§33.16 Connection Formulas

Permalink:: http://dlmf.nist.gov/33.16

§33.16(i) $\mathop{F_{{\ell}}\/}\nolimits$ and $\mathop{G_{{\ell}}\/}\nolimits$ in Terms of $\mathop{f\/}\nolimits$ and $\mathop{h\/}\nolimits$
§33.16(ii) $\mathop{f\/}\nolimits$ and $\mathop{h\/}\nolimits$ in Terms of $\mathop{F_{{\ell}}\/}\nolimits$ and $\mathop{G_{{\ell}}\/}\nolimits$ when $\epsilon>0$
§33.16(iii) $\mathop{f\/}\nolimits$ and $\mathop{h\/}\nolimits$ in Terms of $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ when $\epsilon<0$
§33.16(iv) $\mathop{s\/}\nolimits$ and $\mathop{c\/}\nolimits$ in Terms of $\mathop{F_{{\ell}}\/}\nolimits$ and $\mathop{G_{{\ell}}\/}\nolimits$ when $\epsilon>0$
§33.16(v) $\mathop{s\/}\nolimits$ and $\mathop{c\/}\nolimits$ in Terms of $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ when $\epsilon<0$

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

Notes:: See Seaton (2002a, Eqs. 130, 131).
Keywords:: Coulomb functions: variables $\rho,\eta$
Permalink:: http://dlmf.nist.gov/33.16.i

33.16.1 $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)=\dfrac{(2\ell+1)!\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)}{(-2\eta)^{{\ell+1}}}\mathop{f\/}\nolimits\!\left(1/\eta^{2},\ell;-\eta\rho\right),$

Symbols:: $\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)$ : normalizing constant for Coulomb radial functions, $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : regular Coulomb function, $!$ : $n!$ : factorial, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : 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

33.16.2 $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)=\dfrac{\pi(-2\eta)^{\ell}}{(2\ell+1)!\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)}\mathop{h\/}\nolimits\!\left(1/\eta^{2},\ell;-\eta\rho\right),$

Symbols:: $\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)$ : normalizing constant for Coulomb radial functions, $\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : irregular Coulomb function, $!$ : $n!$ : factorial, $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : 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

where $\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)$ is given by (33.2.5) or (33.2.6).

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

Notes:: These results are generalized from Seaton (2002a, Eqs. 88, 90, 93, 95).
Keywords:: Coulomb functions: variables $r,\epsilon$
Permalink:: http://dlmf.nist.gov/33.16.ii

When $\epsilon>0$ denote

33.16.3 $\tau=\epsilon^{{1/2}}(>0),$

Symbols:: $\epsilon$ : real parameter and $\tau$ : parameter
Referenced by:: §33.16(iv)
Permalink:: http://dlmf.nist.gov/33.16.E3
Encodings:: TeX, pMML, png

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

33.16.4 $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)=\left(\frac{2}{\pi\tau}\frac{1-e^{{-2\pi/\tau}}}{A(\epsilon,\ell)}\right)^{{\ifrac{1}{2}}}\mathop{F_{{\ell}}\/}\nolimits\!\left(-1/\tau,\tau r\right),$

Symbols:: $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : regular Coulomb function, $e$ : base of exponential function, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : 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), §33.21(i)
Permalink:: http://dlmf.nist.gov/33.16.E4
Encodings:: TeX, pMML, png

33.16.5 $\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)=\left(\frac{2}{\pi\tau}\frac{A(\epsilon,\ell)}{1-e^{{-2\pi/\tau}}}\right)^{{\ifrac{1}{2}}}\mathop{G_{{\ell}}\/}\nolimits\!\left(-1/\tau,\tau r\right).$

Symbols:: $\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : irregular Coulomb function, $e$ : base of exponential function, $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : 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

Alternatively, for $r<0$

33.16.6 $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)=(-1)^{{\ell+1}}\left(\frac{2}{\pi\tau}\frac{e^{{2\pi/\tau}}-1}{A(\epsilon,\ell)}\right)^{{\ifrac{1}{2}}}\mathop{F_{{\ell}}\/}\nolimits\!\left(1/\tau,-\tau r\right),$

Symbols:: $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : regular Coulomb function, $e$ : base of exponential function, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : 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

33.16.7 $\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)=(-1)^{{\ell}}\left(\frac{2}{\pi\tau}\frac{A(\epsilon,\ell)}{e^{{2\pi/\tau}}-1}\right)^{{\ifrac{1}{2}}}\mathop{G_{{\ell}}\/}\nolimits\!\left(1/\tau,-\tau r\right).$

