§33.14 Definitions and Basic Properties

Another parametrization of (33.2.1) is given by

33.14.1		$$\frac{d^{2}w}{{dr}^2}+\left(ϵ+\frac{2}{r}-\frac{\mathrm{\ell }(\mathrm{\ell }+1)}{r^2}\right)w=0,$$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: 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		$r$	$=-\eta \rho ,$
		$ϵ$	$=1/{\eta }^2.$
ⓘ Symbols: $r$: real variable, $\rho$: nonnegative real variable, $ϵ$: 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 $\mathrm{\ell }+1$ and $-\mathrm{\ell }$, and an irregular singularity of rank 1 at $r=\mathrm{\infty }$. When $ϵ>0$ the outer turning point is given by

33.14.3		$$r_{\mathrm{tp}}(ϵ,\mathrm{\ell })=\left(\sqrt{1+ϵ\mathrm{\ell }(\mathrm{\ell }+1)}-1\right)/ϵ;$$
ⓘ Defines: $r_{\mathrm{tp}}(ϵ,\mathrm{\ell })$: outer turning point for Coulomb functions Symbols: $\mathrm{\ell }$: nonnegative integer and $ϵ$: 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).

The function $f(ϵ,\mathrm{\ell };r)$ is recessive (§2.7(iii)) at $r=0$, and is defined by

33.14.4		$$f(ϵ,\mathrm{\ell };r)={\kappa }^{\mathrm{\ell }+1}{M}_{\kappa ,\mathrm{\ell }+\frac{1}{2}}\left(2r/\kappa \right)/(2\mathrm{\ell }+1)!,$$
ⓘ Defines: $f(ϵ,\mathrm{\ell };r)$: regular Coulomb function Symbols: $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $!$: factorial (as in $n!$), $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: 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		$$f(ϵ,\mathrm{\ell };r)={(2r)}^{\mathrm{\ell }+1}{\mathrm{e}}^{-r/\kappa }M(\mathrm{\ell }+1-\kappa ,2\mathrm{\ell }+2,2r/\kappa )/(2\mathrm{\ell }+1)!,$$
ⓘ Symbols: $f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $M(a,b,z)$: $={}_{1}F_1(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: 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 $M(a,b,z)$ are defined in §§13.14(i) and 13.2(i), and

33.14.6
ⓘ Defines: $\kappa$: quantity (locally) Symbols: $\mathrm{i}$: imaginary unit, $r$: real variable and $ϵ$: 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 $f(ϵ,\mathrm{\ell };r)$ is obtained. This is a consequence of Kummer’s transformation (§13.2(vii)).

$f(ϵ,\mathrm{\ell };r)$ is real and an analytic function of $r$ in the interval , and it is also an analytic function of $ϵ$ when . This includes $ϵ=0$, hence $f(ϵ,\mathrm{\ell };r)$ can be expanded in a convergent power series in $ϵ$ in a neighborhood of $ϵ=0$ (§33.20(ii)).

For nonzero values of $ϵ$ and $r$ the function $h(ϵ,\mathrm{\ell };r)$ is defined by

33.14.7		$$h(ϵ,\mathrm{\ell };r)=\frac{\mathrm{\Gamma }\left(\mathrm{\ell }+1-\kappa \right)}{\pi {\kappa }^{\mathrm{\ell }}}\left(W_{\kappa ,\mathrm{\ell }+\frac{1}{2}}\left(2r/\kappa \right)+{(-1)}^{\mathrm{\ell }}S(ϵ,r)\frac{\mathrm{\Gamma }\left(\mathrm{\ell }+1+\kappa \right)}{2(2\mathrm{\ell }+1)!}{M}_{\kappa ,\mathrm{\ell }+\frac{1}{2}}\left(2r/\kappa \right)\right),$$
ⓘ Defines: $h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter, $\kappa$: quantity and $S(ϵ,r)$: 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
ⓘ Defines: $S(ϵ,r)$: function (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $r$: real variable and $ϵ$: 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.)

$h(ϵ,\mathrm{\ell };r)$ is real and an analytic function of each of $r$ and $ϵ$ in the intervals and , except when $r=0$ or $ϵ=0$.

