# §33.14 Definitions and Basic Properties

## §33.14(i) Coulomb Wave Equation

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, $\ell$: 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 $\displaystyle r$ $\displaystyle=-\eta\rho,$ $\displaystyle\epsilon$ $\displaystyle=1/\eta^{2}.$ ⓘ 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 $f\left(\epsilon,\ell;r\right)$

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

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, 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 $h\left(\epsilon,\ell;r\right)$

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 and $-\infty<\epsilon<\infty$, except when $r=0$ or $\epsilon=0$.

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

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

 33.14.9 $\displaystyle s\left(\epsilon,\ell;r\right)$ $\displaystyle=(B(\epsilon,\ell)/2)^{1/2}f\left(\epsilon,\ell;r\right),$ $\displaystyle c\left(\epsilon,\ell;r\right)$ $\displaystyle=(2B(\epsilon,\ell))^{-1/2}h\left(\epsilon,\ell;r\right),$ ⓘ 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, $\int$: 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, $\int$: 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

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

 33.14.16 $\displaystyle\mathscr{W}\left\{h,f\right\}$ $\displaystyle=2/\pi,$ $\displaystyle\mathscr{W}\left\{c,s\right\}$ $\displaystyle=1/\pi.$