r,\epsilon

33.13 Complex Variable and Parameters 33.15 Graphics

§33.14 Definitions and Basic Properties

ⓘ

Referenced by:: §1.18(viii), §18.39(iv), §33.22(ii)
Permalink:: http://dlmf.nist.gov/33.14
See also:: Annotations for Ch.33

§33.14(i) Coulomb Wave Equation

ⓘ

Keywords:: Coulomb functions: variables $r,\epsilon$ , Coulomb wave equation, analytic properties, irregular solutions, regular solutions, singularities, turning points
Notes:: See Curtis (1964a, pp. ix–xxv), Seaton (1983), and Seaton (2002a, Eqs. 3, 4 and §2.3).
Permalink:: http://dlmf.nist.gov/33.14.i
See also:: Annotations for §33.14 and Ch.33

Another parametrization of (33.2.1) is given by

33.14.1		$\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}r}^{2}}+\left(\epsilon+\frac{2}{r}-\frac{% \ell(\ell+1)}{r^{2}}\right)w=0,$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\ell$ : nonnegative integer, $r$ : real variable and $\epsilon$ : real parameter Referenced by: §33.23(iii) Permalink: http://dlmf.nist.gov/33.14.E1 Encodings: TeX, pMML, png See also: Annotations for §33.14(i), §33.14 and Ch.33

where

33.14.2	$\displaystyle r$	$\displaystyle=-\eta\rho,$
33.14.2	$\displaystyle\epsilon$	$\displaystyle=1/\eta^{2}.$
ⓘ Symbols: $r$ : real variable, $\rho$ : nonnegative real variable, $\epsilon$ : real parameter and $\eta$ : real parameter Permalink: http://dlmf.nist.gov/33.14.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.14(i), §33.14 and Ch.33

Again, there is a regular singularity at $r=0$ with indices $\ell+1$ and $-\ell$ , and an irregular singularity of rank 1 at $r=\infty$ . When $\epsilon>0$ the outer turning point is given by

33.14.3		$r_{\operatorname{tp}}\left(\epsilon,\ell\right)=\left(\sqrt{1+\epsilon\ell(% \ell+1)}-1\right)\Bigm{/}\epsilon;$
ⓘ Defines: $r_{\operatorname{tp}}\left(\NVar{\epsilon},\NVar{\ell}\right)$ : outer turning point for Coulomb functions Symbols: $\ell$ : nonnegative integer and $\epsilon$ : real parameter Permalink: http://dlmf.nist.gov/33.14.E3 Encodings: TeX, pMML, png See also: Annotations for §33.14(i), §33.14 and Ch.33

compare (33.2.2).

§33.14(ii) Regular Solution $f\left(\epsilon,\ell;r\right)$

ⓘ

Keywords:: Coulomb functions, Coulomb functions: variables $r,\epsilon$ , Whittaker functions, analytic properties, confluent hypergeometric functions, definitions, functions $f(\epsilon,\ell;r),h(\epsilon,\ell;r)$ , relations to other functions
Notes:: See Seaton (2002a, Eqs. 7, 9, 14, 22).
Referenced by:: §33.22(iii)
Permalink:: http://dlmf.nist.gov/33.14.ii
See also:: Annotations for §33.14 and Ch.33

The function $f\left(\epsilon,\ell;r\right)$ is recessive (§2.7(iii)) at $r=0$ , and is defined by

33.14.4		$f\left(\epsilon,\ell;r\right)=\kappa^{\ell+1}M_{\kappa,\ell+\frac{1}{2}}\left(% 2r/\kappa\right)/(2\ell+1)!,$
ⓘ Defines: $f\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : regular Coulomb function Symbols: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $!$ : factorial (as in $n!$ ), $\ell$ : nonnegative integer, $r$ : real variable, $\epsilon$ : real parameter and $\kappa$ : quantity Referenced by: §13.18(vi), (33.14.14) Permalink: http://dlmf.nist.gov/33.14.E4 Encodings: TeX, pMML, png See also: Annotations for §33.14(ii), §33.14 and Ch.33

