§33.5 Limiting Forms for Small $\rho$ , Small $|\eta|$ , or Large $\ell$

Permalink:: http://dlmf.nist.gov/33.5

§33.5(i) Small $\rho$
§33.5(ii) $\eta=0$
§33.5(iii) Small $|\eta|$
§33.5(iv) Large $\ell$

§33.5(i) Small $\rho$

Notes:: See Yost et al. (1936).
Keywords:: Coulomb functions: variables $\rho,\eta$
Permalink:: http://dlmf.nist.gov/33.5.i

As $\rho\to 0$ with $\eta$ fixed,

33.5.1

$\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)\sim\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)\rho^{{\ell+1}},$

${\mathop{F_{{\ell}}\/}\nolimits^{{\prime}}}\!\left(\eta,\rho\right)\sim(\ell+1)\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)\rho^{{\ell}}.$

Symbols:: $\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)$ : normalizing constant for Coulomb radial functions, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : regular Coulomb radial function, $\sim$ : asymptotic equality, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable and $\eta$ : real parameter
A&S Ref:: 14.6.3
Permalink:: http://dlmf.nist.gov/33.5.E1
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33.5.2

$\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)\sim\frac{\rho^{{-\ell}}}{(2\ell+1)\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)},$ $\ell=0,1,2,\dots$ ,

${\mathop{G_{{\ell}}\/}\nolimits^{{\prime}}}\!\left(\eta,\rho\right)\sim-\frac{\ell\rho^{{-\ell-1}}}{(2\ell+1)\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)},$ $\ell=1,2,3,\dots$ .

Symbols:: $\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)$ : normalizing constant for Coulomb radial functions, $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial function, $\sim$ : asymptotic equality, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable and $\eta$ : real parameter
A&S Ref:: 14.6.3
Permalink:: http://dlmf.nist.gov/33.5.E2
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§33.5(ii) $\eta=0$

Notes:: See Hull and Breit (1959, pp. 435–436); also Wheeler (1937).
Keywords:: Coulomb functions: variables $\rho,\eta$
Permalink:: http://dlmf.nist.gov/33.5.ii

33.5.3

$\mathop{F_{{\ell}}\/}\nolimits\!\left(0,\rho\right)=\rho\mathop{\mathsf{j}_{{\ell}}\/}\nolimits\!\left(\rho\right),$

$\mathop{G_{{\ell}}\/}\nolimits\!\left(0,\rho\right)=-\rho\mathop{\mathsf{y}_{{\ell}}\/}\nolimits\!\left(\rho\right).$

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, $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind, $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the second kind, $\ell$ : nonnegative integer and $\rho$ : nonnegative real variable
Permalink:: http://dlmf.nist.gov/33.5.E3
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Equivalently,

33.5.4

$\mathop{F_{{\ell}}\/}\nolimits\!\left(0,\rho\right)=(\pi\rho/2)^{{1/2}}\mathop{J_{{\ell+\frac{1}{2}}}\/}\nolimits\!\left(\rho\right),$

$\mathop{G_{{\ell}}\/}\nolimits\!\left(0,\rho\right)=-(\pi\rho/2)^{{1/2}}\mathop{Y_{{\ell+\frac{1}{2}}}\/}\nolimits\!\left(\rho\right).$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial function, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : regular Coulomb radial function, $\ell$ : nonnegative integer and $\rho$ : nonnegative real variable
A&S Ref:: 14.6.6
Permalink:: http://dlmf.nist.gov/33.5.E4
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For the functions $\mathop{\mathsf{j}\/}\nolimits$ , $\mathop{\mathsf{y}\/}\nolimits$ , $\mathop{J\/}\nolimits$ , $\mathop{Y\/}\nolimits$ see §§10.47(ii), 10.2(ii).