The functions $s(ϵ,\mathrm{\ell };r)$ and $c(ϵ,\mathrm{\ell };r)$ are defined by

33.14.9		$s(ϵ,\mathrm{\ell };r)$	$={(B(ϵ,\mathrm{\ell })/2)}^{1/2}f(ϵ,\mathrm{\ell };r),$
		$c(ϵ,\mathrm{\ell };r)$	$={(2B(ϵ,\mathrm{\ell }))}^{-1/2}h(ϵ,\mathrm{\ell };r),$
ⓘ Defines: $c(ϵ,\mathrm{\ell };r)$: irregular Coulomb function and $s(ϵ,\mathrm{\ell };r)$: regular Coulomb function Symbols: $f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter and $B(ϵ,\mathrm{\ell })$: 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
ⓘ Defines: $B(ϵ,\mathrm{\ell })$: function (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{\ell }$: nonnegative integer, $ϵ$: real parameter and $A(ϵ,\mathrm{\ell })$: 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		$$A(ϵ,\mathrm{\ell })=\prod _{k=0}^{\mathrm{\ell }}(1+ϵk^2).$$
ⓘ Defines: $A(ϵ,\mathrm{\ell })$: function (locally) Symbols: $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer and $ϵ$: 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 $A(ϵ,\mathrm{\ell })$ is

33.14.12		$$A(ϵ,\mathrm{\ell })=\frac{\mathrm{\Gamma }\left(1+\mathrm{\ell }+\kappa \right)}{\mathrm{\Gamma }\left(\kappa -\mathrm{\ell }\right)}{\kappa }^{-2\mathrm{\ell }-1},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{\ell }$: nonnegative integer, $ϵ$: real parameter, $\kappa$: quantity and $A(ϵ,\mathrm{\ell })$: 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 and $\mathrm{\ell }>{(-ϵ)}^{-1/2}$ the quantity $A(ϵ,\mathrm{\ell })$ may be negative, causing $s(ϵ,\mathrm{\ell };r)$ and $c(ϵ,\mathrm{\ell };r)$ to become imaginary.

The function $s(ϵ,\mathrm{\ell };r)$ has the following properties:

33.14.13		$${\int }_0^{\mathrm{\infty }}s({ϵ}_1,\mathrm{\ell };r)s({ϵ}_2,\mathrm{\ell };r)dr=\delta \left({ϵ}_1-{ϵ}_2\right),$$
${ϵ}_1,{ϵ}_2>0$,
ⓘ Symbols: $s(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $dx$: differential of $x$, $\int$: integral, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: 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 ${ϵ}_1,{ϵ}_2>0$ 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 $ϵ=-1/n^2$, $n=\mathrm{\ell }+1,\mathrm{\ell }+2,\mathrm{\dots }$, $s(ϵ,\mathrm{\ell };r)$ is $\exp \left(-r/n\right)$ times a polynomial in $r/n$, and

33.14.14		$$\begin{array}{ll}{\varphi }_{n,\mathrm{\ell }}(r)& ={(-1)}^{\mathrm{\ell }+1+n}{(2/n^3)}^{1/2}s(-1/n^2,\mathrm{\ell };r)\\ & =\frac{{(-1)}^{\mathrm{\ell }+1+n}}{n^{\mathrm{\ell }+2}}{\left(\frac{(n-\mathrm{\ell }-1)!}{(n+\mathrm{\ell })!}\right)}^{1/2}{(2r)}^{\mathrm{\ell }+1}{\mathrm{e}}^{-r/n}{L}_{n-\mathrm{\ell }-1}^{(2\mathrm{\ell }+1)}\left(2r/n\right)\end{array}$$
ⓘ Defines: ${\varphi }_{n,\mathrm{\ell }}(r)$: function (locally) Symbols: $s(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{\ell }$: 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 ${\varphi }_{n,\mathrm{\ell }}(r)$ 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		$${\int }_0^{\mathrm{\infty }}{\varphi }_{m,\mathrm{\ell }}(r){\varphi }_{n,\mathrm{\ell }}(r)dr={\delta }_{m,n}.$$
ⓘ Symbols: ${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$, $\int$: integral, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and ${\varphi }_{n,\mathrm{\ell }}(r)$: function Source: 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 ${\int }_0^{\mathrm{\infty }}{\varphi }_{n,\mathrm{\ell }}^2(r)dr=1$, was rewritten more generally as an orthonormality relation. See also: Annotations for §33.14(iv), §33.14 and Ch.33

Note that the functions ${\varphi }_{n,\mathrm{\ell }}$, $n=\mathrm{\ell },\mathrm{\ell }+1,\mathrm{\dots }$, do not form a complete orthonormal system.

With arguments $ϵ,\mathrm{\ell },r$ suppressed,

33.14.16		$𝒲\left\{h,f\right\}$	$=2/\pi ,$
		$𝒲\left\{c,s\right\}$	$=1/\pi .$
ⓘ Symbols: $c(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $s(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $𝒲$: 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