or equivalently

33.14.5		$f\left(\epsilon,\ell;r\right)=(2r)^{\ell+1}e^{-r/\kappa}M\left(\ell+1-\kappa,2% \ell+2,2r/\kappa\right)/{(2\ell+1)!},$
ⓘ Symbols: $f\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : regular Coulomb function, $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\ell$ : nonnegative integer, $r$ : real variable, $\epsilon$ : real parameter and $\kappa$ : quantity Referenced by: §13.6(vii) Permalink: http://dlmf.nist.gov/33.14.E5 Encodings: TeX, pMML, png See also: Annotations for §33.14(ii), §33.14 and Ch.33

where $M_{\kappa,\mu}\left(z\right)$ and $M\left(a,b,z\right)$ are defined in §§13.14(i) and 13.2(i), and

33.14.6		$\kappa=\begin{cases}(-\epsilon)^{-1/2},&\epsilon<0,r>0,\\ -(-\epsilon)^{-1/2},&\epsilon<0,r<0,\\ \pm\mathrm{i}\epsilon^{-1/2},&\epsilon>0.\end{cases}$
ⓘ Defines: $\kappa$ : quantity (locally) Symbols: $\mathrm{i}$ : imaginary unit, $r$ : real variable and $\epsilon$ : real parameter Referenced by: §33.14(ii), §33.14(iii), §33.14(iv), §33.16(v), §33.19 Permalink: http://dlmf.nist.gov/33.14.E6 Encodings: TeX, pMML, png See also: Annotations for §33.14(ii), §33.14 and Ch.33

The choice of sign in the last line of (33.14.6) is immaterial: the same function $f\left(\epsilon,\ell;r\right)$ is obtained. This is a consequence of Kummer’s transformation (§13.2(vii)).

$f\left(\epsilon,\ell;r\right)$ is real and an analytic function of $r$ in the interval $-\infty<r<\infty$ , and it is also an analytic function of $\epsilon$ when $-\infty<\epsilon<\infty$ . This includes $\epsilon=0$ , hence $f\left(\epsilon,\ell;r\right)$ can be expanded in a convergent power series in $\epsilon$ in a neighborhood of $\epsilon=0$ (§33.20(ii)).

§33.14(iii) Irregular Solution $h\left(\epsilon,\ell;r\right)$

ⓘ

Keywords:: Coulomb functions, Coulomb functions: variables $r,\epsilon$ , Whittaker functions, analytic properties, confluent hypergeometric functions, definitions, functions $f(\epsilon,\ell;r),h(\epsilon,\ell;r)$ , relations to other functions
Notes:: See Seaton (2002a, Eqs. 47, 49, 109, 131).
Referenced by:: §33.22(iii), §33.23(iii), §33.23(iv)
Permalink:: http://dlmf.nist.gov/33.14.iii
See also:: Annotations for §33.14 and Ch.33

For nonzero values of $\epsilon$ and $r$ the function $h\left(\epsilon,\ell;r\right)$ is defined by

33.14.7		$h\left(\epsilon,\ell;r\right)=\frac{\Gamma\left(\ell+1-\kappa\right)}{\pi% \kappa^{\ell}}\left(W_{\kappa,\ell+\frac{1}{2}}\left(2r/\kappa\right)+(-1)^{% \ell}S(\epsilon,r)\frac{\Gamma\left(\ell+1+\kappa\right)}{2(2\ell+1)!}M_{% \kappa,\ell+\frac{1}{2}}\left(2r/\kappa\right)\right),$
ⓘ Defines: $h\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : irregular Coulomb function Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $\ell$ : nonnegative integer, $r$ : real variable, $\epsilon$ : real parameter, $\kappa$ : quantity and $S(\epsilon,r)$ : function Referenced by: §13.18(vi) Permalink: http://dlmf.nist.gov/33.14.E7 Encodings: TeX, pMML, png See also: Annotations for §33.14(iii), §33.14 and Ch.33

where $\kappa$ is given by (33.14.6) and

33.14.8		$S(\epsilon,r)=\begin{cases}2\cos\left(\pi\|\epsilon\|^{-1/2}\right),&\epsilon<0,% r>0,\\ 0,&\epsilon<0,r<0,\\ e^{\pi\epsilon^{-1/2}},&\epsilon>0,r>0,\\ e^{-\pi\epsilon^{-1/2}},&\epsilon>0,r<0.\end{cases}$
ⓘ Defines: $S(\epsilon,r)$ : function (locally) Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $r$ : real variable and $\epsilon$ : real parameter Permalink: http://dlmf.nist.gov/33.14.E8 Encodings: TeX, pMML, png See also: Annotations for §33.14(iii), §33.14 and Ch.33

