33 Coulomb FunctionsVariables $\rho ,\eta$33.9 Expansions in Series of Bessel Functions33.11 Asymptotic Expansions for Large $\rho$

§33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta \right|$

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Permalink:

http://dlmf.nist.gov/33.10

See also:

Annotations for Ch.33

Contents

1. §33.10(i) Large $\rho$
2. §33.10(ii) Large Positive $\eta$
3. §33.10(iii) Large Negative $\eta$

§33.10(i) Large $\rho$

ⓘ

Keywords:

Coulomb functions: variables $\rho ,\eta$, large $\rho$, limiting forms

Notes:

See Yost et al. (1936).

Permalink:

http://dlmf.nist.gov/33.10.i

See also:

Annotations for §33.10 and Ch.33

As $\rho \to \mathrm{\infty }$ with $\eta$ fixed,

33.10.1		$F_{\mathrm{\ell }}(\eta ,\rho )$	$=\sin \left({\theta }_{\mathrm{\ell }}(\eta ,\rho )\right)+o\left(1\right),$
		$G_{\mathrm{\ell }}(\eta ,\rho )$	$=\cos \left({\theta }_{\mathrm{\ell }}(\eta ,\rho )\right)+o\left(1\right),$
ⓘ Symbols: ${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions, $\cos z$: cosine function, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $o\left(x\right)$: order less than, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\sin z$: sine function, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter Referenced by: item 1 Permalink: http://dlmf.nist.gov/33.10.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.10(i), §33.10 and Ch.33

33.10.2		$$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )\sim \exp \left(\pm \mathrm{i}{\theta }_{\mathrm{\ell }}(\eta ,\rho )\right),$$
ⓘ Symbols: ${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions, $\sim$: asymptotic equality, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter A&S Ref: 14.6.5 Permalink: http://dlmf.nist.gov/33.10.E2 Encodings: TeX, pMML, png See also: Annotations for §33.10(i), §33.10 and Ch.33

where ${\theta }_{\mathrm{\ell }}(\eta ,\rho )$ is defined by (33.2.9).

§33.10(ii) Large Positive $\eta$

ⓘ

Keywords:

Coulomb functions: variables $\rho ,\eta$, large $\left|\eta \right|$, limiting forms

Notes:

See Fröberg (1955), Humblet (1984), Humblet (1985, Eqs. 2.10a,b). For (33.10.6) use (33.2.5), (33.2.10), and §5.11(i).

Referenced by:

Erratum (V1.0.19) for Subsections 33.10(ii), 33.10(iii)

Permalink:

http://dlmf.nist.gov/33.10.ii

Errata (effective with 1.0.19):

In the first sentence, originally it was stated that $\rho$ was fixed. This has been corrected to state that $\eta \rho$ is fixed.

Reported 2018-05-15 by Ian Thompson

See also:

Annotations for §33.10 and Ch.33

As $\eta \to \mathrm{\infty }$ with $\eta \rho$ fixed,

33.10.3		$F_{\mathrm{\ell }}(\eta ,\rho )$	$\sim {\displaystyle \frac{(2\mathrm{\ell }+1)!C_{\mathrm{\ell }}\left(\eta \right)}{{(2\eta )}^{\mathrm{\ell }+1}}}{(2\eta \rho )}^{1/2}{I}_{2\mathrm{\ell }+1}\left({(8\eta \rho )}^{1/2}\right),$
		$G_{\mathrm{\ell }}(\eta ,\rho )$	$\sim {\displaystyle \frac{2{(2\eta )}^{\mathrm{\ell }}}{(2\mathrm{\ell }+1)!C_{\mathrm{\ell }}\left(\eta \right)}}{(2\eta \rho )}^{1/2}{K}_{2\mathrm{\ell }+1}\left({(8\eta \rho )}^{1/2}\right).$
ⓘ Symbols: $C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $\sim$: asymptotic equality, $!$: factorial (as in $n!$), $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter A&S Ref: 14.6.7 Permalink: http://dlmf.nist.gov/33.10.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.10(ii), §33.10 and Ch.33

In particular, for $\mathrm{\ell }=0$,

33.10.4		$F_0(\eta ,\rho )$	$\sim {\mathrm{e}}^{-\pi \eta }{(\pi \rho )}^{1/2}{I}_1\left({(8\eta \rho )}^{1/2}\right),$
		$G_0(\eta ,\rho )$	$\sim 2{\mathrm{e}}^{\pi \eta }{\left(\rho /\pi \right)}^{1/2}{K}_1\left({(8\eta \rho )}^{1/2}\right),$
ⓘ Symbols: $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\rho$: nonnegative real variable and $\eta$: real parameter Permalink: http://dlmf.nist.gov/33.10.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.10(ii), §33.10 and Ch.33

33.10.5		$F_0^{\prime }(\eta ,\rho )$	$\sim {\mathrm{e}}^{-\pi \eta }{(2\pi \eta )}^{1/2}{I}_0\left({(8\eta \rho )}^{1/2}\right),$
		$G_0^{\prime }(\eta ,\rho )$	$\sim -2{\mathrm{e}}^{\pi \eta }{\left(2\eta /\pi \right)}^{1/2}{K}_0\left({(8\eta \rho )}^{1/2}\right).$
ⓘ Symbols: $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\rho$: nonnegative real variable and $\eta$: real parameter Permalink: http://dlmf.nist.gov/33.10.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.10(ii), §33.10 and Ch.33

