§33.14 Definitions and Basic Properties

Referenced by:: ¶ ‣ §33.22(ii)
Permalink:: http://dlmf.nist.gov/33.14

§33.14(i) Coulomb Wave Equation
§33.14(ii) Regular Solution $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)$
§33.14(iii) Irregular Solution $\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)$
§33.14(iv) Solutions $\mathop{s\/}\nolimits\!\left(\epsilon,\ell;r\right)$ and $\mathop{c\/}\nolimits\!\left(\epsilon,\ell;r\right)$
§33.14(v) Wronskians

§33.14(i) Coulomb Wave Equation

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

Another parametrization of (33.2.1) is given by

33.14.1 $\frac{{d}^{2}w}{{dr}^{2}}+\left(\epsilon+\frac{2}{r}-\frac{\ell(\ell+1)}{r^{2}}\right)w=0,$

Symbols:: $\frac{df}{dx}$ : 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
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where

33.14.2

$r=-\eta\rho,$

$\epsilon=1/\eta^{2}.$

Symbols:: $r$ : real variable, $\rho$ : nonnegative real variable, $\epsilon$ : real parameter and $\eta$ : real parameter
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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 $\mathop{r_{{\mathrm{tp}}}\/}\nolimits\!\left(\epsilon,\ell\right)=\left(\sqrt{1+\epsilon\ell(\ell+1)}-1\right)\Bigm/\epsilon;$

Defines:: $\mathop{r_{{\mathrm{tp}}}\/}\nolimits\!\left(\epsilon,\ell\right)$ : outer turning point for Coulomb functions
Symbols:: $\ell$ : nonnegative integer and $\epsilon$ : real parameter
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compare (33.2.2).

§33.14(ii) Regular Solution $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)$

Notes:: See Seaton (2002a, Eqs. 7, 9, 14, 22).
Keywords:: Coulomb functions, Coulomb functions: variables $r,\epsilon$ , Whittaker functions, confluent hypergeometric functions
Referenced by:: §33.22(iii)
Permalink:: http://dlmf.nist.gov/33.14.ii

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

33.14.4 $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)=\kappa^{{\ell+1}}\mathop{M_{{\kappa,\ell+\frac{1}{2}}}\/}\nolimits\!\left(2r/\kappa\right)/(2\ell+1)!,$

Defines:: $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : regular Coulomb function
Symbols:: $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $!$ : $n!$ : factorial, $\ell$ : nonnegative integer, $r$ : real variable, $\epsilon$ : real parameter and $\kappa$ : quantity
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or equivalently

33.14.5 $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)=(2r)^{{\ell+1}}e^{{-r/\kappa}}\mathop{M\/}\nolimits\!\left(\ell+1-\kappa,2\ell+2,2r/\kappa\right)/{(2\ell+1)!},$

Symbols:: $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : regular Coulomb function, $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $e$ : base of exponential function, $!$ : $n!$ : factorial, $\ell$ : nonnegative integer, $r$ : real variable, $\epsilon$ : real parameter and $\kappa$ : quantity
Permalink:: http://dlmf.nist.gov/33.14.E5
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where $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ and $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ are defined in §§13.14(i) and 13.2(i), and

33.14.6 $\kappa=\begin{cases}(-\epsilon)^{{-1/2}},&\epsilon<0,r>0,\\ -(-\epsilon)^{{-1/2}},&\epsilon<0,r<0,\\ \pm i\epsilon^{{-1/2}},&\epsilon>0.\end{cases}$

Symbols:: $r$ : real variable, $\epsilon$ : real parameter and $\kappa$ : quantity
Referenced by:: §33.14(ii), §33.14(iii), §33.14(iv), §33.16(v), §33.19
Permalink:: http://dlmf.nist.gov/33.14.E6
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The choice of sign in the last line of (33.14.6) is immaterial: the same function $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)$ is obtained. This is a consequence of Kummer’s transformation (§13.2(vii)).

$\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)$ 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 $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)$ can be expanded in a convergent power series in $\epsilon$ in a neighborhood of $\epsilon=0$ (§33.20(ii)).