(Again, the choice of the ambiguous sign in the last line of (33.14.6) is immaterial.)

$h\left(\epsilon,\ell;r\right)$ is real and an analytic function of each of $r$ and $\epsilon$ in the intervals $-\infty<r<\infty$ and $-\infty<\epsilon<\infty$ , except when $r=0$ or $\epsilon=0$ .

§33.14(iv) Solutions $s\left(\epsilon,\ell;r\right)$ and $c\left(\epsilon,\ell;r\right)$

ⓘ

Keywords:: Coulomb functions: variables $r,\epsilon$ , definitions, functions $s(\epsilon,\ell;r),c(\epsilon,\ell;r)$ , integral representations for Dirac delta
Notes:: See Seaton (2002a, Eqs. 113–115, 122–125). For (33.14.11) and (33.14.12) see Humblet (1985, Eqs. 1.4a,b), Seaton (1982, Eq. 2.4.4).
Referenced by:: §1.17(ii), Erratum (V1.0.24) for Subsection 33.14(iv)
Permalink:: http://dlmf.nist.gov/33.14.iv
Clarification (effective with 1.0.24):: Just below (33.14.9), the phrase “provided that $\ell<(-\epsilon)^{-1/2}$ when $\epsilon<0$ ,” was removed. In (33.14.13), the constraint $\epsilon_{1},\epsilon_{2}>0$ was added. In the line immediately below (33.14.13), it was clarified that $s\left(\epsilon,\ell;r\right)$ is proportional to $\exp\left(-r/n\right)$ times a polynomial in $r/n$ , instead of simply a polynomial in $r$ . In (33.14.14), a second equality was added which expresses $\phi_{n,\ell}(r)$ in terms of Laguerre polynomials. (33.14.15), originally written as a definite integral $\int_{0}^{\infty}\phi_{n,\ell}^{2}(r)\,\mathrm{d}r=1$ , has been rewritten more generally, as an orthonormality relation. A sentence was added immediately below (33.14.15) indicating that the functions $\phi_{n,\ell}$ , $n=\ell,\ell+1,\ldots$ , do not form a complete orthonormal system.
See also:: Annotations for §33.14 and Ch.33

The functions $s\left(\epsilon,\ell;r\right)$ and $c\left(\epsilon,\ell;r\right)$ are defined by

33.14.9	$\displaystyle s\left(\epsilon,\ell;r\right)$	$\displaystyle=(B(\epsilon,\ell)/2)^{1/2}f\left(\epsilon,\ell;r\right),$
33.14.9	$\displaystyle c\left(\epsilon,\ell;r\right)$	$\displaystyle=(2B(\epsilon,\ell))^{-1/2}h\left(\epsilon,\ell;r\right),$
ⓘ Defines: $c\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : irregular Coulomb function and $s\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : regular Coulomb function Symbols: $f\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : regular Coulomb function, $h\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : irregular Coulomb function, $\ell$ : nonnegative integer, $r$ : real variable, $\epsilon$ : real parameter and $B(\epsilon,\ell)$ : function Referenced by: (33.14.14), §33.14(iv), §33.16(iv), §33.16(v), Erratum (V1.0.24) for Subsection 33.14(iv) Permalink: http://dlmf.nist.gov/33.14.E9 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.14(iv), §33.14 and Ch.33

