r,\epsilon

33.15 Graphics 33.17 Recurrence Relations and Derivatives

§33.16 Connection Formulas

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Permalink:: http://dlmf.nist.gov/33.16
See also:: Annotations for Ch.33

§33.16(i) $F_{\ell}$ and $G_{\ell}$ in Terms of $f$ and $h$

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Keywords:: Coulomb functions with variables $r,\epsilon$ , Coulomb functions: variables $\rho,\eta$ , relations to other functions
Notes:: See Seaton (2002a, Eqs. 130, 131).
Permalink:: http://dlmf.nist.gov/33.16.i
See also:: Annotations for §33.16 and Ch.33

33.16.1		$F_{\ell}\left(\eta,\rho\right)=\dfrac{(2\ell+1)!C_{\ell}\left(\eta\right)}{(-2% \eta)^{\ell+1}}f\left(1/\eta^{2},\ell;-\eta\rho\right),$
ⓘ Symbols: $C_{\NVar{\ell}}\left(\NVar{\eta}\right)$ : normalizing constant for Coulomb radial functions, $f\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : regular Coulomb function, $!$ : factorial (as in $n!$ ), $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $\ell$ : 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		$G_{\ell}\left(\eta,\rho\right)=\dfrac{\pi(-2\eta)^{\ell}}{(2\ell+1)!C_{\ell}% \left(\eta\right)}h\left(1/\eta^{2},\ell;-\eta\rho\right),$
ⓘ Symbols: $C_{\NVar{\ell}}\left(\NVar{\eta}\right)$ : normalizing constant for Coulomb radial functions, $h\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : irregular Coulomb function, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial function, $\ell$ : 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 $C_{\ell}\left(\eta\right)$ is given by (33.2.5) or (33.2.6).

§33.16(ii) $f$ and $h$ in Terms of $F_{\ell}$ and $G_{\ell}$ when $\epsilon>0$

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Keywords:: Coulomb functions with variables $\rho,\eta$ , Coulomb functions: variables $r,\epsilon$ , relations to other functions
Notes:: These results are generalized from Seaton (2002a, Eqs. 88, 90, 93, 95).
Permalink:: http://dlmf.nist.gov/33.16.ii
See also:: Annotations for §33.16 and Ch.33

When $\epsilon>0$ denote

33.16.3		$\tau=\epsilon^{1/2}(>0),$
ⓘ Defines: $\tau$ : parameter (locally) Symbols: $\epsilon$ : 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 $A(\epsilon,\ell)$ by (33.14.11) or (33.14.12). Then for $r>0$

33.16.4		$f\left(\epsilon,\ell;r\right)=\left(\frac{2}{\pi\tau}\frac{1-e^{-2\pi/\tau}}{A% (\epsilon,\ell)}\right)^{\ifrac{1}{2}}F_{\ell}\left(-1/\tau,\tau r\right),$
ⓘ Symbols: $f\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : regular Coulomb function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $\ell$ : nonnegative integer, $r$ : real variable, $\epsilon$ : real parameter, $A(\epsilon,\ell)$ : 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		$h\left(\epsilon,\ell;r\right)=\left(\frac{2}{\pi\tau}\frac{A(\epsilon,\ell)}{1% -e^{-2\pi/\tau}}\right)^{\ifrac{1}{2}}G_{\ell}\left(-1/\tau,\tau r\right).$
ⓘ Symbols: $h\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : irregular Coulomb function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial function, $\ell$ : nonnegative integer, $r$ : real variable, $\epsilon$ : real parameter, $A(\epsilon,\ell)$ : 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 $r<0$

