§33.2 Definitions and Basic Properties

33.2.1		$$\frac{d^{2}w}{{d\rho }^2}+\left(1-\frac{2\eta }{\rho }-\frac{\mathrm{\ell }(\mathrm{\ell }+1)}{{\rho }^2}\right)w=0,$$
$\mathrm{\ell }=0,1,2,\mathrm{\dots }$.
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter Referenced by: §33.12(ii), §33.14(i), §33.22(iv), §33.23(iii) Permalink: http://dlmf.nist.gov/33.2.E1 Encodings: TeX, pMML, png See also: Annotations for §33.2(i), §33.2 and Ch.33

This differential equation has a regular singularity at $\rho =0$ with indices $\mathrm{\ell }+1$ and $-\mathrm{\ell }$, and an irregular singularity of rank 1 at $\rho =\mathrm{\infty }$ (§§2.7(i), 2.7(ii)). There are two turning points, that is, points at which $d^{2}w/{d\rho }^2=0$ (§2.8(i)). The outer one is given by

33.2.2		$${\rho }_{\mathrm{tp}}(\eta ,\mathrm{\ell })=\eta +{({\eta }^2+\mathrm{\ell }(\mathrm{\ell }+1))}^{1/2}.$$
ⓘ Defines: ${\rho }_{\mathrm{tp}}(\eta ,\mathrm{\ell })$: outer turning point for Coulomb radial functions Symbols: $\mathrm{\ell }$: nonnegative integer and $\eta$: real parameter Referenced by: §33.12(i), §33.14(i) Permalink: http://dlmf.nist.gov/33.2.E2 Encodings: TeX, pMML, png See also: Annotations for §33.2(i), §33.2 and Ch.33

The function $F_{\mathrm{\ell }}(\eta ,\rho )$ is recessive (§2.7(iii)) at $\rho =0$, and is defined by

33.2.3		$$F_{\mathrm{\ell }}(\eta ,\rho )=C_{\mathrm{\ell }}\left(\eta \right)2^{-\mathrm{\ell }-1}{(\mp \mathrm{i})}^{\mathrm{\ell }+1}{M}_{\pm \mathrm{i}\eta ,\mathrm{\ell }+\frac{1}{2}}\left(\pm 2\mathrm{i}\rho \right),$$
ⓘ Defines: $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function Symbols: $C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{i}$: imaginary unit, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter Referenced by: §13.18(vi), §33.2(ii) Permalink: http://dlmf.nist.gov/33.2.E3 Encodings: TeX, pMML, png See also: Annotations for §33.2(ii), §33.2 and Ch.33

or equivalently

33.2.4		$$F_{\mathrm{\ell }}(\eta ,\rho )=C_{\mathrm{\ell }}\left(\eta \right){\rho }^{\mathrm{\ell }+1}{\mathrm{e}}^{\mp \mathrm{i}\rho }M(\mathrm{\ell }+1\mp \mathrm{i}\eta ,2\mathrm{\ell }+2,\pm 2\mathrm{i}\rho ),$$
ⓘ Symbols: $C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $M(a,b,z)$: $={}_{1}F_1(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter A&S Ref: 14.1.3 Referenced by: §13.6(vii), §33.2(ii), §33.6 Permalink: http://dlmf.nist.gov/33.2.E4 Encodings: TeX, pMML, png See also: Annotations for §33.2(ii), §33.2 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

33.2.5		$$C_{\mathrm{\ell }}\left(\eta \right)=\frac{2^{\mathrm{\ell }}{\mathrm{e}}^{-\pi \eta /2}|\mathrm{\Gamma }\left(\mathrm{\ell }+1+\mathrm{i}\eta \right)|}{(2\mathrm{\ell }+1)!}.$$
ⓘ Defines: $C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\mathrm{\ell }$: nonnegative integer and $\eta$: real parameter A&S Ref: 14.1.7 Referenced by: §33.10(ii), §33.10(iii), §33.16(i) Permalink: http://dlmf.nist.gov/33.2.E5 Encodings: TeX, pMML, png See also: Annotations for §33.2(ii), §33.2 and Ch.33

The choice of ambiguous signs in (33.2.3) and (33.2.4) is immaterial, provided that either all upper signs are taken, or all lower signs are taken. This is a consequence of Kummer’s transformation (§13.2(vii)).

$F_{\mathrm{\ell }}(\eta ,\rho )$ is a real and analytic function of $\rho$ on the open interval , and also an analytic function of $\eta$ when .

The normalizing constant $C_{\mathrm{\ell }}\left(\eta \right)$ is always positive, and has the alternative form

33.2.6		$$C_{\mathrm{\ell }}\left(\eta \right)=\frac{2^{\mathrm{\ell }}{\left((2\pi \eta /({\mathrm{e}}^{2\pi \eta }-1)){\prod }_{k=1}^{\mathrm{\ell }}({\eta }^2+k^2)\right)}^{1/2}}{(2\mathrm{\ell }+1)!}.$$
ⓘ Symbols: $C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer and $\eta$: real parameter Referenced by: §33.13, §33.16(i) Permalink: http://dlmf.nist.gov/33.2.E6 Encodings: TeX, pMML, png See also: Annotations for §33.2(ii), §33.2 and Ch.33

The functions $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ are defined by