Symbols:: $\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : irregular Coulomb function, $e$ : base of exponential function, $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : 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), §33.21(i)
Permalink:: http://dlmf.nist.gov/33.16.E7
Encodings:: TeX, pMML, png

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

Notes:: See Seaton (2002a, Eqs. 104–109).
Keywords:: Coulomb functions, Coulomb functions: variables $r,\epsilon$ , Whittaker functions
Permalink:: http://dlmf.nist.gov/33.16.iii

When $\epsilon<0$ denote

33.16.8 $\nu=1/(-\epsilon)^{{1/2}}(>0),$

Symbols:: $\epsilon$ : real parameter and $\nu$ : parameter
Referenced by:: §33.16(v)
Permalink:: http://dlmf.nist.gov/33.16.E8
Encodings:: TeX, pMML, png

33.16.9

$\zeta _{\ell}(\nu,r)=\mathop{W_{{\nu,\ell+\frac{1}{2}}}\/}\nolimits\!\left(2r/\nu\right),$

$\xi _{\ell}(\nu,r)=\realpart{\left(e^{{i\pi\nu}}\mathop{W_{{-\nu,\ell+\frac{1}{2}}}\/}\nolimits\!\left(e^{{i\pi}}2r/\nu\right)\right)},$

Symbols:: $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function, $\realpart{}$ : real part, $\ell$ : nonnegative integer, $r$ : real variable, $\nu$ : parameter, $\zeta _{\ell}(\nu,r)$ : function and $\xi _{\ell}(\nu,r)$ : function
Referenced by:: §33.16(v), §33.21(i)
Permalink:: http://dlmf.nist.gov/33.16.E9
Encodings:: TeX, TeX, pMML, pMML, png, png

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

33.16.10 $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)=(-1)^{\ell}\nu^{{\ell+1}}\left(-\frac{\mathop{\cos\/}\nolimits\!\left(\pi\nu\right)\zeta _{\ell}(\nu,r)}{\mathop{\Gamma\/}\nolimits\!\left(\ell+1+\nu\right)}+\frac{\mathop{\sin\/}\nolimits\!\left(\pi\nu\right)\mathop{\Gamma\/}\nolimits\!\left(\nu-\ell\right)\xi _{\ell}(\nu,r)}{\pi}\right),$

Symbols:: $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : regular Coulomb function, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : 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), §33.21(i)
Permalink:: http://dlmf.nist.gov/33.16.E10
Encodings:: TeX, pMML, png

33.16.11 $\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)=(-1)^{\ell}\nu^{{\ell+1}}A(\epsilon,\ell)\left(\frac{\mathop{\sin\/}\nolimits\!\left(\pi\nu\right)\zeta _{\ell}(\nu,r)}{\mathop{\Gamma\/}\nolimits\!\left(\ell+1+\nu\right)}+\frac{\mathop{\cos\/}\nolimits\!\left(\pi\nu\right)\mathop{\Gamma\/}\nolimits\!\left(\nu-\ell\right)\xi _{\ell}(\nu,r)}{\pi}\right).$

Symbols:: $\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : irregular Coulomb function, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : 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

Alternatively, for $r<0$

33.16.12 $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)=\frac{(-1)^{\ell}\nu^{{\ell+1}}}{\pi}\left(-\frac{\pi\xi _{\ell}(-\nu,r)}{\mathop{\Gamma\/}\nolimits\!\left(\ell+1+\nu\right)}+\mathop{\sin\/}\nolimits\!\left(\pi\nu\right)\mathop{\cos\/}\nolimits\!\left(\pi\nu\right)\mathop{\Gamma\/}\nolimits\!\left(\nu-\ell\right)\zeta _{\ell}(-\nu,r)\right),$

Symbols:: $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : regular Coulomb function, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : 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

33.16.13 $\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)=(-1)^{\ell}\nu^{{\ell+1}}A(\epsilon,\ell)\mathop{\Gamma\/}\nolimits\!\left(\nu-\ell\right)\zeta _{\ell}(-\nu,r)/\pi.$

Symbols:: $\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : irregular Coulomb function, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\ell$ : 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), §33.21(i)
Permalink:: http://dlmf.nist.gov/33.16.E13
Encodings:: TeX, pMML, png