Also,

33.10.6		${\sigma }_0\left(\eta \right)$	$=\eta (\ln \eta -1)+\frac{1}{4}\pi +o\left(1\right),$
		$C_0\left(\eta \right)$	$\sim {(2\pi \eta )}^{1/2}{\mathrm{e}}^{-\pi \eta }.$
ⓘ Symbols: $C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, ${\sigma }_{\mathrm{\ell }}\left(\eta \right)$: Coulomb phase shift, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $o\left(x\right)$: order less than, $\ln z$: principal branch of logarithm function and $\eta$: real parameter A&S Ref: 14.6.16 Referenced by: §33.10(ii) Permalink: http://dlmf.nist.gov/33.10.E6 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.10(ii), §33.10 and Ch.33

§33.10(iii) Large Negative $\eta$

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Keywords:

Coulomb functions: variables $\rho ,\eta$, large $\left|\eta \right|$, limiting forms

Notes:

See Humblet (1985, Eqs. 4.7a,b). For (33.10.10) use (33.2.5), (33.2.10), and §5.11(i).

Referenced by:

Erratum (V1.0.19) for Subsections 33.10(ii), 33.10(iii)

Permalink:

http://dlmf.nist.gov/33.10.iii

Errata (effective with 1.0.19):

In the first sentence, originally it was stated that $\rho$ was fixed. This has been corrected to state that $\eta \rho$ is fixed.

Reported 2018-05-15 by Ian Thompson

See also:

Annotations for §33.10 and Ch.33

As $\eta \to -\mathrm{\infty }$ with $\eta \rho$ fixed,

33.10.7		$F_{\mathrm{\ell }}(\eta ,\rho )$	$={\displaystyle \frac{(2\mathrm{\ell }+1)!C_{\mathrm{\ell }}\left(\eta \right)}{{(-2\eta )}^{\mathrm{\ell }+1}}}\left({(-2\eta \rho )}^{1/2}{J}_{2\mathrm{\ell }+1}\left({(-8\eta \rho )}^{1/2}\right)+o\left({\left|\eta \right|}^{1/4}\right)\right),$
		$G_{\mathrm{\ell }}(\eta ,\rho )$	$=-{\displaystyle \frac{\pi {(-2\eta )}^{\mathrm{\ell }}}{(2\mathrm{\ell }+1)!C_{\mathrm{\ell }}\left(\eta \right)}}\left({(-2\eta \rho )}^{1/2}{Y}_{2\mathrm{\ell }+1}\left({(-8\eta \rho )}^{1/2}\right)+o\left({\left|\eta \right|}^{1/4}\right)\right).$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $o\left(x\right)$: order less than, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter A&S Ref: 14.6.7 Permalink: http://dlmf.nist.gov/33.10.E7 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.10(iii), §33.10 and Ch.33

In particular, for $\mathrm{\ell }=0$,

33.10.8		$F_0(\eta ,\rho )$	$={(\pi \rho )}^{1/2}{J}_1\left({(-8\eta \rho )}^{1/2}\right)+o\left({\left|\eta \right|}^{-1/4}\right),$
		$G_0(\eta ,\rho )$	$=-{(\pi \rho )}^{1/2}{Y}_1\left({(-8\eta \rho )}^{1/2}\right)+o\left({\left|\eta \right|}^{-1/4}\right).$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $o\left(x\right)$: order less than, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\rho$: nonnegative real variable and $\eta$: real parameter Permalink: http://dlmf.nist.gov/33.10.E8 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.10(iii), §33.10 and Ch.33

33.10.9		$F_0^{\prime }(\eta ,\rho )$	$={(-2\pi \eta )}^{1/2}{J}_0\left({(-8\eta \rho )}^{1/2}\right)+o\left({\left|\eta \right|}^{1/4}\right),$
		$G_0^{\prime }(\eta ,\rho )$	$=-{(-2\pi \eta )}^{1/2}{Y}_0\left({(-8\eta \rho )}^{1/2}\right)+o\left({\left|\eta \right|}^{1/4}\right).$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $o\left(x\right)$: order less than, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\rho$: nonnegative real variable and $\eta$: real parameter Permalink: http://dlmf.nist.gov/33.10.E9 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.10(iii), §33.10 and Ch.33

Also,

33.10.10		${\sigma }_0\left(\eta \right)$	$=\eta (\ln \left(-\eta \right)-1)-\frac{1}{4}\pi +o\left(1\right),$
		$C_0\left(\eta \right)$	$\sim {(-2\pi \eta )}^{1/2}.$
ⓘ Symbols: $C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, ${\sigma }_{\mathrm{\ell }}\left(\eta \right)$: Coulomb phase shift, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $o\left(x\right)$: order less than, $\ln z$: principal branch of logarithm function and $\eta$: real parameter Referenced by: §33.10(iii) Permalink: http://dlmf.nist.gov/33.10.E10 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.10(iii), §33.10 and Ch.33

33.9 Expansions in Series of Bessel Functions33.11 Asymptotic Expansions for Large $\rho$

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