# §33.2 Definitions and Basic Properties

## §33.2(i) Coulomb Wave Equation

 33.2.1 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\rho}^{2}}+\left(1-\frac{2\eta}{\rho}-% \frac{\ell(\ell+1)}{\rho^{2}}\right)w=0,$ $\ell=0,1,2,\dots$.

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

 33.2.2 $\mathop{\rho_{\mathrm{tp}}\/}\nolimits\!\left(\eta,\ell\right)=\eta+(\eta^{2}+% \ell(\ell+1))^{1/2}.$ Defines: $\mathop{\rho_{\mathrm{tp}}\/}\nolimits\!\left(\NVar{\eta},\NVar{\ell}\right)$: outer turning point for Coulomb radial functions Symbols: $\ell$: 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(ii) Regular Solution $\mathop{F_{\ell}\/}\nolimits\!\left(\eta,\rho\right)$

The function $\mathop{F_{\ell}\/}\nolimits\!\left(\eta,\rho\right)$ is recessive (§2.7(iii)) at $\rho=0$, and is defined by

 33.2.3 $\mathop{F_{\ell}\/}\nolimits\!\left(\eta,\rho\right)=\mathop{C_{\ell}\/}% \nolimits\!\left(\eta\right)2^{-\ell-1}(\mp\mathrm{i})^{\ell+1}\mathop{M_{\pm% \mathrm{i}\eta,\ell+\frac{1}{2}}\/}\nolimits\!\left(\pm 2\mathrm{i}\rho\right),$ Defines: $\mathop{F_{\NVar{\ell}}\/}\nolimits\!\left(\NVar{\eta},\NVar{\rho}\right)$: regular Coulomb radial function Symbols: $\mathop{C_{\NVar{\ell}}\/}\nolimits\!\left(\NVar{\eta}\right)$: normalizing constant for Coulomb radial functions, $\mathop{M_{\NVar{\kappa},\NVar{\mu}}\/}\nolimits\!\left(\NVar{z}\right)$: Whittaker confluent hypergeometric function, $\ell$: 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)

or equivalently

 33.2.4 $\mathop{F_{\ell}\/}\nolimits\!\left(\eta,\rho\right)=\mathop{C_{\ell}\/}% \nolimits\!\left(\eta\right)\rho^{\ell+1}e^{\mp\mathrm{i}\rho}\mathop{M\/}% \nolimits\!\left(\ell+1\mp\mathrm{i}\eta,2\ell+2,\pm 2\mathrm{i}\rho\right),$

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

 33.2.5 $\mathop{C_{\ell}\/}\nolimits\!\left(\eta\right)=\frac{2^{\ell}e^{-\pi\eta/2}|% \mathop{\Gamma\/}\nolimits\!\left(\ell+1+\mathrm{i}\eta\right)|}{(2\ell+1)!}.$ Defines: $\mathop{C_{\NVar{\ell}}\/}\nolimits\!\left(\NVar{\eta}\right)$: normalizing constant for Coulomb radial functions Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of exponential function, $!$: factorial (as in $n!$), $\ell$: 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)

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)).

$\mathop{F_{\ell}\/}\nolimits\!\left(\eta,\rho\right)$ is a real and analytic function of $\rho$ on the open interval $0<\rho<\infty$, and also an analytic function of $\eta$ when $-\infty<\eta<\infty$.

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

 33.2.6 $\mathop{C_{\ell}\/}\nolimits\!\left(\eta\right)=\dfrac{2^{\ell}\left((2\pi\eta% /(e^{2\pi\eta}-1))\prod_{k=1}^{\ell}(\eta^{2}+k^{2})\right)^{\ifrac{1}{2}}}{(2% \ell+1)!}.$

## §33.2(iii) Irregular Solutions $\mathop{G_{\ell}\/}\nolimits\!\left(\eta,\rho\right),\mathop{{H^{\pm}_{\ell}}% \/}\nolimits\!\left(\eta,\rho\right)$

The functions $\mathop{{H^{\pm}_{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ are defined by

 33.2.7 $\mathop{{H^{\pm}_{\ell}}\/}\nolimits\!\left(\eta,\rho\right)=(\mp\mathrm{i})^{% \ell}e^{(\pi\eta/2)\pm\mathrm{i}\mathop{{\sigma_{\ell}}\/}\nolimits\!\left(% \eta\right)}\mathop{W_{\mp\mathrm{i}\eta,\ell+\frac{1}{2}}\/}\nolimits\!\left(% \mp 2\mathrm{i}\rho\right),$ Defines: $\mathop{{H^{\NVar{\pm}}_{\NVar{\ell}}}\/}\nolimits\!\left(\NVar{\eta},\NVar{% \rho}\right)$: irregular Coulomb radial functions Symbols: $\mathop{{\sigma_{\NVar{\ell}}}\/}\nolimits\!\left(\NVar{\eta}\right)$: Coulomb phase shift, $\mathop{W_{\NVar{\kappa},\NVar{\mu}}\/}\nolimits\!\left(\NVar{z}\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of exponential function, $\ell$: 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)