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

Notes:: Combine (33.14.9) with (33.16.4)–(33.16.7).
Keywords:: Coulomb functions: variables $r,\epsilon$
Permalink:: http://dlmf.nist.gov/33.16.iv

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

33.16.14

$\mathop{s\/}\nolimits\!\left(\epsilon,\ell;r\right)=(\pi\tau)^{{-1/2}}\mathop{F_{{\ell}}\/}\nolimits\!\left(-1/\tau,\tau r\right),$

$\mathop{c\/}\nolimits\!\left(\epsilon,\ell;r\right)=(\pi\tau)^{{-1/2}}\mathop{G_{{\ell}}\/}\nolimits\!\left(-1/\tau,\tau r\right).$

Symbols:: $\mathop{c\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : irregular Coulomb function, $\mathop{s\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : regular Coulomb function, $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial function, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : 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

Alternatively, for $r<0$

33.16.15

$\mathop{s\/}\nolimits\!\left(\epsilon,\ell;r\right)=(\pi\tau)^{{-1/2}}\mathop{F_{{\ell}}\/}\nolimits\!\left(1/\tau,-\tau r\right),$

$\mathop{c\/}\nolimits\!\left(\epsilon,\ell;r\right)=(\pi\tau)^{{-1/2}}\mathop{G_{{\ell}}\/}\nolimits\!\left(1/\tau,-\tau r\right).$

Symbols:: $\mathop{c\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : irregular Coulomb function, $\mathop{s\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : regular Coulomb function, $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial function, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : 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

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

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).
Keywords:: Coulomb functions, Coulomb functions: variables $r,\epsilon$ , Whittaker functions
Permalink:: http://dlmf.nist.gov/33.16.v

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 $K(\nu,\ell)=\left(\nu^{2}\mathop{\Gamma\/}\nolimits\!\left(\nu+\ell+1\right)\mathop{\Gamma\/}\nolimits\!\left(\nu-\ell\right)\right)^{{-1/2}}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\ell$ : nonnegative integer, $\nu$ : parameter and $K(\nu,\ell)$ : factor
Referenced by:: §33.16(v)
Permalink:: http://dlmf.nist.gov/33.16.E16
Encodings:: TeX, pMML, png

Then for $r>0$

33.16.17

$\mathop{s\/}\nolimits\!\left(\epsilon,\ell;r\right)=\frac{(-1)^{\ell}}{2\nu^{{1/2}}}\left(\frac{\mathop{\sin\/}\nolimits\!\left(\pi\nu\right)}{\pi K(\nu,\ell)}\xi _{\ell}(\nu,r)-\mathop{\cos\/}\nolimits\!\left(\pi\nu\right)\nu^{2}K(\nu,\ell)\zeta _{\ell}(\nu,r)\right),$

$\mathop{c\/}\nolimits\!\left(\epsilon,\ell;r\right)=\frac{(-1)^{\ell}}{2\nu^{{1/2}}}\left(\frac{\mathop{\cos\/}\nolimits\!\left(\pi\nu\right)}{\pi K(\nu,\ell)}\xi _{\ell}(\nu,r)+\mathop{\sin\/}\nolimits\!\left(\pi\nu\right)\nu^{2}K(\nu,\ell)\zeta _{\ell}(\nu,r)\right).$

Alternatively, for $r<0$

33.16.18

$\mathop{s\/}\nolimits\!\left(\epsilon,\ell;r\right)=\frac{(-1)^{{\ell+1}}}{2^{{1/2}}}\left(\frac{\nu^{{3/2}}}{K(\nu,\ell)}\xi _{\ell}(-\nu,r)-\frac{\mathop{\sin\/}\nolimits\!\left(\pi\nu\right)\mathop{\cos\/}\nolimits\!\left(\pi\nu\right)}{\pi\nu^{{1/2}}}K(\nu,\ell)\zeta _{\ell}(-\nu,r)\right),$

$\mathop{c\/}\nolimits\!\left(\epsilon,\ell;r\right)=\frac{(-1)^{\ell}}{\pi(2\nu)^{{1/2}}}K(\nu,\ell)\zeta _{\ell}(-\nu,r).$

§33.16 Connection Formulas

Contents

§33.16(i) and in Terms of and

§33.16(ii) and in Terms of and when

§33.16(iii) and in Terms of when

§33.16(iv) and in Terms of and when

§33.16(v) and in Terms of when

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

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

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

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

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