§33.2 Definitions and Basic Properties

Referenced by:: ¶ ‣ §33.22(ii)
Permalink:: http://dlmf.nist.gov/33.2

§33.2(i) Coulomb Wave Equation
§33.2(ii) Regular Solution $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$
§33.2(iii) Irregular Solutions $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right),\mathop{{H^{{\pm}}_{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)$
§33.2(iv) Wronskians and Cross-Product

§33.2(i) Coulomb Wave Equation

Keywords:: Coulomb functions: variables $\rho,\eta$ , Coulomb wave equation, singularities, turning points
Permalink:: http://dlmf.nist.gov/33.2.i

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

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\ell$ : 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

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 $\ifrac{{d}^{2}w}{{d\rho}^{2}}=0$ (§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:: $\mathop{\rho _{{\mathrm{tp}}}\/}\nolimits\!\left(\eta,\ell\right)$ : 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
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§33.2(ii) Regular Solution $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$

Notes:: See Yost et al. (1936).
Keywords:: Coulomb functions, Coulomb functions: variables $\rho,\eta$ , Whittaker functions, confluent hypergeometric functions
Permalink:: http://dlmf.nist.gov/33.2.ii

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 $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)=\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)2^{{-\ell-1}}(\mp i)^{{\ell+1}}\mathop{M_{{\pm i\eta,\ell+\frac{1}{2}}}\/}\nolimits\!\left(\pm 2i\rho\right),$

Defines:: $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : regular Coulomb radial function
Symbols:: $\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)$ : normalizing constant for Coulomb radial functions, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable and $\eta$ : real parameter
Referenced by:: §33.2(ii)
Permalink:: http://dlmf.nist.gov/33.2.E3
Encodings:: TeX, pMML, png

or equivalently

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

Symbols:: $\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)$ : normalizing constant for Coulomb radial functions, $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $e$ : base of exponential function, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : regular Coulomb radial function, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable and $\eta$ : real parameter
A&S Ref:: 14.1.3
Referenced by:: §33.2(ii), §33.6
Permalink:: http://dlmf.nist.gov/33.2.E4
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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 $\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)=\frac{2^{\ell}e^{{-\pi\eta/2}}|\mathop{\Gamma\/}\nolimits\!\left(\ell+1+i\eta\right)|}{(2\ell+1)!}.$

Defines:: $\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)$ : normalizing constant for Coulomb radial functions
Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential function, $!$ : $n!$ : factorial, $\ell$ : 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

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

Symbols:: $\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)$ : normalizing constant for Coulomb radial functions, $e$ : base of exponential function, $!$ : $n!$ : factorial, $k$ : nonnegative integer, $\ell$ : 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

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

Notes:: See Hull and Breit (1959, pp. 409–410), also Yost et al. (1936).
Keywords:: Coulomb functions, Coulomb functions: variables $\rho,\eta$ , Coulomb phase shift, Whittaker functions, confluent hypergeometric functions
Referenced by:: §33.23(iii), §33.23(iv)
Permalink:: http://dlmf.nist.gov/33.2.iii

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

33.2.7 $\mathop{{H^{{\pm}}_{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)=(\mp i)^{\ell}e^{{(\pi\eta/2)\pm i\mathop{{\sigma _{{\ell}}}\/}\nolimits\!\left(\eta\right)}}\mathop{W_{{\mp i\eta,\ell+\frac{1}{2}}}\/}\nolimits\!\left(\mp 2i\rho\right),$

Defines:: $\mathop{{H^{{\pm}}_{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial functions
Symbols:: $\mathop{{\sigma _{{\ell}}}\/}\nolimits\!\left(\eta\right)$ : Coulomb phase shift, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable and $\eta$ : real parameter
Referenced by:: §33.23(vi)
Permalink:: http://dlmf.nist.gov/33.2.E7
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or equivalently

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

Symbols:: $\mathop{{\theta _{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)$ : phase of Coulomb functions, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $e$ : base of exponential function, $\mathop{{H^{{\pm}}_{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial functions, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable and $\eta$ : real parameter
Referenced by:: §33.13, §33.6
Permalink:: http://dlmf.nist.gov/33.2.E8
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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:: $\mathop{{\theta _{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)$ : phase of Coulomb functions
Symbols:: $\mathop{{\sigma _{{\ell}}}\/}\nolimits\!\left(\eta\right)$ : Coulomb phase shift, $\mathop{\ln\/}\nolimits z$ : 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
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and

33.2.10 $\mathop{{\sigma _{{\ell}}}\/}\nolimits\!\left(\eta\right)=\mathop{\mathrm{ph}\/}\nolimits\mathop{\Gamma\/}\nolimits\!\left(\ell+1+i\eta\right),$

Defines:: $\mathop{{\sigma _{{\ell}}}\/}\nolimits\!\left(\eta\right)$ : Coulomb phase shift
Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : 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

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

$\mathop{{H^{{+}}_{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)=\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)+i\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right),$

$\mathop{{H^{{-}}_{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)=\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)-i\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right).$

Defines:: $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial function
Symbols:: $\mathop{{H^{{\pm}}_{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial functions, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : 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

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, $e^{{\mp i\mathop{{\sigma _{{\ell}}}\/}\nolimits\!\left(\eta\right)}}\mathop{{H^{{\pm}}_{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)$ are analytic functions of $\eta$ when $-\infty<\eta<\infty$ .

§33.2(iv) Wronskians and Cross-Product

Notes:: See Yost et al. (1936).
Keywords:: Coulomb functions: variables $\rho,\eta$
Permalink:: http://dlmf.nist.gov/33.2.iv

With arguments $\eta,\rho$ suppressed,

33.2.12 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{G_{{\ell}}\/}\nolimits,\mathop{F_{{\ell}}\/}\nolimits\right\}=\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{{H^{{\pm}}_{{\ell}}}\/}\nolimits,\mathop{F_{{\ell}}\/}\nolimits\right\}=1.$

Symbols:: $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial function, $\mathop{{H^{{\pm}}_{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial functions, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : regular Coulomb radial function and $\ell$ : 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

33.2.13 $\mathop{F_{{\ell-1}}\/}\nolimits\mathop{G_{{\ell}}\/}\nolimits-\mathop{F_{{\ell}}\/}\nolimits\mathop{G_{{\ell-1}}\/}\nolimits=\ell/(\ell^{2}+\eta^{2})^{{1/2}},$ $\ell\geq 1$ .

Symbols:: $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial function, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : regular Coulomb radial function, $\ell$ : nonnegative integer and $\eta$ : real parameter
A&S Ref:: 14.2.5
Permalink:: http://dlmf.nist.gov/33.2.E13
Encodings:: TeX, pMML, png

§33.2 Definitions and Basic Properties

Contents

§33.2(i) Coulomb Wave Equation

§33.2(ii) Regular Solution

§33.2(iii) Irregular Solutions

§33.2(iv) Wronskians and Cross-Product

§33.2(ii) Regular Solution $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$

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