33.16.6	$\displaystyle f\left(\epsilon,\ell;r\right)$	$\displaystyle=(-1)^{\ell+1}\left(\frac{2}{\pi\tau}\frac{e^{2\pi/\tau}-1}{A(% \epsilon,\ell)}\right)^{\ifrac{1}{2}}F_{\ell}\left(1/\tau,-\tau r\right),$
ⓘ Symbols: $f\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : regular Coulomb function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $\ell$ : nonnegative integer, $r$ : real variable, $\epsilon$ : real parameter, $A(\epsilon,\ell)$ : 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	$\displaystyle h\left(\epsilon,\ell;r\right)$	$\displaystyle=(-1)^{\ell}\left(\frac{2}{\pi\tau}\frac{A(\epsilon,\ell)}{e^{2% \pi/\tau}-1}\right)^{\ifrac{1}{2}}G_{\ell}\left(1/\tau,-\tau r\right).$
ⓘ Symbols: $h\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : irregular Coulomb function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial function, $\ell$ : nonnegative integer, $r$ : real variable, $\epsilon$ : real parameter, $A(\epsilon,\ell)$ : 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

§33.16(iii) $f$ and $h$ in Terms of $W_{\kappa,\mu}\left(z\right)$ when $\epsilon<0$

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Keywords:: Coulomb functions, Coulomb functions: variables $r,\epsilon$ , Whittaker functions, relations to other functions
Notes:: See Seaton (2002a, Eqs. 104–109).
Permalink:: http://dlmf.nist.gov/33.16.iii
See also:: Annotations for §33.16 and Ch.33

When $\epsilon<0$ denote

33.16.8		$\nu=1/(-\epsilon)^{1/2}(>0),$
ⓘ Defines: $\nu$ : parameter (locally) Symbols: $\epsilon$ : 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	$\displaystyle\zeta_{\ell}(\nu,r)$	$\displaystyle=W_{\nu,\ell+\frac{1}{2}}\left(2r/\nu\right),$
33.16.9	$\displaystyle\xi_{\ell}(\nu,r)$	$\displaystyle=\Re\left(e^{\mathrm{i}\pi\nu}W_{-\nu,\ell+\frac{1}{2}}\left(e^{% \mathrm{i}\pi}2r/\nu\right)\right),$
ⓘ Defines: $\zeta_{\ell}(\nu,r)$ : function (locally) and $\xi_{\ell}(\nu,r)$ : function (locally) Symbols: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : 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, $\Re$ : real part, $\ell$ : 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 $A(\epsilon,\ell)$ by (33.14.11) or (33.14.12). Then for $r>0$

33.16.10	$\displaystyle f\left(\epsilon,\ell;r\right)$	$\displaystyle=(-1)^{\ell}\nu^{\ell+1}\left(-\frac{\cos\left(\pi\nu\right)\zeta% _{\ell}(\nu,r)}{\Gamma\left(\ell+1+\nu\right)}+\frac{\sin\left(\pi\nu\right)% \Gamma\left(\nu-\ell\right)\xi_{\ell}(\nu,r)}{\pi}\right),$
ⓘ Symbols: $f\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : regular Coulomb function, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $\ell$ : nonnegative integer, $r$ : real variable, $\epsilon$ : real parameter, $\nu$ : parameter, $\zeta_{\ell}(\nu,r)$ : function and $\xi_{\ell}(\nu,r)$ : 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	$\displaystyle h\left(\epsilon,\ell;r\right)$	$\displaystyle=(-1)^{\ell}\nu^{\ell+1}A(\epsilon,\ell)\left(\frac{\sin\left(\pi% \nu\right)\zeta_{\ell}(\nu,r)}{\Gamma\left(\ell+1+\nu\right)}+\frac{\cos\left(% \pi\nu\right)\Gamma\left(\nu-\ell\right)\xi_{\ell}(\nu,r)}{\pi}\right).$
ⓘ Symbols: $h\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : irregular Coulomb function, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $\ell$ : nonnegative integer, $r$ : real variable, $\epsilon$ : real parameter, $A(\epsilon,\ell)$ : function, $\nu$ : parameter, $\zeta_{\ell}(\nu,r)$ : function and $\xi_{\ell}(\nu,r)$ : 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 $r<0$