§33.14(iii) Irregular Solution $\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)$

Notes:: See Seaton (2002a, Eqs. 47, 49, 109, 131).
Keywords:: Coulomb functions, Coulomb functions: variables $r,\epsilon$ , Whittaker functions, confluent hypergeometric functions
Referenced by:: §33.22(iii), §33.23(iii), §33.23(iv)
Permalink:: http://dlmf.nist.gov/33.14.iii

For nonzero values of $\epsilon$ and $r$ the function $\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)$ is defined by

33.14.7 $\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(\ell+1-\kappa\right)}{\pi\kappa^{\ell}}\left(\mathop{W_{{\kappa,\ell+\frac{1}{2}}}\/}\nolimits\!\left(2r/\kappa\right)+(-1)^{\ell}S(\epsilon,r)\frac{\mathop{\Gamma\/}\nolimits\!\left(\ell+1+\kappa\right)}{2(2\ell+1)!}\mathop{M_{{\kappa,\ell+\frac{1}{2}}}\/}\nolimits\!\left(2r/\kappa\right)\right),$

Defines:: $\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : irregular Coulomb function
Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $!$ : $n!$ : factorial, $\ell$ : nonnegative integer, $r$ : real variable, $\epsilon$ : real parameter, $\kappa$ : quantity and $S(\epsilon,r)$ : function
Permalink:: http://dlmf.nist.gov/33.14.E7
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where $\kappa$ is given by (33.14.6) and

33.14.8 $S(\epsilon,r)=\begin{cases}2\mathop{\cos\/}\nolimits\!\left(\pi|\epsilon|^{{-1/2}}\right),&\epsilon<0,r>0,\\ 0,&\epsilon<0,r<0,\\ e^{{\pi\epsilon^{{-1/2}}}},&\epsilon>0,r>0,\\ e^{{-\pi\epsilon^{{-1/2}}}},&\epsilon>0,r<0.\end{cases}$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $e$ : base of exponential function, $r$ : real variable, $\epsilon$ : real parameter and $S(\epsilon,r)$ : function
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(Again, the choice of the ambiguous sign in the last line of (33.14.6) is immaterial.)

$\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)$ 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 $\mathop{s\/}\nolimits\!\left(\epsilon,\ell;r\right)$ and $\mathop{c\/}\nolimits\!\left(\epsilon,\ell;r\right)$

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).
Keywords:: Coulomb functions: variables $r,\epsilon$
Referenced by:: ¶ ‣ §1.17(ii)
Permalink:: http://dlmf.nist.gov/33.14.iv

The functions $\mathop{s\/}\nolimits\!\left(\epsilon,\ell;r\right)$ and $\mathop{c\/}\nolimits\!\left(\epsilon,\ell;r\right)$ are defined by

33.14.9

$\mathop{s\/}\nolimits\!\left(\epsilon,\ell;r\right)=(B(\epsilon,\ell)/2)^{{1/2}}\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right),$

$\mathop{c\/}\nolimits\!\left(\epsilon,\ell;r\right)=(2B(\epsilon,\ell))^{{-1/2}}\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right),$

Defines:: $\mathop{c\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : irregular Coulomb function and $\mathop{s\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : regular Coulomb function
Symbols:: $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : regular Coulomb function, $\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : irregular Coulomb function, $\ell$ : nonnegative integer, $r$ : real variable, $\epsilon$ : real parameter and $B(\epsilon,\ell)$ : function
Referenced by:: §33.16(iv), §33.16(v)
Permalink:: http://dlmf.nist.gov/33.14.E9
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provided that $\ell<(-\epsilon)^{{-1/2}}$ when $\epsilon<0$ , where

33.14.10 $B(\epsilon,\ell)=\begin{cases}A(\epsilon,\ell)\left(1-\mathop{\exp\/}\nolimits\!\left(-2\pi/\epsilon^{{1/2}}\right)\right)^{{-1}},&\epsilon>0,\\ A(\epsilon,\ell),&\epsilon\leq 0,\end{cases}$

Symbols:: $\mathop{\exp\/}\nolimits z$ : exponential function, $\ell$ : nonnegative integer, $\epsilon$ : real parameter, $B(\epsilon,\ell)$ : function and $A(\epsilon,\ell)$ : function
Permalink:: http://dlmf.nist.gov/33.14.E10
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and

