# §33.14 Definitions and Basic Properties

## §33.14(i) Coulomb Wave Equation

Another parametrization of (33.2.1) is given by

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

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)

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: $\mathop{r_{\mathrm{tp}}\/}\nolimits\!\left(\NVar{\epsilon},\NVar{\ell}\right)$: 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)

compare (33.2.2).

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

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: $\mathop{f\/}\nolimits\!\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$: regular Coulomb function Symbols: $\mathop{M_{\NVar{\kappa},\NVar{\mu}}\/}\nolimits\!\left(\NVar{z}\right)$: 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) Permalink: http://dlmf.nist.gov/33.14.E4 Encodings: TeX, pMML, png See also: Annotations for 33.14(ii)

or equivalently

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

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 $\kappa=\begin{cases}(-\epsilon)^{-1/2},&\epsilon<0,r>0,\\ -(-\epsilon)^{-1/2},&\epsilon<0,r<0,\\ \pm\mathrm{i}\epsilon^{-1/2},&\epsilon>0.\end{cases}$ Defines: $\kappa$: quantity (locally) Symbols: $r$: 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)

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

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: $\mathop{h\/}\nolimits\!\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$: irregular Coulomb function Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\mathop{M_{\NVar{\kappa},\NVar{\mu}}\/}\nolimits\!\left(\NVar{z}\right)$: Whittaker confluent hypergeometric function, $\mathop{W_{\NVar{\kappa},\NVar{\mu}}\/}\nolimits\!\left(\NVar{z}\right)$: 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)

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

 33.14.8 $S(\epsilon,r)=\begin{cases}2\mathop{\cos\/}\nolimits\!\left(\pi|\epsilon|^{-1/% 2}\right),&\epsilon<0,r>0,\\ 0,&\epsilon<0,r<0,\\ e^{\pi\epsilon^{-1/2}},&\epsilon>0,r>0,\\ e^{-\pi\epsilon^{-1/2}},&\epsilon>0,r<0.\end{cases}$ Defines: $S(\epsilon,r)$: function (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $\mathrm{e}$: base of exponential function, $r$: 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)

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

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

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

provided that $\ell<(-\epsilon)^{-1/2}$ when $\epsilon<0$, where

 33.14.10 $B(\epsilon,\ell)=\begin{cases}A(\epsilon,\ell)\left(1-\mathop{\exp\/}\nolimits% \!\left(-2\pi/\epsilon^{1/2}\right)\right)^{-1},&\epsilon>0,\\ A(\epsilon,\ell),&\epsilon\leq 0,\end{cases}$ Defines: $B(\epsilon,\ell)$: function (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\exp\/}\nolimits\NVar{z}$: 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)

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)

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

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

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 $\int_{0}^{\infty}\mathop{s\/}\nolimits\!\left(\epsilon_{1},\ell;r\right)% \mathop{s\/}\nolimits\!\left(\epsilon_{2},\ell;r\right)\mathrm{d}r=\mathop{% \delta\/}\nolimits\!\left(\epsilon_{1}-\epsilon_{2}\right),$

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 $\phi_{n,\ell}(r)=(-1)^{\ell+1+n}(2/n^{3})^{1/2}\mathop{s\/}\nolimits\!\left(-1% /n^{2},\ell;r\right)$ Defines: $\phi_{n,\ell}(r)$: function (locally) Symbols: $\mathop{s\/}\nolimits\!\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$: regular Coulomb function, $\ell$: nonnegative integer and $r$: real variable Referenced by: §33.22(v) Permalink: http://dlmf.nist.gov/33.14.E14 Encodings: TeX, pMML, png See also: Annotations for 33.14(iv)

satisfies

 33.14.15 $\int_{0}^{\infty}\phi_{n,\ell}^{2}(r)\mathrm{d}r=1.$

## §33.14(v) Wronskians

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

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