or equivalently

 33.2.8 $\mathop{{H^{\pm}_{\ell}}\/}\nolimits\!\left(\eta,\rho\right)=e^{\pm\mathrm{i}% \mathop{{\theta_{\ell}}\/}\nolimits\!\left(\eta,\rho\right)}(\mp 2\mathrm{i}% \rho)^{\ell+1\pm\mathrm{i}\eta}\mathop{U\/}\nolimits\!\left(\ell+1\pm\mathrm{i% }\eta,2\ell+2,\mp 2\mathrm{i}\rho\right),$

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

 33.2.9 $\mathop{{\theta_{\ell}}\/}\nolimits\!\left(\eta,\rho\right)=\rho-\eta\mathop{% \ln\/}\nolimits\!\left(2\rho\right)-\tfrac{1}{2}\ell\pi+\mathop{{\sigma_{\ell}% }\/}\nolimits\!\left(\eta\right),$ Defines: $\mathop{{\theta_{\NVar{\ell}}}\/}\nolimits\!\left(\NVar{\eta},\NVar{\rho}\right)$: phase of Coulomb functions Symbols: $\mathop{{\sigma_{\NVar{\ell}}}\/}\nolimits\!\left(\NVar{\eta}\right)$: Coulomb phase shift, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function, $\ell$: 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)

and

 33.2.10 $\mathop{{\sigma_{\ell}}\/}\nolimits\!\left(\eta\right)=\mathop{\mathrm{ph}\/}% \nolimits\mathop{\Gamma\/}\nolimits\!\left(\ell+1+\mathrm{i}\eta\right),$ Defines: $\mathop{{\sigma_{\NVar{\ell}}}\/}\nolimits\!\left(\NVar{\eta}\right)$: Coulomb phase shift Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\mathop{\mathrm{ph}\/}\nolimits$: phase, $\ell$: 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)

the branch of the phase in (33.2.10) being zero when $\eta=0$ and continuous elsewhere. $\mathop{{\sigma_{\ell}}\/}\nolimits\!\left(\eta\right)$ is the Coulomb phase shift.

$\mathop{{H^{+}_{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ and $\mathop{{H^{-}_{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ are complex conjugates, and their real and imaginary parts are given by

 33.2.11 $\displaystyle\mathop{{H^{+}_{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ $\displaystyle=\mathop{G_{\ell}\/}\nolimits\!\left(\eta,\rho\right)+\mathrm{i}% \mathop{F_{\ell}\/}\nolimits\!\left(\eta,\rho\right),$ $\displaystyle\mathop{{H^{-}_{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ $\displaystyle=\mathop{G_{\ell}\/}\nolimits\!\left(\eta,\rho\right)-\mathrm{i}% \mathop{F_{\ell}\/}\nolimits\!\left(\eta,\rho\right).$ Defines: $\mathop{G_{\NVar{\ell}}\/}\nolimits\!\left(\NVar{\eta},\NVar{\rho}\right)$: irregular Coulomb radial function Symbols: $\mathop{{H^{\NVar{\pm}}_{\NVar{\ell}}}\/}\nolimits\!\left(\NVar{\eta},\NVar{% \rho}\right)$: irregular Coulomb radial functions, $\mathop{F_{\NVar{\ell}}\/}\nolimits\!\left(\NVar{\eta},\NVar{\rho}\right)$: regular Coulomb radial function, $\ell$: 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)

As in the case of $\mathop{F_{\ell}\/}\nolimits\!\left(\eta,\rho\right)$, the solutions $\mathop{{H^{\pm}_{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ and $\mathop{G_{\ell}\/}\nolimits\!\left(\eta,\rho\right)$ are analytic functions of $\rho$ when $0<\rho<\infty$. Also, $e^{\mp\mathrm{i}\mathop{{\sigma_{\ell}}\/}\nolimits\!\left(\eta\right)}\mathop% {{H^{\pm}_{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ are analytic functions of $\eta$ when $-\infty<\eta<\infty$.

## §33.2(iv) Wronskians and Cross-Product

With arguments $\eta,\rho$ suppressed,

 33.2.12 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{G_{\ell}\/}\nolimits,\mathop{F_{% \ell}\/}\nolimits\right\}=\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{{H^{% \pm}_{\ell}}\/}\nolimits,\mathop{F_{\ell}\/}\nolimits\right\}=1.$
 33.2.13 $\mathop{F_{\ell-1}\/}\nolimits\mathop{G_{\ell}\/}\nolimits-\mathop{F_{\ell}\/}% \nolimits\mathop{G_{\ell-1}\/}\nolimits=\ell/(\ell^{2}+\eta^{2})^{1/2},$ $\ell\geq 1$.