where

33.14.10		$B(\epsilon,\ell)=\begin{cases}A(\epsilon,\ell)\left(1-\exp\left(-2\pi/\epsilon% ^{1/2}\right)\right)^{-1},&\epsilon>0,\\ A(\epsilon,\ell),&\epsilon\leq 0,\end{cases}$
ⓘ Defines: $B(\epsilon,\ell)$ : function (locally) Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $\ell$ : nonnegative integer, $\epsilon$ : real parameter and $A(\epsilon,\ell)$ : function Permalink: http://dlmf.nist.gov/33.14.E10 Encodings: TeX, pMML, png See also: Annotations for §33.14(iv), §33.14 and Ch.33

and

33.14.11		$A(\epsilon,\ell)=\prod_{k=0}^{\ell}(1+\epsilon k^{2}).$
ⓘ Defines: $A(\epsilon,\ell)$ : function (locally) Symbols: $k$ : nonnegative integer, $\ell$ : nonnegative integer and $\epsilon$ : real parameter Referenced by: §33.14(iv), §33.16(ii), §33.16(iii), §33.19, §33.20(iii) Permalink: http://dlmf.nist.gov/33.14.E11 Encodings: TeX, pMML, png See also: Annotations for §33.14(iv), §33.14 and Ch.33

An alternative formula for $A(\epsilon,\ell)$ is

33.14.12		$A(\epsilon,\ell)=\frac{\Gamma\left(1+\ell+\kappa\right)}{\Gamma\left(\kappa-% \ell\right)}\kappa^{-2\ell-1},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\ell$ : nonnegative integer, $\epsilon$ : real parameter, $\kappa$ : quantity and $A(\epsilon,\ell)$ : function Referenced by: §33.14(iv), §33.16(ii), §33.16(iii), §33.16(v), §33.19, §33.20(iii) Permalink: http://dlmf.nist.gov/33.14.E12 Encodings: TeX, pMML, png See also: Annotations for §33.14(iv), §33.14 and Ch.33

the choice of sign in the last line of (33.14.6) again being immaterial.

When $\epsilon<0$ and $\ell>(-\epsilon)^{-1/2}$ the quantity $A(\epsilon,\ell)$ may be negative, causing $s\left(\epsilon,\ell;r\right)$ and $c\left(\epsilon,\ell;r\right)$ to become imaginary.

The function $s\left(\epsilon,\ell;r\right)$ has the following properties:

33.14.13		$\int_{0}^{\infty}s\left(\epsilon_{1},\ell;r\right)s\left(\epsilon_{2},\ell;r% \right)\,\mathrm{d}r=\delta\left(\epsilon_{1}-\epsilon_{2}\right),$
$\epsilon_{1},\epsilon_{2}>0$ ,
ⓘ Symbols: $s\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : regular Coulomb function, $\delta\left(\NVar{x-a}\right)$ : Dirac delta (or Dirac delta function), $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ell$ : nonnegative integer, $r$ : real variable and $\epsilon$ : real parameter Referenced by: §13.10(vi), §13.23(v), §33.14(iv), Erratum (V1.0.24) for Subsection 33.14(iv) Permalink: http://dlmf.nist.gov/33.14.E13 Encodings: TeX, pMML, png Addition (effective with 1.0.24): The constraint $\epsilon_{1},\epsilon_{2}>0$ was added. See also: Annotations for §33.14(iv), §33.14 and Ch.33

where the right-hand side is the Dirac delta (§1.17). When $\epsilon=-1/n^{2}$ , $n=\ell+1,\ell+2,\dots$ , $s\left(\epsilon,\ell;r\right)$ is $\exp\left(-r/n\right)$ times a polynomial in $r/n$ , and