33.2.7		$$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )={(\mp \mathrm{i})}^{\mathrm{\ell }}{\mathrm{e}}^{(\pi \eta /2)\pm \mathrm{i}{\sigma }_{\mathrm{\ell }}\left(\eta \right)}{W}_{\mp \mathrm{i}\eta ,\mathrm{\ell }+\frac{1}{2}}\left(\mp 2\mathrm{i}\rho \right),$$
ⓘ Defines: $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions Symbols: ${\sigma }_{\mathrm{\ell }}\left(\eta \right)$: Coulomb phase shift, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter Referenced by: §13.18(vi), §33.23(vi) Permalink: http://dlmf.nist.gov/33.2.E7 Encodings: TeX, pMML, png See also: Annotations for §33.2(iii), §33.2 and Ch.33

or equivalently

33.2.8		$$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )={\mathrm{e}}^{\pm \mathrm{i}{\theta }_{\mathrm{\ell }}(\eta ,\rho )}{(\mp 2\mathrm{i}\rho )}^{\mathrm{\ell }+1\pm \mathrm{i}\eta }U(\mathrm{\ell }+1\pm \mathrm{i}\eta ,2\mathrm{\ell }+2,\mp 2\mathrm{i}\rho ),$$
ⓘ Symbols: ${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions, $U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter Referenced by: §13.6(vii), §33.13, §33.6 Permalink: http://dlmf.nist.gov/33.2.E8 Encodings: TeX, pMML, png See also: Annotations for §33.2(iii), §33.2 and Ch.33

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

33.2.9		$${\theta }_{\mathrm{\ell }}(\eta ,\rho )=\rho -\eta \ln \left(2\rho \right)-\frac{1}{2}\mathrm{\ell }\pi +{\sigma }_{\mathrm{\ell }}\left(\eta \right),$$
ⓘ Defines: ${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions Symbols: ${\sigma }_{\mathrm{\ell }}\left(\eta \right)$: Coulomb phase shift, $\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter A&S Ref: 14.5.5 Referenced by: §33.10(i), §33.11 Permalink: http://dlmf.nist.gov/33.2.E9 Encodings: TeX, pMML, png See also: Annotations for §33.2(iii), §33.2 and Ch.33

and

33.2.10		$${\sigma }_{\mathrm{\ell }}\left(\eta \right)=\mathrm{ph}\mathrm{\Gamma }\left(\mathrm{\ell }+1+\mathrm{i}\eta \right),$$
ⓘ Defines: ${\sigma }_{\mathrm{\ell }}\left(\eta \right)$: Coulomb phase shift Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $\mathrm{\ell }$: nonnegative integer and $\eta$: real parameter A&S Ref: 14.5.6 Referenced by: §33.10(ii), §33.10(iii), §33.13, §33.2(iii), §33.25, §5.20 Permalink: http://dlmf.nist.gov/33.2.E10 Encodings: TeX, pMML, png See also: Annotations for §33.2(iii), §33.2 and Ch.33

the branch of the phase in (33.2.10) being zero when $\eta =0$ and continuous elsewhere. ${\sigma }_{\mathrm{\ell }}\left(\eta \right)$ is the Coulomb phase shift.

$H_{\mathrm{\ell }}^{+}(\eta ,\rho )$ and $H_{\mathrm{\ell }}^{-}(\eta ,\rho )$ are complex conjugates, and their real and imaginary parts are given by

33.2.11		$H_{\mathrm{\ell }}^{+}(\eta ,\rho )$	$=G_{\mathrm{\ell }}(\eta ,\rho )+\mathrm{i}{F}_{\mathrm{\ell }}(\eta ,\rho ),$
		$H_{\mathrm{\ell }}^{-}(\eta ,\rho )$	$=G_{\mathrm{\ell }}(\eta ,\rho )-\mathrm{i}{F}_{\mathrm{\ell }}(\eta ,\rho ).$
ⓘ Defines: $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function Symbols: $\mathrm{i}$: imaginary unit, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter Permalink: http://dlmf.nist.gov/33.2.E11 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.2(iii), §33.2 and Ch.33

As in the case of $F_{\mathrm{\ell }}(\eta ,\rho )$, the solutions $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$ are analytic functions of $\rho$ when . Also, ${\mathrm{e}}^{\mp \mathrm{i}{\sigma }_{\mathrm{\ell }}\left(\eta \right)}{H}_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ are analytic functions of $\eta$ when .

With arguments $\eta ,\rho$ suppressed,

33.2.12		$$𝒲\left\{G_{\mathrm{\ell }},F_{\mathrm{\ell }}\right\}=𝒲\left\{H_{\mathrm{\ell }}^{\pm },F_{\mathrm{\ell }}\right\}=1.$$
ⓘ Symbols: $𝒲$: Wronskian, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function and $\mathrm{\ell }$: nonnegative integer A&S Ref: 14.2.4 Referenced by: §33.23(v) Permalink: http://dlmf.nist.gov/33.2.E12 Encodings: TeX, pMML, png See also: Annotations for §33.2(iv), §33.2 and Ch.33

33.2.13		$$F_{\mathrm{\ell }-1}{G}_{\mathrm{\ell }}-F_{\mathrm{\ell }}{G}_{\mathrm{\ell }-1}=\mathrm{\ell }/{({\mathrm{\ell }}^2+{\eta }^2)}^{1/2},$$
$\mathrm{\ell }\ge 1$.
ⓘ Symbols: $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer and $\eta$: real parameter A&S Ref: 14.2.5 Permalink: http://dlmf.nist.gov/33.2.E13 Encodings: TeX, pMML, png See also: Annotations for §33.2(iv), §33.2 and Ch.33