r>0$

33.16.14		$s(ϵ,\mathrm{\ell };r)$	$={(\pi \tau )}^{-1/2}{F}_{\mathrm{\ell }}(-1/\tau ,\tau r),$
		$c(ϵ,\mathrm{\ell };r)$	$={(\pi \tau )}^{-1/2}{G}_{\mathrm{\ell }}(-1/\tau ,\tau r).$
ⓘ Symbols: $c(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $s(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $\pi$: the ratio of the circumference of a circle to its diameter, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter and $\tau$: parameter Permalink: http://dlmf.nist.gov/33.16.E14 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.16(iv), §33.16 and Ch.33

Alternatively, for

33.16.15		$s(ϵ,\mathrm{\ell };r)$	$={(\pi \tau )}^{-1/2}{F}_{\mathrm{\ell }}(1/\tau ,-\tau r),$
		$c(ϵ,\mathrm{\ell };r)$	$={(\pi \tau )}^{-1/2}{G}_{\mathrm{\ell }}(1/\tau ,-\tau r).$
ⓘ Symbols: $c(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $s(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $\pi$: the ratio of the circumference of a circle to its diameter, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter and $\tau$: parameter Permalink: http://dlmf.nist.gov/33.16.E15 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.16(iv), §33.16 and Ch.33

When denote $\nu$, ${\zeta }_{\mathrm{\ell }}(\nu ,r)$, and ${\xi }_{\mathrm{\ell }}(\nu ,r)$ by (33.16.8) and (33.16.9). Also denote

33.16.16		$$K(\nu ,\mathrm{\ell })={\left({\nu }^2\mathrm{\Gamma }\left(\nu +\mathrm{\ell }+1\right)\mathrm{\Gamma }\left(\nu -\mathrm{\ell }\right)\right)}^{-1/2}.$$
ⓘ Defines: $K(\nu ,\mathrm{\ell })$: factor (locally) Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{\ell }$: nonnegative integer and $\nu$: parameter Referenced by: §33.16(v) Permalink: http://dlmf.nist.gov/33.16.E16 Encodings: TeX, pMML, png See also: Annotations for §33.16(v), §33.16 and Ch.33

Then for $r>0$

33.16.17		$s(ϵ,\mathrm{\ell };r)$	$={\displaystyle \frac{{(-1)}^{\mathrm{\ell }}}{2{\nu }^{1/2}}}\left({\displaystyle \frac{\sin \left(\pi \nu \right)}{\pi K(\nu ,\mathrm{\ell })}}{\xi }_{\mathrm{\ell }}(\nu ,r)-\cos \left(\pi \nu \right){\nu }^{2}K(\nu ,\mathrm{\ell }){\zeta }_{\mathrm{\ell }}(\nu ,r)\right),$
		$c(ϵ,\mathrm{\ell };r)$	$={\displaystyle \frac{{(-1)}^{\mathrm{\ell }}}{2{\nu }^{1/2}}}\left({\displaystyle \frac{\cos \left(\pi \nu \right)}{\pi K(\nu ,\mathrm{\ell })}}{\xi }_{\mathrm{\ell }}(\nu ,r)+\sin \left(\pi \nu \right){\nu }^{2}K(\nu ,\mathrm{\ell }){\zeta }_{\mathrm{\ell }}(\nu ,r)\right).$
ⓘ Symbols: $c(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $s(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter, $\nu$: parameter, ${\zeta }_{\mathrm{\ell }}(\nu ,r)$: function, ${\xi }_{\mathrm{\ell }}(\nu ,r)$: function and $K(\nu ,\mathrm{\ell })$: factor Referenced by: §33.16(v), item 2 Permalink: http://dlmf.nist.gov/33.16.E17 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.16(v), §33.16 and Ch.33

Alternatively, for

33.16.18		$s(ϵ,\mathrm{\ell };r)$	$={\displaystyle \frac{{(-1)}^{\mathrm{\ell }+1}}{2^{1/2}}}\left({\displaystyle \frac{{\nu }^{3/2}}{K(\nu ,\mathrm{\ell })}}{\xi }_{\mathrm{\ell }}(-\nu ,r)-{\displaystyle \frac{\sin \left(\pi \nu \right)\cos \left(\pi \nu \right)}{\pi {\nu }^{1/2}}}K(\nu ,\mathrm{\ell }){\zeta }_{\mathrm{\ell }}(-\nu ,r)\right),$
		$c(ϵ,\mathrm{\ell };r)$	$={\displaystyle \frac{{(-1)}^{\mathrm{\ell }}}{\pi {(2\nu )}^{1/2}}}K(\nu ,\mathrm{\ell }){\zeta }_{\mathrm{\ell }}(-\nu ,r).$
ⓘ Symbols: $c(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $s(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter, $\nu$: parameter, ${\zeta }_{\mathrm{\ell }}(\nu ,r)$: function, ${\xi }_{\mathrm{\ell }}(\nu ,r)$: function and $K(\nu ,\mathrm{\ell })$: factor Referenced by: §33.16(v), item 2 Permalink: http://dlmf.nist.gov/33.16.E18 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.16(v), §33.16 and Ch.33