33.14.14		$\phi_{n,\ell}(r)=(-1)^{\ell+1+n}(2/n^{3})^{1/2}s\left(-1/n^{2},\ell;r\right)=% \frac{(-1)^{\ell+1+n}}{n^{\ell+2}}\left(\frac{(n-\ell-1)!}{(n+\ell)!}\right)^{% 1/2}(2r)^{\ell+1}{\mathrm{e}}^{-r/n}L^{(2\ell+1)}_{n-\ell-1}\left(2r/n\right)$
ⓘ Defines: $\phi_{n,\ell}(r)$ : function (locally) Symbols: $s\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : regular Coulomb function, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\ell$ : nonnegative integer and $r$ : real variable Referenced by: (33.14.14), (33.14.15), §33.14(iv), §33.22(v), Erratum (V1.0.24) for Equation (33.14.15), Erratum (V1.0.24) for Subsection 33.14(iv) Permalink: http://dlmf.nist.gov/33.14.E14 Encodings: TeX, pMML, png Addition (effective with 1.0.24): For the second equation in (33.14.14) first, via (33.14.9) and (33.14.4), express $\phi_{n,\ell}(r)$ in terms of Whittaker functions and then use (13.18.17). See also: Annotations for §33.14(iv), §33.14 and Ch.33

satisfies

33.14.15		$\int_{0}^{\infty}\phi_{m,\ell}(r)\phi_{n,\ell}(r)\,\mathrm{d}r=\delta_{m,n}.$
ⓘ Symbols: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ell$ : nonnegative integer, $r$ : real variable and $\phi_{n,\ell}(r)$ : function Proof sketch: Derivable from (33.14.14) by combining it with Dunkl (2003, Theorem 2.2). Referenced by: §13.10(vi), §13.23(v), §33.14(iv), Erratum (V1.0.24) for Equation (33.14.15), Erratum (V1.0.24) for Subsection 33.14(iv) Permalink: http://dlmf.nist.gov/33.14.E15 Encodings: TeX, pMML, png Clarification (effective with 1.0.24): This equation, originally written as a definite integral $\int_{0}^{\infty}\phi_{n,\ell}^{2}(r)\,\mathrm{d}r=1$ , was rewritten more generally as an orthonormality relation. See also: Annotations for §33.14(iv), §33.14 and Ch.33

Note that the functions $\phi_{n,\ell}$ , $n=\ell,\ell+1,\ldots$ , do not form a complete orthonormal system.

§33.14(v) Wronskians

ⓘ

Keywords:: Coulomb functions: variables $r,\epsilon$ , Wronskians
Notes:: See Seaton (2002a, Eqs. 51, 116).
Permalink:: http://dlmf.nist.gov/33.14.v
See also:: Annotations for §33.14 and Ch.33

With arguments $\epsilon,\ell,r$ suppressed,

33.14.16	$\displaystyle\mathscr{W}\left\{h,f\right\}$	$\displaystyle=2/\pi,$
33.14.16	$\displaystyle\mathscr{W}\left\{c,s\right\}$	$\displaystyle=1/\pi.$
ⓘ Symbols: $c\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : irregular Coulomb function, $f\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : regular Coulomb function, $h\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : irregular Coulomb function, $s\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : regular Coulomb function, $\mathscr{W}$ : Wronskian and $\pi$ : the ratio of the circumference of a circle to its diameter Permalink: http://dlmf.nist.gov/33.14.E16 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.14(v), §33.14 and Ch.33

33.13 Complex Variable and Parameters 33.15 Graphics

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§33.14 Definitions and Basic Properties

Contents

§33.14(i) Coulomb Wave Equation

§33.14(ii) Regular Solution $f\left(\epsilon,\ell;r\right)$

§33.14(iii) Irregular Solution $h\left(\epsilon,\ell;r\right)$

§33.14(iv) Solutions $s\left(\epsilon,\ell;r\right)$ and $c\left(\epsilon,\ell;r\right)$

§33.14(v) Wronskians

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§33.14 Definitions and Basic Properties

Contents

§33.14(i) Coulomb Wave Equation

§33.14(ii) Regular Solution f⁡(ϵ,ℓ;r)

§33.14(iii) Irregular Solution h⁡(ϵ,ℓ;r)

§33.14(iv) Solutions s⁡(ϵ,ℓ;r) and c⁡(ϵ,ℓ;r)

§33.14(v) Wronskians

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§33.14(ii) Regular Solution $f\left(\epsilon,\ell;r\right)$

§33.14(iii) Irregular Solution $h\left(\epsilon,\ell;r\right)$

§33.14(iv) Solutions $s\left(\epsilon,\ell;r\right)$ and $c\left(\epsilon,\ell;r\right)$