33.5.5

$\mathop{F_{{0}}\/}\nolimits\!\left(0,\rho\right)=\mathop{\sin\/}\nolimits\rho,$

$\mathop{G_{{0}}\/}\nolimits\!\left(0,\rho\right)=\mathop{\cos\/}\nolimits\rho,$

$\mathop{{H^{{\pm}}_{{0}}}\/}\nolimits\!\left(0,\rho\right)=e^{{\pm i\rho}}.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $e$ : base of exponential function, $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : 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, $\mathop{\sin\/}\nolimits z$ : sine function and $\rho$ : nonnegative real variable
Permalink:: http://dlmf.nist.gov/33.5.E5
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33.5.6 $\mathop{C_{{\ell}}\/}\nolimits\!\left(0\right)=\frac{2^{\ell}\ell!}{(2\ell+1)!}=\frac{1}{(2\ell+1)!!}.$

Symbols:: $\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)$ : normalizing constant for Coulomb radial functions, $!!$ : $n!!$ : double factorial, $!$ : $n!$ : factorial and $\ell$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/33.5.E6
Encodings:: TeX, pMML, png

§33.5(iii) Small $|\eta|$

Keywords:: Coulomb functions: variables $\rho,\eta$
Permalink:: http://dlmf.nist.gov/33.5.iii

33.5.7 $\mathop{{\sigma _{{0}}}\/}\nolimits\!\left(\eta\right)\sim-\EulerConstant\eta,$ $\eta\to 0$ ,

Symbols:: $\mathop{{\sigma _{{\ell}}}\/}\nolimits\!\left(\eta\right)$ : Coulomb phase shift, $\EulerConstant$ : Euler’s constant, $\sim$ : asymptotic equality and $\eta$ : real parameter
Permalink:: http://dlmf.nist.gov/33.5.E7
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where $\EulerConstant$ is Euler’s constant (§5.2(ii)).

§33.5(iv) Large $\ell$

Notes:: See Yost et al. (1936) and Biedenharn et al. (1955). For (33.5.9) combine the second formula for $(2\ell+1)!!$ given in §33.1 with (5.11.7).
Keywords:: Coulomb functions: variables $\rho,\eta$
Permalink:: http://dlmf.nist.gov/33.5.iv

As $\ell\to\infty$ with $\eta$ and $\rho$ ( $\neq 0$ ) fixed,

33.5.8

$\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)\sim\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)\rho^{{\ell+1}},$

$\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)\sim\dfrac{\rho^{{-\ell}}}{(2\ell+1)\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)},$

Symbols:: $\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)$ : normalizing constant for Coulomb radial functions, $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : irregular Coulomb radial function, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : regular Coulomb radial function, $\sim$ : asymptotic equality, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable and $\eta$ : real parameter
A&S Ref:: 14.6.3
Referenced by:: §33.18, §33.23(iv)
Permalink:: http://dlmf.nist.gov/33.5.E8
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33.5.9 $\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)\sim\dfrac{e^{{-\pi\eta/2}}}{(2\ell+1)!!}\sim e^{{-\pi\eta/2}}\dfrac{e^{\ell}}{\sqrt{2}(2\ell)^{{\ell+1}}}.$

Symbols:: $\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)$ : normalizing constant for Coulomb radial functions, $!!$ : $n!!$ : double factorial, $e$ : base of exponential function, $\sim$ : asymptotic equality, $\ell$ : nonnegative integer and $\eta$ : real parameter
Referenced by:: §33.23(iv), §33.5(iv)
Permalink:: http://dlmf.nist.gov/33.5.E9
Encodings:: TeX, pMML, png

§33.5 Limiting Forms for Small , Small , or Large

Contents

§33.5(i) Small

§33.5(ii)

§33.5(iii) Small

§33.5(iv) Large

§33.5 Limiting Forms for Small $\rho$ , Small $|\eta|$ , or Large $\ell$

§33.5(i) Small $\rho$

§33.5(ii) $\eta=0$

§33.5(iii) Small $|\eta|$

§33.5(iv) Large $\ell$