33.16.12		$f\left(\epsilon,\ell;r\right)=\frac{(-1)^{\ell}\nu^{\ell+1}}{\pi}\left(-\frac{% \pi\xi_{\ell}(-\nu,r)}{\Gamma\left(\ell+1+\nu\right)}+\sin\left(\pi\nu\right)% \cos\left(\pi\nu\right)\Gamma\left(\nu-\ell\right)\zeta_{\ell}(-\nu,r)\right),$
ⓘ Symbols: $f\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : regular Coulomb function, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $\ell$ : nonnegative integer, $r$ : real variable, $\epsilon$ : real parameter, $\nu$ : parameter, $\zeta_{\ell}(\nu,r)$ : function and $\xi_{\ell}(\nu,r)$ : 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		$h\left(\epsilon,\ell;r\right)=(-1)^{\ell}\nu^{\ell+1}A(\epsilon,\ell)\Gamma% \left(\nu-\ell\right)\zeta_{\ell}(-\nu,r)/\pi.$
ⓘ Symbols: $h\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : irregular Coulomb function, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\ell$ : nonnegative integer, $r$ : real variable, $\epsilon$ : real parameter, $A(\epsilon,\ell)$ : function, $\nu$ : parameter and $\zeta_{\ell}(\nu,r)$ : 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

§33.16(iv) $s$ and $c$ in Terms of $F_{\ell}$ and $G_{\ell}$ when $\epsilon>0$

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Keywords:: Coulomb functions with variables $\rho,\eta$ , Coulomb functions: variables $r,\epsilon$ , relations to other functions
Notes:: Combine (33.14.9) with (33.16.4)–(33.16.7).
Permalink:: http://dlmf.nist.gov/33.16.iv
See also:: Annotations for §33.16 and Ch.33

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

33.16.14	$\displaystyle s\left(\epsilon,\ell;r\right)$	$\displaystyle=(\pi\tau)^{-1/2}F_{\ell}\left(-1/\tau,\tau r\right),$
33.16.14	$\displaystyle c\left(\epsilon,\ell;r\right)$	$\displaystyle=(\pi\tau)^{-1/2}G_{\ell}\left(-1/\tau,\tau r\right).$
ⓘ Symbols: $c\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : irregular Coulomb function, $s\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : regular Coulomb function, $\pi$ : the ratio of the circumference of a circle to its diameter, $G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial function, $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $\ell$ : nonnegative integer, $r$ : real variable, $\epsilon$ : 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 $r<0$

33.16.15	$\displaystyle s\left(\epsilon,\ell;r\right)$	$\displaystyle=(\pi\tau)^{-1/2}F_{\ell}\left(1/\tau,-\tau r\right),$
33.16.15	$\displaystyle c\left(\epsilon,\ell;r\right)$	$\displaystyle=(\pi\tau)^{-1/2}G_{\ell}\left(1/\tau,-\tau r\right).$
ⓘ Symbols: $c\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : irregular Coulomb function, $s\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : regular Coulomb function, $\pi$ : the ratio of the circumference of a circle to its diameter, $G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial function, $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $\ell$ : nonnegative integer, $r$ : real variable, $\epsilon$ : 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

§33.16(v) $s$ and $c$ in Terms of $W_{\kappa,\mu}\left(z\right)$ when $\epsilon<0$

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Keywords:: Coulomb functions, Coulomb functions: variables $r,\epsilon$ , Whittaker functions, relations to other functions
Notes:: See Seaton (2002a, Eqs. 119–121). For (33.16.17) and (33.16.18) combine (33.14.6), (33.14.9)–(33.14.12), (33.16.10)–(33.16.13), and (33.16.16).
Permalink:: http://dlmf.nist.gov/33.16.v
See also:: Annotations for §33.16 and Ch.33

When $\epsilon<0$ denote $\nu$ , $\zeta_{\ell}(\nu,r)$ , and $\xi_{\ell}(\nu,r)$ by (33.16.8) and (33.16.9). Also denote