33.14.11 $A(\epsilon,\ell)=\prod _{{k=0}}^{\ell}(1+\epsilon k^{2}).$

Symbols:: $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\epsilon$ : real parameter and $A(\epsilon,\ell)$ : function
Referenced by:: §33.14(iv), §33.16(ii), §33.16(iii), §33.19, §33.20(iii)
Permalink:: http://dlmf.nist.gov/33.14.E11
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An alternative formula for $A(\epsilon,\ell)$ is

33.14.12 $A(\epsilon,\ell)=\frac{\mathop{\Gamma\/}\nolimits\!\left(1+\ell+\kappa\right)}{\mathop{\Gamma\/}\nolimits\!\left(\kappa-\ell\right)}\kappa^{{-2\ell-1}},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : 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
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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 $\mathop{s\/}\nolimits\!\left(\epsilon,\ell;r\right)$ and $\mathop{c\/}\nolimits\!\left(\epsilon,\ell;r\right)$ to become imaginary.

The function $\mathop{s\/}\nolimits\!\left(\epsilon,\ell;r\right)$ has the following properties:

33.14.13 $\int _{0}^{\infty}\mathop{s\/}\nolimits\!\left(\epsilon _{1},\ell;r\right)\mathop{s\/}\nolimits\!\left(\epsilon _{2},\ell;r\right)dr=\mathop{\delta\/}\nolimits\!\left(\epsilon _{1}-\epsilon _{2}\right),$

Symbols:: $\mathop{s\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : regular Coulomb function, $\mathop{\delta\/}\nolimits\!\left(x-a\right)$ : Dirac delta (or Dirac delta function), $dx$ : differential of $x$ , $\int$ : integral, $\ell$ : nonnegative integer, $r$ : real variable and $\epsilon$ : real parameter
Permalink:: http://dlmf.nist.gov/33.14.E13
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where the right-hand side is the Dirac delta (§1.17). When $\epsilon=-1/n^{2}$ , $n=\ell+1,\ell+2,\dots$ , $\mathop{s\/}\nolimits\!\left(\epsilon,\ell;r\right)$ is $\mathop{\exp\/}\nolimits\!\left(-r/n\right)$ times a polynomial in $r$ , and

33.14.14 $\phi _{{n,\ell}}(r)=(-1)^{{\ell+1+n}}(2/n^{3})^{{1/2}}\mathop{s\/}\nolimits\!\left(-1/n^{2},\ell;r\right)$

Symbols:: $\mathop{s\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : regular Coulomb function, $\ell$ : nonnegative integer, $r$ : real variable and $\phi _{{n,\ell}}(r)$ : function
Referenced by:: §33.22(v)
Permalink:: http://dlmf.nist.gov/33.14.E14
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satisfies

33.14.15 $\int _{0}^{\infty}\phi _{{n,\ell}}^{2}(r)dr=1.$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\ell$ : nonnegative integer, $r$ : real variable and $\phi _{{n,\ell}}(r)$ : function
Permalink:: http://dlmf.nist.gov/33.14.E15
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§33.14(v) Wronskians

Notes:: See Seaton (2002a, Eqs. 51, 116).
Keywords:: Coulomb functions: variables $r,\epsilon$
Permalink:: http://dlmf.nist.gov/33.14.v

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

33.14.16

$\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{h\/}\nolimits,\mathop{f\/}\nolimits\right\}=2/\pi,$

$\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{c\/}\nolimits,\mathop{s\/}\nolimits\right\}=1/\pi.$

Symbols:: $\mathop{c\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : irregular Coulomb function, $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : regular Coulomb function, $\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : irregular Coulomb function, $\mathop{s\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : regular Coulomb function and $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian
Permalink:: http://dlmf.nist.gov/33.14.E16
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§33.14 Definitions and Basic Properties

Contents

§33.14(i) Coulomb Wave Equation

§33.14(ii) Regular Solution

§33.14(iii) Irregular Solution

§33.14(iv) Solutions and

§33.14(v) Wronskians

§33.14(ii) Regular Solution $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)$

§33.14(iii) Irregular Solution $\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)$

§33.14(iv) Solutions $\mathop{s\/}\nolimits\!\left(\epsilon,\ell;r\right)$ and $\mathop{c\/}\nolimits\!\left(\epsilon,\ell;r\right)$