33.16.16		$K(\nu,\ell)=\left(\nu^{2}\Gamma\left(\nu+\ell+1\right)\Gamma\left(\nu-\ell% \right)\right)^{-1/2}.$
ⓘ Defines: $K(\nu,\ell)$ : factor (locally) Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\ell$ : 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	$\displaystyle s\left(\epsilon,\ell;r\right)$	$\displaystyle=\frac{(-1)^{\ell}}{2\nu^{1/2}}\left(\frac{\sin\left(\pi\nu\right% )}{\pi K(\nu,\ell)}\xi_{\ell}(\nu,r)-\cos\left(\pi\nu\right)\nu^{2}K(\nu,\ell)% \zeta_{\ell}(\nu,r)\right),$
33.16.17	$\displaystyle c\left(\epsilon,\ell;r\right)$	$\displaystyle=\frac{(-1)^{\ell}}{2\nu^{1/2}}\left(\frac{\cos\left(\pi\nu\right% )}{\pi K(\nu,\ell)}\xi_{\ell}(\nu,r)+\sin\left(\pi\nu\right)\nu^{2}K(\nu,\ell)% \zeta_{\ell}(\nu,r)\right).$
ⓘ Symbols: $c\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : irregular Coulomb function, $s\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : regular Coulomb function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $\ell$ : nonnegative integer, $r$ : real variable, $\epsilon$ : real parameter, $\nu$ : parameter, $\zeta_{\ell}(\nu,r)$ : function, $\xi_{\ell}(\nu,r)$ : function and $K(\nu,\ell)$ : 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 $r<0$

33.16.18	$\displaystyle s\left(\epsilon,\ell;r\right)$	$\displaystyle=\frac{(-1)^{\ell+1}}{2^{1/2}}\left(\frac{\nu^{3/2}}{K(\nu,\ell)}% \xi_{\ell}(-\nu,r)-\frac{\sin\left(\pi\nu\right)\cos\left(\pi\nu\right)}{\pi% \nu^{1/2}}K(\nu,\ell)\zeta_{\ell}(-\nu,r)\right),$
33.16.18	$\displaystyle c\left(\epsilon,\ell;r\right)$	$\displaystyle=\frac{(-1)^{\ell}}{\pi(2\nu)^{1/2}}K(\nu,\ell)\zeta_{\ell}(-\nu,% r).$
ⓘ Symbols: $c\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : irregular Coulomb function, $s\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : regular Coulomb function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $\ell$ : nonnegative integer, $r$ : real variable, $\epsilon$ : real parameter, $\nu$ : parameter, $\zeta_{\ell}(\nu,r)$ : function, $\xi_{\ell}(\nu,r)$ : function and $K(\nu,\ell)$ : 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

33.15 Graphics 33.17 Recurrence Relations and Derivatives

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§33.16 Connection Formulas

Contents

§33.16(i) Fℓ and Gℓ in Terms of f and h

§33.16(ii) f and h in Terms of Fℓ and Gℓ when ϵ>0

§33.16(iii) f and h in Terms of Wκ,μ⁡(z) when ϵ<0

§33.16(iv) s and c in Terms of Fℓ and Gℓ when ϵ>0

§33.16(v) s and c in Terms of Wκ,μ⁡(z) when ϵ<0

§33.16(i) $F_{\ell}$ and $G_{\ell}$ in Terms of $f$ and $h$

§33.16(ii) $f$ and $h$ in Terms of $F_{\ell}$ and $G_{\ell}$ when $\epsilon>0$

§33.16(iii) $f$ and $h$ in Terms of $W_{\kappa,\mu}\left(z\right)$ when $\epsilon<0$

§33.16(iv) $s$ and $c$ in Terms of $F_{\ell}$ and $G_{\ell}$ when $\epsilon>0$

§33.16(v) $s$ and $c$ in Terms of $W_{\kappa,\mu}\left(z\right)$ when $\epsilon<0$